Defending the Axioms
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Defending the Axioms:
On the Philosophical
Foundations of
Set Theory
Penelope Maddy
1
3
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For the Cabal
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Contents
Preface
Introduction
ix
1
I. The Problem
1. An historical reversal
2. How applied mathematics became pure
3. Where we are now
2
3
6
27
II. Proper Method
1. The meta-philosophy
2. Some examples from set-theoretic practice
3. Proper set-theoretic method
4. The challenge
38
38
41
52
55
III. Thin Realism
1. Introducing Thin Realism
2. What Thin Realism is not
3. Thin epistemology
4. The objective ground of Thin Realism
5. Retracing our steps
60
61
64
71
77
83
IV. Arealism
1. Introducing Arealism
2. Mathematics in application
3. What Arealism is not
4. Comparison with Thin Realism
5. Thin Realism/Arealism
88
88
89
96
99
103
viii conte nts
V. Morals
1. Objectivity in mathematics
2. Robust Realism revisited
3. More examples from set-theoretic practice
4. Intrinsic versus extrinsic
113
114
117
123
131
Bibliography
Index
138
147
Preface
This question of how set-theoretic axioms are properly defended has
been with me for some time. ‘Believing the axioms’ ([1988]) catalogs
all the actual arguments I could find, in the literature or in conversation, as a preliminary step toward the project of determining which are
cogent, which not, and why. Realism in Mathematics ([1990]) is an
attempt to turn away the objection that questions independent of the
standard axioms—which obviously includes all new axiom candidates—have no answers; there I propose a more naturalistic variant
of Gödel’s Robust Realism. Though some take me to task for apostasy,
I soon despaired of this position, for three reasons: it relies on a Quine/
Putnam indispensability argument that I couldn’t continue to endorse;
arguments for and against axiom candidates that seem compelling
don’t fit well with the metaphysics; and most importantly, just as a
fundamentally naturalistic perspective counts against criticizing a bit of
mathematics on the basis of extra-mathematical considerations, it
counts just as heavily against supporting a bit of mathematics on the
basis of extra-mathematical considerations.
Since then, Naturalism in Mathematics ([1997]) takes a more strictly
naturalistic approach to the methodological question of how arguments for or against set-theoretic principles should be evaluated,
attempting to separate that question from traditional philosophical
issues of truth and existence. Second Philosophy ([2007]) lays out the
broader philosophical background that seemed to me necessary for a
return to those traditional questions; }IV.4 of that book contains a brief
sketch of what the resulting answers might look like. The goal of the
current book, then, is to fill in and develop those sketchy answers.
An early version of Chapter I appeared in the inaugural issue of
the Review of Symbolic Logic ([2008]). The Association for Symbolic
Logic and Cambridge University Press have generously permitted its
re-appearance here.
x pre face
Finally, I’m grateful to many people for help of various kinds during
the course of this project, among them Jeremy Avigad, John Burgess,
Justin Clarke-Doane, Michael Ernst, Matthew Glass, Jeremy Heis,
Juliette Kennedy, Peter Koellner, Michael Liston, David Malament,
Patricia Marino, Tony Martin, Colin McLarty, Bennett McNulty,
A. J. Packman, John Rapalino, Erich Reck, Brian Rogers, Jeffrey
Roland, Stewart Shapiro, John Steel, Jamie Tappenden, Scott Tidwell,
Clinton Tolley, Mark Wilson, and the members of my winter 2009
seminar. I’m also indebted, as always, to Peter Momtchiloff of Oxford
University Press. And finally, my thanks again to David, for his
friendship and encouragement along the way.
P.M.
Irvine, California
May 2010
Introduction
Mathematics, as we all know, depends on proofs. And proofs, as we
all know, have to begin somewhere, from some fundamental assumptions. In contemporary pure mathematics, the axioms of set theory
are particularly salient, for reasons traced in Chapter I. In this case,
the Euclidean ideal of postulates that are simply obvious or selfevident can’t be the whole story, which raises two basic questions:
what are the proper methods for defending set-theoretic axioms?
and, why are these the proper methods? The first of these is the
subject of Chapter II. Addressing the second requires engagement
with the troublesome ontological and epistemological issues that
have dogged the philosophy of mathematics from its beginnings. In
Chapter III and IV, I describe and explore two apparently conflicting
stands on these issues, not so much to recommend either one, but with
an eye to suggesting that the question of which is correct has less bite
than it might appear. In the end, my hope is to shift attention away from
these elusive matters of truth and existence, and to direct it toward the
type of mathematical objectivity emphasized in the opening section of
Chapter V. (Though set theory is the focus in these pages, I believe the
source of objectivity traced there is also at work in other branches of
pure mathematics.) The concluding sections of Chapter V return, at last,
to the question of set-theoretic method and draw some concrete morals
for the project of defending the axioms.
I
The Problem
The subject of this book is contemporary set theory, its methods and
its subject matter: what are set theorists doing? how are they managing to do it? To understand the nature and force of these questions,
we first need to appreciate how we came to the point we now
occupy, how pure mathematics arose out of applied mathematics
and how set theory developed from there. From this perspective,
we can then address the questions of how the progress of pure
mathematics is guided, of which mathematical entities and proof
techniques are legitimate, of what constraints our methods must properly satisfy.
As a start, we need to recognize that the relationship between pure
and applied mathematics hasn’t been static over the centuries, that
the historical rise of pure mathematics has coincided with a gradual
shift in our understanding of how mathematics works in application
to the world. In some circles today, it’s held that historical developments of this sort simply represent changes in fashion, or in social
arrangements, in governments, in power structures, or some such
thing, but I resist the full force of this way of thinking, clinging to the
old-school notion that we’ve gradually learned more about the world
over time, that our opinions on these matters have improved, and
that seeing how we reached the point we now occupy may help us
avoid falling back into old philosophies that are now no longer
viable.
In that spirit, it seems to me that once we focus on the general
question of how mathematics relates to science, one observation is
immediate: the march of the centuries has produced an amusing
reversal of philosophical fortunes. Let me begin there.
the problem
3
1. An historical reversal
In the beginning, that is, in Plato, mathematical knowledge was
sharply distinguished from ordinary perceptual belief about the
world. According to Plato’s metaphysics, mathematics is the study of
eternal and unchanging abstract Forms1 while science is uncertain and
changeable opinion about the world of mere becoming. Indeed, by
Plato’s lights, of the two, only mathematics deserves to be called
‘knowledge’ at all! Of course if sense perception can’t give us knowledge, if mathematics is not about perceivable things, then Plato owes
us an account of how we ordinary humans achieve this wonderful
insight into the properties of the abstract world of Forms. Plato’s
answer is that we don’t actually acquire mathematical knowledge at
all; rather we recollect it from a time before birth, when our souls,
unencumbered by physical bodies, were free to commune with the
Forms, and not just the mathematical ones, either—also Truth, Beauty,
Justice, The Good, and so on.2 Whatever appeal this position may have
held for the ancient Greeks, it won’t begin to satisfy a contemporary,
scientifically-minded philosopher. But in any case, Plato’s view on the
relative standing of mathematics and science is unambiguous: mathematics is the highest form of knowledge; science is mere opinion.
Of course ‘science’ for the Greeks wasn’t what we call ‘science’ today;
between the ancients and ourselves looms the Scientific Revolution, the
beginning of experimental natural science as we know it. Pioneers of the
new science like Galileo took mathematics to be central to God’s design
of the universe, as in this famous passage from 1610:
[Nature] is written in that great book which ever lies before our eyes—I mean
the universe—but we cannot understand it if we do not first learn the
language and grasp the symbols in which it is written. The book is written
in the mathematical language, and the symbols are triangles, circles, and other
geometric figures, without whose help it is impossible to comprehend a single
word of it; without which one wanders in vain through a dark labyrinth.
1
In fact, for various reasons, Plato may posit a kind of intermediate abstract entity, called a
‘mathematical’, between the Forms and the physical world. For discussion, see Wedberg
[1955].
2
See e.g. Silverman [2003].
4 th e problem
(as quoted by Kline [1972], pp. 328–329. See also Machamer [1998a],
pp. 64–65, for a slightly different translation.)
In the case of free fall, for example, Galileo notes that many accounts of
its causes have been proposed, but he rejects this inquiry:
Such fantasies, and others like them, would have to be examined and resolved,
with little gain. (Galileo [1638], p. 202)
His idea is that we should concentrate on finding and testing a purely
mathematical description of motion, which he goes on to do:
It suffices our Author that we understand him to want us to investigate and
demonstrate some attributes of a motion so accelerated . . . whatever be the
cause of its acceleration . . . that in equal times, equal additions of speed are
made. (op. cit.)
The universe operates according to mathematical laws, which we can
uncover by bringing mathematics to bear on our observations.3
Notice the dramatic shift here: mathematics isn’t placed above
science; rather the two have become one. In the words of the mathematical historian Morris Kline:
mathematics became the substance of scientific theories. . . . The upshot . . . was
a virtual fusion of mathematics . . . and science. (Kline [1972], pp. 394–395)
The great thinkers of that time—from Descartes and Galileo to
Huygens and Newton—did mathematics as science and science as
mathematics without any effort to separate the two.
Appealing as this picture may be, it’s not the way we tend to see
things today. What with the various shocks dealt in the interim to our
3
It should be noted that this bold proposal, later adopted by Newton (see below), didn’t
meet with universal approval. Cf. Kline [1972], pp. 333–334: ‘First reactions to this principle
of Galileo are likely to be negative. Description of phenomena in terms of formulas hardly
seems to be more than a first step. It would seem that the true function of science had really
been grasped by the Aristotelians, namely, to explain why phenomena happened. Even
Descartes [who reacted against the Aristotelians] protested Galileo’s decision to seek descriptive formulas. He said, “Everything that Galileo says about bodies falling in empty space is
built without foundation: he ought first to have determined the nature of weight”. Further,
said Descartes, Galileo should reflect on ultimate reasons’. Looking forward to Newton and
his successors, Kline holds that ‘Galileo’s decision to aim for description was the deepest and
most fruitful idea that anyone has had about scientific methodology’.
the problem
5
intuitive sense of how an orderly mathematical universe should behave,
a starker empiricism has come to replace the rationalistic tendencies of
an earlier, simpler time. So, for example, early in the 20th century, the
logical positivists of the famous Vienna Circle held that the meaning of
any scientific assertion could be reduced in some way or other to the
conditions under which it would be empirically verified. This idea did
admirable service in eliminating much apparently empty philosophical
talk as meaningless—Carnap classifies Heidegger’s claim that ‘the Nothing itself nothings’ as a pseudo-statement4—but it also threatened to do
the same for mathematics, an outcome that certainly didn’t appeal to the
self-described scientific philosophers of the Circle! Their solution was to
view mathematics as purely linguistic, as true by the conventions of
language, as telling us nothing contentful about the world.5 At the same
time, those like Gödel who continued to maintain that mathematics
provides substantial information about properly mathematical objects
were confronted with a variation of Plato’s problem: if the cognitive
machinery of human beings works as we think it does, how can we gain
knowledge of non-spatiotemporal, acausal entities? 6 In all this, experimental natural science is taken as the paradigm of well-grounded
knowledge and mathematics is called into question when it doesn’t
clearly measure up.
This is the reversal of philosophical fortunes alluded to a moment
ago: for Plato, mathematics is perfect knowledge and science is mere
opinion; for the pioneers of the scientific revolution, mathematics and
science are one; for many contemporary philosophers, science is the
best knowledge we have and the status of mathematics is problematic.
The movement from the cross-over point—when science and mathematics were identified—to our current state coincides roughly with the
rise of pure mathematics, with the separation of mathematics from its
worldly roots. Perhaps we can better understand where we are now if
we reconsider how mathematics came to be peeled away from natural
science in this way. This is a complex story, of course, but I hope to
4
5
6
See Carnap [1932], p. 69.
See Carnap [1950].
See e.g. Gödel [1964] and Benacerraf [1973].
6 th e problem
draw out three of its individual strands, ranging from the more mathematical perspective to the more scientific.7
2. How applied mathematics became pure
I think the first of these strands, primarily a mathematician’s-eye-view,
is fairly familiar. Kline describes the situation this way:
. . . Descartes, Newton, Euler, and many others believed mathematics to be
the accurate description of real phenomena . . . they regarded their work as the
uncovering of the mathematical design of the universe. (Kline [1972], p. 1028)
Over the course of the 19th century, this picture changed dramatically:
. . . gradually and unwittingly mathematicians began to introduce concepts
that had little or no direct physical meaning. (Kline [1972], p. 1029)
Citing the rise of negative numbers, complex numbers, n-dimensional
spaces, and non-commutative algebras, he remarks that ‘mathematics
was progressing beyond concepts suggested by experience’, but that
‘mathematicians had yet to grasp that their subject . . . was no longer,
if it ever had been, a reading of nature’ (Kline [1972], p. 1030). By
mid-century, the tide had turned:
. . . after about 1850, the view that mathematics can introduce and deal with . . .
concepts and theories that do not have immediate physical interpretation . . .
gained acceptance. (Kline [1972], p. 1031)
This movement continued with the study, for example, of abstract
algebras, pathological functions, and transfinite numbers. The heady
new view of mathematics that accompanied this change is perhaps best
expressed by Cantor:
Mathematics is entirely free in its development . . . The essence of mathematics
lies in its freedom. (as quoted in Kline [1972], p. 1031)
7
I should admit that I’m no historian myself. As will become obvious, I draw heavily on
the work of various real scholars in what follows.
the problem
7
This sentiment appears in the thinking of many of the most innovative
mathematicians of the late 19th century; today, it is standard orthodoxy. Mathematics progresses by its own lights, independent of ties to
the physical world. Legitimate mathematical concepts and theories
need have no direct physical interpretation.
A careful, systematic analysis of these developments would explain
how mathematicians gradually came to see themselves as free to
investigate whatever concepts or structures or theories seem of sufficient mathematical interest or importance. Detailed historical studies
would illustrate how particular mathematical inquiries are motivated
by particular mathematical goals and values. To take just one example,
consider the development of the concept of an abstract group.8
Though we now recognize the role of substitution groups and their
subgroups in Galois’s work around 1830, Galois himself never isolated
the concept; that was left to Cayley some 20 years later. The surprise is
that Cayley’s version passed unnoticed, as did Dedekind’s a decade
later still, simply because there weren’t enough examples of groups to
make the notion useful. It wasn’t until the 1870s, when many diverse
examples of groups had been identified—in Galois theory, number
theory, geometry, and the theory of differential equations—that the
idea of an abstract group caught on and flourished. Only at that point
did it begin to serve a clear mathematical purpose: it calls attention to
similarities between a broad range of otherwise quite dissimilar structures; it provides an elaborate and detailed general theory that can be
applied in different contexts; and it produces illuminating diagnoses of
the features responsible for particular phenomena (‘that x has feature y
isn’t due to its idiosyncrasies z or v or w, but only to its group
structure’). Incidentally, it wasn’t until the 1920s that group theory
entered physics, where it is now a central theme.
This, then, is the first strand to the story of how mathematics came
to separate from natural science—the pursuit of various purely mathematical goals gradually led mathematicians to new studies not motivated by their immediate application to the world—but again I think
this is only part of the story. Also worth recalling is a second familiar
8
For more, see Wussing [1969], Stillwell [2002], chapter 19, or the quick survey in [2007],
}IV.3.
8 th e problem
thread, most apparent in the evolution of attitudes toward geometry.
In the beginning, it seems even Euclid found the parallel postulate less
obvious than the rest of his fundamental assumptions; after him, generations of geometers attempted to prove it from the others. By 1800,
several mathematicians held that the parallel postulate cannot be
proved, that alternative geometries are logically consistent, but nevertheless that Euclidean geometry is the true theory of actual space.
The pivotal figure in this story is Gauss, whose efforts to prove the
parallel postulate eventually led him, in his words, ‘to doubt the truth
of geometry itself’ (see Kline [1972], p. 872). We’ve probably all heard
the sometimes-disputed tale of Gauss measuring the sum of the angles
of a triangle formed by three mountain tops, intending to test Euclid,
only to conclude that the disparity fell within the margins of experimental error. In any case, it’s beyond dispute that Gauss considered
alternative geometries to be candidates for application to the physical
world; the full flowering of this idea came when Gauss set the foundations of geometry as a topic for the qualifying exam of his student,
Riemann. Of course it was Riemannian geometry that the mathematician Grossman recommended when Einstein consulted him in 1912.
With the confirmation of General Relativity some years later,
Euclidean geometry could no longer be regarded as true of physical
space, but mathematicians were reluctant to classify it as straightforwardly false. Instead, they distinguished physical space from abstract
mathematical space, or rather, from a full range of different abstract
mathematical spaces, and Euclidean geometry was seen as true in some
and false in others among these. (Resnik ([1997], p. 130) calls this sort
of move a ‘Euclidean rescue’.) At that point, it became natural to
regard mathematicians as providing a well-stocked warehouse of abstract structures from which the natural scientist is free to select
whichever tool best suits his needs in representing the world.9
This, then, is the second strand in the story of how mathematics
pulled away from science. This time applications are involved, as they
weren’t in the first, purely mathematical strand, though the tale of
Euclidean rescue is still one visible primarily from the mathematician’s
9
See [2007], }}IV.2.iii and IV.4 for further discussion and references.
the problem
9
point of view: mathematical theories are protected from empirical
falsification by positing a special realm of abstracta about which they
remain true. The moral for the natural scientist, for the application of
mathematics, is that we now have a wide variety of mathematical
options and that it may take delicate empirical investigation to determine which works best for a given application. Nevertheless, the
mathematics that is successfully applied might still be regarded as the
native language of the Book of Nature, just as Galileo understood it so
long ago.
The third and final strand I’d like to touch on here, perhaps less
familiar than the two rehearsed so far, originates directly from the
point of view of the natural scientist. We’ve so far considered aspects of
the historical rise of pure mathematics, but over roughly the same
period there was a profound shift in the common understanding of
how applied mathematics relates to the world. To get a feel for this,
let’s return to the Scientific Revolution, to Galileo’s heir, Sir Isaac
Newton.
Like Galileo, Newton views the world as mathematical in design;
following Galileo in method also, he sets out to give a mathematical
description of gravitational force without concern for the mechanism
that produces it.10 In a famous passage at the end of the Principia, he
writes:
I have explained the phenomena of the heavens and of our sea by the force of
gravity, but I have not yet assigned a cause to gravity. . . . I do not feign
hypotheses. For whatever is not deduced from the phenomena must be called a
hypothesis; and hypotheses, whether metaphysical or physical, or based on occult
quantities, or mechanical, have no place in experimental philosophy. . . . it is
enough that gravity really exists and acts according to the laws that we have set
forth. (Newton [1687], p. 943)
This move was especially liberating given the context: Newtonian
gravity was not a natural fit for the prevailing Cartesian picture of
action by contact forces, not to mention that the Cartesian doctrine of
10
Cf. Kline [1972], p. 334: ‘we should note how completely Galileo’s program was
accepted by giants such as Newton’.
10
the problem
vortices in the plenum was highly problematic in its own right.11
Newton champions the mathematical description of the motions we
can straightforwardly observe—what he calls ‘manifest qualities’—
over hypothetical explanations of those motions in terms of hidden
causes.12
Altogether, Newton paints a remarkable mathematical portrait of
the universe: from the general laws of motion and the law of universal
gravitation, he explains the motions of the planets, the action of the
tides, the trajectories of comets, the shape of the earth, and more,13 and
he does so while inventing the required mathematics. The only weakness in all this is his mathematical conservatism. Unlike Leibniz,14 who
developed the rudiments of the calculus at roughly the same time,
Newton didn’t think of his mathematical techniques as constituting a
general theory or method: rather than devising all-purpose algorithms,
he was content to solve one individual problem after another.15 Worse
11
See e.g. Slowik [2005], for more on Descartes’s physics. Cf. Smith [2002], pp. 141–142:
‘As Newton well realized . . . no hypothetical contact mechanism seems even imaginable to
effect “attractive” forces among particles of matter generally. The Scholium [Newton [1687],
pp. 588–589] thus occurs at the point where adherents to the mechanical philosophy would
start viewing Newton’s reasoning as “absurd” (to use the word Huygens chose privately).
The Scholium attempts to carry the reader past this worry, but not by facing the demand for a
contact mechanism head-on. Instead, Newton warns that he is employing mathematically
formulated theory in physics in a new way, with forces treated abstractly, independently of
mechanism’.
12
Cf. Shapiro [2002], p. 228: ‘Newton believed that by formulating his theories phenomenologically, in terms of experimentally observed properties, or principles deduced from
them, without any causal explanations (hypotheses) of those properties, he could develop a
more certain science’.
13
Cf. Newton [1687], p. 382: ‘the basic problem of philosophy [i.e., natural science]
seems to be to discover the forces of nature from the phenomena of motions and then to
demonstrate the other phenomena from these forces. . . . we derive from celestial phenomena
the gravitational forces by which bodies tend toward the sun and toward the individual
planets. Then the motions of the plants, the comets, the moon, and the sea are deduced from
these forces’.
14
For these methodological contrasts, see e.g. Cohen and Smith [2002a], pp. 20–22, Hall
[2002], Kline [1972], pp. 378–380.
15
Cf. Truesdell [1981], p. 98: ‘Newton’s Principia . . . is a monument of human achievement; it deserves the admiration and esteem of everyone. Should an engineer study it with a
view to using its contents to determine the motion of a capsule projected into space, he
would be gravelled. Motions there are in abundance, but no general equations. Each motion
furnishes a new problem and is treated by itself. Examples there are, but no algorism:
towering concepts and a magnificent approach, certainly, but no method’.
the problem
11
yet for the progress of British mathematics was Newton’s insistence on
purely synthetic geometric methods. Though he sometimes used
Descartes’s analytic techniques, especially early on, he came to regard
them as merely heuristic; the justifications in the Principia are overwhelmingly geometric. Kline writes:
Newton did not really believe that he had departed from Greek geometry.
Though he used algebra and coordinate geometry, which were not to his
taste, he thought his underlying methods were but natural extensions of pure
geometry. (Kline [1972], p. 384)16
For Newton, as for Galileo, the language of the universe was synthetic
geometry,17 and British mathematicians loyally, if ill-advisedly,
continued in Newton’s footsteps.18
16
Incidentally, Stein notes Newton’s claim, in the preface to Principia, that the principles
of geometry are ‘obtained from other fields’ and that ‘geometry is founded on mechanical
practice’ (Newton [1687], p. 382) and remarks ‘I am not aware of any other mathematician or
philosopher of the seventeenth century who expressed such a view . . . Gauss appears to have
been the first mathematician of stature (after Newton) to have come—and only after a
struggle—to hold seriously the view that the grounds of geometry are empirical’ (Stein
[1990], pp. 30, 44).
17
Cf. Guicciardini [2002], p. 323: ‘According to the Galilean tradition the Book of
Nature is written in geometric terms. Newton endorsed this tradition’.
18
See e.g. Kline [1972], pp. 380–381: Because of the priority dispute between Newton
and Leibniz, ‘the English and Continental mathematicians ceased exchanging ideas. Because
Newton’s major work and first publication on the calculus, the Principia, used geometrical
methods, the English continued to use mainly geometry for about a hundred years after his
death. The Continentals took up Leibniz’s analytical methods and extended and improved
them. These proved to be far more effective; so not only did the English mathematicians fall
behind, but mathematics was deprived of contributions that some of the ablest minds might
have made’. When the British finally began to import Continental analysis in the early 19th
century, the principles of Leibniz’s dy notation were called ‘d-ism’, as opposed to ‘dot-age’
for the use of Newton’s y (see Kline [1972], p. 622). See also Smith and Wise [1989], pp.
151–152: ‘The [British] reformers originally saw it as their mission to bring the most powerful
techniques of mathematical analysis . . . to the moribund centres of mathematical nonlearning in Britain. Of greatest immediate importance was replacing the cumbersome system
of dots in the Newtonian fluxional notation with the d’s of Leibnizian differentials. Symbols
nearly as political as they were mathematical, the d’s represented youth and progress in the
modern age. Babbage later claimed that in 1812, as the Memoirs of the Analytical Society neared
publication, he had suggested a more apt title: “The Principle of pure D-ism in opposition to
the Dot-age of the University”. His pun on deism suggests the ideology of natural law that he
espoused and how it could be embedded in the symbols of mathematics. Dr Thomson, while
no deist in the religious sense, agreed . . . that the “inferiority” of dot-age had been “a
principle cause of the small progress made in later times by British mathematicians”’.
12
the problem
Leibniz labored under no such restrictions. As Kline describes him, he
Was a man of vision who thought in broad terms, like Descartes. He saw the
long-term implications of the new ideas and did not hesitate to declare that a
new science was coming to light. (Kline [1972], p. 384)
Leibniz’s algebraic, symbolic approach proved far more flexible and
effective than Newton’s synthetic geometry, and his Continental
followers were the ones to expand and improve the methods of the
calculus. The historians Cohen and Smith write:
It was left to individuals within the Leibnizian tradition to recast the Principia
into the symbolic calculus. What became clear in this process was the
superiority of purely symbolic methods . . . With this realization the fundamental step in problems of physics ceased being one of finding an adequate
geometric representation of the quantities involved, and instead became one
of formulating appropriate differential equations. (Cohen and Smith [2002a],
p. 22)19
This was the job of the 18th century, carried out largely on the
Continent.
This time the key figure is Euler. What we know as Newton’s laws
of motion—including the famous F=ma—were actually formulated by
Euler in the mid-1700s, but that’s just the tip of the iceberg:
Euler’s mathematical productivity is incredible. His major mathematical fields
were the calculus, differential equations, analytic and differential geometry of
curves and surfaces, the theory of numbers, series, and the calculus of variations. This mathematics he applied to the entire domain of physics. He created
analytic mechanics (as opposed to the older geometrical mechanics) and the
subject of rigid body mechanics. He calculated the perturbative effect of
celestial bodies on the orbit of a planet and the paths of projectiles in resisting
media. . . . He investigated the bending of beams and calculated the safety load
of a column. . . . He was the first to treat the vibrations of light analytically and
19
Cf. Truesdell [1968], pp. 92–93: ‘Except for certain simple if important special problems, Newton gives no evidence of being able to set up differential equations of motion for
mechanical systems. . . . As we shall see, a large part of the literature of mechanics for sixty
years following the Principia searches various principles with a view to finding the equations
of motion for the systems Netwon had studied and for other systems nowadays though of as
governed by “Newtonian” equations’.
the problem
13
to deduce the equation of motion taking into account the dependence on the
elasticity and density of the ether . . . The fundamental differential equations
for the motion of an ideal fluid are his. (Kline [1972], pp. 401–402)20
Euler’s collected works now run to nearly 90 volumes.21
In fact the 18th century was tremendously productive in all these
areas; perhaps it’s not surprising that along the way the foundational
difficulties experienced by Newton and Leibniz were only exacerbated
as mathematical analysis expanded and deepened. The momentous
shift away from Newton’s geometric methods toward the symbolic
techniques of Leibniz not only opened the way to bold new developments at breathtaking speed, but also served to relax the level of
rigor downwards from the high standard traditionally associated with
the Greeks and their followers; Lacroix remarks ‘Such subtleties as the
Greeks worried about we no longer need’.22 Kline describes the
situation this way:
Eighteenth-century thinking was certainly loose and intuitive. Any delicate
questions of analysis, such as the convergence of series and integrals, the
interchange of the order of differentiation and integration, the use of differentials of higher order, and questions of existence of integrals and solutions of
20
See also Kline [1972], pp. 402–403: ‘Euler did not open up new branches of mathematics. But no one was so prolific or could so cleverly handle mathematics; no one could
muster and utilize the resources of algebra, geometry, and analysis to produce so many
admirable results. Euler was superbly inventive in methodology and a skilled technician. One
finds his name in all branches of mathematics: there are formulas of Euler, polynomials of
Euler, Euler constants, Euler integrals, and Euler lines’.
21
Several of these volumes include scholarly introductions by Truesdell who writes
([1968], p. 106): ‘Euler was the dominating theoretical physicist of the eighteenth century.
His work is undervalued in the usual, vague historical works. . . . The great bulk of [his]
publication is not the only impediment to a just historical estimate of what he did. He put
most of mechanics into its modern form; from his books and papers, if indirectly, we have all
learned the subject, and his way of doing things is so clear and natural as to seem obvious. In
fact, it was he who made mechanics simple and easy, and for the straightforward it is
unnecessary to give references. In return, the scientist of today who consults Euler’s later
writings will find them perfectly modern, while other works of that period require effort and
some historical generosity to be appreciated’. Much more could be said in Euler’s praise, but
let me just add this stunning fact: ‘of the entire corpus of research on mathematics, theoretical
physics, and engineering mechanics published from 1726 to 1800’ his writings ‘alone account
for approximately one third’ (Calinger [1975], p. 211).
22
This appears in the 1810 preface to his three-volume compendium of 18th-century
differential and integral calculus. See Kline [1972], p. 618, for the reference.
14
the problem
differential equations, were all but ignored. That the mathematicians were
able to proceed at all was due to the fact that the rules of operation were clear.
Having formulated the physical problems mathematically, the virtuosos got to
work, and new methodologies and conclusions emerged. . . . How could the
mathematicians have dared merely to apply rules and yet assert the reliability
of their conclusions? (Kline [1972], p. 617)
The answer lies in the virtual identification of mathematics and natural
science inherited from Galileo and Newton:
Their technical skill was unsurpassed; it was guided, however, not by sharp
mathematical thinking but by intuitive and physical insights. (Kline [1972],
p. 400)
The physical meaning of the mathematics guided the mathematical steps and
often supplied partial arguments to fill in nonmathematical steps. The
reasoning was in essence no different from a proof of a theorem of geometry,
wherein some facts entirely obvious in the figure are used even though no
axiom or theorem supports them. Finally, the physical correctness of the
conclusions gave assurance that the mathematics must be correct. (Kline
[1972], p. 617)
Notice that if the result is a method that we don’t quite recognize as
mathematical, it also isn’t what we normally think of as physical,
either.
For our purposes, though, these subtleties are less important than
the clear line of influence tracing back to Galileo and Newton. Euler
remarks:
The generality I here take on . . . reveals to us the true laws of Nature in all
their brilliance. (as quoted by Truesdell [1981], p. 113)
Kline summarizes that for the 18th century, ‘mathematics was simply
unearthing the mathematical design of the universe’ (Kline [1972],
p. 619). Clifford Truesdell, the great practitioner and historian of
rational mechanics, remarks that its statements
are called phenomenological, because they represent the immediate phenomena
of experience, not attempting to explain them in terms of corpuscles or other
inferred (or hypothesized) quantities. (Truesdell [1960], p. 22)
the problem
15
Thus we find both key elements from Galileo and Newton continuing
here: the conviction that mathematical theories truly represent the
underlying mathematical structure of the world, and endorsement of
these theories as describing phenomena directly, without appeal to
theoretical hidden causes.
This same world view carries forward into the 19th century as well,
perhaps best exemplified by Fourier’s ground breaking work on the
dynamics of heat. In the opening sentences of what has been called his
great ‘mathematical poem’,23 first published in 1822, Fourier announces that
Primary causes are unknown to us; but are subject to simple and constant laws,
which may be discovered by observation, the study of them being the object
of natural philosophy.
Heat, like gravity, penetrates every substance of the universe, its rays occupy
all of space. The object of our work is to set forth the mathematical laws which
this element obeys. (Fourier [1822], p. 1)24
In the following paragraph he makes explicit his connection to Galileo
and Newton.25 At the end he concludes:
The chief results of our theory are the differential equations of the movement of heat in solid or liquid bodies, and the general equation which
relates to the surface. The truth of these equations is not founded on any
physical explanation of the effects of heat. In whatever manner we please to
imagine the nature of this element[26] we shall always arrive at the same
equations, since the hypothesis which we form must represent the general
and simple facts from which the mathematical laws are derived. (Fourier
[1822], p. 464)
23
See Smith and Wise [1989], p. 149, for references.
See also Fourier [1822], p. 7: ‘Profound study of nature is the most fertile source
of mathematical discoveries. . . . it is . . . a sure method of forming analysis itself ’.
25
The only other predecessor mentioned is Archimedes, whose name often appears in
such contexts, e.g., Kline ([1972], p. 401) refers to Euler as ‘the man [of the eighteenth
century] who should be ranked with Archimedes, Newton and Gauss’. Machamer [1998a]
makes the case for Galileo’s debt to Archimedes.
26
‘Whether we regard it as a distinct material thing which passes from one part of space to
another, or whether we make heat consist simply in the transfer of motion’.
24
16
the problem
Once again we see the conviction that nature is mathematical and that
its laws can be directly observed and hold true whatever mechanisms
may underlie them.
As it happens, a lively controversy arose between Fourier and
Poisson, whose rival theory of heat appeared in 1835.27 Poisson’s
objections actually first appear in a 1808 paper of Laplace, written in
response to an early version of Fourier’s work, where they trace to this
Laplacian credo:
I have wanted to establish that the phenomena of nature reduce in the final
analysis to action ad distans from molecule to molecule, and that the consideration of these actions ought to serve as the basis of the mathematical theory of
these phenomena. (As quoted by Smith and Wise [1989], p. 160)
Poisson took this approach to heat, positing a complex microstructure
of molecules and caloric fluid,28 summing (integrating) over their
interactions and eventually generating a differential equation that
differs from Fourier’s by an extra term. This he embraced as representing a novel prediction of the theory—that conductivity varies with
absolute temperature—subject to experimental test.
Fourier also begins with molecules, but instead of devising a theory
to test, he works directly from experiment in the first place: first the
observation that a warmer body loses heat to a cooler body at a rate
proportional to the temperature difference; second that radiant heat
doesn’t penetrate a thin foil. He concludes that
If two molecules of the same body are extremely near, and are at unequal
temperatures, that which is the most heated communicates directly to the
other during one instant a certain quantity of heat; which quantity is proportional to the extremely small difference of the temperatures. (Fourier [1822],
pp. 456–457)
27
Here I follow Wise [1981], pp. 23–29, and Smith and Wise [1989], pp. 155–162.
Cf. Smith and Wise [1989], p. 160: ‘an explicit model of the relation between
ponderable molecules and caloric fluid in a solid, incorporating both the free caloric radiated
from the molecules and responsible for temperature, and the bound or latent caloric involved
in changes of phase. His model attributed to the radiating molecules the full complexity of
observable objects, including radiation to finite distances, radiation rates between molecules
proportional to finite temperature differences, and a correction factor depending on absolute
temperature to account for possible non-linearity’.
28
the problem
17
and
The layers in contact are the only ones which communicate their heat
directly . . . There is no direct action except between material points extremely
near. (Fourier [1822], p. 460)
By a subtle shift of the term ‘molecule’ from a physical body to an
infinitesimal volume, Fourier establishes his differential equation—
which he regards as ‘rest[ing] on observations alone rather than on
any hypothesis as to the true nature of radiated heat’ (Wise [1981],
p. 25)—so for all his talk of ‘molecules’, Fourier in fact treats heat as a
continuous flow. From Fourier’s perspective, Poisson’s novel prediction can be easily accommodated at a certain point in his derivation if
the purported variability is in fact observed; until then, its presence in
the equation is an inappropriate extension of theory beyond experimental fact.
Given the stark contrast between their methodologies, Laplace and
Poisson’s objections to Fourier are predictable. Most fundamentally,
the charge is that Fourier is masking the underlying physics: the
differential heat equation is not the literal truth, but an approximation
(first a finite sum over the molecules involved is treated as an integral,
then the integral equation is transformed into the familiar differential
equation).29 Of course, for Fourier, the differential equation is fundamental, a direct representation of observed behavior. A similar disagreement concerns the transition between the object under
consideration and its environment; the historian Norton Wise puts it
this way:
Because temperature to Laplace was a density of caloric, it could never change
abruptly. Fourier, however, treated the boundary as a surface of no thickness
across which the temperature jumped between internal and external values.
[For Laplace,] an acceptable analysis would require a boundary of some
thickness within which temperature changed gradually to the external value.
(Wise [1981], p. 26)
29
Cf. Wise [1981], p. 28: ‘For Laplace and Poisson . . . the differential equations were not
the fundamental representation of the physics; they were to be established only as transformations of integral equations, where the integrals represented physical sums over effects of
isolated sources’.
18
the problem
Given the unsatisfactory state of caloric theory, Fourier’s position
may be seen as running parallel to Newton’s rejection of Descartes’s
vortices.
The denouement of this long story springs from a momentous
scientific development that gathered energy over the course of the
19th century and culminated in the opening decades of the 20th. For all
the talk of ‘corpuscles’ or ‘atoms’ or ‘molecules’ throughout the time
period we’ve been reviewing, there was no responsible view of their
nature—tiny infinitely hard ball-bearings,30 mathematical points,31
infinitesimal volumes,32?—and the hypothesis of underlying, invisible,
discrete structure of matter richly deserved its Newtonian expulsion
from experimental natural science. But this began to change with
Dalton’s experimental work in the first decade of the 19th century.
It’s often surprising to those of us with little background in the history
of science that the discipline we now know as chemistry was such a
relative late-comer. While Galileo was studying free fall and Newton
was writing the Principia, chemistry was largely alchemy;33 the beginning of ‘modern’ chemistry is typically set around 1750, linked to the
work of Lavoisier,34 and a viable atomic theory of chemical combination began only with Dalton, roughly contemporary with Fourier.
Dalton proposed that a sample of an element consists of many
identical atoms of constant weight, that the atoms of different elements
are of different weight, that these atoms remain unchanged through
chemical reactions, and that a chemical compound is composed of
many identical molecules, each of which is composed of atoms of its
30
See e.g. Smith and Wise [1989], pp. 155–156: ‘Newton justified his programme through
the “analogy of nature”, arguing that whatever held true for all observable objects has also to
hold for their unobservable parts . . . Since all observable objects possessed the qualities of
extension, hardness, impenetrability, mobility, inertia, and gravitation, so also did its parts;
they were infinitely hard atoms of finite size that attracted one another with a force varying as
the inverse square of the distance between their centres, like perfect planets or marbles’. Of
course such a hypothesis would not appear in Newton’s official experimental science (see
Shapiro [2002], p. 228).
31
As in Laplace, and before him, Boscovich. See Smith and Wise [1989], p. 156.
32
As we’ve seen in Fourier.
33
Boyle (1627–1691) was an exception (see Partington [1957], pp. 66–77). Newton was
not; the so-called ‘other Newton’ spent much time and energy on alchemy (see Cohen and
Smith [2002a], pp. 23–29).
34
See Idhe [1964], chapter 3; Partington [1957], chapter VII.
the problem
19
constituent elements. This simple hypothesis immediately explained
the known laws of chemical combination and then swept through
chemistry during the first half of the 19th century; by 1860 stable
atomic weights had been measured and confirmed, and ‘the atom
[came] into general acceptance as the fundamental unit of chemistry’
(Idhe [1964], p. 257). Beginning in mid-century, atomic theory spread
into physics with the kinetic theory of Maxwell and Boltzmann. By
the end of the 1800s, the atomic theory was a well-developed scientific
hypothesis with considerable empirical support.35
Meanwhile, the 19th century also saw the invention and development of classical thermodynamics by Carnot and Clausius; its familiar
second law states that entropy never decreases in a closed system, for
example, that spilt milk doesn’t spontaneously return to the glass. The
physical chemist Jean Perrin sees Carnot as heir to Galileo, describing
his ‘inductive’ method this way:
Each of these principles [of thermodynamics] has been reached by noting
analogies and generalising the results of experience, and our lines of reasoning
and statements of results have related only to objects that can be observed and
to experiments that can be performed. . . . in the doctrine . . . there are no
hypotheses. (Perrin [1913], p. vii)
Another physical chemist of the period, Pierre Duhem, begins his account
of the history with Newton, cites Fourier with approval, and concludes
that late 19th-century physicists have been ‘led . . . gradually back to the
sound doctrines Newton had expressed so forcefully’ (Duhem [1906],
p. 53). Here we find a true successor to the purely phenomenological
methods we’ve been tracing; the late 19th-century descendants of the
Galileo/Newton/Euler/Fourier line regarded thermodynamics as ‘the
very epitome of a scientific method of analogy and classification . . . the
apex of inductively derived . . . science’ (Nye [1972], p. 34).36
Thus the familiar battle lines were drawn, between those who
explain phenomena by appeal to hidden structures and those who
describe what they see in terms of differential equations—except that
For further discussion and references, see [1997], pp. 135–142, [2007], }IV.5.
Cf. Einstein [1949], p. 33: ‘Classical thermodynamics is the only physical theory of
universal content concerning which I am convinced that, within the framework of the
applicability of its basic concepts, it will never be overthrown’.
35
36
20 the problem
this time the hypothesis in question, atomic theory, is more highly
developed and empirically successful than ever before. Those opposed
to hypotheses raised the familiar objection that atomic theory goes
beyond experience; indeed it was argued that atoms are in principle
inaccessible to empirical test.37 To this general offense, one more
specific was added, namely, that it conflicts with the favored theory,
with classical thermodynamics:
[atomic theory] robbed Carnot’s principle of its claim to rank as an absolute
truth and reduced it to the mere expression of a very high probability. (Perrin
[1913], p. 86)
According to kinetic theory, the spilt milk might spontaneously
reassemble in the glass, though this is highly unlikely.
The conflict between the atomists and their thermodynamical
opponents was so acute that Einstein, in one of his remarkable series
of papers in 1905, sets out
to find facts which would guarantee as much as possible the existence of atoms
of definite finite size. (Einstein [1949], p. 47)
He does so by re-deriving kinetic theory (because he was ‘not acquainted with earlier investigations of Boltzmann and Gibbs’ (op. cit.))
and predicting in mathematical detail the behavior of ‘bodies of microscopically-visible size suspended in a liquid’ (Einstein [1905], p. 1).
For all his efforts, Einstein was skeptical that experiments of the
required precision were possible,38 but Perrin, a brilliant experimentalist, was in fact able ‘to prepare spherules of measurable radius’
(Perrin [1913], p. 114) and to confirm Einstein’s predictions in a series
of experiments on Brownian Motion carried out around 1910. Poincaré, along with other leading skeptics, was immediately converted,
declaring to a 1912 conference that ‘the atom of the chemist is now a
reality’.39 And this consensus has only grown stronger since.
37
Cf. Perrin [1913], p. 15: ‘It appeared to them more dangerous than useful to employ
a hypothesis deemed incapable of verification’.
38
See Nye [1972], p. 135.
39
As quoted in Nye [1972], p. 157. The two well-known opponents of atomic theory
who weren’t converted—Duhem and Mach—both died in 1916. Wilson [2006], pp. 356–
369, 654–659, argues that both were led to their anti-atomism in part because of real
the problem
21
The upshot of this transformation, for our purposes, is an equally
profound change in the understanding of how mathematics relates to the
world. Before his conversion, Poincaré mused on the consequences that
would follow if the kinetic theory should turn out to be correct:
Physical law will then take an entirely new aspect; it will no longer be solely a
differential equation. (as quoted in Nye [1972], p. 38)
Much as Laplace and Poisson insisted—though decidedly not for the
reasons they gave!—the differential equation for heat flow is actually
an approximation, an idealization, what Richard Feynman describes as
‘a smoothed-out imitation of a really much more complicated microscopic world’ (Feynman et al [1964], p. 12–12). The same goes for all
the wonderful episodes of applied mathematics developed by Euler
and his successors, and indeed, contemporary applied mathematicians
take great care to determine precisely when various idealizations and
simplifications that underlie their central differential equations can be
counted both beneficial and benign.
To take just one example, consider the case of fluid dynamics. D. J.
Tritton, the author of one recent textbook, observes:
The equations concern physical and mechanical quantities, such as velocity,
density, pressure, temperature, which will be supposed to vary continuously
from point to point throughout the fluid. How do we define these quantities
at a point? To do so we have to make what is known as the assumption of the
applicability of continuum mechanics or the continuum hypothesis[40]. We
suppose that we can associate with any volume of liquid, no matter how
small, those macroscopic properties that we associate with the fluid in
bulk. . . . Now we know that this assumption is not correct if we go right
down to molecular scales. We have to consider why is it nonetheless
plausible to formulate the equations on the basis of the continuum hypothesis. (Tritton [1988], p. 48)
difficulties in the foundations of classical mechanics. Duhem also had religious motivations:
he held that religious revelation, not physical science, is the proper source of information
about underlying metaphysics (see Duhem [1906], appendix).
40
Of course this is just an amusing terminological coincidence, not an unexpected
appearance of the set-theoretic continuum hypothesis!
22 the problem
Suppose, for example, that our model41 assigns a temperature to every
point in a three-dimensional volume. In fact we know that temperature is
an average energy over a group of molecules; the ‘temperature’ of our
fluid point will function properly only if it successfully stands in for a small
volume. Too small a volume will contain only a few molecules that come
and go at random, so its average energy will be subject to large fluctuations; too large a volume will include areas of significantly different
average energies. ‘The applicability of the continuum hypothesis depends
on there being a significant plateau’ between these two extremes:
One may regard [the intermediate volume] as being an infinitesimal distance
so far as macroscopic effects are concerned, and formulate the equations (as
differential equations implicitly involving the limit of small separations) ignoring the behavior on still smaller length scales. (Tritton [1988], p. 50)
There is more to it than that, but this gives the flavor of the applied
mathematician’s task.
But, even if the vaunted differential equations of Euler’s analytical
mechanics and the observationally perfect laws of thermodynamics are
now regarded as ‘smoothed-out imitation[s] of a really much more
complicated microscopic world’ (Feynman et al [1964], p. 12–12),
perhaps there remains room for literal description where continuum
mathematics seems more fundamental. Unfortunately, there is little
comfort from this quarter: consider, for example, the difficulties with
the self-energy of a point particle in classical electrodynamics42 or the
uncertainties about the small-scale structure of relativistic spacetime.43
It seems our best hope actually lies in the opposite direction, in the
discrete44—in kinetic theory or statistical mechanics—where the underlying microstructure is taken seriously and the phenomenological
principles of earlier theories are in some sense recovered from it.
41
Philosophers of science use the term ‘model’ in many senses; see Emch and Liu [2002],
}1.3, for a bewildering survey. I use the term simply for an abstract mathematical object,
ultimately (we might as well say) for a set (whose existence is presumably provable from the
axioms of set theory). The ‘temperature’ assigned here is just a real number.
42
See [1997], pp. 147–149, for discussion and references.
43
See [1997], pp. 149–151, for discussion and references.
44
In [2007], Part III and }IV.2.ii, I argue that a rudimentary logic and elementary
arithmetic are literally true of many aspects of the ordinary macro-world. The focus here is
on the status of more advanced scientific theorizing.
the problem
23
The first triumph of kinetic theory is the derivation of the Ideal Gas
Law.45 Of course, this proof begins with a ‘gas’ made up of point
masses that don’t interact with each other and engage only in perfectly
elastic collisions with the walls of the container. This is clearly an
abstract model, not a description of real gases, for which the Ideal
Gas Law holds only under special conditions (at low densities and high
temperatures) and even then only as a good approximation. The
model can be improved by replacing the point masses with tiny hard
spheres of finite radius and allowing for forces acting between them;
the result is van der Waals’s equation.46 Still, a van der Waals gas
remains an abstract model, if one somewhat closer in structure to a
real gas than the ideal gas model. This process can be continued,47 but
in the end the behavior of the gas elements is governed by quantum
mechanics. Unfortunately, despite the stunning success of quantum
mechanics as a predictive device, we still have no firm grasp of what
worldly features underlie its various mathematical constructs.48
The promise of literal description is perhaps greater in elementary
statistical mechanics, where the reasoning is almost purely combinatorial. Whatever the small discrete elements of an isolated49 mole of gas
are actually like, they can be properly described as occupying the left or
the right half of the box that contains them.50 Suppose I’m interested
45
For textbook treatment, see e.g. McQuarrie and Simon [1997], }16–1, or Engel and
Reid [2006], }16.1.
46
For textbook treatment, see e.g. McQuarrie and Simon [1997], }16–2, 16–7, or Brown
et al [2006], }10.9.
47
Cf. McQuarrie and Simon [1997], p. 648: ‘There are more sophisticated equations of
state (some containing more than 10 parameters!) that can reproduce the experimental data to
a high degree of accuracy over a large range of pressure, density and temperature’, or Engel
and Reid [2006], p. 9: ‘there are other more accurate equations of state that are valid over a
wider range than the van der Waals equation. Such equations of state include up to 16
adjustable substance-specific parameters’.
48
See [2007], }}III.4, III.6, for discussion and references.
49
Perfect thermal isolation is itself an idealization, of course, but let me set this aside. We
might imagine that a small error factor has been included—so we only assume the box to be
very well insulated—and that the error is too small to affect the main conclusions.
50
Determinate particle locations give way to probability densities over such locations in
ordinary quantum theory, and further mysteries arise in attempts to formulate a relativistic
quantum mechanics of particles (see Malament [1996] for discussion). Here I’m assuming that
whatever our ultimate understanding of the structure of our mole of gas in the small, it will
somehow reproduce enough particle structure to underwrite this reasoning from elementary
statistical mechanics.
24 the problem
in what portion of the Avogadro’s number of molecules is located in
the left half of the box. I can straightforwardly calculate how many
divisions of the individual molecules into one half or the other—how
many ‘micro-states’—are configurations with, say, one-third of the
molecules in the left half of the box. I can also calculate how many
micro-states there are all together, and I can compute the ratio of the
first number to the second. All this is literal description of the situation.
If I now assume that all the micro-states are equally likely, this ratio
I’ve computed is the probability that one-third of the molecules are
now in the left half of the box. With a few more such calculations,
I begin to realize that this is dramatically less likely than something
closer to a 50/50 split. Assuming my probabilistic assumption is correct, this is still straightforwardly literal, a matter of combinatorial fact
about any collection of this large but finite size. It’s comparable in
status to 2þ2=4 as a description of the total number of fruits that result
when we collect two apples and two oranges on the table.51
So far so good. But fairly soon I find myself wanting to compute the
most likely state directly. To do this, I consider a function f from the set
of configurations to the finite numbers, where f(x molecules in the left
side of the box) = the number of micro-states that place x molecules in
the left side of the box. The most likely state is the one where f reaches
a peak, that is to say, where the derivative of f is 0. But applying the
methods of the differential calculus to f doesn’t really make sense,
because the domain of f is finite. To make this work, I treat the domain
of f as a continuous variable—and now I’ve taken leave of literal
representation.
Ludwig Boltzmann, who pioneered this line of thought, apparently
had finitistic leanings; he cautions his readers against forgetting that the
underlying basis for his mathematical treatment lies in finite collections:
The concepts of the integral and differential calculus, cut loose from any
[finitary] atomic representation, are purely metaphysical. (Quoted with references and discussion in Emch and Liu [2002], p. 239)
In practical terms, the move from discrete sets to continuous mathematics actually takes place even earlier in our simple reasoning. The
51
See [2007], }IV.2.ii, for more on the status of elementary arithmetic.
the problem
25
trouble is that determining the number of micro-states involves computing N!, where N is Avogadro’s number (approximately 6 x 1023);
given that 100! is already 9.3 x 10157, the N! here is clearly out of
feasible computational range. The standard solution is to work with
ln(N!) and to use what’s called Stirling’s approximation: N x ln(N) N. This move is justified because ln(N!) is the sum of the Rln(n)’s as n
N
varies from 1 to N, and if N is large, this is very close to 1 lnðxÞdx,
52
which yields Stirling’s approximation. So computational practicality
counsels the move from large finite to continuous even before the
need for derivatives. I should note that this phenomenon—finite
collections that are treated more successfully with infinitary, indeed
continuum mathematics—isn’t at all special to statistical mechanics: in
garden-variety statistics, discrete phenomena like household incomes
or numbers of correct responses are routinely treated as continuous
variables for similar reasons.53
In any case, this style of reasoning in statistical mechanics leads to
Boltzmann’s definition of entropy, the statistical version of the second
law of thermodynamics, the statistical explanation of why real-world
processes seem irreversible when the underlying mechanics is reversible,54 and more practically, to the physical chemist’s ability to predict
the direction of chemical reactions.55 Efforts to justify the mathematical methods employed in these accounts involve departures from literal
description of a more conceptual nature than those discussed so far: for
example, in ergodic theory, one considers, among other things, the
evolution of a system as time goes to infinity; in the theory of the
thermodynamic limit, one takes a limit as the number of molecules in
the system goes to infinity.56 So beyond its first baby-steps, statistical
mechanics also departs substantially from literal description.
52
See e.g. McQuarrie and Simon [1997], pp. 809–815, or Arfken [1985], pp. 555–558,
for textbook discussions. This method appears in introductory treatments, e.g., Halliday,
Resnick, and Walker [2005], pp. 552–553, or Engel and Reid [2006], pp. 284–285.
53
For a textbook example, see Rice [2007], p. 60.
54
See e.g. Emch and Liu [2002], pp. 106–112.
55
For textbook treatments, see Brown et al [2006], chapter 19, or McQuarrie and Simon
[1997], chapters 20–22.
56
See Uffink [2007], }}6.1 and 6.3, respectively.
26
the problem
Under the circumstances, I think it’s fair to characterize the work of
contemporary scientists as presenting mathematical models of physical
systems much as Duhem describes them in this passage:57
When a physicist does an experiment, two very distinct representations of the
instrument on which he is working fill his mind: one is the image of the
concrete instrument that he manipulates in reality; the other is a schematic
model of the same instrument, constructed with the aid of symbols supplied by
theories; and it is on this ideal and symbolic instrument that he does his
reasoning, and it is to it that he applies the laws and formulas of physics.
(Duhem [1906], pp. 155–156)
A manometer, for example, is
On the one hand, a series of glass tubes, solidly connected to one another . . .
filled with a very heavy metallic liquid called mercury by the chemists; on the
other hand, a column of that creature of reason called a perfect fluid in
mechanics, and having at each point a certain density and temperature defined
by a certain equation of compressibility and expansion. (Duhem [1906],
pp. 156–157)
There would be little point to such talk if the relation between the
model and the physical system were a straightforward isomorphism but
the story just recounted shows that this is not true for the ubiquitous
differential equations of applied mathematics and (with minor exceptions58) we don’t appear to be in a position to make any stronger claims
about subsequent theories that use other types of mathematics. In fact,
the exact structure of those relations varies from case-to-case, as does
our level of understanding of them. When we represent a cannon ball
as a perfect sphere, the lengths, times, angles and forces involved as real
numbers, the local surface of the earth as flat, and so on, in order to
57
There is some irony in the appeal to Duhem here, as he was one of the last antiatomists. Obviously his embrace of Newton’s ‘sound doctrines’ didn’t include the idea that
mathematics correctly describes reality; Duhem [1906], pp. 133–134, actually harkens back to
Poisson’s objections to Fourier, e.g., ‘the body studied is geometrically defined; its sides are
true lines without thickness’. In fact, Duhem is a fascinating case: he opposed those who
formed hypotheses, but he also opposed those who took differential equations literally,
perhaps for reasons of the sort noted in footnote 39. This drove him to an unappealing
fictionalism about natural science (see below).
58
That is, for the likes of ‘2+2=4’ and elementary statistical mechanics.
the problem
27
determine where a given ball, fired with a given force, will land, we
have a fairly good idea of at least some of our departures from literal
truth and why they are admissible. When we represent spacetime as a
continuous manifold, we aren’t entirely sure whether or not this
constitutes a literal truth, though our well-informed hunch is that
even if it is an idealization, it’s a good one—much as Euclidean
geometry is a good approximation to the truth in most ordinary
cases. But the fact remains that the mathematics has been peeled
away from the science; the actual claim the scientist makes about the
world is that it is probably, at least approximately, similar in structure
to the mathematical model in certain respects, and that the idealizations involved are beneficial and benign for the purposes at hand.
We’ve now viewed the rise of pure mathematics from several
vantage points. First, we’ve seen how the study of many pure mathematical concepts, structures and theories arose simply because mathematicians began to pursue a range of peculiarly mathematical goals
with no immediate connection to applications. Second, we’ve seen
how Euclidean geometry, once unblushingly regarded as the true
theory of physical space, became the study of one among many abstract
mathematical spaces. Third and finally, we’ve seen how our best
mathematical accounts of physical phenomena aren’t the literal truths
Newton took them for, but free-standing abstract models that resemble the world in ways that are complex and sometimes not fully
understood. Paradoxical as it may sound, it now appears that even
applied mathematics is pure.
3. Where we are now
This story, the story of how applied mathematics became pure, has its
morals for our understanding of mathematics in both its pure and
applied forms. One clear moral for mathematics in application is that
we aren’t in fact uncovering the underlying mathematical structures
realized in the world; rather, we’re constructing abstract mathematical
models and trying our best to make true assertions about the ways in
which they do and don’t correspond to the physical facts. There are
28 the problem
rare cases where this correspondence is something like isomorphism—
we’ve touched on elementary arithmetic and the simple combinatorics
of beginning statistical mechanics, and there are probably others, like
the use of finite group theory to describe simple symmetries—but
most of the time the correspondence is something more complex,
and all too often it’s something we simply don’t yet understand: we
don’t know the small-scale structure of spacetime or the physical
structures that underlie quantum mechanics. And even this leaves
out the additional approximations and accommodations required to
move from the initial mathematical model to actual predictions. Stirling’s approximation is among the simpler of such ad hoc fixes.59
This sort of thing leads some philosophers to a despair, to something
along the lines of this from Duhem:
Our physical theories do not pride themselves on being explanations; our
hypotheses are not assumptions about the very nature of material things. Our
theories have as their sole aim the economical condensation and classification
of experimental laws. (Duhem [1906], p. 219)[60]
Agreement with experiment is the sole criterion of truth for a physical theory. (Duhem
[1906], p. 21)
59
See Wilson [2006], p. 26, for his ‘lesson of applied mathematics’: ‘Why do predicates
behave so perversely? . . . I believe the answer rests largely at the unwelcoming door of
Mother Nature. The universe in which we have been deposited seems disinclined to render
the practical description of the macroscopic bodies around us especially easy. . . . Insofar as we
are capable of achieving descriptive successes within a workable language . . . we are frequently
forced to rely upon unexpectedly roundabout strategies to achieve these objectives. It is as if
the great house of science stands before us, but mathematics can’t find the keys to its front
door, so if we are to enter the edifice at all, we must scramble up backyard trellises, crawl
through shuttered attic windows and stumble along half-lighted halls and stairwells’. See also
p. 452.
60
‘The diverse principles or hypotheses of a theory are combined together according to
the rules of mathematical analysis. . . . The magnitudes on which [the theorist’s] calculations
bear are not claimed to be physical realities, and the principles he employs in his deductions
are not given as stating real relations among those realities; therefore it matters little whether
the operations he performs do or do not correspond to real or conceivable physical transformations. All that one has the right to demand of him is that his syllogisms be valid and his
calculations accurate. . . . Thus a true theory is not a theory which gives an explanation of
physical appearances in conformity with reality; it is a theory which represents in a satisfactory manner a group of experimental laws. A false theory is not an attempt at explanation
based on assumptions contrary to reality; it is a group of propositions which do not agree with
experimental laws’ (Duhem [1906], pp. 20–21). See also Duhem [1906], p. 266.
the problem
29
Here the whole of theoretical science is regarded as a black box whose
sole purpose and achievement is to catalog experimental outcomes.
Just as the hope of a simple isomorphism between abstract model and
worldly structure seems too optimistic, this fictionalist dismissal of any
descriptive content seems too pessimistic, an over-reaction to the
complexities of scientific modeling. Though we may well be at a
complete loss as to how quantum mechanics relates to the world,
this is hardly true of the many other cases we’ve touched on; in kinetic
theory, in fluid dynamics, in practical chemistry, we have a fair idea of
what our models are on to, of where they are deficient, and of why
they work well nonetheless.
Still, one chastening conclusion does seem warranted: it appears
unlikely that any general, uniform account of how mathematics applies
to the world could cover the wide variety of cases. To take just a few of
the examples we’ve noted along the way, the point particle model of
an ideal gas works effectively for dilute gases because the occupied
volume is negligibly small compared to the total volume;61 the ‘continuum hypothesis’ works effectively in fluid dynamics because there is
a suitable ‘plateau’ between volumes too small to have stable temperature and volumes too large to have uniform temperature; the billiard
ball model of a van der Waal gas works effectively because actual
molecules do have fairly stable ‘effective radii’.62 Textbook examples
could be multiplied:
We should discuss how good an approximation [a harmonic oscillator] is for a
vibrating diatomic molecule. . . . Although the harmonic-oscillator potential
may appear to be a terrible approximation to the experimental curve [of
internuclear potential], note that it is, indeed, a good approximation in the
region of the minimum. This region is the physically important region for
many molecules at room temperature.[63] The harmonic oscillator will be a
good approximation for vibrations with small amplitudes. (McQuarrie and
Simon [1997], pp. 163–164)
61
For textbook treatment, see e.g. Engel and Reid [2006], pp. 149–150.
For textbook discussion, see e.g. McQuarrie and Simon [1997], }16–7.
63
‘Although the harmonic oscillator unrealistically allows the displacement to vary from
0 to +1, these large displacements produce potential energies so large that they do not often
occur in practice.’
62
30
the problem
Treatments of applied mathematics also include careful analyses of
when it’s appropriate to replace a discrete variable with a continuous
one:
The only difference between these last two equations is that the summation
over discrete values of the variable . . . has been replaced by integration over
the range of the variable. The preceding comparison demonstrates that the
continuous approximation is close to the exact result given by summation. In
general, if the differences between the values the function can assume are small
relative to the domain of interest, then treating a discrete variable as continuous is appropriate. This issue will become critical when the various energy levels
of an atom or molecule are discussed. Specifically, the approximation . . . will be
used to treat translational and rotational states from a continuous perspective
where direct summation is impractical. In the remainder of this text, situations
in which the continuous approximation is not valid will be carefully noted.
(Engel and Reid [2006], p. 290. See also p. 326)
Given the diversity of the considerations raised to delimit and defend
these various mathematizations, anything other than a patient case-bycase approach would appear unpromising.
But our primary focus here is on pure mathematics, and I think our
historical survey teaches us an important methodological lesson about
its pursuit. What we’ve traced is a more-or-less simultaneous rise of
pure mathematics and re-evaluation of applied mathematics. Before all
this, back in Newton’s or Euler’s day,64 the methods of mathematics
and the methods of science were one and the same; if the goal is to
uncover the underlying structure of the world, if mathematics is simply
the language of that underlying structure, then the needs of celestial
mechanics (for Newton) or rational mechanics (for Euler) are the
needs of mathematics. From this perspective, the correctness of a
new mathematical method—say the infinitary methods of the calculus
64
In fairness to Euler, I should note that Truesdell ([1981], pp. 120–121) writes: ‘Today
we look upon classical physics as providing us with mathematical models for the behavior of
physical objects. We use these models with great caution, for we are deeply aware of their
limitations . . . The Bernoullis had no idea that they were dealing with models; like Galileo,
they thought that nature herself spoke in mathematics. Euler in his middle life began to
perceive how much the mathematician replaced nature by his own conceptions’. I don’t
know what aspect of Euler’s thought Truesdell has in mind here.
the problem
31
or the expanded notion of function65—is established by its role in
application. In contrast, after the developments we’ve been tracing,
mathematics has been freed to pursue inquiries without application,
it’s encouraged to stock the warehouses with structures and leave the
choices to the natural scientists, and even the mathematical constructs
that do function in application do so with a new autonomy as freestanding abstract models. In this brave new world, where can the pure
mathematician turn for guidance? How can we properly determine if a
new sort of entity is acceptable or a new method of proof reliable?
What constrains our methodological choices?
These were among the pressing questions faced by the mathematical
community at the end of the 19th century. In Kline’s words,
The circle within which mathematical studies appeared to be enclosed at the
beginning of the century was broken at all points, and mathematics exploded
into a hundred branches. (Kline [1972], p. 1023)
This dramatic expansion brought with it the realization that for some
time mathematics had ‘rested on an empirical and pragmatic basis’, that
a ‘rapidly increasing mass of mathematics . . . rested on the loose foundations of the calculus’, that even in geometry mathematicians had
overlooked ‘inadequacies in proofs’ and ‘been gullible and relied upon
intuition’ (Kline [1972], pp. 1024–1025). All this produced a renewed
emphasis on rigor, the central tool of which was axiomatization, along
the lines of Hilbert’s axioms for geometry and Dedekind’s axioms for
the real numbers. Kline continues,
Mathematics . . . was by the end of the nineteenth century a collection of
structures each built on its own system of axioms. (Kline [1972], p. 1038)
. . . the axiomatic method . . . permitted the establishment of the logical foundations of many old and newer branches of mathematics, . . . revealed precisely
what assumptions underlie each branch and made possible the comparison and
clarification of the relationships of various branches. (Kline [1972], p. 1027)
65
In his treatment of the wave equation, D’Alembert required the initial shape of a
vibrating string to be given by an analytic expression (so as to be twice differentiable), but
Euler insisted that functions shouldn’t be limited in this way, because the most common
initial conditions—for example, a plucked string—don’t satisfy this requirement. The result
was a major revision of the notion of function, a firm push toward the modern set-theoretic
conception. See [1997], pp. 116–123, for discussion and references.
32
the problem
But it was widely appreciated that simply laying down a list of axioms
isn’t enough to establish that they succeed in describing a genuine
structure.66 So, for example, the coherence of the axioms for nonEuclidean geometries had been demonstrated by modeling them in
Euclidean geometry, and Euclidean geometry itself could be modeled
using the real numbers, but the buck has to stop somewhere.
What emerged gradually—in the theory of functions, in Dedekind’s
constructions of the reals, in the foundations of arithmetic and elsewhere—is that set theory provides a natural arena in which to interpret
the myriad structural descriptions of mathematics, to settle which are
and aren’t coherent.67 Unfortunately, early 20th-century set theory was
itself subject to considerable controversy arising from the discovery of
the paradoxes, Zermelo’s controversial proof of the well-ordering theorem, and a number of other uncertainties. Gregory Moore writes:
In the wake of Russell’s paradox, published in 1903, it was even uncertain
what constituted a set. Moreover, Zermelo’s proof itself raised a number of
methodological questions: Was it legitimate to define a set . . . in terms of a
totality of which [it] was a member . . . ? . . . Did the class . . . of all ordinals
invalidate Zermelo’s proof and entangle it in Burali-Forti’s paradox? Most
important of all, was his Axiom of Choice true? Was it a law of logic? Should
one postulate simultaneous, independent choices in preference to successive,
66
Nowadays we’d tend to say that the axioms must be consistent, but our contemporary
use of that term presupposes a developed account of syntax, semantics, and their interrelations that wasn’t available in 1900 (e.g., the completeness theorem wasn’t proved until
Gödel’s doctoral dissertation of 1930). Structuralists in the philosophy of mathematics tend
to use the less precise-sounding word ‘coherent’ (see e.g. Shapiro [1997], pp. 95, 132–136,
Parsons [2008], p. 114)—a structure exists if its description is coherent—and I follow them in
the next sentence.
67
Following up the previous footnote, Shapiro ([1997], p. 136) allows that ‘in mathematics as practiced’, set theory is used to settle questions of which axiomatizations or implicit
definitions are and aren’t coherent, and thus of which structures do and don’t exist. In
contrast, Awodey ([2004], pp. 58, 60) suggests that ‘Every mathematical theorem is of the
form “if such-and-such is the case, then so-and-so holds” . . . Of course many theorems do not
literally have this form, but every theorem has some conditions under which it obtains . . .
There is usually no question about whether such conditions are ever satisfied; rather, like
axiomatic definitions, they serve to specify the range of application of the subsequent statement’. On this basis, he denies the need for a foundation of the sort set theory has been able to
provide. Perhaps this sounds more appealing now, when the worries of 1900 have long since
been quieted; one forgets the mathematical forces that led the mathematicians of 1900 to feel
that need in the first place.
the problem
33
dependent ones? Did the cardinality of the set of choices affect the validity of
the Axiom, so that the Denumerable Axiom was true but not the Axiom of
Choice in general? . . . All these questions echoed a broader problem which
had rarely been enunciated explicitly: What methods were permissible in
mathematics? Must such methods be constructive? If so, what constituted a
construction? What did it mean to say that a mathematical object existed?
(Moore [1982], p. 85)
Clearly set theory itself stood in need of the rigorizing benefits of
axiomatization, which Zermelo set out to provide.
In his 1908 presentation of the first system of axioms for set theory,
Zermelo recognizes its special role:
Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions ‘number’, ‘order’ and ‘function’, taking
them in their pristine, simple form, and to develop thereby the logical
foundations of all arithmetic and analysis; it thus constitutes an indispensable
component of the science of mathematics. (Zermelo [1908b], p. 200)
But despite its importance,
The very existence of this discipline seems to be threatened by certain contradictions, or ‘antinomies’, that can be derived from its principles . . . to which
no entirely satisfactory solution has yet been found. (op. cit.)
The simple motivating idea that any objects can be collected into a set
‘requires some restriction’, so he aims
To seek out the principles required for establishing the foundations of this
mathematical discipline . . . we must, on the one hand, restrict these principles
sufficiently to exclude all contradictions and, on the other, take them sufficiently wide to retain all that is valuable in this theory. (op. cit.)
The result was a list of axioms that was eventually developed and
extended into the now-standard system Zermelo-Fraenkel with
Choice or ZFC.
In the century since Zermelo’s first effort, set theory has solidified its
role as the backdrop for classical mathematics. Questions of the form—
is there a structure or a mathematical object like this?—are answered
by finding an instance or a surrogate within the set-theoretic hierarchy.
Questions of the form—can such-and-such be proved or disproved?—
34
the problem
are answered by investigating what follows or doesn’t follow from the
axioms of set theory. This isn’t to say that set theory shows us what
numbers or functions really are; all set theory need claim is that certain
sets can do all the mathematical jobs required of numbers or functions. It
also isn’t to claim that we’re somehow more certain of the coherence of
ZFC than, say, that of Peano Arithmetic, or that all mathematics
could or should be done using exclusively set-theoretic methods.68
What set theory does is provide a generous, unified arena to which all
local questions of coherence and proof can be referred. In this way, set
theory furnishes us with a single tool that can give explicit meaning to
questions of existence and coherence; make previously unclear concepts
and structures precise; identify perfectly general fundamental assumptions that play out in many different guises in different fields; facilitate
interconnections between disparate branches of mathematics now all
uniformly represented; formulate and answer questions of provability
and refutability; open the door to new strong hypotheses to settle old
open questions; and so on. In this philosophically modest but mathematically rich sense, set theory can be said to found contemporary pure
mathematics.69
The remaining question, then, is how set theory itself is properly
done. What kinds of considerations count for or against a given assumption or method? Over the years, some have argued for a return to
justifications based in natural science, for example, in the case of the
Axiom of Choice. Those familiar with the history70 will recall that
Zermelo’s introduction of the Axiom in 1904 set off a fierce debate
between its proponents and its opponents. This dispute had many
strands—some metaphysical, some practical—but one feature of the
Axiom that particularly troubled many mathematicians was the so-called
68
e.g., it’s well-recognized that category-theoretic formulations are more natural than settheoretic for many mathematical purposes, but this doesn’t distinguish category theory from
various other branches of mathematics with their own vocabularies and techniques. What does
distinguish category theory is the claim that it can provide a foundation of the sort described in
the text, an alternative to set-theoretic foundations, but it remains to be seen whether or not it
can do this effectively for the whole of mathematics (including higher set theory). In any case,
what matters here is that set theory was developed at least partly to do this job, and that this fact
has consequences for its proper practice.
69
See [1997], }I.2, for more discussion of the nature of set-theoretic foundations.
70
See Moore [1982].
the problem
35
Banach-Tarski paradox: using the Axiom, a sphere can be decomposed
into finitely many parts and those parts reassembled into two spheres of
the same size as the original.71 (The key is that the parts into which the
sphere is cut are non-measurable, which is where Choice comes in.) This
conclusion seems obviously absurd from a physical point of view, so
Banach and Tarski’s result is sometimes taken as evidence against the
Axiom of Choice, as something akin to a false prediction.72
If we’ve assimilated the lesson of the third strand of our story of
how applied mathematics became pure, the strand that highlights the
role of abstract models that resemble the world only partially and
within certain limits, then the reply to this argument against Choice
is straightforward: if physical regions aren’t literally modeled by sets
of ordered triples of real numbers, then we can’t assume that all
consequences of our mathematical theory of those sets will hold for
those regions; therefore, we can’t conclusively draw our false empirical
conclusion. The realization that our model departs from reality might
lead us to modify our model, or it might, as in this case, simply lead us
to apply our model with more care, making sure not to rely too heavily
on its more esoteric aspects. What’s perhaps slightly less obvious is that
the full force of the third thread isn’t needed to defend Choice here;
the second thread is enough, the one leading to the well-stocked
warehouses of abstract structures.
To see this, ignore the third thread and suppose for the sake of
argument that physical regions are literally modeled by subsets of ℝ3.
Then the line of thought goes like this: working in a set theory with the
Axiom of Choice, we perform the Banach-Tarski construction, which
is physically ridiculous; we conclude that Choice has been empirically
disconfirmed. But isn’t it at least as reasonable to conclude that the full
power set of ℝ3 was a poor choice as a model for physical regions?73
Indeed, given the many internal mathematical considerations in favor of
71
Banach and Tarski proved this theorem in 1924. See Wagon [1985] for discussion.
Fraenkel, Bar-Hillel, and Levy [1973], p. 83, give references to this sort of reaction from
Borel and others. They also note that Hausdorff had a similarly ‘paradoxical’ result in 1914,
ten years before Banach-Tarski.
73
Again, as above, I’m not suggesting that this is what we actually do in our full
appreciation of the third strand. For most purposes, it isn’t worth complicating our model
in this way; it’s more practical just to use it with care.
72
36
the problem
the Axiom,74 wouldn’t it be considerably more reasonable to conclude
that physical regions are more effectively modeled by measurable subsets
of ℝ3? If, for example, our set theory includes sufficient large cardinals,
we might count Banach-Tarski as a good reason to model physical space
in L(ℝ), where all sets of reals are measurable.75 In any case, the
suggestion is that what’s disconfirmed is the claim that regions are best
modeled by P(ℝ3), not the Axiom of Choice.76 Once we have those
well-stocked warehouses, any candidate for an empirical confirmation
or disconfirmation of the mathematics is more reasonably viewed as
confirming or disconfirming the claim that the best model has been
identified. If this is right, then perhaps the added force of the third
thread, where we relinquish the dream of literal modeling, is more
consequential for the practice of science—cautioning us against regarding all questions about our mathematical models as real physical issues—
than it is for the practice of mathematics.
In any case, I think by now it’s clear that considerations from
applications are quite unlikely to prompt mathematicians to restrict
the range of abstract structures they admit. It’s just possible that as-yetunimagined pressures from science will lead to profound expansions of
the ontology of mathematics, as with Newton and Euler, but this
seems considerably less likely than in the past, given that contemporary
set theory is explicitly designed to be as inclusive as possible.77 More
likely, pressures from applications will continue to influence which
parts of the set-theoretic universe we attend to, as they did in the case
of Dirac’s delta function;78 in contemporary science, for example, the
74
See e.g. [1997], pp. 54–57, for summary and references.
L(ℝ) is the smallest model of ZF containing all the real numbers; the existence of a
supercompact cardinal implies that all its sets of reals are Lebesgue measurable. See Jech
[2003], pp. 650–653, or Kanamori [2003], }32. ( Jech’s hypothesis is actually the existence of a
superstrong cardinal, but the existence of a supercompact is considered the more natural
hypothesis—a generalization of measurability—and it implies the existence of a superstrong.
I come back to supercompacts in V.3.i.) For more, see II.2.iv.
76
See Urquhart [1990], p. 152, for a similar conclusion in the context of Field’s nominalization project.
77
For that matter, any hint of an important structure unrealized in the universe of sets
would most likely motivate such an expansion, whether the structure originated in science
or not!
78
The delta function takes the value zero except at x=0, but its overall integral is one.
This is impossible, of course, because the so-called function is zero almost everywhere,
75
the problem
37
needs of quantum field theory and string theory have both led to the
study of new provinces of the set-theoretic universe.79
All this simply reinforces the conclusion that we should no longer
expect science to provide the sort of methodological guidance for
mathematics that it once did, and returns us to the challenge of
isolating and understanding the proper methods of set theory. This
question becomes especially acute when we discover that we cannot
simply rest content with the time-honored axioms of ZFC. Powerful
as it is—powerful enough to provide surrogates for all classical mathematical objects and to prove versions of all classical mathematical
theorems—ZFC isn’t powerful enough to settle various natural questions that arise in set theory and the various branches of mathematics:80
are all projective point sets Lebesgue-measurable? are all Whitehead
groups free? is every uncountable set of real numbers equinumerous
with the set of all real numbers (the famous Continuum Problem)? To
settle these matters, we must add new assumptions to Zermelo’s list,
and the question of how to do so properly cannot be avoided.
By this long and circuitous route, we’re brought at last to an
appreciation of the mathematical importance of these two fundamental
questions: what are the proper methods of set theory, and why? I touch
on the first of these briefly in the next chapter, aiming to bring sharper
focus to the special features and difficulties of the second. The remainder of the book then addresses that matter of ‘why’—what is set theory
that these are the correct ways of going about it?81
which guarantees an integral of 0. Dirac noted that it could nevertheless be used ‘for
practically all purposes of quantum mechanics without getting incorrect results’ (quoted by
Pais in Pais et al [1998], p. 7). (This function actually played a role as early as the 1890s in
Heaviside’s operational calculus. See van der Pol and Bremmer [1950], pp. 1–5, 62–66, for the
history.) Efforts to rigorize the delta function eventually led to Schwartz’s theory of distributions or generalized functions (see van der Pol and Bremmer [1950], p. 84, or Zemanian
[1965]).
79
See e.g. Emch [1972], Witten [1998].
For more on these questions, see [1997], }I.3, Eklof [1976].
81
The exclusively methodological study of [1997] set this question aside (see pp. 200 –203).
After an unavoidable detour through [2007], I now return to it.
80
II
Proper Method
We’re faced with a stark challenge: what are the proper methods for
set theory and why? The goal of this chapter to address the first half
of this challenge, and to refine our appreciation of what the second
half requires of us. But first we need a clear sense of the philosophical
perspective from which I propose to approach these matters.
1. The meta-philosophy
Imagine a simple inquirer who sets out to discover what the world is
like, the range of what there is and its various properties and behaviors.
She begins with her ordinary perceptual beliefs, gradually develops
more sophisticated methods of observation and experimentation, of
theory construction and testing, and so on; she’s idealized to the extent
that she’s equally at home in all the various empirical investigations,
from physics, chemistry, and astronomy to botany, psychology, and
anthropology. She believes that ordinary physical objects are made up
of atoms, that plants live and grow by photosynthesis, that humans use
language to describe the world to one another, that social groups tend
to behave in certain ways, and so on. She also believes that she and her
fellow inquirers are engaged in a highly fallible, but partly and potentially successful exploration of the world, and like anything else, she
looks into the matter of how and why the methods she and others use
in their inquiries work when they do and don’t work when they don’t;
in these ways, she gradually improves her methods as she goes.
In the course of these investigations, our inquirer begins to notice
that logic and arithmetic are essential tools in her efforts to understand
prope r method
39
the world, and she eventually sees that the calculus, higher analysis, and
much of contemporary pure mathematics are also invaluable for getting at the behaviors she studies and for formulating her explanatory
theories. This gives her good reason to pursue mathematics herself,
as part of her investigation of the world, but she also recognizes that it
is developed using methods that appear quite different from the sort
of observation, experimentation, and theory formation that guide the
rest of her research. This raises questions of two general types. First, as
part of her continual evaluation and assessment of her methods of
investigation, she will want an account of the methods of pure mathematics; she will want to know how best to carry on this particular type
of inquiry. Second, as part of her general study of human practices, she
will want an account of what pure mathematics is: what sort of activity
is it? what is the nature of its subject matter? how and why does it
intertwine so remarkably with her empirical investigations?
In this humdrum way, by entirely natural steps, our inquirer has
come to ask questions typically classified as philosophical.1 She doesn’t
do so from some special vantage point outside of science, but as an
active participant, entirely from within. Meta-philosophical positions
that advocate this sort of approach are often called ‘naturalistic’, but
one cautionary note: our inquirer doesn’t believe as she does because
‘science says so’, as some naturalists would have it,2 but on perfectly
ordinary grounds—this experiment, that well-confirmed theory.
Though we might use the rough-and-ready term ‘scientific’ to describe
her behavior, no specific characterization of what counts as ‘scientific’
is in play here, nothing beyond the perfectly ordinary story of beginning from observation, developing more refined methods, and so on.
Indeed, any attempt at a once-and-for-all characterization of our inquirer’s methods would run counter to the ever-improving, open-ended
1
Earlier on, she would have paused to ask others, e.g., what is the nature of logical truth
and elementary arithmetic? For discussion, see [2007], Part III and }IV.2.ii.
2
Compare Burgess and Rosen’s characterization of ‘naturalism’: ‘The naturalists’ commitment is at most to the comparatively modest proposition that when science speaks with a
firm and unified voice, the philosopher is either obliged to accept its conclusions or to offer
what are recognizably scientific reasons for resisting them’ (Burgess and Rosen [1997], p. 65).
My inquirer doesn’t decide to place her faith in something called ‘science’; she is simply one
of those speaking with a firm voice, on the basis of the evidence.
40 prope r method
nature of her project. So I’m not advocating any meta-philosophical
doctrine or principle to the effect that we should ‘trust only science’; I’m
simply describing this inquirer, counting on you to get the hang of how
she would approach the various traditionally philosophical questions
we’re interested in, and hoping that you find this exercise as illuminating
as I do.
To round out this quick portrait, consider the contrast with philosophy understood as starting either before science begins or after all
scientific evidence is in, that is, philosophy as an entirely independent
enterprise. Notice that if such a philosophical undertaking intends to
correct science, or even to justify it in some way, then it isn’t effectively
separated from our inquirer’s sphere of interest: working without any
litmus test for ‘science’ or ‘non-science’, she will view it as a potential
part of her own project, out to revise or buttress her methods; faced with
such a proposal, she will want to know the grounds on which the
criticism or confirmation is based and to evaluate these grounds on her
own terms.3 To be truly autonomous, a philosophical enterprise would
grant that science is perfectly in order for scientific purposes, but insist
that there are other, extra-scientific purposes for which different methods
are appropriate.
So, for example, Bas van Fraassen holds that familiar experimental
evidence establishes the existence of unobservables (like atoms) for
scientific purposes, but that, from a philosophical or epistemic standpoint, such beliefs can never be justified. Faced with such a challenge,
our inquirer is simply baffled: all her good evidence has been declared
irrelevant, ‘merely scientific’; she’s asked to justify her belief in atoms
on other grounds for unfamiliar purposes, but she’s given no working
understanding of what those purposes are and what methods are appropriate in their service. Philosophy undertaken in such complete isolation
from science and common sense is often called ‘First Philosophy’, so I call
our inquirer a Second Philosopher.4
Let’s now focus on the Second Philosopher’s investigation of mathematics. Since we’re mainly interested in her treatment of set theory,
3
e.g., she will test the merits of Descartes’s Method of Doubt, intended to found science
more firmly (see [2007], }I.1).
4
For more, see [2007], especially Part I and }IV.1, and [2010].
prope r method
41
let’s skip lightly over the story of how her narrowly applied sense of the
subject gradually gives way to the full pursuit of pure mathematics,
recapitulating the dramatic developments surveyed in Chapter I. By
that point, the great expansion in the range of mathematical structures
studied and the variety of methods used left some uncertainty about
which of these were legitimate, and as set theory developed, it gradually
took on the role of providing a unified account of the world of classical
mathematics and its fundamental assumptions. When our Second
Philosopher is faced with contemporary set theory, we’ve seen that
questions of two types arise. The first group is methodological: what
are the proper grounds on which to introduce sets, to justify settheoretic practices, to adopt set-theoretic axioms? The second group
is more traditionally philosophical: what sort of activity is set theory?
how does set-theoretic language function? what are sets and how do
we come to know about them? Given that the Second Philosopher will
want to pursue set theory, along with her other inquiries, the most
immediate problem will be the methodological one—how am I to
proceed?—so it makes sense to begin there. To get a feel for the forces
at work, let’s review some concrete examples.
2. Some examples from set-theoretic practice
(i) Cantor’s introduction of sets
In the early 1870s, Cantor was engaged in a straightforward project
in analysis: generalizing a theorem on representing functions by trigonometric series.5 Having shown that such a representation is unique if
the series converges at every point in the domain, Cantor began to
investigate the possibility of allowing for exceptional points, where
the series fails to converge to the value of the represented function. It
turned out that uniqueness is preserved despite finitely many exceptional points, or even infinitely many exceptional points, as long as
these are arranged around a single limit point, but Cantor realized that
See Dauben [1979], chapter 2, Ferreirós [2007], }}IV.4.3 and V.3.3, for historical context
and references.
5
42 prope r method
it extends even further. To get at this extension, he moved beyond the
set of exceptional points and its limit points to what he called ‘the
derived set’:
It is a well determined relation between any point in the line and a given set P,
to be either a limit point of it or no such point, and therefore with the pointset P the set of its limit points is conceptually co-determined; this I will denote
P 0 and call the first derived point-set of P. (As translated and quoted in Ferreirós
[2007], p. 143)
Once this new set, the first derived set, P 0 , is in place, the same
operation can be applied again: with P 0 , the set of its limit points is
‘conceptually co-determined’; this P 00 is the second derived set of the
original P; and so on. Cantor then proved that if the n-th derived set of
the set of exceptional points is empty for some natural number n, then
the representation is unique.6
Of course there had been talk of point sets before Cantor navigated
this line of thought, but here for the first time a point set is regarded as
an entity in its own right, susceptible to the operation of taking its
derived set. In the words of the historian José Ferreirós,
What is really original in this contribution is that Cantor does not consider
limit points in isolation, so to say, as Weierstrass had done, but makes a step
toward a set-theoretical perspective. As a result, ‘set derivation’ is conceived as
an operation on sets. (Ferreirós [2007], p. 143)
Cantor went beyond the customary approach to analysis within the Berlin
school, with its close attention to explicit analytic representations . . . He presented the notion of limit point as an abstract one, and took the crucial steps of
considering sets of limit points and forming the derived set of a point-set. In so
doing, Cantor was turning toward an abstract approach to mathematics that
employed the language of sets. (Ferreirós [2007], p. 159)
From a methodological point of view, what’s happened is that a new
type of entity—a set—has been introduced as an effective means
toward an explicit and concrete mathematical goal: extending our
understanding of trigonometric representations.
Ferreirós ([2007], p. 160) notes that uniqueness continues to hold if the Æ-th derived set is
empty for some transfinite ordinal Æ, but Cantor apparently never made this extension.
6
prope r method
43
(ii) Dedekind’s introduction of sets
Also in the 1870s, Dedekind made early use of what we now recognize
as sets, this time in algebra, in his theory of ideals. One central strand of
this methodologically rich story7 involves Dedekind’s decision
to replace the ideal number of Kummer, which is never defined in its own
right, but only as a divisor of actual numbers . . . by a noun for something
which actually exists. (Dedekind writing in 1877; see Avigad [2006], p. 172,
for translation and references)
This ‘something which actually exists’, the ideal number, Dedekind
identifies with the set of numbers Kummer would have taken it to
divide. In his illuminating survey of Dedekind’s thought during the
development of his theory of ideals, Jeremy Avigad remarks:8
The insistence on treating [sets] of numbers . . . as objects in their own right
[has] important methodological consequences: it encourages one to speak of
‘arbitrary’ [sets], and allows one to define operations on them in terms of their
behavior as sets . . . in a manner that is independent of the way in which they
are represented.
Among Dedekind’s goals were general arguments in representation-free
terms that would then ‘ “explain” why calculations with and properties
of the objects do not depend on these choices of representations’.
By downplaying the underlying computational algorithms highlighted
by Kummer, Dedekind
Sends the strong message that such algorithms are not necessary, i.e. that one
can have a fully satisfactory theory that fails to provide them. This paves the
way to more dramatic uses of non-constructive reasoning.
Here again, sets are being introduced in service of explicit mathematical
desiderata—representation-free definitions, abstract (non-constructive)
reasoning—though Dedekind’s vision is broader than the above-cited
example from Cantor:
7
See Avigad [2006], Ferreirós [2007], chapters III, IV and VII, for more on Dedekind’s
thought.
8
All quotations in this paragraph come from Avigad [2006], pp. 173–174.
44 prope r method
It is striking that Dedekind adopts the use of [infinite sets] without so much as
a word of clarification or justification. That is, he simply introduced a style of
reasoning that was to have decisive effects on future generations, without
fanfare.
The mathematical fruitfulness of Dedekind’s innovations was dramatically demonstrated as abstract algebra went on to thrive in the hands of
Noether and her successors.9
The same drive toward new numbers as actual objects with representation-free characterizations is on display in Dedekind’s theory of
the real numbers. Here Dedekind’s goal is to provide a ‘perfectly
rigorous foundation for the principles of infinitesimal analysis’,10 and
in particular, to remove the ‘geometric evidence . . . [that] can make no
claim to being scientific’. He takes (some version of ) the theorem—a
non-empty set of reals with an upper bound has a least upper bound11—
to be ‘a more or less sufficient foundation’ for the subject; since the
calculus is held to deal with ‘continuous quantities’, he reasons, one
would expect to find the proof of this theorem resting on ‘an explanation
of this continuity’; instead one finds ‘appeal . . . to geometric representations or to representations suggested by geometry’. Dedekind sets out,
therefore, ‘to discover [the] true origin’ of the theorem, and thereby
‘secure a real definition of the essence of continuity’.
The result, of course, is his elegant definition of continuity in terms
of cuts and his theory of real numbers, each an infinite set of rationals.12
As Ferreirós notes,
The title and the whole exposition emphasize the fact that the definition of
continuity, and the proof of the continuity of [the reals], constitute the core of
the matter. This is a way of presenting the whole issue that differs from the
expositions of Weierstrass, Heine and Cantor. Dedekind writes that only a
precise definition of continuity will offer a sound foundation for ‘the investigation of all continuous domains’. (Ferreirós [2007], p. 133)
9
See McLarty [2006].
All quotations in this paragraph come from Dedekind [1872], p. 767.
11
Dedekind’s version is ‘every magnitude which grows continually, but not beyond all
limits, must certainly approach a limiting value’ (Dedekind [1872], p. 767).
12
He had already defined the integers and rationals in terms of natural numbers (see
Ferreirós [2007], p. 219).
10
prope r method
45
Weierstrass and Cantor employ infinitistic ‘constructs’ that were usual in
analysis, series and sequences respectively; Dedekind chooses to rely on a
new means for ‘construction’. The resulting theory is simpler in that every real
number corresponds to only one, or at most two, cuts, while it corresponds
to many . . . sequences . . . or . . . series. (Ferreirós [2007], p. 131)
So here again we see Dedekind preferring definitions that aren’t tied to
particular representations (like series or sequences), while pursuing
broader mathematical goals (a general theory of continuity).
Another important mathematical goal, also clearly present in this
work on real numbers, is the pursuit of rigor: ‘In science nothing
capable of proof ought to be believed without proof ’ (Dedekind
[1888], p. 790). This declaration opens Dedekind’s account of the
natural numbers, a third venue for his appeal to sets. Here he officially
lays out his background set theory and goes on to develop his account
of the natural numbers. In all these cases, we find Dedekind introducing sets in the service of explicit mathematical goals: a representationfree, non-constructive abstract algebra; a rigorous characterization of
continuity to serve as a foundation for analysis and a more general
study of continuous structures; a rigorous characterization of the
natural numbers and resulting foundation for arithmetic.
(iii) Zermelo’s defense of his axiomatization
Turning from the introduction of sets to the adoption of axioms about
them, we find Zermelo in 1908 with a range of motives. Locally, he
hopes to quiet the controversy over his proof of the well-ordering
theorem from the Axiom of Choice by proving it again from a weaker
and more precise list of assumptions.13 More globally (as noted in }I.3),
he sees his efforts as part of the Hilbertian foundational project:14
Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions ‘number’, ‘order’, and ‘function’, taking
them in their pristine, simple form, and to develop thereby the logical
foundations of all arithmetic and analysis; thus it constitutes an indispensable
component of the science of mathematics. (Zermelo [1908b], p. 200)
13
14
See Moore [1982], pp. 143–160.
See Ebbinghaus [2007], pp. 76–79.
46 prope r method
He notes the poor prospects for a compelling and fruitful definition
of ‘set’ that could found set theory—something comparable, say, to
Dedekind’s definition of ‘continuity’ and its role in founding analysis—so
he concludes that
There is at this point nothing left for us to do but to proceed in the opposite
direction and, starting from set theory as it is historically given, to seek out
the principles required for establishing the foundations of this mathematical
discipline. In solving the problem we must, on the one hand, restrict these
principles sufficiently to exclude all contradictions and, on the other, take
them sufficiently wide to retain all that is valuable in this theory. (Zermelo
[1908b], p. 200)
He goes on to propose the first axiomatization of set theory, in seven
axioms, including Choice; he then develops ‘the entire theory created
by Cantor and Dedekind’ (op. cit.) on this basis and demonstrates that
the known paradoxes cannot be so generated.
Of particular interest for our purposes are his reflections on the
proper methods for justifying axioms. Presumably their foundational
success counts in favor of his axioms as a whole, but when pressed on
the Axiom of Choice in particular, Zermelo distinguishes evidence of
two sorts:
That this axiom . . . has frequently been used, and successfully at that, in the most
diverse fields of mathematics, especially in set theory, by Dedekind, Cantor,
F. Bernstein, Schoenflies, J. König, and others is an indisputable fact . . . Such an
extensive use of a principle can be explained only by its self-evidence . . . No matter
if this self-evidence is to a certain degree subjective—it is surely a necessary
source[15] of mathematical principles. (Zermelo [1908a], p. 187)
The claim is that Choice is ‘intuitively evident’ (op. cit.), as revealed in
the informal practice of set theorists; we might now express this by
saying it is part of the informal ‘concept of set’. But, as we’ve seen,
Zermelo despairs of defining this concept with a precision adequate to
the development of set theory.
Instead he appeals to a second standard of evidence that can be
‘objectively decided’, namely ‘whether the principle is necessary for
15
Could Zermelo literally mean ‘source’ here, leaving open the question of justificatory
force? Cf. }V.4, below.
prope r method
47
science’ (op. cit.). Here he lists various outstanding problems that can be
resolved on the assumption of Choice, and concludes
So long as the relatively simple problems mentioned here remain inaccessible
[without Choice], and so long as, on the other hand, the principle of choice
cannot be definitely refuted, no one has the right to prevent the representatives of productive science from continuing to use this ‘hypothesis’—as
one may call it for all I care—and developing its consequences to the greatest
extent, especially since any possible contradiction inherent in a given point of
view can be discovered only in that way. . . . principles must be judged from
the point of view of science, and not science from the point of view of
principles fixed once and for all. (Zermelo [1908a], p. 189)
This mode of defense goes beyond the observation that his axioms allow
the derivation of set theory as it currently exists and the foundational
benefits thereof; Zermelo here counts the mathematical fruitfulness of
his axioms, their effectiveness and promise, as points in their favor.
Gödel also recognized the importance of such evidence, for example,
in this well-known passage:
Even disregarding the intrinsic necessity of some new axiom, and even in case
it has no intrinsic necessity at all, a probable decision about its truth is possible
also in another way, namely, inductively by studying its ‘success’. Success here
means fruitfulness in consequences, in particular in ‘verifiable’ consequences,
i.e., consequences demonstrable without the new axiom, whose proofs with
the help of the new axiom, however, are considerably simpler and easier
to discover, and make it possible to contract into one proof many different
proofs. . . . There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such
powerful methods for solving problems . . . that, no matter whether or not
they are intrinsically necessary, they would have to be accepted at least in the
same sense as any well-established physical theory. (Gödel [1964], p. 261)
It has become customary to describe these two rough categories of
justification as ‘intrinsic’—self-evident, intuitive, part of the ‘concept
of set’, and such like—and ‘extrinsic’—effective, fruitful, productive.
(iv) The case for determinacy
To round off this list of examples, we should consider a contemporary
case. Determinacy hypotheses came in for serious study beginning in
48 prope r method
the 1960s16 as part of a broader search for new principles that
might settle the problems in analysis17 and set theory18 left open by
the now-standard descendent of Zermelo’s system, Zermelo-Fraenkel
with Choice (ZFC). Assuming Choice, as is now standard, not all sets
of reals can be determined,19 but the assumption that all projective sets
of reals20 are determined—Projective Determinacy (PD)—is an attractive fall back. Perhaps more natural is the assumption that the full Axiom
of Determinacy21 holds in some appealing model of ZF; ADL(ℝ) asserts
the determinacy of all sets of reals in L(ℝ), the smallest inner model
containing all the real numbers. L(ℝ) includes the projective sets, so
ADL(ℝ) implies PD.
In his 1980 state-of-the-art compendium on the subject, Moschovakis
observed that ‘no one claims direct intuitions . . . either for or against
determinacy hypotheses’, that ‘those who have come to favor these
hypotheses as plausible, argue from their consequences’ (Moschovakis
[1980], p. 610). At that time, he concluded:
At the present state of knowledge only few set theorists accept [ADL(ℝ)] as
highly plausible and no one is quite ready to believe it beyond a reasonable
doubt; and it is certainly possible that someone will simply refute [it] in ZFC.
On the other hand, it is also possible that the web of implications involving
determinacy hypotheses and relating them to large cardinals will grow steadily
until it presents such a natural and compelling picture that more will succumb.
(Moschovakis [1980], pp. 610–611)
See, e.g., Kanamori [2003], }27.
e.g., the Lebesgue measurability of projective sets (see footnote 20).
18
e.g., of course, the Continuum Hypothesis (see footnote 40).
19
If A is a set of real numbers between 0 and 1 (imagine these uniquely represented as
infinite sequences of 0s and 1s), then the game G(A) consists of two players alternately
choosing a 0 or a 1; if the resulting real is in A, player I wins, otherwise II wins. A is said to be
determined if one of the players has a winning strategy.
20
Borel sets are obtained from open sets of reals by taking complements and countable
unions. An analytic set is the projection onto the x-axis of a Borel subset of the plane; a
co-analytic set is the complement of an analytic set. (Borel sets can also be characterized as
those that are both analytic and co-analytic.) The projective sets are obtained from the
Borel sets by repeated application of projection and complementation. The determinacy of
Borel sets is provable in ZFC, though this result, due to Martin, wasn’t established until
1974.
21
i.e., the assumption that all sets of reals are determined.
16
17
prope r method
49
Here Moschovakis displays impressive foresight, as more have succumbed in recent decades, on the basis of new discoveries.
In telegraphic summary, the current evidence for determinacy falls
roughly into four classes:
1. ADL(ℝ) generates a rich theory of definable sets of reals with many of
the virtues identified by Gödel.22 In Steel’s words,
The theory . . . based on [ADL(ℝ)]23 contains answers to all the questions about
projective sets from classical descriptive set theory. The theory of projective
sets one gets in this way extends in a natural way the theory of low-level
projective sets developed by the classical descriptive set theorists using only
ZFC; indeed, in retrospect, much of the classical theory can be seen as based
on open determinacy, which is provable in ZFC. Virtually nothing about sets
in the projective hierarchy beyond the first few levels can be decided in ZFC
alone, but . . . PD . . . yield[s] a deep and powerful extension of the classical
theory to the full projective hierarchy . . . By placing the classical theory in this
broader context, we have understood it better. (Steel [2000], p. 428)
Steel goes on to note that a rival theory of projective sets derived from
Gödel’s Axiom of Constructibility (V=L) ‘is certainly not the sort of
theory that looks useful to Analysts’. Furthermore, the V=L theory
isn’t lost if one adopts a theory containing PD, because it can be
regarded as ‘a wonderful, useful part of the first-order theory of L’,
a move that can’t be duplicated for the PD theory if one adopts V=L
(Steel [2000], p. 429).24
2. Moschovakis’s ‘web of implications . . . relating [ADL(ℝ)] to large
cardinals’ has indeed ‘grown steadily’. Large cardinal axioms themselves enjoy some intrinsic support, most fundamentally by reflecting
the intuitive idea that the cumulative hierarchy of sets goes on forever,
but it must be admitted that this sort of evidence is less compelling for
the larger large cardinals central to the developments we’re tracing
here.25 Still, the success of the inner model program provides evidence
22
And ADL(ℝ) is necessary for this theory: it’s actually implied by its consequences for
definable sets (see Koellner [2006], pp. 170, 174).
23
Steel actually mentions large cardinal hypotheses that imply ADL(ℝ) and hence PD (see
(2) below).
24
For an attempt to spell out an argument against V=L along these lines, see [1997], }III.6.
25
I come back to this point in }V.3.i. Indeed, the entire case for determinacy is re–visited
in }V.3.
50 prope r method
for the consistency of the large cardinals it has reached,26 and the
hierarchy of large cardinals has emerged as a remarkably effective
measure of the consistency strength of hypotheses going beyond
ZFC.27
The relationship between large cardinals and determinacy was dramatically illuminated in the decade following Moschovakis’s book, when
Martin, Steel and Woodin, building on work of Foreman, Magidor and
Shelah, showed that ADL(ℝ) follows from the existence of large cardinals.28
Koellner describes the current situation this way:
The case for axioms of definable determinacy is further strengthened by the
fact that they are implied by large cardinal axioms . . . Conversely, definable
determinacy implies (inner models of ) large cardinals . . . Ultimately, we see
that definable determinacy is equivalent to the existence of certain inner models
of large cardinals axioms. (Koellner [2006], pp. 170, 173)
Thus ADL(ℝ) inherits the intrinsic and extrinsic evidence for large
cardinals, and large cardinals, in turn, gain extrinsic support by implying the determinacy-based account of projective sets.
3. Returning to the hierarchy of consistency strengths, a striking
phenomenon has emerged. As Steel puts it,
any natural theory of consistency strength at least that of PD actually implies
PD. For example, the Proper Forcing Axiom [PFA] implies PD. So does
the existence of a homogeneous saturated ideal on ø1. (Neither of these
propositions has anything to do with PD on its surface.) (Steel [2000], p. 428)
26
See Steel [2000], p. 426: ‘these canonical inner models admit a systematic, detailed,
“fine structure theory” much like Jensen’s theory of L. Such a thorough and detailed
description of what a universe satisfying H might look like . . . provides good evidence that H
is indeed consistent’. Also Martin and Steel [1988], p. 6583: ‘the full and detailed description of
such a model gives some evidence of consistency . . . A hidden inconsistency . . . should emerge
quickly in the theory of ’ the canonical inner model.
27
See Steel [2000], pp. 425–427: ‘these axioms have proved crucial to organizing and
understanding the family of possible extensions of ZFC. Of course, there is nothing like a
systematic classification of all the possible extensions of ZFC, but there is more order here
than one might suspect . . . Thus it seems that the consistency strengths of all natural extensions of ZFC are wellordered, and the large cardinal hierarchy provides a sort of yardstick
which enables us to compare these consistency strengths’.
28
See Kanamori [2003], }32, for discussion and references.
prope r method
51
On the same theme, Koellner cites ‘a method—Woodin’s core model
induction’—which
can be used to show that virtually every natural mathematical theory of sufficiently strong consistency strength actually implies ADL(ℝ). Here are two
representative examples: . . . there is an ø1-dense ideal on ø1 . . . [and] PFA . . . .
These two theories are incompatible and yet both imply ADL(ℝ). There are
many other examples. For instance, the axioms of Foreman [1998] . . . also imply
ADL(ℝ). Definable determinacy is inevitable in that it lies in the overlapping
consensus of all sufficiently strong natural mathematical theories. (Koellner
[2006], pp. 173–174)
Given the long-standing foundational goal of set theory and the openendedness of contemporary pure mathematics, we have good grounds
to seek theories of ever-higher consistency strength. If all reasonable
theories past a certain point imply ADL(ℝ), this constitutes a strong
argument in its favor.
4. We’ve seen that sufficiently large cardinals, via determinacy, settle all
questions the classical analysts thought to ask about projective sets, but
we might wonder if other such questions are left unresolved. Our best
method for demonstrating independence is forcing, and in the presence of large cardinals, forcing cannot succeed in showing a question
about projective sets to be independent.29 This means that if anything
is left unresolved, this can’t be shown by forcing; the independence
involved would have to be a new and unfamiliar variety. It can also be
shown, conversely, that (in the presence of very modest large cardinals)
determinacy is required for this so-called ‘generic completeness’.30
Given that we want our theory of sets to be as decisive as possible,
within the limitations imposed by Gödel’s theorems, generic completeness would appear a welcome feature of determinacy theory.
The current case for determinacy has blossomed so impressively that
many would agree with Woodin’s assessment: ‘Projective determinacy
is the correct axiom for the projective sets’ (Woodin [2001], p. 575).
29
If there is a proper class of Woodin cardinals, then L(ℝ) is elementarily equivalent to
the L(ℝ) in any forcing extension. (See Koellner [2006], p. 171. Cf. Steel [2000], p. 430.)
30
If there is a proper class of inaccessibles, and L(ℝ) is elementarily equivalent to the L(ℝ)
in any forcing extension, then ADL(ℝ). (See Koellner [2006], p. 173.)
52 prope r method
3. Proper set-theoretic method
Assuming these examples are typical, the Second Philosopher hoping
to undertake an investigation of sets has access to a rich array of
methods, both for introducing them in the first place and for determining their extent and their properties thereafter. We’ve seen that
sets were posited in the first place in the service of explicit mathematical goals: a broader condition on uniqueness of trigonometric representations, representation-free definitions for abstract algebra, an
account of continuity to found analysis and launch a general study of
continuous structures, a foundation for arithmetic. In broad overview,
these goals range from relatively local problem-solving, to providing
foundations, to more open-ended pursuit of promising mathematical
avenues. The arguments subsequently offered for and against settheoretic hypotheses and axioms take many of the same forms: the
Axiom of Choice solves an array of problems, helps found the theory
of sets and promises more ‘productive science’ (Zermelo [1908a],
p. 189); determinacy hypotheses settle the questions once raised by
the early analysts and produce a rich and deep extension of the classical
theory; V=L presents an unproductive alternative theory of projective
sets and conflicts with set theory’s foundational role by restricting the
universe of sets.31 And so on. Given what set theory is intended to do,
relying on considerations of these sorts is a perfectly rational way to
proceed: embrace effective means toward desired mathematical ends.32
Now actual discussions surrounding the introduction of mathematical objects or extensions of our assumptions about them often involve
more speculative or typically philosophical material. To take just one
example, Dedekind’s paper on the natural numbers includes his belief
that they are ‘free creations of the human mind’ (Dedekind [1888],
p. 791). Given the wide range of views mathematicians tend to hold on
these matters, it seems unlikely that the many analysts, algebraists, and
set theorists ultimately led to embrace sets would all agree on any single
conception of the nature of mathematical objects in general, or of sets
31
32
This is the line of thought developed in [1997], }III.6.
I set intrinsic justifications aside until Chapter V.
prope r method
53
in particular; the Second Philosopher concludes that such remarks
should be treated as colorful asides or heuristic aides, but not as part
of the evidential structure of the subject. What matters for her methodological purposes is that all concerned do feel the force of the kinds
of considerations we’ve been focusing on here; these are the shared
convictions that actually drive the practice.33
This study of actual set-theoretic methods also confirms the Second
Philosopher’s initial impression that this is an inquiry governed by norms
distinct from familiar observation, theory-formation and testing: for
example, she isn’t accustomed to embracing new entities to increase her
expressive powers (as in Cantor) or to encourage definitions of a certain
desirable kind (as in Dedekind), or to rejecting a theory because it
produces less interesting consequences (as with the alternative to determinacy’s theory of projective sets that results from V=L). She might reasonably wonder if her more familiar, tried-and-true methods could be called
upon to underwrite, supplement or even correct these new approaches.
On examination, though, she concludes that the answer here is no.
Ordinary perceptual cognition is most likely involved in our grasp of
elementary arithmetic,34 but she recognizes that this connection to the
physical world has long since been idealized away in the infinitary
structures of contemporary pure mathematics. Though Quine has
argued that mathematical claims are empirically confirmed by a less
direct route, this position appears to her to rest on accounts of science,
mathematics, and the relations between them that don’t accurately
reflect the true features of these practices.35 Though she appreciates
that providing tools for empirical science remains one of the central
goals of pure mathematics, she also realizes that science no longer
shapes the ontology or fundamental assumptions of mathematics as it
once did in the days of Newton or Euler. Finally, cases like group
theory—which was considered useless and nearly dropped from the
curriculum at Princeton just years before it entered physics as an
33
For more, see [1997], }III.4, also the terminological mea culpa of [2007], p. 349,
footnote 12.
34
See [2007], IV.2.ii.
35
In [1997], }}II.6–7, I argue that Quine’s case is flawed (see also [2007], pp. 314–317).
Chapter I above presents an account of the role of mathematics in application that runs starkly
counter to Quine’s. This comes up again in }IV.2. See also }III.5, footnote 44.
54 prope r method
essential tool36—such cases convince her that any effort to reign in the
broad range of goals pursued by pure mathematicians would be unwise. So she’s faced with an array of new methods for justifying claims,
methods that appear to be both rational and autonomous.
If all she ultimately cared about were answering questions of the first
type—what are the proper set-theoretic methods?—she’d now be done,
but our Second Philosopher will also ask questions of the second type,
beginning with the stark: are these methods reliable? Do they successfully
track the existence of sets and their properties and relations? Of course
she’s familiar with questions of this form: she has an ongoing investigation of how ordinary perception gives her information about the
medium-sized objects in the world around her, of when it’s likely to fail
her, of what she can do to increase the accuracy of her perceptual beliefs;
she examines the efficacy of our instrumental means of detecting the small
parts of matter; she devises double-blinds to reduce the risk of misleading
experimental results, and so on. In all these familiar cases, she employs her
usual methods to evaluate how humans, as described in biology, physiology, psychology, evolutionary theory, and so on, come to know the
world, as described in physics, chemistry, geology, astronomy, and so on.
The case of set theory is much the same: she’s begun by observing
and lightly categorizing the methods set theorists actually use to justify
the introduction of sets and the elaboration of our theory of them, and
she’s asked whether or not these procedures lead to accurate beliefs.
This is not the sort of question that can be answered within set theory
or pure mathematics proper; even proof theory, our examination of
the reliability of our proving techniques, is actually a piece of applied
mathematics,37 and we’re now concerned with the less formal types of
argumentation offered in support of the assumptions from which those
proofs begin. Just as in other cases, we are, in effect, standing within
empirical science, asking a question about a particular human practice:
do its methods track the truth about its subject matter? We aren’t
asking whether or not a certain statement of set theory is true in the
sense of asserting that it holds in V;38 we’re asking whether set theory
36
See [2007], pp. 330–331, 347, for discussion and references.
Cf. Burgess [1992].
38
By ‘V ’ here, I intend the universe of sets that set theory is investigating, which we
learn, in the course of that investigation, takes the form of stages VÆ, one for each ordinal Æ.
37
prope r method
55
as a whole should be viewed as a body of truths, alongside physics,
astronomy, and botany.
This raises a prior question: should set theory be understood as
describing a subject matter, as attempting to deliver truths about it?
As we’ve seen, the Second Philosopher differs from Quine in rejecting
the idea that the mathematics used in application is justified by ordinary empirical evidence along with the physical theory in which it is
embedded. If she’s to conclude that pure mathematics is a body of
truths, her case for this will presumably rest more loosely on the way it
is intertwined with empirical science. For now, let me leave a bookmark at this point, to return to it in Chapter IV. For now, let’s assume
that the Second Philosopher is justified in regarding set theory as a
body of truths, and since she sees no reason to take its existence claims
at other than face value,39 she’s also justified in believing that sets exist.
Though she’s viewing the practice from her external, scientific perspective, as a human activity, she sees no opening for the familiar tools
of that perspective to provide supports, correctives, or supplements to
the actual justificatory practices of set theory. She has no grounds to
question the very procedures that do such a good job of delivering
truths, so she concludes that the proper methods to employ, the
operative supports and correctives, are the ones that set theory itself
provides; she concludes that the methods of set theory are reliable
guides to the facts about sets. Of course this doesn’t mean that every
set-theoretic argument is correct, only that methods of this general sort
are the right ones and that particular such arguments can be properly
supported or critiqued only by more arguments of the same kind.
4. The challenge
To this point, then, the Second Philosopher has settled the first group
of questions to her satisfaction. She takes the proper methods for
39
I don’t have in mind here any general case for the reliability of surface syntax, e.g., of
the sort proposed in Wright [1992] (see [2007], }II.5, for further discussion and references).
It’s just that the Second Philosopher sees no reason to think that set-theoretic claims say
anything other than what they appear to say. I touch on Wright’s minimalism in }III.2 below.
56
prope r method
introducing sets, for adding new axioms to our theory of them, to be
methods of the sort we’ve been rehearsing: sets are legitimately posited
as effective means toward various mathematical goals (in analysis,
algebra, foundations, and elsewhere); axioms are defended by a careful
balance of detailed considerations, both intrinsic and extrinsic. She’s
also made a start on questions of the second sort: she’s concluded that
set-theoretic methods are rational, autonomous, and generally reliable,
that sets exist, and that set theory is (largely) a body of truths about
them. But this is just the beginning; she still wonders: what sort of
activity is set theory? how does set-theoretic language function? what
are sets and why are these the proper methods for finding out about
them? These questions would need answers in any case, but they can
appear more pressing in the face of natural statements like the Continuum Hypothesis40 (CH) that can’t be settled on the basis of the current
axioms. Is CH nevertheless a legitimate question with a determinate
answer? This might seem to hinge on our account of the subject matter
of set theory.
It was this question, in fact, that inspired Gödel to his well-known
Platonistic metaphysics:
The set-theoretic concepts and theorems describe some well-determined
reality, in which Cantor’s conjecture must be either true or false. Hence its
undecidability from the axioms being assumed today can only mean that these
axioms do not contain a complete description of that reality. Such a belief is by
no means chimerical, since it is possible to point out ways in which the
decision of a question, which is undecidable from the usual axioms, might
nevertheless be obtained. (Gödel [1964], p. 260)
He goes on to describe intrinsic and extrinsic methods of justifying
new axioms. The basic idea here—that the legitimacy of CH is to be
defended by appeal to some sort of objective reality in which it is either
true or false—this basic idea can be fleshed out in a number of different
ways: the objective reality might be the cumulative hierarchy of sets
(as in Gödel [1964]), a set-theoretic structure (as in Shapiro [1997]),
40
Despite the amusing terminological convergence, this obviously isn’t the ‘continuum
hypothesis’ from fluid dynamics touched on in }I.2, but (in one form) the claim that every
infinite set of reals is either countable or of the same size as the full set of reals.
prope r method
57
the concept of set (as in Gödel [1951]), some purely modal facts (as
in Hellman [1989]), and there are doubtless other possibilities. An
analogy may or may not be drawn between the set-theoretic reality
described by set theory and the physical reality described by natural
science (as in Gödel [1944], [1964], or my [1990]). Let me call metaphysical positions of this general type ‘Robust Realism’.
As noted in }I.1, there’s a well-known objection to views of this
sort, familiar from Benacerraf [1973]: how can human beings, with the
sorts of cognitive capacities we understand ourselves to enjoy, manage
to gain reliable information about the world of sets? Benacerraf’s
particular statement of the question rested on philosophical theories
since defunct, but the issue has not evaporated. Burgess and Rosen
trace its resilience to a ‘perennially powerful’ picture (Burgess and
Rosen [2005], p. 534):
Reality is . . . a system connected by causal relations and ordered by causal
laws, containing entities ranging from the diverse inorganic creations and
organic creatures that we daily observe and with which we daily interact,
to the various unobservable causes of observable reactions that have been
inferred by scientific theorists . . . [Robust Realists] hold that outside, above,
and beyond all this (and here one gestures expansively to the circumambient
universe) there is another reality, teaming with entities radically unlike concrete entities—and causally wholly isolated from them. . . . between [us] and
the other world . . . there is a great gulf fixed . . . Surely [Robust Realists] owe
us a detailed explanation of how anything we do here can provide us with
knowledge of what is going on over there, on the other side of the great gulf.
(Burgess and Rosen [1997], p. 29)
Robust Realists (including those noted above) have attempted to meet
this challenge, but no one has managed to satisfy all parties involved.
My own, second-philosophical concerns about Robust Realism
begin even before this familiar epistemological challenge can take
hold. To see how, consider any one of the set-theoretic arguments
rehearsed above; for concreteness, let’s take Cantor’s case for positing
sets in his study of trigonometric representations. Setting aside the
murky talk of the ‘conceptual co-determination’ of P’s derived set—
safely regarded as the sort of thing that’s not widely shared, or even
understood, by the range of practitioners—we have a straightforward
58
prope r method
means-ends argument for this introduction of sets as objects: they
allow Cantor to formulate and prove a stronger theorem on uniqueness. Of course this advantage could be outweighed by accompanying
disadvantages—if these sets were difficult to work with or led to
contradictions or some such thing—but no apparent downsides are
in evidence here. Given the Second Philosopher’s evaluation of proper
set-theoretic method, this would seem a clear example of good
grounds for introducing sets, of good mathematical evidence for
their existence.
So far so good. But if the Robust Realist is right, if the goal of set
theory is to describe an independently-existing reality of some kind,
then it appears that Cantor’s evidence needs supplementation, and not
supplementation of the same sort, like adding in Dedekind’s grounds
and so on, but supplementation of an entirely different kind: we need
an account of how the fact that sets serve this or that particular
mathematical goal makes it more likely that they exist. Without this
account we have no way of ruling out the possibility that reality is sadly
uncooperative, that much as we’d like to use sets in our mathematical
pursuits, they just don’t happen to exist. To the Second Philosopher,
this hesitation seems misplaced: why should perfectly sound mathematical reasoning require supplementation? Hasn’t something gone
wrong when rational mathematical methods are called into question in
this way?
The situation gets worse when we contemplate what the Robust
Realist’s supplementation would require. If the world of sets that set
theory hopes to describe is entirely objective, perhaps analogous to our
familiar physical world, then it’s hard to see, for example, how the fact
that ADL(ℝ) is implied by all natural hypotheses of sufficiently high
consistency strength is evidence in its favor. Granted, we have good
reason to pursue axioms with higher and higher consistency strength,
so as to maximize the interpretive power of set theory and thus further
its foundational goal, but how does our desire that set theory play this
role make it any more likely that it’s capable of doing so? The same can
be said about each of our many instances of preferring a set theory with
the sorts of mathematical advantages we’ve been surveying. The physical
world, famously, cares little for our theoretical preferences, as the failure
of Euclidean Geometry and the mysteries of Quantum Mechanics
prope r method
59
amply demonstrate. Why should we expect the set-theoretic universe to
be any more cooperative than the physical world?
The familiar Benacerrafian challenge suggests that the abstract character of the objects of set theory poses a formidable obstacle to the sort
of supplementary epistemological account the Robust Realist requires.
In contrast, the Second Philosopher’s inclination is to think that no
such supplementary account should be required in the first place: if
Robust Realism questions the cogency of apparently sound mathematical reasoning, her guess is that the fault lies with Robust Realism,
not the tried-and-true ways of set theory. The problem, then, is how
to answer the second group of questions—about the nature of settheoretic activity, about its subject matter and the reliability of its
methods—in a way that both respects the actual methods of set theory
and preserves the legitimacy of the pursuit of new axioms and a
solution to the continuum problem.
III
Thin Realism
What we’re after is a satisfying form of realism without the shortcomings of the Robust versions. Within the set-theoretic community
we’ve been focused on, hints of this sort of position turn up in various
remarks and observations of John Steel:
Realism in set theory is simply the doctrine that there are sets . . . Virtually
everything mathematicians say professionally implies there are sets. . . . As a
philosophical framework, Realism is right but not all that interesting. (FOM
posting 15 January 1998)1
I take what ‘mathematicians say professionally’ as an appeal to the
ordinary set-theoretic methods we’ve been focused on here. Steel
nods toward Robust Realism—
Both proponents and opponents [of realism] sometimes try to present it as
something more intriguing than it is, say by speaking of an ‘objective world of
sets’. (FOM posting 15 January 1998)
—and he entertains the possibility that his mundane realism is
something different
Whether this is Gödelian naı̈ve realism I don’t know. (FOM posting
30 January 1998)
1
All excerpts from Steel’s FOM postings are quoted with permission. As readers of [2005]
or [2007], }IV.4, will know, the line of thought developed in this chapter was originally
inspired by some remarks of Steel and John Burgess (see below). The position as it now stands
can’t be attributed to either of them without putting quite a few words in their mouths, but I
remain grateful to them both for pushing me in this direction when the only realism I could
imagine was Robust. Apparently related thoughts turn up in Liston [2004] and Tait [1986]
and [2001], but see footnotes 10 and 16, and 6, 17, 27, respectively.
thin realism 61
Speaking some years later, he concludes that it is not, and recommends
A more sophisticated realism, one accompanied by some self-conscious,
metamathematical considerations related to meaning and evidence in mathematics. (Steel [2004], p. 2)
This is exactly what our second-philosophical investigations have led
us to hope for: a version of realism that genuinely accounts for the
nature of set-theoretic language and practice, that respects the actual
structure of set-theoretic justifications. The goal of this chapter is to
develop such a position.
1. Introducing Thin Realism
Recall (from }II.1) that the Second Philosopher faces two types of
questions about set theory: first, what are its proper methods? second,
what sort of activity is set theory, what is its subject matter, and why are
these the proper methods? These aren’t strictly mathematical questions;
she poses them in her capacity as an empirical scientist examining a
particularly salient human practice. To this point, she has answered
questions of the first type—the actual set-theoretic methods she’s
cataloged are the proper methods, both rational and autonomous—
and she’s made a start on questions of the second type: she’s concluded
that sets exist, that set theory is a body of truths about them, and that
set-theoretic methods are reliable guides in this inquiry.2 She’s now
faced with the challenge of explaining what makes these methods
reliable, of what sets must be like for this to be so. Under the circumstances, the Second Philosopher is naturally inclined to entertain the
simplest hypothesis that accounts for the data: sets just are the sort of
thing set theory describes; this is all there is to them; for questions about
sets, set theory is the only relevant authority.
From this narrow beginning we can draw some immediate
consequences. For example, Steel points out that
Recall (from }II.3) that examination of the precise nature of her justification for the
reliability, truth, and existence claims has been postponed to Chapter IV.
2
62 thin realism
none of [what mathematicians say professionally] is about their causal relations
to anything . . .
from which he concludes that
sets do not depend causally on us (or anything else, for that matter). (FOM
posting 15 January 1998)
Since set theory tells us nothing about sets being dependent on us as
subjects, or enjoying location in space or time, or participating in
causal interactions, it follows that sets are abstract in the familiar
ways. John Burgess sums up this particular sentiment nicely,
One can justify classifying mathematical objects as having all the negative
properties that philosophers describe in a misleadingly positive-sounding way
when they say that they are abstract [acausal, non-spatiotemporal, etc.]. But
beyond this negative fact, and the positive things asserted by set theory, I don’t
think there is anything more that can be or needs to be said about ‘what sets
are like’.3
In contrast with the entities posited by the various rich metaphysical
and epistemological theories of the Robust Realists—which all go well
beyond ‘the positive things asserted by set theory’—these sets will
seem rather insubstantial; this realism is ‘thin’, as opposed to robust.
What’s happened here is that the second-philosophical Thin Realist, in the ordinary course of her investigations, has made a surprising
discovery: in addition to the familiar concrete objects she’s been
studying so far, there are also objective, non-spatiotemporal, acausal
sets; not only the methods of set theory, but the things themselves are
new. Of course the Robust Realist could say these same words; both
our realists now take set theory to aim at describing the properties of an
objectively-existing, non-spatiotemoral, acausal reality, so some care
must be taken to distinguish the new version from the old. As a start,
recall how Gödel appeals to Robust Realism to justify his claim that
CH has a determinate truth value, despite its independence from ZFC.
How will our new realist view the case of CH? Her analysis is simpler:
‘CH or not-CH’ is a theorem, established by her best methods as a fact
3
Personal communication, 24 April 2002, quoted with permission.
thin realism 63
about V;4 therefore CH is either true or false there.5 For the Robust
Realist, this appeal to classical logic isn’t enough; for him, without a
guarantee that the logic tracks the metaphysics, the possibility remains
open that this theorem is incorrect.6 In contrast, the Thin Realist holds
the set-theoretic methods are the reliable avenue to the facts about sets,
that no external guarantee is necessary or possible.7 So the fundamental
diagnostic is this: the Robust Realist requires a non-trivial account of
the reliability of set-theoretic methods, an account that goes beyond
what set theory tells us; for the Thin Realist, set theory itself gives
the whole story; the reliability of its methods is a plain fact about what
sets are.
The key here is that the second-philosophical Thin Realist begins
from her confidence in the authority of set-theoretic methods when it
comes to determining what’s true and false about sets, and from the
observation that her more familiar methods appear irrelevant; she
concludes that the call for a non-trivial, external epistemology is
misplaced. But why couldn’t the Robust Realist, faced with Benacerraf-style challenges, make the same move? The answer is that the
4
Again, by ‘V ’ I mean the universe of sets that set theory is investigating, which we learn,
in the course of that investigation, takes the form of stages VÆ, one for each ordinal Æ. Here
I’m presupposing that the second-philosophical set theorist is moved by various mathematical considerations, conspicuously the desire for set theory to serve as a foundation (in the
sense of }I.3), to seek a unified theory of this single universe V. (See e.g. [1997], pp. 208–9,
[2007], p. 354.)
5
This isn’t to say there’s any guarantee that set-theoretic methods will eventually tell us
which one it is. Set-theoretic methods are the only reliable source of information about sets;
sets just are the sort of thing these methods tell us about; but they could have features that, for
one reason or another, we miss, or even have no way of getting at. See footnote 40 below.
6
Tait appears to take up the new realist position—‘that CH is either true or false amounts
to nothing more than the application of the law of excluded middle to it’ (Tait [2001],
p. 96)—but he apparently loses the courage of his convictions two pages later, declaring that
‘until we determine it, CH is indeed indeterminate’. See footnote 17 below.
7
It’s conceivable that future mathematical developments could convince the Thin Realist
that classical logic shouldn’t be applied in set theory in this way, for example, that set theory
isn’t the study of a single universe of sets after all. If, to take a speculative example, we should
eventually come upon conflicting, equally attractive theories of sets, and if we were unable in
principle to regard them all as describing parts of a larger unified theory of sets, then we might
well conclude that set theory has failed at its foundational goal: no one theory of sets is able to
encompass all of contemporary mathematics. For now I think it’s fair to say there’s no
compelling evidence that this is likely to be the outcome of set-theoretic investigation, so I
ignore it in the text. (For more, see [2007], pp. 387–389, Koellner [2006], }5.2., [2009a], }5.)
64 thin realism
Robust Realist wants more than Thin Realism can offer; for example,
as Charles Parsons says of Gödel, he wants an ‘objectively determinate
answer’ for CH, ‘where this is to mean more than that our logic
incorporates the law of excluded middle’.8 Of course the Thin Realist
will protest that she does mean more—she takes the correctness of
classical logic for set-theoretic inquiry to show that CH has an objective and determinate truth value—but for the Robust Realist, this is
still too thin; he wants a full-bodied metaphysical theory of what sets
are that will ratify CH in a more substantive way, and to get this, he
needs a non-trivial epistemology.9 So the Thin Realist’s minimal
theory to account for the methodological data—her simple assumption
about what sets are like—will not satisfy the Robust Realist.
2. What Thin Realism is not
As so far described, Thin Realism risks conflation with more familiar
theories, so let me pause a moment to draw some contrasts. The close
connection between sets and set theory, between sets and set-theoretic
methods, has so far been treated as a brute fact. Of course we’d like a
more fundamental explanation for this connection, but that impulse
can easily lure us beyond the austerity of Thin Realism, beyond what
set theory tells us. To illustrate, let me consider three typical examples.
One way of thinking about the Second Philosopher’s approach to
understanding the subject matter of set theory is that she’s asking: what
must sets be like in order that we can know about them in these ways
(that is, by the methods of set theory)? To the philosophical ear, this
formulation recalls the iconic Kantian question: what must the world
be like in order that we can know it as we do (that is, partly a priori)?
Kant’s answer, famously, is that the world we experience is partly
constituted by our human modes of cognition (the forms of intuition
and the pure categories), so we can know a priori that this world will
8
Parsons [2004], p. 62. See also Parsons [1995], p. 71: ‘The widespread impression that
Gödel was not just affirming CH v ~CH, i.e., allowing the application of the law of the
excluded middle here, seems to me correct’.
9
See }V.2 for examples.
thin realism 65
consist of spatiotemporal objects standing in causal relations. Among
philosophers sympathetic to a position like Thin Realism, there’s a
temptation to slip into an analogous stance: sets are constituted by our
set-theoretic practices; that’s why we can be confident that those
practices track the facts about sets.10
The Kantian story of the world of experience is unacceptable to the
Second Philosopher because it involves an explicitly extra-scientific
mode of inquiry.11 In Kant’s terms, space, time and causation are
empirically real, but transcendentally ideal; this means that the empirical
enquirer—that is, the scientist, the Second Philosopher—is correct to
regard spatiotemporality and causality as objective features of the world,
known a priori, but the transcendental inquirer—that is, the Kantian
critical philosopher—sees that these features are contributions of our
particular form of cognition, and thus, when regarded transcendentally,
only subjective or ideal.12 This qualifies Kant as a First Philosopher as
the term is used here: from the empirical viewpoint, these ordinary
beliefs about the world are entirely in order; from the transcendental
viewpoint, they aren’t. Once again, the Second Philosopher finds
her beliefs challenged and all her usual forms of evidence set aside as
irrelevant; once again, she doesn’t understand the higher purpose
involved or the methods appropriate to it. Of course, to be fair, the
Second Philosopher doesn’t feel the pressure Kant did to account for the
a priori truth of geometry!
But the situation with set theory is different, because the Second
Philosopher does recognize a legitimate form of inquiry outside of set
theory, namely, the ordinary empirical inquiry in which she begins her
investigations. As we’ve seen, it’s from this point of view that she
reaches her decision to undertake the study of sets, and from this point
10
In describing his ‘Thin-blooded Platonism’, Liston [2004], p. 154, cites with approval a
passage from Stalnaker [1988], p. 119: ‘The existence of numbers is just constituted by the fact
that there is a legitimate practice involving discourse with a certain structure, and that certain
of the products of this discourse meet the standards of correctness that it sets’. (Stalnaker isn’t
endorsing this idea.)
11
For more, see [2007], }I.4.
12
This transcendental psychology must be sharply distinguished from ordinary empirical
psychology: the latter could at best tell us how we must perceive the world, not how the
world must be; science all too often reveals that our natural ways of cognizing don’t match
reality.
66 thin realism
of view that she conducts her examination of proper set-theoretic
method and the nature of sets and set theory. Presumably it would also
be from this point of view that she would determine, if she were to
determine, that sets are constituted by set-theoretic methods. The
trouble with such a determination isn’t that it’s first-philosophical—it
isn’t. Obviously it also isn’t the same as Thin Realism: sets are no longer
objective, independent entities, but rather dim projections of our practices; there is considerably more to them than what set theory tells us.
Recognizing that it’s different from Thin Realism, we should pause
to ask if this Kantian suggestion might serve our Second Philosopher as a
viable alternative. Recall that her discomfort with Robust Realism arose
(in }II.4) from its demand that ordinary, apparently rational mathematical methods be supplemented with evidence tied to its non-trivial
metaphysics, especially as it seems inevitable that the required supplementation will not in fact be forthcoming: from the Robust Realist’s
perspective, the various considerations surveyed in }II.2 look like so
much wishful thinking. Given that our proposed idealism views sets as
somehow constituted by our methods, it would presumably claim to
ratify many of the set-theoretic arguments that Robust Realism calls
into question; though the Second Philosopher regards this kind of
supplementation as unnecessary and even dubious, at least it wouldn’t
immediately undercut what she regards as rational argumentation. Still,
it seems likely that this idealism, however it’s spelled out, would have to
face the possibility that set-theoretic methods will fail to settle CH, and
in this way our idealist, like the Robust Realist, would want more than
the Thin Realist’s straightforward confidence that CH is either true or
false in V. Here, once again, a potential challenge to the cogency of an
apparently rational set-theoretic method—in particular, use of excluded
middle—would be grounded on an extra-mathematical metaphysics.
The relevant ontology this time would be idealistic rather than realistic,
but no less objectionable for that.
Another tempting line of thought is to explain the fact that sets
are what set theory describes by appeal to Carnapian13 themes: the
13
I have in mind here the Carnap of the popular imagination, especially as in Carnap
[1950]. The real Carnap is considerably more complex and elusive (see [2007], }I.5, for
discussion and references).
thin realism 67
connection between sets and set-theoretic methods is ‘analytic’ or
‘conceptual’.14 In the bold strokes of Carnap’s well-known ‘Empiricism, semantics and ontology’ (Carnap [1950]), the linguistic framework for Xs includes the vocabulary and the evidential rules necessary
for talk of Xs; these framework principles fix the ‘meaning of X’ or
delineate the ‘concept of X’; working within the framework, we
establish what’s true and what’s false about Xs by confirming or
disconfirming statements on the basis of the framework principles. In
this way, the connection between Xs and the evidential methods
enshrined as rules of the framework are deemed ‘analytic’ or ‘conceptual’; they are part of the language we use in order to talk about Xs
at all.
The general notion of a linguistic framework is extremely flexible:
there’s a linguistic framework for ordinary physical objects, which
Carnap calls ‘the thing language’; there’s one for numbers, one for
sets, and presumably one for poltergeists, as well. The key to Carnap’s
solution, or dissolution, of ontological questions is that matters of truth
or falsity only make sense within a framework, where there are determinate evidential rules by means of which they can be settled, so one
can’t coherently ask, prior to the adoption of a framework, whether
or not the things it features exist. One can ask, however, whether
or not it would be pragmatically useful to adopt the corresponding
framework, to adopt the corresponding linguistic conventions. So, for
example, it would be useful for many purposes to move from the thing
language to the thing-and-number language. The mistake would be to
imagine that this move is only allowed if one can first establish that
numbers exist.
From a second-philosophical point of view, this account is least
compelling for the thing language. To begin with, we never consciously adopt it; as Carnap notes, ‘we all have accepted the thing
language early in our lives as a matter of course’ (Carnap [1950],
p. 243). He hopes to recover something of the flavor of the framework
14
Burgess seems sympathetic to some such line of thought, especially in his [2004a] and
[2004b], but as far as I can tell, he doesn’t actually endorse it. I used the terminology of
‘conceptual truth’ myself (in [2005] and [2007], }IV.4), but now regret doing so, for reasons
about to emerge.
68 thin realism
story by suggesting that we do consciously decide not to switch to ‘a
language of sense-data or other “phenomenal” entities’ (op. cit.), but
given that no one has ever succeeded in devising such a language, this
seems an empty gesture. At least as troubling is the status of the
evidential rules; as part of the framework, they are themselves conceptual and known a priori by anyone working within the framework. The
idea that what counts as good evidence for the existence of some
physical object is a fact known a priori seems to cohere best with
verification-based theories of meaning or a priori accounts of confirmation, both of which appealed to Carnap at various times in his career, but
must appear problematic to a contemporary Second Philosopher.
So let’s set empirical frameworks aside, return to our familiar secondphilosophical point of view—beginning from perception, corrected and
confirmed by careful observation and experimentation, and so on—and
see if we can understand the thin-realist embrace of set theory as the
adoption of a new linguistic framework in something like Carnap’s
sense. The answer is clearly no: the Thin Realist doesn’t regard the
embrace of ‘the concept of set’, the adoption of the set-theoretic
framework, as a purely pragmatic matter of linguistic convention; she
takes Cantor and Dedekind to have discovered the existence of these
mathematical objects, not to have come to a convenient decision. Just
as sets aren’t constituted by our methods, as in the Kantian line, settheoretic truth isn’t constituted by the set-theoretic framework;15 its
evidential rules allow us to get at the truth about sets, but that’s due to
our simple hypothesis about what sets themselves are, not some special
kind of truth.
So this Carnapian idea is clearly not the same as Thin Realism, but
again we should ask: might it be an attractive alternative for the Second
Philosopher? Might the Second Philosopher regard set-theoretic practice as taking place within a specified linguistic framework that implicitly defines the concept of set? On this picture, presumably Cantor and
15
Liston seems tempted by this move: ‘truth is just correctness according to the standards
set by the conception and the practice’ (Liston [2004], p. 155). Epistemic or pragmatic
accounts of truth hold little appeal for the Second Philosopher (see [2007], }I.7). I’ve also
argued that Thin Realism need not be linked with disquotational truth, that it’s compatible
with a second-philosophical version of the correspondence theory (see [2007], }IV.4,
pp. 370–377).
thin realism 69
Dedekind gave us reason to adopt this framework, and we’re now
busily investigating within it. The problem comes when we try to
specify what exactly would be enshrined in a linguistic framework for
‘the concept of set’. The simplest proposal would be to include the settheoretic axioms and classical logic. The trouble with this suggestion is
that it fails to capture one of the elements of set-theoretic practice
we’re most eager to describe and assess: the addition of new axioms.
On this view, one axiom wouldn’t be selected over another for
compelling set-theoretic reasons—these are all internal to the framework—but as a pragmatic, conventional decision to move from one
linguistic framework to another.16 Obviously this is not a suitable path
for the Second Philosopher.
We might try broadening the characterization of the set-theoretic
framework to include not just the current axioms, but all intrinsic
justifications as well, since they are after all intended to spell out features
implicit in the ‘concept of set’. But even if we could successfully corral
what seems an open-ended range of intrinsic considerations, this approach would still exclude from set-theoretic practice all the justifications we tend to classify as extrinsic, and this is no more acceptable to the
Second Philosopher than excluding intrinsic justifications. Finally, we
could try to add even these to a still-looser characterization of a settheoretic framework, but then both intrinsic and extrinsic justifications
would be lumped together as ‘conceptual’, obscuring a distinction
important to our understanding of the practice. All in all, this seems an
unpromising avenue for the Second Philosopher.17
16
Though he doesn’t mention Carnap, this seems to me to capture the essence of Tait’s
position: ‘the question of mathematical truth or existence becomes well-defined only with
the introduction of the axioms’ (Tait [2001], p. 89) and ‘the intuitions or dialectical considerations that may lead us to accept a new axiom’ (p. 98) don’t justify that axiom. This
would explain the failure of nerve in footnote 6: viewed inside the framework, CH has a
determinate truth value (as a consequence of classical logic); viewed outside the framework,
it’s indeterminate (because the axioms of the framework are too weak to settle it).
17
Of course, the topic here is the pursuit of pure mathematics, and of set theory in
particular. If we address instead a natural scientist’s adoption of a given bit of mathematics, the
Second Philosopher will agree with Carnap that this decision need not wait on any question
of abstract ontology or epistemology (see Chapter I and }IV.2, below). Still, she would
hesitate to dismiss the whole process as purely pragmatic—there are descriptive goals
involved—and absent the holism of Carnap and Quine, there’s no temptation to think any
sort of ‘truth by convention’ is involved.
70 thin realism
A third train of thought that might be confused with Thin Realism
arises from Wright’s minimalism about truth.18 Like the Thin Realist,
Wright’s minimalist believes in sets because he takes set-theoretic
existence claims, understood at face value, to be true. The stark differences come out in the reasons the two offer for embracing the premises
they share. Wright’s minimalist takes set theory to be a body of
truths because it enjoys certain syntactic resources and displays wellestablished standards of assertion that our set-theoretic claims can be
seen to meet; the idea is that a minimalist truth predicate can be defined
for any such discourse in such a way that statements assertable by its
standards come out true. In contrast, the Thin Realist take set theory to
be a body of truths, not because of some general syntactic and structural
features it shares with other discourses, but because of its particular
relations with the defining empirical inquiry from which she begins.19
Similarly the minimalist takes set-theoretic claims at face value,
rejecting efforts to re-interpet them as saying something other than
they seem to say, because he rejects the general notion of covert logical
structure, because he trusts surface syntax on principle. Here again the
Thin Realist has no blanket position on the matter; she simply sees no
grounds on which to argue that set-theoretic claims in particular say
anything other than what they appear to say.20 So despite superficial
similarities, the order of argument is quite different. The minimalist
begins from the well-behaved syntax and agreed-upon methods of settheoretic discourse, moves by a general construction to an appropriate
truth predicate and the truth of set-theoretic existence claims, and
finally to the existence of sets via the general reliability of surface
syntax. The Thin Realist begins from the role of mathematics in
science, moves from there to the truth of set theory and its existence
claims, and finally to the existence of sets via the lack of evidence that
those existence claims stand in need of re-interpretation. So the Thin
Realist is not arguing as a minimalist would. In fact, the minimalist’s
e.g., Wright [1992]. See [2007], }II.5, for further discussion and references.
Again, I come back to this point in the next chapter.
20
Some would argue for nominalistic reinterpretations on epistemological grounds.
See }IV.3 for a second-philosophical take on this line of thought.
18
19
thin realism 71
guiding ideas about truth and syntax are hardly congenial to the
second-philosophical point of view.
So idealist or conventionalist or minimalist elaborations like these
hold little appeal for our second-philosophical Thin Realist. She’s
left with the claim that sets simply are the things that set theory
describes, and it’s hard to see how any further explanation could be
given without introducing an external metaphysics that goes beyond
what set theory tells us. Still, I think there is a bit more for the Thin
Realist to say. Let me start with epistemology, then circle back to the
metaphysics.
3. Thin epistemology
So far, the only epistemology Thin Realism has offered is what follows
from the bare claim that sets are known by set-theoretic methods. This
might seem to bring any epistemological conversation to an abrupt
halt, but in fact I think a few illuminating morals can be drawn.
For the sake of contrast, let’s return for a moment to Kant, this time
to the finer points of his theory of cognition. For Kant, any judgment
arises from the cooperative efforts of two faculties: the passively receptive sensibility and the actively spontaneous understanding.21 What the
sensibility passively receives is the so-called ‘matter’ of cognition; the
sensibility automatically forms this inchoate matter spatiotemporally,
into an intuition orderly enough for the understanding to apply its
concepts. For our purposes, the key idea is that the contributions of
two faculties are quite different: the sensibility achieves an immediate,
direct connection to the object of cognition; the understanding relates
to that object only mediately, by means of features or properties.22
Pure mathematics, on Kant’s account, is an investigation of precisely
those spatiotemporal forms of intuition supplied by the sensibility. It
21
Hence the oft-quoted remark: ‘Without sensibility no object would be given to us, and
without understanding none would be thought. Thoughts without content are empty,
intuitions without concepts are blind’ (A51/B75).
22
Cf. A320/B376–7: ‘an objective perception is a cognition . . . [This] is either an
intuition or a concept . . . The former is immediately related to the object and is singular;
the latter is mediate, by means of a mark, which can be common to several things’.
72 thin realism
isn’t entirely clear whether Kant thinks that pure mathematics is
without sense until referred to an empirical object or that the close
relationship between pure and empirical intuition makes pure intuition enough to ground a real cognition,23 but what’s beyond doubt is
that some sort of conceptually unmediated input is required for a real
judgment of any kind.
Here our second-philosophical Thin Realist begins not from a
theory of what judgment must be, but from her conviction that the
way to find out about sets is to do set theory. For her, the fact that settheoretic methods give us knowledge of sets by appeal to an array of
mathematical considerations, without a role for perception or any
other such mechanism, is enough to establish that there are objects
we can know without the sort of unmediated connection provided by
Kant’s sensible intuition. (This is just to turn the familiar worry about
Robust Realism on its head, to argue directly that, for example, the
perception-like epistemology proposed in my [1990] is unnecessary.24)
So set theory does tell us something more about epistemology: our settheoretic knowledge doesn’t require immediate cognition of sets.
In fact, I think we can squeeze out a bit more along these lines.
Recall Cantor and Dedekind’s grounds for introducing sets in the first
23
At least it’s not entirely clear to me. Kant says: ‘Sensible intuition is either pure intuition
(space and time) or empirical intuition of that which, through sensation, is immediately
represented as real in space and time. Through determination of the former we can acquire a
priori cognitions of objects (in mathematics), but only as far as their form is concerned . . .
whether there can be things that must be intuited in this form is still left unsettled.
Consequently all mathematical concepts are not by themselves cognitions, except insofar as
one presupposes that there are things that can be presented to us only in accordance with the
form of that pure sensible intuition’ (B147). See also A239–240/B299: ‘Although all these
principles [of mathematics], and the representation of the object with which this science
occupies itself, are generated in the mind completely a priori, they would still not signify
anything at all if we could not always exhibit their significance in appearances (empirical
objects) . . . Mathematics fulfills this requirement by means of the construction of the figure’
or, in the case of number, ‘in the fingers, in the beads of an abacus, or in strokes and points
that are placed before the eyes’.
24
Gödel ([1964], p. 268) also speaks of ‘something like a perception . . . of the objects of
set theory’, but he also denies that this need be ‘conceived of as a faculty giving an immediate
knowledge of the objects concerned’. This may sound like a thin-realist denial that any
immediate access is required, but Gödel actually suggests that something else—other than
‘the objects concerned’—is immediately given. He appears to have in mind something akin
to Kant’s pure categories, but now he’s strayed into precisely the features of Kant’s account
that aren’t immediate. I won’t try to sort this out here.
thin realism 73
place; both rested on the effectiveness of sets in performing certain
specific mathematical jobs in analysis, algebra, and foundations. As
we’ve seen, grounds of this type are still offered for new sets. Of course
any such effectiveness must be weighed against other desiderata: it
would be overridden, for example, if the added sets rendered our
overall mathematical theory inconsistent;25 less dramatically, its importance would be tempered if it somehow blocked the pursuit of
other worthy goals. But whatever the outcome of this calculation, it
remains true that the evidence for the existence of sets is all and only
linked to their mathematical virtues, to the mathematical jobs they are
able to perform.
Now suppose that the overwhelming mathematical virtues of sets
have been established, as I think we’d all agree that they have. Does
there remain a remote possibility that sets don’t exist or that they’re
wildly different than we think they are? Our analysis of the epistemological situation implies that we would never come to know this, that
no corrective to set-theoretic method would be forthcoming. But I’m
asking a metaphysical question about sets themselves: could it be a sad
metaphysical fact that set-theoretic methods—however reasonable,
however apparently reliable—are entirely wrong?
Some writers sympathetic to positions in the vicinity of Thin Realism have drawn a parallel here with radical skepticism about the external
world: any concern about the ultimate reliability of set-theoretic
methods is no better founded, they claim, than the skeptic’s concern
about the ultimate reliability of our perceptual beliefs.26 From a secondphilosophical point of view, this seems to me to under-estimate the
strength of the set-theoretic case. To see this, consider for a moment
the challenge posed by the external world skeptic: he asks me to defend
25
As noted in Chapter I, footnote 78, the delta function was used in physical science for
some time, despite its obvious inconsistency. Indeed it’s often good policy to allow unrigorous methods to proceed for a time without too much hindrance, with the hope of domesticating them in the future. (Wilson [2006] calls this a period of ‘semantic agnosticism’.) This
has happened in the history of mathematics as well, e.g., in the early days of the calculus, and
it might conceivably happen even in set theory. Still, given that one of set theory’s leading
motivations is to provide the sort of rigorous foundation that mathematics sorely needed in
the early 20th century (as described in }I.3), the demand for immediate consistency is much
stronger here than it is elsewhere.
26
See e.g. Burgess and Rosen [2005], pp. 522–523, Tait [1986], pp. 63–65.
74 thin realism
my belief that there’s a rose in my garden when I’m looking straight at it
under optimal conditions. I can answer this challenge, of course, if I’m
allowed to appeal to my well-confirmed theories of optics, of my eye
and my brain, of human belief-forming mechanisms, and so on; on this
basis I can defend the reliability of my current perception. But this isn’t
what the skeptic wants. By appeal to colorful hypotheses like the Evil
Demon—who somehow fits me with a full range of false experiences of
a non-existent external world—by entertaining such fanciful hypotheses the skeptic in effect challenges me to defend my belief in the
rose without appealing to any of my usual ways of defending beliefs,
because these are all undercut by the possibility of the Demon. This
challenge is one I don’t know how to meet, but there’s nothing
ill-formed about it; I wouldn’t mind having an answer to it myself,
that is, a way of defending all my methods entirely from scratch, ex
nihilo. So my second-philosophical response to the skeptic’s challenge
as he intends it isn’t to argue that I can meet it directly or to argue that
it’s in some way illegitimate; my response is rather that, because
the challenge doesn’t arise naturally out of my ordinary ways of finding
out about the world, my inability to meet it doesn’t undermine the
reasonableness of my belief in that rose.27
Now compare this to the imagined radical challenge to the reliability of our set-theoretic methods. We’ve seen that sets are in fact
introduced when they have important mathematical virtues and
don’t produce inconsistencies or pre-empt any (or enough) other
mathematical desiderata. The skeptic asks if it might not be—however
unlikely it must inevitably seem to us—that all such evidence is
misleading, that despite the positing of sets having all the payoffs we
could ask, sets don’t exist or do exist but don’t have the properties we
think they do. This isn’t a matter of our capacity to make mistakes
about some particular piece of evidence, or even to be globally
misguided about where the mathematical payoffs truly lie. The question is whether impeccable evidence about the mathematical merits of
sets could still be entirely and undetectably unreliable, just as Evil
27
This is too quick, of course, but perhaps it can be taken on faith for present purposes.
For more, see [2007], }I.2, and [2010].
thin realism 75
Demon-generated experiences would be entirely and undetectably
unreliable.28
I think the Thin Realist’s answer to this question is that it couldn’t,
that to suggest that set theory could enjoy all these virtues and sets still
not exist or be radically different than they seem is to misunderstand
the nature of set theory and its subject matter. Recall the Thin Realist’s
credo: sets are the things set theory tells us about. Though set theory
doesn’t now tell us the size of the continuum, and, for all we know,
may never get around to settling that question, still, what’s needed to
mount a skeptical challenge is far more radical than this: there has to be
the Evil Demon-like possibility that virtually everything set theory
tells us about sets is wrong. Not that its purported theorems are
fallacious or its analyses of the mathematical benefits of sets are distorted, rather that this whole way of doing things is unreliable. But for
the Thin Realist, sets simply are the sort of things we can find out
about in these ways. From this point of view, there is no room for a
radical epistemological gap between sets and set-theoretic methods;
the skeptical challenge here, unlike the case of the external world, is
simply ill-formed.
I suspect this point is related to the Thin Realist’s discovery that sets
can be known without any immediate cognition of them. That unmediated link to a subject matter is what allows us to identify ‘that
which I’m linked to’ while possibly being all wrong in my beliefs about
it.29 Kant provides a prime illustration: for him, that ‘matter’ involved
in our judgments, which comes to us unbidden, is what makes our
cognition into cognition of the world, even though, on his view, we
can only know that world as it is experienced by knowers like us, not
28
In discussions of external world skepticism, it’s often assumed that we have incorrigible
access to our own experiences, that I can’t be wrong about it appearing to me that there’s a
rose in my garden; the issue is the inference from here to the existence of the rose. In the settheoretic case, there’s no temptation to think that we can’t be mistaken about the evidence—
by getting the logical inferences wrong, by failing to notice the disadvantages, etc.—but again
what’s at issue is the inference from the evidence: granting that we’re right about the payoffs
of positing sets, could we still go wrong when we infer that they exist?
29
An aside: cognitive scientists note that human perceptual experience actually begins
from a primitive system that simply tracks an object before any properties are attributed to it
(think of ‘it’s a bird, it’s a plane, it’s Superman’). See [2007], pp. 255–257, for discussion and
references.
76 thin realism
as it is in itself. The corresponding idea for the Second Philosopher
is that her experience could be entirely wrong; she can’t rule out
the possibility of an Evil Demon arranging things so that ‘that which
generates my experience’ is radically different from what she takes
it to be. But for the second-philosophical Thin Realist, there is no
perception-like direct access to sets and none is needed. This means
there is no primitive connection with a subject matter that
could underlie the possibility that these things (the ones with which I’m
primitively connected) are radically different from all I think I know
about them.
This contrast between the external world case—where the radical
skeptic’s challenge is coherent (though less damaging than he imagines)—and the set-theoretic case—where the radical skeptical challenge
is incoherent—might serve as a criterion of the distinction between
abstract objects and concrete ones.30 After all, the close connection
between the practice of set theory and the nature of its objects is what
makes sets so different from ordinary physical objects: though we’re
capable of learning about physical things, we don’t regard this as a
trivial consequence of what physical objects are; we give a detailed
explanation of how we’re able to do this, beginning with an analysis of
the general reliability and various shortcomings of our human perceptual apparatus. If this contrast is right, it might help undermine the
‘perennially powerful’ picture sketched by Burgess and Rosen ([2005],
p. 534). The second-philosophical Thin Realist easily recognizes
something not unlike their ‘cosmos’—
a system . . . containing entities ranging from the diverse inorganic creations
and organic creatures that we daily observe and with which we daily interact,
to the various unobservable causes of observable reactions that have been
inferred by scientific theorists (Burgess and Rosen [1997], p. 29)
—but she harbors no in-principle objection to expanding this ontology when the evidence points that way. Her investigations lead her to
believe that there are ‘entities radically unlike concrete entities’, but, as
we’ve seen, to deny that between us and these sets ‘there is a great gulf
30
This apparently differs from the various characterizations surveyed in Burgess and
Rosen [1997], pp. 16–25.
thin realism 77
fixed’ (op. cit.). She thinks the Thin Realist has provided an account
‘of how anything we do here can provide us with knowledge of what
is going on over there’ (op. cit.), an account that rests on what sets are,
not on a direct cognition of sets ‘on the other side of the . . . great wall’
(op. cit.).
4. The objective ground of Thin Realism
It’s time to take stock. The Thin Realism on offer guarantees the
objectivity of set-theoretic truth and existence, respects the actual
methods of set theory, recognizes a determinate truth value for CH,
and raises no difficult epistemological problem. It not only squares
with the Second Philosopher’s austere and hard-nosed scientism, it
actually seems to arise naturally from it. It might be just as well to quit
while we’re ahead, but I think it would be disingenuous to ignore a
nagging worry that it’s all too easy, that it rests on some sleight of hand.
Connecting sets and set-theoretic methods so intimately continues to
invite the suspicion that sets aren’t fully real, that they’re a kind of
shadow play thrown up by our ways of doing things, by our mathematical decisions. The position would be considerably more compelling if it offered some explanation of why sets are this way, but any step
in that direction, in the direction of an underlying account of sets that
explains this fact, seems to lead us inevitably beyond what set theory
tells us about sets, into the realm of Robust Realism and the like.
In fact, I think something can be offered that draws the sting from
this nagging doubt, but it won’t take quite the form expected. What
we want is a sense of what sets are that explains why these methods
track them. What I think we can get, from the Thin Realist’s perspective, is a sense of an objective reality underlying both the methods and
the sets that illuminates the intimate connection between them. Perhaps this will be enough.
Let me come at the question by asking what objective reality
underlies and constrains set-theoretic methods, what objective reality
it is that set-theoretic methods track. The simple answer, of course, is
that they track the truth about sets, but our goal is to find out more
78 thin realism
about what sets are, without going beyond what set theory tells us, and
our hope is that asking the question this way might help. So, what
constrains our methods? Part of the answer lies in the ground of
classical logic,31 but our interest here is in the mathematical features.
To get at these, let me draw one last comparison to Kant.
In his discussions of mathematical truth, Kant draws his now-familiar distinction between analytic and synthetic: in an analytic judgment,
the predicate ‘is (covertly) contained in’ the concept of the subject; in a
synthetic judgment, the predicate ‘lies . . . outside’ the concept of the
subject (A6/B10). The concept of a triangle, for example, is defined
by us; since we ‘deliberately made it up’ (A729/B757), we can know
what belongs to it, that is, we can know trivial analytic truths like ‘all
triangles are three-sided’. In contrast, no amount of meditating on the
concept of triangle will reveal to us that the three interior angles of a
triangle are equal to two right angles; for this we need to construct a
triangle—in our imagination or on the page—draw a line through the
apex parallel to the base and reason from there (cf. A716/B744). How
does this process take us beyond the concept to something synthetic?
Kant’s answer is that the constructions involved are shaped by the
structure of our underlying spatial form of sensibility, either in pure
intuition (when we construct in our visual imagination) or in empirical
intuition (when we draw an actual diagram). Because of this ‘shaping’,
the argument tracks more than just what’s built into the concept; the
derivation is also constrained by the nature of space itself, which, as we
know, Kant thought to be Euclidean.
Of course this picture of geometric knowledge hasn’t survived
subsequent progress in logic, mathematics, and natural science. I
present it here because I think it provides a helpful analogy for what
I want to suggest in the case of set theory. Think of it this way. Kant is
out to explain what underlies the proof of this geometric theorem,
what makes it a proof; his answer is: not just the concept of triangle,
not just logical consequence, but also the nature of the underlying
space. We’re out to explain what underlies the justificatory methods of
set theory, what makes considerations like those sketched in }II.2 good
31
For discussion of the ground of logical truth, see [2007], Part III.
thin realism 79
reasons to believe what we believe. Part of the answer, for the intrinsic
justifications, may be that they spell out what’s implicit in our ‘concept
of set’, but the bulk of the justifications that interest us are extrinsic.32
What validates them? What takes us beyond mere logical connections
and allows us to track something more? And what is this ‘something
more’? We’re looking for the counterpart to Kant’s intuitive space.
Before trying to answer these questions for set theory, let’s first
consider another type of case in which we go beyond the logical,
namely, in mathematical concept-formation. In the logical neighborhood of any central mathematical concept, say the concept of a group,
there are innumerable alternatives and slight alterations that simply
aren’t comparable in their mathematical importance. Logic does nothing to differentiate these one from another, assuming they are all
consistently defined, but ‘group’ stands out from the crowd as getting
at the important similarities between structures in widely differing
areas of mathematics and allowing those similarities to be developed
into a rich and fruitful theory. In ways that the historians of mathematics spell out in detail, ‘group’ effectively opens the door to deep
mathematics in ways the others don’t.33 So what guides our conceptformation, beyond the logical requirement of consistency, is the way
some logically possible concepts track deep mathematical strains that
the others miss.
Of course there are stark differences between group theory and set
theory, because the two pursuits have different goals. Group theory
aims to draw together a wide variety of diverse structures that share
mathematically important features; it’d be counter-productive to require that all groups be commutative (or not), because there are deep
structural similarities between commutative and non-commutative
groups that it’s mathematically fruitful to trace. Set theory, on the
other hand, aims at least in part to provide a single foundational arena
for all classical mathematics, so it strives to develop a unified theory
that’s as decisive as possible.34 This is why the set-theoretic theorem
I come back to intrinsic justifications in }}V.3 and V.4.
e.g., see Wussing [1969], Stillwell [2002], chapter 19, or the quick survey in [2007],
}IV.3.
34
See [2007], pp. 351–355.
32
33
80 thin realism
‘CH or not-CH’ has a different significance from the group-theoretic
theorem ‘(x)(y)(x þ y=y þ x) or not-(x)(y)(x þ y=y þ x)’: the set theorist is describing the single structure V, and learns that one of CH or
not-CH holds there;35 the group theorist is describing the features
common to a wide range of structures, and learns that each is either
commutative or not.
Still, there are overarching similarities. Set-theoretic concepts are
formed in response to set-theoretic goals just as the concept ‘group’
was formed in response to algebraic goals. In large cardinal theory, for
example, we can trace the conceptual progression from the superstrong cardinal to the Shelah cardinal to the Woodin cardinal, which
turned out to be the optimal notion for the purposes at hand,36 or
the gradual migration of the concept of measurable cardinal from its
origins in measure theory to the mathematically rich context of elementary embeddings.37 Of course the set-theoretic cases we’ve been
concerned with involve not definitions but existence assumptions—
like the introduction of sets in the first place or the addition of large
cardinals—and new hypotheses—like determinacy—but in these
cases, too, far more than consistency is at stake: these favored candidates differ from alternatives and near-neighbors in that they track
what we might call the topography of mathematical depth. This
topography stands over and above the merely logical connections
between statements, and furthermore, it is entirely objective:38 just as
it’s not up to us which bits of pure mathematics best serve the needs of
natural science, just as it’s not up to us that it would be counterproductive to insist that all ‘groups’ be commutative, it’s also not up to
us that appealing to sets and transfinite ordinals allows us to capture
facts about the uniqueness of trigonometric representations, that the
Axiom of Choice takes an amazing range of different forms and plays a
fundamental role in many different areas, that large cardinals arrange
themselves into a hierarchy that serves as an effective measure of
35
36
Though recall footnote 7.
See Kanamori [2003], p. 461.
See Kanamori [2003], }}2 and 5.
38
Perhaps it’s worth recalling (from [2007], Part III) that the Second Philosopher regards
logic as robustly true in any situation with a certain minimal structuring, and that V enjoys
this sort of structuring (see [2007], p. 382). So the network of logical implications that
underlies the mathematically deep strains is itself objective.
37
thin realism 81
consistency strength, that determinacy is the root regularity property
for projective sets and interrelates with large cardinals, and so on. These
are the facts that play a role analogous to Kant’s Euclidean space, the
facts that constrain our set-theoretic methods, and these facts, unlike
Kant’s, are not traceable to ourselves as subjects.
A generous variety of expressions is typically used to pick out the
phenomenon I’m after here: mathematical depth, mathematical fruitfulness, mathematical effectiveness, mathematical importance, mathematical productivity, and so on. (I have been and will continue to use
such terms more or less interchangeably.) One point worth emphasizing is that the notion in question is not being offered up as a candidate
for conceptual analysis or some such thing. To begin with, I doubt
that an attempt to give a general account of what ‘mathematical
depth’ really is would be productive; it seems to me the phrase is
best understood as a catch-all for the various kinds of special virtues we
clearly perceive in our illustrative examples of concept-formation and
axiom choice.39 But even if I’m wrong about this, even if something
general can be said about what makes this or that bit of mathematics
count as important or fruitful or whatever, I would resist the claim that
this ‘something general’ would provide a more fundamental justification for the mathematics in question: our second-philosophical analysis
strongly suggests that the context-specific justifications we’ve been
considering so far are sufficient on their own, that they neither need
nor admit supplementation from another source.
It also bears repeating that judgments of mathematical depth are not
subjective: I might be fond of a certain sort of mathematical theorem,
but my idiosyncratic preference doesn’t make some conceptual or
axiomatic means toward that goal into deep or fruitful or effective
mathematics; for that matter, the entire mathematical community
could be blind to the virtues of a certain method or enamored of a
merely fashionable pursuit without changing the underlying facts of
which is and which isn’t mathematically important.40 This is what
39
This is why I spend so much time rehearsing these various cases, to give the reader a feel
for what ‘mathematical depth’ looks like.
40
In particular, we might never come to formulate or to appreciate the virtues of some settheoretic axiom that would settle CH. In this way, our Thin Realist might never come to know
whether CH is true or false, despite its having a determinate truth value. (Cf. footnote 5 above.)
82 thin realism
anchors our various local mathematical goals. Cantor may have wished
to expand his theorem on the uniqueness of trigonometric representations, but if this theorem hadn’t formed part of a larger enterprise of
real mathematical importance, his one isolated result wouldn’t have
constituted such compelling evidence for the existence of sets; similarly the overwhelming case for Dedekind’s innovations depends in large
part on the subsequent successes of the abstract algebra they helped
produce. The key here is that mathematical fruitfulness isn’t defined as
‘that which allows us to meet our goals’, irrespective of what these
might be; rather, our mathematical goals are only proper insofar as
satisfying them furthers our grasp of the underlying strains of mathematical fruitfulness. In other words, the goals are answerable to the
facts of mathematical depth, not the other way ’round.41 Our interests
will influence which areas of mathematics we find most attractive or
compelling, just as our interests influence which parts of natural
science we’re most eager to pursue, but no amount of partiality or
neglect from us can make a line of mathematics fruitful if it isn’t, or
fruitless if it is.42
Thus we’ve answered our leading question: the objective ‘something
more’ that our set-theoretic methods track is these underlying contours
of mathematical depth. Of course the simple answer—they track sets—
is also true, so what we’ve learned here is that what sets are, most
fundamentally, is markers for these contours, what they are, most
fundamentally, is maximally effective trackers of certain strains of
mathematical fruitfulness. From this fact about what sets are, it follows
that they can be learned about by set-theoretic methods, because settheoretic methods, as we’ve seen, are all aimed at tracking particular
instances of effective mathematics. The point isn’t, for example, that
‘there is a measurable cardinal’ really means ‘the existence of measurable
cardinals is mathematically fruitful in ways x, y, z (and this advantage
isn’t outweighed by accompanying disadvantages)’; rather, the fact of
measurable cardinals being mathematically fruitful in ways x, y, z (and
41
I’m grateful to Matthew Glass for pressing me to clarify this point.
Here at last are grounds on which to reject the nihilism of footnote 9 on p. 198 of
[1997], and even the tempered version in [2007], pp. 350–351. If mathematicians wander off
the path of mathematical depth, they’re going astray, even if no one realizes it.
42
thin realism 83
these advantages not being outweighed by accompanying disadvantages) is evidence for their existence. Why? Because of what sets are:
repositories of mathematical depth. They mark off a mathematically
rich vein within the indiscriminate network of logical possibilities.
Notice also that our conclusion about radical skepticism is reinforced. Any particular extrinsic justification may fail to meet its mark,
for reasons ranging from a straightforward error in what follows from
what to a deep misconception about the true mathematical values in
play. We can be uncertain whether or not a given set-theoretic posit
will pay off, and therefore uncertain about whether or not it exists, but
if it does pay off, there’s no longer any room for doubt; we can be
uncertain that we’re getting at the deepest and most fruitful theory of
sets, and therefore uncertain about whether or not our axiom candidate is true, but if we are succeeding, there’s no further room to doubt
that we’re learning about sets. This is what defeats an Evil Demonstyle concern: the Demon might somehow induce in me all the
experiences I’d have if there were an external world without there
actually being such a world, but he can’t present a set-theoretic posit
that does a maximally-efficient job of tracking mathematical fruitfulness and yet doesn’t exist—because the posit just is the sort of thing
that does this sort of job.
So there is a well-documented objective reality underlying Thin
Realism, what I’ve been loosely calling the facts of mathematical depth.
The fundamental nature of sets (and perhaps all mathematical objects) is
to serve as means for tapping into that well; this is simply what they are.
And since set-theoretic methods are themselves tuned to detecting
these same contours, they’re perfectly suited to telling us about sets;
they lie beyond the reach of even the most radical skepticism. This,
I suggest, is the core insight of Thin Realism.
5. Retracing our steps
Before continuing on the Second Philosopher’s journey, I think it’s
worth pausing for a moment to dissect the step-by-step structure of her
progress to this point. In our schematic description, she first encounters
mathematics as something of a black box that issues forth useful
84 thin realism
expressive machinery and effective techniques for exploring and manipulating it.43 I propose to examine her train of thought from there by
comparing it with a more transparent case.
Let’s imagine the Second Philosopher in her lab, engaged in some
cognitive investigation of chimpanzees. The work is going smoothly,
but the chimps themselves don’t seem entirely well; they’re a bit
lethargic, without good appetite. When our Second Philosopher airs
her concern about this in the common lunch room, the local botanist
asks where these chimps hail from. Given this information, she retires
to her workroom and returns with a plastic bag of processed pellets.
The Second Philosopher feeds these to her subjects and they thrive.
Here botany is functioning as a black box, issuing forth useful
advice. Given the all-encompassing curiosity of the Second Philosopher, she will want to know how this black box works. The botanist
carefully explains how she and her fellows have collected plant samples
from around the world, how they’ve studied and probed these to form
an initial classification, how they experimented with growing conditions and hybridization, and so on. With her resulting expertise, our
helpful botanist knew the prevalent plants in the chimps’ ancestral
habitat, knew which of these their forebears were likely to have
consumed, knew that one of the plants she herself was studying was
a close relative, and guessed that feeding the pellets she’d prepared
from them for other purposes might well appeal to the Second Philosopher’s chimps. Hearing about how the botanists have conducted
their inquiries, the Second Philosopher can easily appreciate the structure of their discipline and the rationality of their methods of observation, experimentation, theory-formation, and testing. Based on her
own experiences in other areas and in her more general methodological studies, she may even be able to offer some help to the botanist,
perhaps some more refined thinking about experimental design
or even, to stretch a bit, some information about subtle perceptual
distortions that could throw off some of her observations.
43
A more direct approach would be to describe the Second Philosopher as developing
the mathematics she needs as she goes along, as e.g. Newton did, then eventually finding her
way into pure mathematics. I adopt the approach in the text to highlight the key points of
novelty that arise in the case of mathematics.
thin realism 85
I hope all this seems entirely straightforward given our developed
understanding of the Second Philosopher. Now let’s return to the
case of mathematics. Impressed by the black box effectiveness of
mathematics, the Second Philosopher begins as she did in the case of
botany, by investigating the actual methods mathematicians have used
to generate the relevant concepts, techniques, proofs, etc. As we saw in
examples from Cantor, Dedekind, Zermelo, and the determinacy
theorists, she’s able to appreciate that the methods used are rational
given the goals being pursued, and that the goals themselves are natural
and productive. Based on her previous efforts, can she go on to offer
advice and corrections to the mathematician as she did with the
botanist? We’ve seen (in }II.3) that the two cases diverge at this
point, that for mathematics the answer is no: perception and experimentation are irrelevant to modern pure mathematics; its connections
with applications do not provide the kind of methodological guidance
they once did. The Second Philosopher concludes that pure mathematics, unlike botany, is autonomous.
Now comes the matter of reliability, and here again the two cases
diverge. The botanist’s goal is simply to learn about the world’s plant
life, so the Second Philosopher’s assessment of the rationality of her
methods involves a straightforward assessment of their reliability.
Some might hold that the mathematician’s goal is simply to learn
about a realm of mathematical things, but the various cases examined
by the Second Philosopher don’t actually sound like this: Cantor wants
to formulate a stronger theory on the uniqueness of trigonometric
representations; Dedekind hopes for a fruitful abstract algebra; the
determinacy theorists develop a theory of projective sets that’s mathematically richer than the alternative that follows from V=L. There’s
no clear appeal to a mathematical reality in any of this—only to various
mathematical benefits—so the Second Philosopher’s assessment of the
means-ends rationality in terms of those goals doesn’t involve direct
consideration of their reliability.
When the issue of reliability is raised, it brings with it the prior
question of whether or not set theory should be viewed as an attempt
to correctly describe a subject matter at all. In }}II.3 and III.1, we
temporarily took for granted that the close interconnection of mathematics with the Second Philosopher’s ongoing empirical studies gives
86 thin realism
her good reason to regard it as a body of truths, and it’s a small step from
the truth of its existential claims to the existence of a subject matter. The
autonomous methods of set theory have produced this body of truths,
and this provides persuasive evidence of their reliability. To explain
why this is so, the second-philosophical Thin Realist forms her simple
hypothesis about what sets are. Finally, she traces the truth of this
hypothesis, the source of this fact about sets, to the strains of mathematical depth that the sets mark and the methods track (in }III.4).
I think we can now see how the Second Philosopher is led to a form
of realism so different from the familiar Robust variety. The key is her
starting point, so firmly rooted in the practical details of the actual
mathematics. Philosophers often begin from a more elevated perspective; rather than examining the day-to-day practices, they content
themselves with classifying mathematics as a non-empirical, a priori
discipline, concerning a robust abstract ontology, then begin to wonder how we could possibly come to know such things, how what
mathematicians actually do could have any connection to the subject
matter they’re attempting to describe. In this way, ‘the great gulf ’ is
fixed. Roughly put, they begin with the metaphysics and are led to
confusion about the methods. In contrast, the Second Philosopher
begins with the methods, finds them good, then devises a minimal
metaphysics to suit the case.44
Notice that this elevated brand of philosophizing sometimes makes
an appearance—I would say an unwelcome appearance—in connection with those day-to-day practices. The way I’ve told the story here,
the Second Philosopher follows a clear line of mathematical development that recapitulates the methodologically sound innovations of
Cantor and Dedekind in the late 19th century. Historically, however,
various lines of constructivist and predicativist thought developed
around the same time. Now it nearly always makes good mathematical
sense to investigate how much one can do with how few resources—
such inquiries often generate useful insights and more precise techniques—so there’s every reason to pursue constructive and predicative
analysis side-by-side with the more familiar classical approach, and all
44
This is the approach endorsed in [1997], pp. 200–202. (Here by the ‘metaphysics’ of a
human linguistic practice, I just mean an account of its subject matter.)
thin realism 87
this can be understood as taking place within the vast arena afforded by
set theory. But there are also less tolerant forms of constructivism and
predicativism that go on to deny the legitimacy of the stronger methods of set theory. It’s hard to see a purely mathematical reason to take
this stand, to reject the Cantor-Dedekind approach outright. Obviously the classical theory involves stronger hypotheses, of higher
consistency strength, and in that sense it’s more risky, but this hardly
gives us reason to decide ahead of time that this avenue is unworthy of
investigation.
What, then, could the motivation be? I suspect one answer45 is that
some common features of set-theoretic method—like the notion of an
arbitrary subset or the Axiom of Choice—are taken to derive their
justification from some version of Robust Realism.46 Then, in light of
the familiar epistemological objections to Robust Realism,47 the practice of set theory itself is called into question. On this approach, the
justification—or lack of justification—for mathematical methods is
based on a metaphysical account of its subject matter. From the Second
Philosopher’s point of view, this gets things backwards: the order of
justification goes the other way ‘round, from the math to the metaphysics, not the metaphysics to the math. From her point of view,
metaphysical considerations of this sort shouldn’t be allowed to restrict
the free pursuit of pure mathematics—and, in fact, they haven’t.
Thus, Thin Realism. Let’s now approach questions of the second
type—about the nature of set-theoretic activity and its subject matter—
along an entirely different avenue.
45
Another might be that we should pursue only the mathematics that’s directly needed
for natural science, and that non-constructive or impredicative mathematics is not. This
sentiment is often based on a Quinean holism that sees mathematics in application as
confirmed along with the rest of our overall web of belief, but leaves the remaining pure
mathematics without justification. This picture of the relations between pure mathematics
and natural science, mentioned in passing in }II.3, is undercut by the considerations of
Chapter I. See [1997], }}II.6–7, [2007], pp. 314–317, for more direct engagement with
Quine’s holism. The general suggestion that only applied mathematics is worthy comes up
again in }IV.2.
46
See e.g. Feferman [1987], pp. 44–45.
47
Rehearsed in }II.4. These various schools of non-classical thought then replace Robust
Realism with some other external metaphysics that supports only more restricted methods.
One irony here is that Robust Realism doesn’t seem to support our actual set-theoretic
methods (again see }II.4).
IV
Arealism
The discussion of Thin Realism in the last chapter was predicated on
the assumption that the Second Philosopher has good reason to regard
pure mathematics in general, and set theory in particular, as a body of
truths. Along the way, I set aside the question of precisely what that
good reason is. I come back to this central question below, but first I’d
like to explore an alternative picture, one that does without that key
assumption. After sketching such a view and contrasting it with near
neighbors, I consider its relations to Thin Realism and, then, at last,
take up the question of truth.
1. Introducing Arealism
To return to the Second Philosopher’s starting point, she begins her
investigation of the world with ordinary perception, graduates to more
sophisticated forms of observation, theory-formation, and testing,
improving her methods as she goes; eventually she turns to mathematical methods, and from there, to the pursuit of mathematics itself.
Recapitulating the developments of the 19th century, she finds her
mathematical inquiries broadening to include structures and methods
without immediate application, which eventually leads her to set
theory along the path of Cantor, Dedekind, Zermelo, and the rest.
Now there’s no doubt that she has clear motivation to pursue pure
mathematics, but the question before us is whether or not she has good
grounds to regard it as a body of truths. When she notices that its
methods are quite different, that its claims aren’t supported by her
familiar observation, experimentation, theory-formation, and so on,
arealism
89
but by the sorts of intrinsic and extrinsic arguments canvassed in }II.2,
might she not simply conclude that whatever its merits, pure mathematics isn’t in the business of uncovering truths?
But if he’s not uncovering truths, then what is the pure mathematician doing? For the case of set theory, we’ve got a sense of the answer:
among many other things, Cantor is extending our grasp of trigonometric representations; Dedekind is pushing towards abstract algebra;
Zermelo is providing an explicit foundation for a mathematically
important practice; contemporary set theorists are trying to solve the
continuum problem.1 Just as the concept of group is tailored to the
mathematical tasks set for it, the development of set theory is constrained by its own particular range of mathematical goals, both local
and global. Mightn’t the Second Philosopher rest content with this
description? Set theory is the activity of developing a theory of sets that
will effectively serve a concrete and ever-evolving range of mathematical purposes. Such a Second Philosopher would see no reason to think
that sets exist or that set-theoretic claims are true—her well-developed
methods of confirming existence and truth aren’t even in play here—
but she does regard set theory, and pure mathematics with it, as a
spectacularly successful enterprise, unlike any other.2 Let’s call this
position Arealism.3
2. Mathematics in application
Whatever reason the Thin Realist may have to count pure mathematics as true, it must rest somehow on the role of mathematics in
empirical science, so we need to ask: can the Arealist account for the
1
And, lest we forget, much of pure mathematics is still consciously aimed at the goal of
providing tools for empirical science.
2
In particular, its complex interrelations with natural science mark it off from other
human endeavors—astrology, theology—whose methods also differ from those usual to the
Second Philosopher. See [1997], pp. 203–205, [2007], pp. 345–347, and more below.
3
I noted earlier (}III.2, footnote 15) that Thin Realism doesn’t require a disquotational
theory of truth. Perhaps it’s worth noting here that Arealism is compatible with disquotationalism: the Arealist isn’t straightforwardly asserting the claims of set theory, so isn’t
committed to their truth.
90
arealism
application of mathematics without regarding it as true? We’ve seen
(in Chapter I) that contemporary pure mathematics works in application by providing the empirical scientist with a wide range of abstract
tools; the scientist uses these as models—of a cannon ball’s path or the
electromagnetic field or curved spacetime—which he takes to resemble the physical phenomena in some rough ways, to depart from it in
others; indeed often enough, in fundamental theories, we aren’t sure
exactly how the correspondence plays out in detail. The applied
mathematician labors to understand the idealizations, simplifications
and approximations involved in these deployments of his abstract
structures; he strives as best he can to show how and why a given
model resembles the world closely enough for the particular purposes
at hand. In all this, the scientist never asserts the existence of the
abstract model; he simply holds that the world is like the model is
some respects, not in others. For this, the model need only be welldescribed, just as one might illuminate a given social situation by
comparing it to a imaginary or mythological one, marking the similarities and dissimilarities.
Michael Liston worries that this line of thought is not enough to
account for the role of mathematics in physical science.4 His concern is
that there is more to the use of mathematics in such cases than the
description of an abstract model sufficiently similar to the physical
phenomena. As we’ve just noted, scientists invariably pursue an account of how far the similarities extend and why, as part of assuring
themselves that the model is well-suited to its job: some of these
assurances will involve physical information, as the use of van der
Waals’s equation is justified by the fact that actual molecules have
stable ‘effective radii’; others will involve mathematics, as the reliability
of Stirling’s approximation rests on the Robservation that the sum of
ln(n), as n varies from 1 to N, approaches 1N ln(x)dx when N is large. In
some cases, we still have no satisfactory answer to this type of question:
for example, we suspect that our abstract quantum mechanics must
4
See Liston [2007], p. 4. This may be related to what Resnik [1997], chapter 3, has called
his ‘pragmatic indispensability argument’, but I’m not sure. Liston himself cites Wilson
[2006], presumably in connection with examples like the discussion of Euler’s Method (pp.
116–117, 163–165, 212–217, 573–575) (see below).
arealism
91
resemble the world in the small in some way or other, because it makes
such amazingly accurate predictions, but we have no account of what
the underlying physical structures are like, which aspects of the model
reflect them, and why this representation is so effective. Still, despite
occasional setbacks, we always seek this type of account, very often
with success, and we continue to do so even for quantum mechanics.
Liston’s point is that giving these accounts—what he calls ‘reliability
explanations’—often requires substantial mathematics, well beyond
what goes into the original modeling. So, for example,
Solutions to numerous important physical problems require the determination
of a function satisfying a differential equation . . . Sometimes . . . the existence
of a solution . . . can be established by directly solving the equations. . . . where
direct methods fail, the existence of a solution must be established indirectly,
generally by constructing a sequence of functions that converges to a limit
function that satisfies the [equation]. Moreover, the solution . . . very often
cannot be evaluated by analytic methods, and scientists must rely on discrete
variable or finite element numerical methods to approximate the solution. . . .
The numerical methods often provide our only way of extracting an actual
solution. (Liston [2004], p. 146)
In such cases, these highly developed mathematical theories of indirect
solution and approximation are essential to our treatment of the
physical problem; they help us find solutions where this is possible,
and ‘provide us with . . . valuable qualitative information about the
solutions’ (op. cit.) where it isn’t. Liston observes that:
Mathematical physicists rely on the theories presupposed in proving the
existence of solutions and approximating them. It is difficult to see how
they could do this while adding the . . . disclaimer, ‘But, you know, I don’t
believe any of the mathematics I’m using’. (op. cit.)
For this reason, Liston argues, it isn’t enough that the Arealist can
account for the role of abstract mathematical models as means of
describing phenomena; she also needs to account for the role of
mathematical reasoning in determining the features of those models.5
5
Liston’s actual target in these passages is Mark Balaguer’s fictionalism, but I assume he
has something similar in mind in his [2007], p. 4.
92 arealism
If she’s to believe her abstract models have certain properties, mustn’t
she believe the mathematics used to establish those properties?6
Of course Liston is quite right that we aren’t and shouldn’t be satisfied
to regard our mathematical model as a black box with a good track
record, and to point out that some of our reliability explanations, like
the case of Stirling’s approximation, involve mathematics. But let’s look
a bit more closely, focusing on the kind of case he highlights. The
original differential equation for our physical problem is proposed as an
abstract model of some physical phenomenon: we know something
about the rate of change of some quantity—the temperature, the location, the density—and we want to figure out the quantity itself—the
temperature or location as a function of time, the density as a function of
position. To get at this, we replace the actual situation with an abstract
model: time is represented as a continuous real variable, space as ℝ3, the
temperature, location and density as real-valued functions; what we
know about the rate of change is then formulated as a differential
equation. This mathematical structure—the abstract model satisfying
the stated equation—doesn’t exist in a vacuum; it’s embedded in a
rich mathematical universe—V if you like—and it has all the properties
the advanced methods reveal—for example, it has a solution that can be
approximated in a certain way—as part of its identity as a mathematical
object. Liston’s challenge is this: we form a simple abstract model; why
should we believe all the extra things that advanced mathematics tells us
about our model? I’d put it somewhat differently: the abstract model has
all those features to start with, as part of its mathematical pedigree; the
question is why—given a model whose simple features track the world
reasonably well—why should we expect its mathematically esoteric
features to continue to track the world reasonably well? It seems what
we need to know isn’t so much that the advanced mathematics is true,
but that the more esoteric features it reveals will continue to be effective
in modeling the world.
6
It’s worth noting that Liston takes the truth of the background mathematics to be
necessary but not sufficient: ‘Robust Realism will not help: why should beliefs about a freefloating mathematical realm give us moral certainty of the practical success of our calculations?’ (Liston [2007], p. 4). Presumably the point is that, by itself, the truth of these
mathematical beliefs wouldn’t make their applicability any less mysterious.
arealism
93
This is undoubtedly an important question. The short answer is that
the new features revealed by the full force of our pure mathematics
often aren’t effective modelers: to recall an example from }I.3, though
point sets in ℝ3 are well-known to be good stand-ins for spatial
regions, we shouldn’t expect the Banach-Tarski construction to be
physically realizable. So we need to focus more closely on the kinds of
examples that interest Liston. Often the particular methods he discusses
got their start in Euler’s day, when the relations between physics and
mathematics were quite different than they are now (see }I.2): new
mathematical structures were being added in response to the needs of
science, and physical intuition was a central guide. Consider, for
example, Euler’s Method, an approximation procedure familiar to
readers of Mark Wilson:
There is a venerable computational technique called Euler’s method of finite
differences that will estimate our cannon ball’s instantaneous . . . deceleration
using an averaged change of speed considered over, say, ¼ second stretches of
time . . . This routine allows us to calculate a succession of numerical values
which, if graphed and connected together by straight lines, generally provides
a reasonable broken line facsimile to our cannon ball’s path. (Wilson [2006],
p. 117)
This procedure is entirely natural from a physical point of view:
The matrix of numerical data assembled by this syntactic routine provides us
with an excellent stage by stage ‘image’ of our ball’s flight . . . Our symbolic
calculations ‘walk along’ at discrete stages with our cannon ball . . . indeed,
Euler’s procedure is commonly called a ‘marching method’ for that very
reason. (Wilson [2006], p. 164)
The unwelcome surprise is that Euler’s Method doesn’t always work;
if we represent the cannon ball with a slightly modified pair of
differential equations, Euler’s Method tells us that the ball will never
fall to the ground!7 Subsequent mathematical work uncovers the
trouble: Euler’s Method provides a good approximation to the true
solution if the differential equations in question satisfy the Lipschitz
condition, and these modified equations do not.
7
See Wilson [2006], pp. 214–215.
94 arealism
Here we have a clear illustration of what Liston means by a ‘reliability explanation’: we’ve proved that Euler’s Method is reliable when
our equation meets the Lipschitz condition. Liston might imagine us
building a bridge or some such thing. We model the situation with
some differential equations but find we can’t solve them directly. Can
we trust Euler’s Method to keep us close enough to the true solution;
can we trust it enough to give the go-ahead to those poised to begin
pouring concrete? We then confirm that our equations satisfy the
Lipschitz condition. If we’re to allow this fact to underwrite our
trust in the Method, don’t we have to believe the mathematics
involved in proving the relevant theorems?
Before answering yes, think again about what those theorems have
done. Have they guaranteed that our computations will be physically
reliable? No, what they’ve shown is something purely abstract; they’ve
shown that if you have a differential equation with a certain nice
feature, then Euler’s Method will generate outcomes fairly close to
the actual function that satisfies your equation. This, as I’ve described
it, is a property of the equation, as a mathematical object. In deciding
whether or not to green-light the cement mixers, what we need to
know is whether or not that equation, in all its Euler’s Methodratifying glory, is a good model for the relevant features of our building
situation. If we trust Euler’s Method, go ahead, and the bridge falls
down, it isn’t because the mathematics used to prove those theorems
about Euler’s Method isn’t true; it’s because the original equation we
selected to model our bridge wasn’t a good model, after all. What the
reliability argument tells us isn’t that our original equation is a good
model or that our bridge won’t fall down; it tells us that if the original
equation is a good model, then the bridge won’t fall down (at least not
because the use of Euler’s Method led us astray!). In ordinary engineering contexts these days, our ways of determining the appropriate
equations have a long history, plausible physical underpinnings, and a
strong track record, so it’s perfectly reasonable to trust Euler’s Method
when the equation is of the right type.
The case of the Axiom of Choice and the Banach-Tarski paradox is
quite different. As noted in Chapter I, the set-theoretic methods involved, the set-theoretic goals in play, are largely the product of contemporary pure mathematics, a long way from the physically-inspired
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95
approaches employed by Euler. Though we could construct a bogus
‘reliability argument’ that tells us an object can be disassembled and
recombined into two objects the same size as the original, of course
that wouldn’t make it so. The trouble isn’t that the set theory used to
prove the relevant theorems isn’t true; it’s that the theorems rely on
aspects of point sets that don’t correspond well with physical regions.
Unlike the case of Euler’s Method—which is well-grounded in our
understanding of the physical situation and enjoys a long history of
empirical success—what’s involved in the set-theoretic case is the finestructure of pathological sets of real numbers (that is, sets that don’t have a
coherent ‘size’, even by the generous standards of Lebesgue measure). If
we draw too heavily on these features of our mathematical model of
space, we will be led astray.
Now it might be tempting to suggest that this unfettered contemporary mathematics be sequestered in some way, set aside as unreliable—that we carve out from the vast reach of mathematical lore the
sounder body of physically-inspired studies and restrict our serious
attention, and perhaps our NSF grants and university professorships,
to its pursuit. The trouble is that bits of unfettered pure mathematics
have turned out to be profoundly applicable, and there’s no way to
predict ahead of time which these will be.8 (As noted in }II.3, group
theory is a famous example.) The practice of contemporary pure
mathematics answers to many desiderata, including but not limited
to foreseeable applications, but I think no honest observer, even one
primarily interested in mathematics only for the sake of application,
would think it prudent to rein it in.
This raises an obvious question: how is it that mathematics pursued
for purely mathematical reasons ends up serving the needs of empirical
science? why should the theorems of group theory be reliable guides to
the behavior of subatomic particles? why should some troublesome,
mathematically esoteric aspect of an otherwise effective mathematical
model turn out—most unexpectedly!—to have a physical correlate?!
(Here the well-worn examples are radio waves and positrons.9) This,
I think, is the sort of thing that’s really bothering Liston: just because
8
9
See [2007], pp. 329–343, for more on the topic of this paragraph and the two following.
See [2007], pp. 332–333.
96 arealism
the aspects of our model we’ve been attending to work well, why
should we expect some newly revealed mathematical feature of that
same model to continue to work well? There are good answers to this
question for many familiar tools of applied mathematics, good reasons,
for example, to regard an equation’s satisfying the Lipschitz condition
as evidence that Euler’s Method will be reliable. But why should we
expect these new features to work well when they arise from a notion
of ‘correctness [that is] a function of furthering internal goals of pure
mathematics’ (Liston [2007], p. 4)?
What we have here is one sub-problem of Wigner’s famous ‘miracle
of applied mathematics’: why does mathematics generated for mathematics’ sake end up being successfully applied?10 On closer examination of the contexts in which pure mathematics does and doesn’t work
in applications, I’m not sure its successes are as miraculous as all that,11
but for present purposes it’s enough to notice that what’s at issue here,
perplexing as it may be, doesn’t hinge on the truth (or not) of the
mathematics involved, but on the fact that it is generated in pursuit of
purely mathematical goals.
3. What Arealism is not
Assuming then that the truth (or not) of mathematics is irrelevant to
explaining its role in scientific application, it appears that Arealism is
open to our Second Philosopher: she notes that mathematics is successful on its own terms and immensely useful to science, but since it
isn’t confirmed by her usual methods, even by her need to explain the
role it plays in her empirical theorizing, she concludes that she has no
grounds on which to regard its objects as real or its claims as truths. In
philosophical taxonomy, the standard term for someone who doesn’t
believe in abstract objects is ‘nominalist’. If we limit attention to
mathematical abstracta, the Arealist would seem to qualify, but, at
least as ‘nominalism’ is usually conceived in contemporary philosophy
10
See Wigner [1960].
Again, see [2007], pp. 329–343. In his writings on ‘the miracle’, Liston [2000] seems to
agree. See e.g. [2007], p. 340, footnote 58.
11
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97
of mathematics, this way of talking seems to me to invite misunderstanding.
To see how, recall that contemporary nominalism began with
Goodman and Quine’s annunciation of
a philosophical intuition that cannot be justified by appeal to anything more
ultimate . . .
namely,
We do not believe in abstract entities. . . . We renounce them altogether.
(Goodman and Quine [1947], p. 105)
In Burgess and Rosen’s characterization:
Nominalism (as understood in contemporary philosophy of mathematics)
arose toward the mid-century . . . It arose . . . among philosophers, and to this
day is motivated largely by the difficulty of fitting orthodox mathematics into
a general philosophical account of the nature of knowledge. (Burgess and
Rosen [1997], p. vii)
What’s at work here is the picture of the ‘great gulf’ introduced in
}II.4: to avoid nominalism, one must
explain in detail how anything we do and say on our side of the great wall
separating the cosmos of concreta from the heaven of abstracta can provide us
with knowledge of the other side. (Burgess and Rosen [1997], p. 41)
Various familiar ideas on the nature of knowledge in concrete cases,
like the causal theory of knowledge and its successors, are floated to
highlight the severe obstacles that stand in the way of such an explanation. These elements provide the raw materials for a perfectly general,
in-principle argument against abstracta of all kinds.
I hope and trust it’s clear that this is not a portrait of the secondphilosophical Arealist. She doesn’t come to her investigations with any
a priori prejudice against abstract objects or with any preconceptions
about what knowledge must be like that would seem to rule out
knowledge of sets. She doesn’t argue that set-theoretic knowledge
is problematic or impossible on principle; she simply surveys the
evidence at hand and concludes that it doesn’t confirm the existence
of sets or the truth of our theory of them. So if Arealism is to be
98 arealism
considered a version of nominalism, it certainly isn’t the ‘stereotypical’
variety (Burgess and Rosen [1997], p. 29).
Another popular characterization of the view that there are no
mathematical objects comes under the rubric of ‘fictionalism’. As
Hartry Field has put it, ‘the sense in which’ a mathematical claim
is true is pretty much the same as the sense in which ‘Oliver Twist lived in
London’ is true: the latter is true only in the sense that it is true according to a
certain well-known story, and the former is true only in that it is true according to
standard mathematics. (Field [1989], p. 3)
Presumably the fictionalist would say that Cantor and Dedekind told
the opening chapters in the story of set theory, that Zermelo organized
it, that contemporary set theorists are extending it. Of course, as we’ve
seen, the story line is closely constrained, not just by logical consequence and consistency, but (the fictionalist continues) the story line in
good fiction-writing must also hew to demanding standards (say of
realistic description and psychology).
Drawing an analogy in this way between fiction-writing and the
practice of set theory has the merit of providing a clear illustration of
how one can legitimately make assertions, make truth and existence
claims, ‘within a story’, while denying them in a broader context.12
Beyond this, perhaps further exploration of the similarities and dissimilarities will shed light on set theory—the Arealist has no cause or
grounds to rule this out—but I confess to some skepticism. The central
challenge is to delineate and defend the proper ways of extending what
the fictionalist calls the ‘set-theoretic story’, but calling it that, rather
that just ‘set theory’, doesn’t appear to advance our understanding on
this point. It’s no doubt difficult to spell out what makes one way of
extending a fictional story better than another, as the labors of literary
critics demonstrate, but it seems unlikely that any insight that might be
gleaned on that topic would translate in any useful way to set theory.
Or vice versa. If this is right, then the value of the fictionalist analogy is
limited, and keeping it at the forefront of our thinking about set theory
might tempt us to impose categories and judgments foreign to our
subject and to ignore important features without correlates in fiction.
12
Thanks to Patricia Marino for pressing this point.
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99
For reasons like these, it seems to me best to push all the popular
analogies to the background—math is like science, math is like a game,
along with math is like fiction—and to study set theory directly, on its
own terms. Such an Arealist simply understands herself as developing a
theory of sets, guided, as we’ve seen, by various concrete set-theoretic
norms and goals and values.
Finally, various strains of formalism or if-thenism also deny that
mathematics is in the business of discovering truths about abstracta;
they take their lead from mathematicians who say, ‘I’m just figuring
out what follows from what’. Again, I hope it’s clear that the Arealist
would say no such thing: though mathematicians are often engaged in
proving one thing from another, they obviously don’t regard any
starting point, even any consistent starting point, as equally worthy
of investigation; if one characterizes set-theoretic practice as that of
deriving theorems in one or another axiomatic setting, one ignores the
very features of that practice that have been my Arealist’s focus,
namely, the forces that shape the concepts and assumptions of the
setting itself.13 A more sophisticated if-thenist would admit that mathematics is more than a matter of determining what follows from what,
that mathematicians are also engaged in forming those concepts and
selecting those assumptions, and would then assume responsibility for
explaining how this process is constrained, what principles should
guide it and why. While the term ‘if-thenism’ invites an overly narrow
focus on logical connections, there doesn’t appear to be any difference
of substance between this more subtle if-thenist and the Arealist.
4. Comparison with Thin Realism
If Arealism doesn’t quite fit the standard profiles of nominalism or
fictionalism or simple if-thenism, one position from which it would
13
Recall from }III.2 the related problem for an account of set-theoretic practice in terms
of Carnapian linguistic frameworks: enshrining any fixed principles as implicitly defining ‘the
concept of set’ seemed false to the open-ended nature of the practice. Essentially the same
problem arises when the if-thenist decides what to put in the antecedent to the conditionals.
Burgess [unpublished] argues persuasively that the same problem arises for many versions of
structuralism.
100
arealism
seem to stand unambiguously apart is Thin Realism. After all, the Thin
Realist holds that sets exist and set theory is a body of truths, and the
Arealist denies both. But despite their disagreements over truth and
existence, the Thin Realist and the Arealist are indistinguishable at the
level of method. On grounds like those that motivated Cantor and
Dedekind, both would elect to introduce sets into their pursuit of pure
mathematics; both would regard Zermelo’s defenses of his axioms as
persuasive; both would follow the path of contemporary set theorists
on determinacy and large cardinals. This methodological agreement
reflects a deeper metaphysical bond: the objective facts that underlie
these two positions are exactly the same, namely, the topography of
mathematical depth brought to light in }III.4. For the Thin Realist,
sets are the things that mark these contours; set-theoretic methods are
designed to track them. For the Arealist, these same contours are what
motivate and guide her elaboration of the theory of sets; she can go
wrong as easily as the Thin Realist if she fails to detect the genuine
mathematical virtues in play. For both positions, the development of
set theory responds to an objective reality—and indeed to the very
same objective reality.
What separates the Arealist from the Thin Realist, then, doesn’t lie
in their set-theoretic practices or what underlies them; in that respect,
the two are indistinguishable. Where they differ is in their secondphilosophical reflections on the human undertaking called ‘set theory’.
They would agree precisely on what counts as proper grounds for
adding a new large cardinal axiom to the theory of sets; they would
disagree only on the Thin Realist’s added assertion that these grounds
confirm the existence of the large cardinal in question and the truth of
the corresponding axiom. Notice that it isn’t an ordinary set-theoretic
claim of existence or truth that’s at issue here: the Arealist like the Thin
Realist will formulate the axiom in existential form and call it ‘true’ in
the sense of holding in V. Their disagreement takes place not within
set theory, but in the judgments they form as they regard set-theoretic
language and practice from an empirical perspective and ask secondphilosophical versions of the traditional philosophical questions, questions in the second group we’ve been considering.
At this point, we have two apparently second-philosophical positions in play; how is the Second Philosopher to adjudicate between
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Thin Realism and Arealism? This returns us at last to the problem
we’ve set aside twice: on what grounds does the Thin Realist judge
that set theory is a body of truths? Given that she rejects the usual
Quinean arguments, given that she endorses the Arealist’s account of
how mathematics works in application, the Second Philosopher’s case
for Thin Realism will have to rest more loosely on the interconnections of mathematics with empirical science: she recognizes that pure
mathematics arose out of a subject very closely tied to our study of the
physical world; she regards the project of providing a rich array of
structures for the contemporary scientist as one of the overarching
goals of mathematical practice; she well appreciates that contemporary
pure mathematics continues to find its way into scientific applications,
sometimes along deliberately anticipated paths, and sometimes along
wholly unexpected ones. Thus mathematics, whatever its idiosyncrasies, appears as an integral part of her overall enterprise (as opposed to
astrology, theology, etc., which are idiosyncratic without playing a
part in that enterprise).14 On this picture, the Second Philosopher
pursues mathematics in a spirit continuous with her other inquiries:
some of its methods, like logical deduction and means-ends reasoning,
are familiar; others, like Cantor’s, Dedekind’s, Zermelo’s, and the
determinacy theorists’, are unfamiliar, but taken to be rational and
reliable along the lines we’ve been following.
Thus the divergence between the second-philosophical Arealist and
the second-philosophical Thin Realist comes down to this: as the
Second Philosopher conducts her inquiry into the way the world is,
beginning with her ordinary methods of perception and observation,
theory-formation and testing, she’s eventually faced with the effectiveness of pure mathematics and elects to add it to her ever-growing list of
investigations; she also recognizes that the appropriate methods are
different and that the objects studied are different; the point at issue
hinges on what she concludes from this. If the new objects seem a bit
odd—non-spatiotemporal, acausal, etc.—but still enough like the
old—singular bearers of properties, etc.—, if the new methods seem
a bit odd, but still of-a-piece with the old, then she concludes that she’s
14
See [1997], pp. 203–205, [2007], pp. 345–347.
102 arealism
made a surprising discovery, that the world includes abstracta as well as
concreta. If, on the other hand, she regards the new methods and
would-be objects as sharply discontinuous with what came before, she
has no grounds for thinking pure mathematics is true, so she concludes
that this new practice—valuable as it is—isn’t in the business of
developing a body of truths. So, which is it? Is pure mathematics just
another inquiry among many or it is a different sort of thing that’s
immensely helpful to the others? Are the grounds cited by Cantor,
Dedekind, Zermelo, and the determinacy theorists just more evidence
of an unexpected sort, or are they the trademarks of a different sort of
activity altogether?
I think this understanding of the disagreement between the Thin
Realist and the Arealist further illuminates the contrast between Arealism and more familiar forms of nominalism. To see this, consider
David Lewis’s well-known credo:
Renouncing [sets] means rejecting mathematics. That will not do. Mathematics is an established, going concern. Philosophy is shaky as can be. To
reject mathematics for philosophical reasons would be absurd. If we philosophers are sorely puzzled by the [sets] that constitute mathematical reality, that’s
our problem. We shouldn’t expect mathematics to go away to make our life
easier. Even if we reject mathematics gently—explaining how it can be a most
useful fiction . . . —we still reject it, and that’s still absurd. . . . How would
you like the job of telling the mathematicians that they must change their
ways . . . ? (Lewis [1991], pp. 58–59)
The Arealist doesn’t reject mathematics or recommend that mathematicians change their ways; as we’ve seen, the Arealist is indistinguishable from the Thin Realist as far as the practice of mathematics is
concerned. Furthermore, as we’ve also seen, the Arealist isn’t denying
the truth of mathematics on general philosophical grounds, as Lewis
implies. She isn’t ‘puzzled’ by mathematical existence; she takes the
Thin Realist’s position to be coherent, even in some ways attractive.
She just doesn’t see that the evidence supports it.
All this is familiar, but another element is suggested here: that the
Arealist is disagreeing with what the mathematician says. In one sense,
I’ve suggested that this isn’t right: the Arealist is speaking as a Second
Philosopher proposing an account of human mathematical activity; she
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103
isn’t questioning the propriety of any aspect of the practice, including
its assertions in the shape of existence claims. Of course, it’s also clear
that mathematicians themselves have philosophical views about the
nature of mathematics, about what those assertions mean—recall (from
}II.3) Dedekind’s remark about ‘free creations of the human mind’—
and this part of what mathematicians believe can come into conflict
with views like Arealism and Thin Realism. The Arealist doesn’t
disagree with what mathematicians say qua mathematicians, but
when they branch out into questions of truth and existence external
to mathematics proper—what is the nature of human mathematical
activity? what is its subject matter and how do we come to know about
it? and so on—then she reserves her right to differ.
The larger point here is that the fine structure of what particular
mathematicians mean or intend by their mathematical assertions—for
example, whether or not they think like Dedekind or make their
claims with ‘mental reservation or purpose of evasion’15—these psychological matters are no more relevant to the correctness or incorrectness of Thin Realism or Arealism than a physicist’s personal
views—for example, that he’s discovering the acts of God or merely
organizing the course of our sense-data—would be to the correctness
or incorrectness of some analysis of the status of general relativity or
quantum mechanics. What matters, from our second-philosophical
point of view, isn’t what the practitioners think about these issues in
their heart-of-hearts, but where the evidence leads. The Thin Realist
gives one answer and the Arealist another.
5. Thin Realism/Arealism
So who’s right, where does the evidence lead—to Thin Realism or
Arealism? The way I’ve been describing the Second Philosopher’s
intellectual journey, it apparently leaves the Thin Realist with the
challenge of explaining exactly how the role of pure mathematics in
science supports her view that it is a body of truths about objectively
15
Burgess [2004a], p. 54, brings this up in his discussion of fictionalism.
104 arealism
existing things, exactly what justifies her assumption that the methods
of pure mathematics, distinctive as they are, should be regarded as partand-parcel of her ever-evolving approach to finding out about the
world. But there are other ways of describing that second-philosophical
journey that might seem to reshape the debate.
Consider, as a start, Burgess and Rosen’s characterization of the
‘stereotypical anti-nominalist’:
We come to philosophy16 believers in a large variety of mathematical and
scientific theories—not to mention many deliverances of everyday common
sense—that are up to their ears in suppositions about entities nothing like
concrete bodies we can see or touch, from numbers to functions and sets . . .
To be sure, we also come . . . prepared to submit all our . . . beliefs to critical
examination and to revise them if good reasons for doing so emerge. (Burgess
and Rosen [1997], p. 34)
A Second Philosopher could be described along these lines. She begins
in our contemporary world-view, where pure mathematics is ‘considered . . . the very model of a progressive and brilliantly successful cognitive endeavor’ (Burgess and Rosen [1997], p. 211). Burgess and
Rosen’s
thorough-going naturalist would take the fact that abstracta are customary and
convenient for the mathematical (as well as other) sciences to be sufficient to
warrant acquiescing in their existence. (Burgess and Rosen [1997], p. 212)
Our Second Philosopher may be a bit pickier than that—requiring, for
example, the kinds of detailed support offered by Cantor, Dedekind
and the rest—but the underlying idea is simple: this second-philosophical starting point already includes pure mathematics as a body of truths
alongside physics, chemistry, botany, and so on.
Of course some of the things we tend to believe, from the gambler’s
fallacy to spiritualism, will succumb to critical examination, but what
grounds would dictate that ‘the mathematical sciences . . . be expelled
from the circle of “sciences” ’, what grounds could there be for
‘marginalizing some sciences (the mathematical) and privileging others
16
Speaking second-philosophically, we’d say: ‘to the examination of human mathematical
activity . . . ’.
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105
(the empirical)’, for ‘abridging the roll of sciences’ (Burgess and Rosen
[1997], pp. 211–212)? The stereotypical nominalist would offer such
grounds, but we’ve seen that the Second Philosopher comes to the
table with no prejudice against abstracta, no general theory of what
knowledge must be, and thus apparently no reason to change her
initial opinion about the truth of pure mathematics, her opinion that
mathematical evidence is evidence. When she comes to inquire into
the nature of the human mathematical practice, she elaborates Thin
Realism as in the previous chapter.
Since Chapter II, we’ve been investigating what the Second Philosopher would do or say in this situation or that, but we’ve now returned
to the question of how Second Philosophy itself should be characterized. My rhetoric there, and subsequently, has been consistently of the
bottom-up variety; the Second Philosopher begins with her ordinary
perceptual beliefs and refines her methods from there, expanding her
reach into all areas of what we might call ‘natural science’ and even
‘social science’: physics, chemistry, astronomy; biology, botany, mineralogy; psychology, linguistics, and the study of human inquiry itself.
One of these extensions is into pure mathematics, where she may well
wonder if its methods serve to establish truth and existence or actually
do something else; this is what seemed to place a further explanatory
demand on the Thin Realist and made Arealism appear as an attractive
possibility. But if we imagine the Second Philosopher along these new
lines inspired by Burgess and Rosen, starting with a complex body of
beliefs and deciding which to reject, then pure mathematics is included
from the start, and no piecemeal defense, beginning from perception
and leading to mathematics, would seem to be required. From this
point of view, Thin Realism may appear more natural than Arealism.
It’s hard not to think that one must be right and the other wrong, that
either sets exist or they don’t, that set theory is a body of truths or it isn’t,
that either the considerations cited by Cantor, Dedekind, Zermelo, and
the determinacy theorists are confirming evidence or they aren’t. But
perhaps this tempting position is in fact incorrect, perhaps our strong
conviction otherwise rests on what Mark Wilson calls, in his typically
colorful style, ‘tropospheric complacency’: we tend to think that our
concepts—in our case ‘true’, ‘exist’, ‘evidence’, ‘believe’, ‘know’—mark
fully determinate features or attributes, that there is a determinate fact of
106 arealism
the matter as to where they apply and where they don’t, that this is so
even for questions we haven’t yet been able to settle one way or the
other. Wilson’s case against this picture is one thread running through his
massive Wandering Significance (Wilson [2006]); it rests largely on a series
of fascinating and down-to-earth examples. To get a feel for how these
examples go, let’s look at two of them.
First, consider ice. Surely we all know what ice is—it’s frozen
water—but Wilson takes us in for a closer look:
Water, in fact, represents a notoriously eccentric substance, capable of forming
into a wide range of peculiar structures. (Wilson [2006], p. 55)
He goes on to quote a recent textbook on the subject, which describes
‘ice cousins’,
the clathrate hydrates . . . Like ice polymorphs, they are crystalline solids,
formed by water molecules, but hydrogen-bonded in such a way that polyhedral cavities of different sizes are created that are capable of accommodating
certain kinds of ‘guest’ molecules. (Quoted by Wilson [2006], p. 55)
Wilson remarks that
The author doesn’t regard the clathrate structure as true ice . . . but is it clear
that our everyday conception of ice requires—as opposed to accepts—this
distinction? (I, for one, had never thought about such matters at all.) (Wilson
[2006], pp. 55–56)
It gets worse: there are in fact more than a dozen ways that water can
form into a solid. In one case, if one cools water quickly enough, the
result lacks crystalline structure and more closely resembles ordinary
glass. Wilson asks
Should this glass-like stuff qualify as a novel form of ‘ice’ or not? Our chemist
will presumably say ‘no’ because the stuff is not crystalline but many of us would
perhaps put a higher premium on its apparent solidity. (Wilson [2006], p. 56)
In fact other chemists do happily call this ‘an amorphous type of ice’
(Caro [1992], p. 99).17 And so on.
17
Wilson doesn’t cite this passage in his discussion of ‘ice’, but he does quote Caro’s book
when he treats the relations between ‘water’ and ‘H2O’ (Wilson [2006], pp. 428–429).
arealism
107
Is there a right and a wrong answer here? Our everyday use of the
word ‘ice’ clearly correlates with an objective feature of the world, the
substance chemists call ‘ice Ih’ or hexagonal crystalline ice. So ‘ice’
definitely doesn’t apply to liquid water or to sand or to window glass.
But does it apply to amorphous ice—is amorphous ice really ice?
Wilson’s thought is that nothing in our ordinary use or understanding
of the term ‘ice’, indeed nothing in the underlying chemical facts that
we subsequently discover about the many ways water can form into a
solid—in short, nothing in our heads, in our language, or in the world
will force either answer to this question.18 And notice that this isn’t a
version of the well-known Kripkesteinian challenge: what makes 1002
rather than 1004 the right continuation of þ2 after 1000? We have
here not the hyperbolic doubt of a radical skeptic, but real life cases
‘where the underlying directivities seem genuinely unfixed’ (Wilson
[2006], p. 39).
A second example is more fanciful, but still quite compelling for all
that. Imagine the inhabitants of an isolated island; imagine they’ve
never seen an airplane until one passes overhead and crashes in their
midst. They might quite naturally regard it as a bird, regard themselves
as having learned, unexpectedly, that the world includes a type of bird
very different from the ordinary birds they’re familiar with, a great
silver bird made of metal. Now imagine the story again, except that
this time the plane crashes undetected and the islanders discover it in
the jungle with the stranded crew taking shelter in the fuselage. This
time, the islanders might reasonably regard it as a house, might well
regard themselves as having discovered a new and unusual type of house.
Is there any temptation here to think that one group is wrong and the
other right? It seems clear that nothing in their pre-airplane concepts of
‘bird’ and ‘house’ or the corresponding worldly resemblances is enough
18
A similar theme turns up in Austin [1940], pp. 67–68: ‘Suppose that I live in harmony
and friendship for four years with a cat: and then it delivers a philippic. We ask ourselves,
perhaps, “Is it a real cat? or is it not a real cat?” “Either it is, or it is not, but we cannot be sure
which.” Now actually, that is not so: neither “It is real cat” nor “it is not a real cat” fits the facts
semantically: each is designed for other situations than this one . . . Ordinary language breaks
down in extraordinary cases . . . no doubt an ideal language would not break down, whatever
happened . . . In ordinary language . . . words fail us. If we talk as though an ordinary [language]
must be like an ideal language, we shall misrepresent the facts’.
108 arealism
to determine this, that either option is open to them as a consistent and
defensible extension of the earlier concepts, that their choice is determined by sheer historical contingency. But notice:
neither set of alternative [islanders] has any psychological reason to suspect that
they have not followed the preestablished conceptual contents of their words
‘bird’ and ‘house’. ( Wilson [2006], p. 36)
Here we see the psychological force of tropospheric complacency in its
purest form.19
Could it be that a similar brand of complacency is at work in the case
of the Second Philosopher faced with pure mathematics? The line of
thought we’ve been following contrasts two ways of describing the
Second Philosopher: if she starts from scratch, slowly accumulating
true statements, gradually adding to the stock of her ontology, then she
will be faced with justifying the leap to the existence of sets and the
truth of set theory; if she starts from a body of accepted doctrine, and
employs her critical faculties to eliminate those entities and statements
that aren’t well-supported, she will find no clear grounds on which to
remove her seal of approval from sets and set theory. Depending on
her starting point, the Second Philosopher comes to these opposing
conclusions, in each case equally convinced of their faithfulness to
original concepts of ‘evidence’, ‘object’, ‘truth’, ‘existence’, and so on,
but the difference of starting point is surely a contingency no deeper
than the accident of how the islanders happen to first encounter the
airplane.
Appealing as it may be,20 the analogy here is imperfect. Consider
this from Wilson, speaking of islanders:
The key ingredient in our fictional tale lies in its attention to the enlargement of
linguistic application: specifically, to the latitude displayed when a usage previously confined to a limited application silently expands into some wider
domain. . . . we can profitably picture these circumstances as representing a
circumstance where we prolong our usage from one neighborhood of local
application into another. In the [islander] case, two competing continuations are
19
See Wilson [2006], pp. 34–37, for more on the islanders, or [2007], pp. 186–188, for a
somewhat more complete summary.
20
It appealed to me in [2007], pp. 385–386.
arealism
109
available whereby the old usage might plausibly enlarge to take proper
account of aircraft. (Wilson [2006], p. 37)
Other examples also involve such linguistic extensions into new arenas, in our case, the extension of ‘ice’ to unusual temperatures and
pressures. To treat our case analogously, we’d imagine ourselves with
an established use of ‘true’, ‘exists’, and the rest in empirical contexts,
now facing the question of how to properly extend it to the new arena
of pure mathematics—but this is to presuppose the bottom-up starting
point, to beg the question against the top-down point of view, from
which there’s no call for any extending, because pure mathematics is
already regarded as true. So the problem doesn’t appear to lend itself to
Wilsonian dissolution, at least not in quite this way.
We’ve been exploring the idea that the answer to our question—which
position on pure mathematics emerges from the second-philosophical
approach, Thin Realism or Arealism?—may hinge on whether we characterize Second Philosophy bottom-up or top-down. Indeed it may well
be that something like this does lie behind a not-insignificant portion of
the rhetoric of this debate. But in fact I think that the bottom-up/topdown contrast is a red herring, that the true point at issue is both less
sophisticated and more fundamental. To see this, consider again the topdown story.
We imagine ourselves as we are, immersed in a vast system of beliefs,
adding to it and subtracting from it as we go. Some of those subtractions result from the kind of critical examination Burgess and Rosen
acknowledge: we root out superstition, improve our experimental
techniques, reject what’s ill-confirmed. Just as our investigations of
optics, of the structure of the eye, and of cognitive processing assure us
that perception is largely reliable under certain conditions, our cleareyed evaluation of astrological evidence finds it lacking. Now what
happens when we turn our critical attention on the methods of
pure mathematics? Burgess and Rosen’s stereotypical anti-nominalist
responds primarily to their stereotypical nominalist’s in-principle epistemological case against abstracta, and the Second Philosopher is in
agreement here that this case is ineffective. But this won’t be the end of
her examinations, any more than the rejection of some superficial case
against perception ends her efforts to assess its reliability. After setting
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arealism
aside the stereotypical nominalist’s general concerns, the Second Philosopher will still want to know whether or not the considerations
cited by Cantor, Dedekind, Zermelo, and the determinacy theorists
properly count as evidence for the truth of their claims and the
existence of sets.
In response, Burgess and Rosen’s stereotypical anti-nominalist observes that ‘abstracta are customary and convenient for the mathematical (as well as other) sciences’ and decries ‘the marginalizing [of ] some
sciences (the mathematical) and privileging [of ] others (the empirical)’
(Burgess and Rosen [1997], pp. 211–212). This may seem to rely on a
more substantive characterization of ‘science’ than we’ve got—the
introduction of the Second Philosopher as a character was largely
motivated, after all, by the lack of such necessary and sufficient conditions—but I don’t think it must. My bottom-up characterization of
the Second Philosopher’s inquiries has repeatedly appealed to a suggestive list: in her hands, perceptual belief and ordinary common sense
gradually lead to the pursuit of studies from physics, chemistry, and
astronomy, to botany, psychology, and linguistics. I see no reason the
top-down characterization couldn’t use the same device, in which case
the claim needn’t be that pure mathematics is a science, but that it
belongs on the list with physics, botany, and the rest.
Which brings us to this question: given that the methods of pure
mathematics differ from those of the other entries on that list far more
than the other entries differ amongst themselves, on what grounds does
the top-down Second Philosopher ratify her previously-unexamined
judgment that it belongs in this company, nonetheless? As far as I can
determine, the only available answer rests on the basic idea that the roots
of pure mathematics in physics, astronomy, engineering, the way it now
intertwines with these subjects and with others on the list, gives us
reason to regard it as one of them. And here, quite unexpectedly, we
find ourselves speaking in the precisely same terms as the bottom-up
Second Philosopher when she defends Thin Realism with the same sort
of appeal to the physical roots of pure mathematics and its continued
interconnections with the empirical sciences. So the same considerations arise either way, whether we’re contemplating an extension of our
usage or evaluating the propriety of an existing usage; the contrast of
arealism
111
bottom-up with top-down characterizations of Second Philosophy
turns out to be irrelevant.
Thus we’re returned to an embarrassingly simple question: does
the history and current practice of pure mathematics qualify it as just
another item on the list with physics, chemistry, biology, sociology,
geology, and so on? In fact, this is just another way of posing our
original question: do honorifics like ‘true’, ‘exist’, ‘evidence’, ‘confirm’—indisputably at home in those other studies—belong in pure
mathematics as well? We’ve examined how pure mathematics arose
out of our empirical study of the world, how it remains intensely
important as a tool for that study, even in parts that weren’t expressly
developed for that purpose; we’ve noted how it continues to be
inspired by the descriptive and inferential needs of the natural and
social sciences. If all this is taken to establish it as a body of truths,
we’ve seen how the Thin Realist explicates the ground of that truth
and how mathematical evidence manages to track it. But we’ve also
seen how the Arealist gives a plausible account of pure mathematics
as a deep and vital undertaking that happens not to aim at producing
truths.
What I want to suggest now, indeed at last to claim, is that a path has
opened to a simpler Wilsonian dissolution: our central questions—is
pure mathematics of-a-piece with physics, astronomy, psychology,
and the rest? is it a body of truths? do its methods confirm its
claims?—these questions have no more determinate answers than ‘is
amorphous ice really ice?’ Once we understand the various ways in
which water can solidify, how these processes are affected by temperature, pressure, and other factors, how the various structures generated
are similar and how they’re different, there’s nothing more to know;
we can reflect these facts in either way of speaking, or, to put it the
other way around, neither way of speaking comes into conflict with
the facts. Some version of tropospheric complacency—our tendency
to overestimate the determinateness of our concepts—might well
leave us convinced of the exclusive correctness of one or the other—
it must be ice because it’s solid! it can’t be ice because it’s not
crystalline!—but we’ve seen that this psychological confidence is
often baseless, and also largely harmless.
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arealism
Likewise, once we understand how pure mathematics developed,
how it now differs from empirical sciences,21 once we understand the
many ways in which it remains intertwined with those sciences, how
its methods work and what they are designed to track—once we
understand all these things, what else do we need to know? Or better,
what else is there to know? Just as robins are birds and bungalows are
houses, physics and botany are sciences, but this isn’t enough to settle
the status of downed airplanes and pure mathematics. Just as amorphous ice can be classified as ice or as ice-like, mathematics can be
classified as science or as science-like—and nothing in the world makes
one way of speaking right and the other wrong.
If this is right, then we, more self-aware than the islanders, should
recognize that there is no substantive fact to which our decision
between Thin Realism and Arealism must answer. The application
of ‘true’ and ‘exists’ to the case of pure mathematics isn’t forced upon
us—as it would be if Thin Realism were right and Arealism wrong—
nor is it forbidden—as it would be if Arealism were right and Thin
Realism wrong. Rather, the two idioms are equally well-supported by
precisely the same objective reality: those facts of mathematical depth.
These facts are what matter, what make pure mathematics the distinctive discipline that it is, and that discipline is equally well described as
the Thin Realist does or as the Arealist does. Once we see this, we can
feel free to employ either mode of expression, as we choose—even to
move back and forth between them at will.
The proposal, then, comes to this: Thin Realism and Arealism
are equally accurate, second-philosophical descriptions of the nature
of pure mathematics. They are alternative ways of expressing the
very same account of the objective facts that underlie mathematical
practice.
21
In case there’s any lingering doubt, I’m not assuming we have a characterization of
‘science’ or ‘empirical science’; I’m using the term as short-hand for the familiar list of
activities we’ve been talking about.
V
Morals
At this point, I imagine that some readers sympathetic to the general
themes of Thin Realism may recoil at the barren world of Arealism,
and likewise that some hard-nosed Arealists may disapprove the ontological excesses of Thin Realism. Sadly for such partisans, our secondphilosophical analysis gives them little encouragement. To secure a
Thin Realism unsullied by association with Arealism, one would
need a sound argument for the truth of mathematics, and we’ve seen
strong reasons for thinking that the most promising avenue to such
an argument—the effectiveness of mathematics in application—isn’t
conclusive. Likewise, the staunch advocate of an Arealism decoupled
from Thin Realism would need an equally effective, and apparently
equally elusive, argument against the truth of mathematical claims.
Though the analysis also suggests that those preferring one idiom
to the other are free to speak as they choose without fear of error, it
denies them any exclusive rights.
In the end, then, the Second Philosopher is led to a position that
is neither a whole-hearted endorsement of Thin Realism nor an
unequivocal assent to Arealism: she recognizes instead that there is
no substantive difference between the two, that either way of describing the underlying constraints of the practice is admissible. In this final
chapter, I hope to draw a few morals from this conclusion, with a look
at a traditional line of thought on mathematical objectivity, a glance
back at Robust Realism, and finally, a return to the original problem of
defending the axioms, in particular, to the interrelations of intrinsic
and extrinsic justifications.
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morals
1. Objectivity in mathematics
One way of posing perhaps the central philosophical question about
mathematics arises from the pure phenomenology of the practice, from
what it feels like to do mathematics. Anything from solving a homework
problem to proving a new theorem involves the immediate recognition
that this is not an undertaking in which anything goes, in which we may
freely follow our personal or collective whims; it is, rather, an objective
undertaking par excellence. Part of the explanation for this objectivity
lies in the inexorability of the various logical connections,1 but that can’t
be the whole story; an if-thenist effort to treat mathematics simply as a
matter of what follows from what will capture the claim that the Peano
axioms logically imply 2þ2=4, that some set-theoretic axioms imply the
fundamental theorem of calculus, but miss 2þ2=4 and the fundamental
theorem themselves. Another way of putting this is to say that we don’t
form our mathematical concepts or adopt our fundamental mathematical assumptions willy-nilly, that these practices are highly constrained.2
But, one asks (as this whole book has asked), by what?
Various versions of Robust Realism constitute perhaps the most
popular response to this challenge: what constrains our practices here,
what makes our choices right or wrong, is a world of abstracta that
we’re out to describe. This idea is nicely expressed by Moschovakis:
The main point in favor of the realistic approach to mathematics is the
instinctive certainty of most everybody who has ever tried to solve a problem
that he is thinking about ‘real objects’, whether they are sets, numbers, or
whatever. (Moschovakis [1980], p. 605)
Often enough, this sentiment is accompanied by a loose analogy
between mathematics and natural science:
We can reason about sets much as physicists reason about elementary particles
or astronomers reason about stars. (Moschovakis [1980], p. 606)3
1
For discussion of the ground of logical truth, see [2007], Part III.
This is the problem touched on in }III.2 in connection with Carnap, and in }IV.3 in
connection with if-thenism.
3
Cf. Gödel [1944], p. 128: ‘It seems to me that the assumption of such objects [‘classes and
concepts . . . conceived as real objects . . . existing independently of our definitions and
2
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115
In keeping with our close observation of the experience itself, it
seems only right to admit that mathematics is, if anything, more tightly
constrained than the physical sciences. We tend to think that mathematics doesn’t just happen to be true, it has to be true.
Those wishing to avoid Robust Realism for one reason or another
often appeal to a sentiment famously expressed by Kreisel—or perhaps
I should say ‘apparently expressed’, as no clear published source is
known to me.4 Dummett’s paraphrase goes like this:
What is important is not the existence of mathematical objects, but the
objectivity of mathematical statements. (Dummett [1981], p. 508)
Putnam casts the idea in terms of realism:
The question of realism, as Kreisel long ago put it, is the question of the
objectivity of mathematics and not the question of the existence of mathematical objects. (Putnam [1975], p. 70)
Shapiro makes the connection explicit:
. . . there are two different realist themes. The first is that mathematical objects
exist independently of mathematicians, and their minds, languages, and so on.
Call this realism in ontology. The second theme is that mathematical statements
have objective truth-values independent of the minds, languages, conventions, and so forth, of mathematicians. Call this realism in truth-value. . . . The
traditional battles in philosophy of mathematics focused on ontology. . . .
Kreisel is often credited with shifting attention toward realism in truthvalue, proposing that the interesting and important questions are not over
constructions’] is quite as legitimate as the assumption of physical bodies and there is quite as
much reason to believe in their existence. They are in the same sense necessary to obtain a
satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of
our sense perceptions’. Also Gödel [1964], p. 268: ‘the question of the objective existence of
the objects of mathematical intuition . . . is an exact replica of the question of the objective
existence of the outer world’.
4
Dummett [1978], p. xxviii, identifies the source as something ‘Kreisel remarked in a
review of Wittgenstein’, but if the passage in question is the one pinpointed by Linnebo
[2009]—namely Kreisel [1958], p. 138, footnote 1—it’s hard not to agree with Linnebo that it
‘is rather less memorable than Dummett’s paraphrase’. (The relevant portion of the note in
question reads: ‘Incidentally, it should be noted that Wittgenstein argues against a notion of a
mathematical object . . . but, at least in places . . . not against the objectivity of mathematics’.)
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morals
mathematical objects, but over the objectivity of mathematical discourse. (Shapiro [1997], p. 37)5
On this approach, our mathematical activities are constrained not by
an objective reality of mathematical objects, but by the objective truth
or falsity of mathematical claims, which traces in turn to something
other than an abstract ontology (say to modality, to mention just one
prominent example6).
I bring this up because it seems to me that the Second Philosopher
has arrived at a position that does Kreisel one better. If Thin Realism
and Arealism are equally accurate, second-philosophical descriptions
of the nature of pure mathematics, just alternative ways of expressing
the very same account of the objective facts that underlie mathematical
practice, then we have here a form of objectivity in mathematics that
doesn’t depend on the existence of mathematical objects or the truth
of mathematical statements, or even on the non-existence of mathematical objects or the rejection of mathematical claims. This form of
objectivity is, as you might say, post-metaphysical. Though it doesn’t
involve truths about a mathematical ontology, it does involve an array of
facts like the sort of thing we roughly express by saying that the concept
of group opens up a lot of deep mathematics. Indeed, it seems fair to say
that the objectivity of such facts is more robust than thin: an Evil Demon
could deceive us wholesale about what follows from what or about
where the deepest, most fruitful strains lie. Still, the coherence of the
radical skeptical challenge isn’t enough to revive Benacerraf-style
worries: though it’s hard to see why our set-theoretic methods should
track the truth about the Robust Realist’s ontology, they’re clearly welldesigned (the Demon aside) to track set-theoretic depth.
To return to the phenomenology from which we began, I suggest
that this account of the objective underpinning of mathematics—the
phenomenon of mathematical fruitfulness—is closer to the actual constraint experienced by mathematicians than any sense of ontology,
epistemology or semantics; what presents itself to them is the depth,
the importance, the illumination provided by a given mathematical
5
6
See also Shapiro [2000], p. 29, and [2005], p. 6.
See e.g. Hellman [1989].
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117
concept, theorem, or method. A mathematician may blanch and stammer, unsure of himself, when confronted with questions of truth and
existence, but on judgments of mathematical importance and depth he
brims with conviction. For this reason alone, a philosophical position
that puts this notion center stage should be worthy of our attention.
I certainly don’t pretend to have given a satisfactory account of
mathematical depth—what I’ve said remains uncomfortably metaphorical—nor do I imagine that giving such an account would be an easy
task, but if questions of ontology and truth are red herrings, as the
present analysis suggests, then I can at least hope to shift attention away
from a misplaced worry and to focus it instead on the challenge of
understanding the phenomenon that in fact drives the practice of pure
mathematics.
2. Robust Realism revisited
Still, faced with the interchangeability of Thin Realism and Arealism,
I suspect some may be tempted to throw up their hands and contemplate a return to the familiar terrain of Robust Realism. But now, what
exactly makes a realism Robust rather than Thin? A few such contrasts
emerge from discussions in Chapter III: the Robust Realist demands
more than the Thin Realist’s appeal to classical logic to defend the
legitimacy of the continuum problem; presumably the Robust Realist,
unlike the Thin Realist, would be inclined to regard a radical skeptical
challenge to our mathematical knowledge as running parallel to the
more familiar challenge to our knowledge of the external world.
Underlying observations like these, we’ve isolated the fundamental
diagnostic: the Robust Realist requires a non-trivial account of the
reliability of mathematical methods, or more precisely, an account that
goes beyond what mathematics itself tells us.7 The Thin Realist, on the
other hand, thinks that mathematics itself gives us the whole story.
7
On this generous characterization, some versions of truth-value realism (to use Shapiro’s
term) would count as Robust despite their lack of ontology, providing the truths in question
are understood in such a way as to require a non-trivial epistemology. e.g., Hale ([1996])
argues that Hellman’s modalism faces a serious epistemological challenge.
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morals
This diagnostic easily assigns the Robust label to views that appeal to
‘perception-like’ mathematical intuition. One flagrant example is my
own set-theoretic realism ([1990]), where the intuition involved is
explicitly linked to ordinary perception, but there are others: for
example, Shapiro ([1997]) proposes an epistemology that begins in
human pattern recognition. In addition to more naturalistic views
like these, the range of possibilities also includes descendants of the
sort of intellectual intuition one finds in Descartes’s clear and distinct
perception. Some readings of Gödel’s ‘something like a perception . . . of
the objects of set theory’ ([1964], p. 268) fall in this category, though these
remarks of Gödel’s are notoriously difficult to interpret (see e.g. Parsons
[1995]). One clear contemporary example is James Brown’s platonistic
position:
We can intuit mathematical objects and grasp mathematical truths. Mathematical
entities can be ‘seen’ or ‘grasped’ with ‘the mind’s eye’ . . . The main idea is
that we have a kind of access to the mathematical realm that is something like
our perceptual access to the physical realm. . . . My bold conjecture . . . is this:
Some ‘pictures’ are not really pictures, but rather are windows to Plato’s heaven. . . . As
telescopes help the unaided eye, so some diagrams are instruments (rather than
representations) which help the unaided mind’s eye. (Brown [1999], pp. 13, 39)
Another advocate of ‘the mind’s eye’ (Katz [1998], p. 39) is Jerrold
Katz:
Our reason is an appropriate instrument for determining how things must be
in [the realm of abstract objects] . . . our rationalist epistemology . . . posits basic
ratiocinative knowledge of evident properties of abstract objects. . . . we have a
case of seeing—though not with our eyes . . . intuition . . . is . . . an immediate,
i.e., noninferential, purely rational apprehension of the structure of an abstract
object. (Katz [1998], pp. 39, 42, 43, 44)
In all these cases, the realism on offer is clearly Robust. And for
all of them, the Second Philosopher’s concerns remain as presented in
}II.4:8 the Robust Realist’s epistemological stories seem to deny the
autonomous authority of set-theoretic methods, to require external
8
Though the discussion in }II.4 takes place under the temporary assumption that the
Second Philosopher has good reason to regard set theory as a body of truths, these concerns
don’t rest on that premise.
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119
supplementation for mathematical reasoning that looks entirely
rational on its own terms; in addition, there seems to be a mis-match
between the focus of so many of those set-theoretic justifications on
the concrete mathematical advantages of the hypotheses in question and
the Robust Realist picture of tracking a set-theoretic reality potentially
as uncooperative with our theorizing as the physical world. Still, not all
versions of Robust Realism rely on a direct perceptual or intellectual
connection with the objects of mathematics, and I think it’s worth
examining one particularly intriguing example.
The account I have in mind derives from Tyler Burge’s fascinating
reconstruction of Frege’s epistemology.9 Frege holds that arithmetical
judgments ultimately rest on logic, but what founds logic itself?
He saw the foundation as consisting of primitive logical truths, which may be
used as axioms . . . they are in need of no justification from any other principles. (Burge [1998], p. 317)
Frege accepted the traditional rationalist account of knowledge of the relevant
primitive truths . . . This account . . . maintained that the basic truths . . . of
logic are self-evident. (Burge [1992], p. 299)
In agreement with his traditional forebears, Frege ‘did not regard selfevidence as subjective or psychological obviousness’ (Burge [2005],
p. 61), nor did he regard it as infallible. Instead
He took logical laws to be objectively self-evident and to be subjectively obvious
only to a mind that adequately understands them. (Burge [2005], pp. 61–62)
Such understanding can be incomplete, which is how we can come to
err.
Where Burge’s Frege differs dramatically from the tradition is in his
rejection of the idea that full understanding consists in a kind of
immediate quasi-perceptual insight.
In analyzing inferences Frege is concerned that appeals to self-evidence not be
allowed to obscure the formal character of the inferences, which can be found
only by rigorous logical analysis. This analysis is arrived at not primarily by
9
See Burge [1984], [1992], and especially [1998], plus pp. 61–68 of the introduction to the
collection, Burge [2005], in which these papers are reprinted. For present purposes, it doesn’t
matter if this is really Frege’s position.
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consulting unaided intuition, but by surveying inferential patterns in actual
scientific-mathematical reasoning. (Burge [1998], p. 340)
So the bedrock here is ordinary mathematical (and scientific) practice,
which merits our confidence because
All these concepts have been developed in science and have proved their
fruitfulness . . . fruitfulness is the acid test of concepts, and scientific workshops
the true field of study for logic. (Frege [1880/1], p. 33, cited in Burge [1998],
p. 340)
In the case of pure mathematics,
Justification derives from . . . considerations of simplicity, duration, fruitfulness, and power in pure mathematical practice . . . Frege emphasizes that
pure mathematical practice works. It produces a community of agreement
through finding some systems ‘better’, ‘simpler’, ‘more enduring’. (Burge
[1998], p. 341)
With mathematics itself as a starting point,
Frege’s method was to reason to logical structure by observing patterns of
judgments and patterns of inferences—and then postulating formal structures
that would account for these patterns. While this method makes use of intuitions
about deductive validity, it has at least as much kinship to theory construction as
to intuitive mathematical reflection. (Burge [1998], pp. 340–341)
By this process, Frege uncovers his logical system; he
Repeatedly appeals to advantages, to simplicity, and to the power of his
axioms in producing proofs of widely recognized mathematical principles, as
recommendations for his logical axioms. (Burge [1998], p. 339)
Here Burge notes that Frege’s use of this extrinsic style of axiom
defense anticipates Zermelo’s.
So far so good, but what’s become of the original idea of selfevidence? As Burge understands him, Frege regards these extrinsic
considerations as a ‘secondary, fallible, non-demonstrative’ (Burge
[1998], p. 355) form of evidence:
Our justification for believing [mathematical propositions] is partly pragmatic—
we find their place in mathematical practice secure through long usage, through
advantages of simplicity, plausibility and fruitfulness, and through applications to
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121
non-mathematical domains. A deeper justification for believing in these propositions lies in finding their place in a logicist proof structure, by understanding the
grounds within this structure that support them (if they are non-basic) or by
understanding the self-evidently true basic principles. (Burge [1998], p. 345)
Thus we seem to be back where we started, in search of an account
of this elusive self-evidence. Frege is well aware that his purportedly selfevident logical laws are often not subjectively obvious to his colleagues:
We find Frege recommending to those who are skeptical of his logical system
that they get to know it from the inside. He thinks that familiarity with the
proofs themselves will engender more confidence in his basic principles.
(Burge [1998], p. 339)
The basic principles gain something from our seeing what obviously correct
consequences they have and from recognizing ‘advantages’ of simplicity,
sharpness and the like. (Burge [1998], p. 344)
But this is no help, because once again we’ve moved away from
self-evidence and toward extrinsic supports:
What do the basic principles gain from our seeing their consequences and our
realizing their various ‘advantages’? If they are indeed axioms, they can be
recognized as true ‘independently of other truths’. The sort of justification
that derives from understanding them and recognizing their truth through this
understanding needs no further justificatory help from reflecting on their
consequences or the advantages of the system in which they are embedded.
(Burge [1998], p. 345)
So once again, the question is how we gain this particular sort of
sharpened understanding that allows us to recognize the self-evidence
of the basic laws.
This is the point at which the true originality of Burge’s Frege
comes into focus. Full understanding is not a product of a typically
rationalistic, perception-like, clear-and-distinct insight; instead Frege
offers his own original and
deep conception of what goes into adequate understanding. This conception
rests on his method of finding logical structure through studying patterns of
inference. Coming to an understanding of logical structure is necessary to full
understanding of a thought. And understanding logical structure derives from
122 morals
seeing what structures are most fruitful in accounting for the patterns of
inference that we reflectively engage in. (Burge [1998], p. 354)
So in addition to their decidedly inferior evidential contribution,
extrinsic considerations help us achieve the requisite understanding.
They serve
not to justify the first principles (except in a secondary, inductive way which
will be overshadowed, given full understanding) but to engender full understanding of them. One might recognize the truth of axioms independently
of other truths only in so far as one fully understands the axioms. But
understanding them depends not only on understanding Frege’s elucidatory
remarks about the interpretation of his symbols, but also on understanding
their logical structure—their power to entail other truths, and their reasongiving priority. This latter understanding is not independent of reasoning that
connects them to other truths. All full understanding involves discursive
elements, even if recognition of the truth of the axiom is, given sufficient
understanding, ‘immediate’. (Burge [1998], pp. 354–355)
Thus for Burge’s Frege, the ultimate force of the extrinsic merits of
basic axioms is not justification, but elucidation; they enable us to
appreciate those axioms’ self-evidence.
Now what would a set-theoretic version of this idea look like? Our
efforts to justify or discredit an axiom candidate would involve careful
examination of the very same kinds of considerations we’ve been focused
on since Chapter II. The difference would be that the various welcome
outcomes laid out by Cantor, Dedekind, Zermelo, Gödel, and the contemporary supporters of determinacy and large cardinals—all these would
not be regarded as evidence, or at least not as primary evidence, but as
clarifications of the true nature of our set-theoretic claims, clarifications
that ultimately allow us to appreciate their truth ‘immediately’, to recognize their self-evidence. Because it offers a non-trivial account of the
reliability of set-theoretic methods that goes beyond what set theory tells
us, this qualifies as a version of Robust Realism, but a version without a
direct quasi-perceptual or intellectual access to mathematical objects.
The Second Philosopher originally balked at Robust Realism because
it takes the ordinary justifications of set theory to be incomplete, in need
of supplementation, and because the nature of the supplementation
required by the most familiar varieties—especially those based on
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some sort of analogy between mathematics and natural science, between
the set-theoretic universe and the physical world, between mathematical
intuition and sense perception—because these most familiar varieties
seemed likely to come into conflict with the apparently reasonable
methods in actual use. In this the Burge-inspired variety is different:
the methods favored by practicing set theorists are happily and
completely embraced, so no question of potential conflict arises. Still,
significant supplementation is required, in the form of that non-trivial
epistemology: we need an account of how the process of coming to
appreciate the extrinsic merits of an axiom candidate manages to bring
our subjective sense of obviousness into line with actual self-evidence,
and thus, with the truth.
Whatever the significance of this extra epicycle in the Fregean context,10 it’s hard to see what work the machinery of ‘full understanding’
and ‘self-evidence’ does in the set-theoretic context, so perhaps the
Second Philosopher can be forgiven for thinking that the various mathematical advantages she identifies show just what they appear to show:
that the concept or hypothesis or method in question is getting at some
deep mathematics. The post-metaphysical Objectivist described in the
previous section would view the evidence this way, allowing for both
Thin Realist and Arealist modes of expression. It remains intriguing that
at least one approach to providing a more nuanced version of Robust
Realism ends up producing a position that in some ways resembles the
Second Philosopher’s. Still, in addition to the fundamental disagreement
over the need for a non-trivial epistemology, the two also diverge in their
understanding of the relative importance of intrinsic and extrinsic evidence—the topic to which we now turn.
3. More examples from set-theoretic practice
Finally, after this multi-chaptered excursion into the metaphysics and
epistemology of set theory, with our post-metaphysical Objectivism
10
Frege is concerned with logical truth. For the record, I take Robust Realism to be
appropriate in that case, which is to say that I take a non-trivial metaphysics with a non-trivial
epistemology to be in order (see [2007], Part III).
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now firmly on the table at last, I’d like to return in this section and
the next to the mechanics of set-theoretic practice, in particular, to the
rough classification of set-theoretic justifications into intrinsic and
extrinsic, first raised in }II.2.11 The hope is that what we’ve learned
in the interim might help us understand the varieties and the comparative virtues of the two sorts of considerations. In by-now-familiar
second-philosophical form, we should begin with some examples.
i. Intrinsic justifications
When a principle is defended in terms customarily classified as intrinsic,
various descriptors typically appear: the principle is intuitive, selfevident, obvious; it’s part of the meaning of the word ‘set’; it’s implicit
in the very concept of set; and so on. Of course, each of these glosses
raises its own suite of questions. These days, I think that the most
common idea is the last-mentioned—implicit in the concept of set—
and that the concept of set intended is the iterative conception. This
well-known picture of the set-theoretic universe was introduced by
Zermelo ([1930]) and subsequently entered into the fabric of set-theoretic pedagogy and practice; following Gödel’s description,12 sets are
understood as generated in a series of stages (‘a set is something obtainable . . . by iterated application of the operation “set of ”’); at each stage
we take every possible combination of the available elements, ‘no
matter whether we can define it in a finite number of words (so that
random sets are not excluded)’; this process is iterated indefinitely into
the transfinite (so, for example, ‘the totality of sets obtained by finite
iteration is considered to be itself a set and a basis for further applications
of the operation “set of”’). Of course the language of ‘generation’ and
‘application’ is regarded as metaphorical, merely a colorful way of
describing the structure of the set-theoretic universe, V.
11
Terms like ‘justification’ and ‘evidence’ may seem better suited to the Thin Realist
idiom, but the Arealist can also speak of justifying the addition of a new axiom or of giving
evidence for its suitability. Likewise, the Objectivist can justify or give evidence for the
mathematical effectiveness of the hypothesis.
12
See Gödel [1964], p. 259.
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Some of the simpler set-theoretic axioms follow unproblematically
from this concept. For example, if a and b are sets, then there must be
stages sa and sb, respectively, at which a and b first appear; if s is the later
of these two stages, then the pair set {a, b} will appear at the next stage
after s. Here we have a straightforward intrinsic justification for the
Pairing Axiom. The Axiom of Infinity is also contained in the stipulation that the series of stages extends into the transfinite.
A slightly less immediate, but still typically intrinsic case concerns
small large cardinals. For example, an uncountable inaccessible cardinal
Œ is one that towers over its predecessors in two ways: you can’t climb
up to Œ in fewer than Œ steps (unlike Aø, which can be reached in only
A0 steps: A0, A1, A2 , . . . ); and if a set has fewer than Πelements, then
so does its power set (unlike A1: A0 is less than A1, but its power set has
at least A1 members). If Œ is inaccessible, then VŒ is a model of ZFC—
it’s big enough to satisfy the axioms of Power Set and Replacement—
but we know from Gödel’s incompleteness theorem that no consistent
system of this sort can prove its own consistency, so ZFC (if consistent)
can’t prove the existence of an inaccessible cardinal.13 But the iterative
conception presumably involves the idea that the succession of stage
after stage shouldn’t stop at some fixed point, and asserting the existence of an inaccessible is a way of saying it keeps going on and on,
even after it exhausts the operations of Power Set and Replacement.
This constitutes an intrinsic argument for the first large cardinal axiom:
the Axiom of Inaccessibles.
Intrinsic justifications for larger large cardinals involve less direct and
less transparent connections with the concept of set.14 To get a feel for
this, consider one of the large cardinal notions central to determinacy
theory: the supercompact cardinal. As with any large cardinal, positing
a supercompact can be viewed as a way of assuring that the stages go
on and on; for example, below any supercompact cardinal Πthere are
Œ measurable cardinals, and below any measurable cardinal º, there are
13
As Tony Martin remarks, this can actually be seen without appeal to Gödel’s theorem:
suppose ZFC implies that there is an inaccessible; let Πbe the smallest, then VΠis a model of
ZFC + ‘there is no inaccessible’; contradiction.
14
A customary mark of a ‘large’ large cardinal is inconsistency with V=L. Indeed
implying that V6¼L is usually counted in favor of these cardinals, but this strong extrinsic
evidence is beside the point here.
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º inaccessible cardinals. But in the case of inaccessibles (and other small
large cardinals), this connection with the indefinite extendibility of the
sequence of stages is direct: we consider some processes, like applications of Power Set and Replacement, and posit that there is a stage after
all those generated by these processes. In the case of supercompacts
(and measurables), the connection is less direct: they imply the existence of cardinals that have this intrinsically supported feature.
Attempts to derive supercompacts by more direct intrinsic reasoning
often appeal to other general principles regarded as implicit in the
concept of set, for example, resemblance.15 The leading idea here is
that the class of stages is so large as to be extraordinarily rich; indeed
there are so many of them that some will be indistinguishable from
others. It’s then argued that this resemblance between two stages can
be spelled out in terms of a non-trivial elementary embedding of one
stage into another, and that from there one can generate a supercompact.16 Arguments along these lines are vulnerable at various
points, and their connection with the concept of set is more tenuous
than in the earlier examples.17 Much of the case for supercompacts is
actually extrinsic, arising from their role in determinacy theory.
ii. Extrinsic justifications
As we’ve seen (in }II.2.iii), the first extrinsic justification for a settheoretic axiom was Zermelo’s case for the Axiom of Choice, based on
its effectiveness for solving problems in set theory and analysis. Zermelo himself lists seven such problems, but subsequent progress has
uncovered many, many more.18 Those versed in set theory need only
contemplate the theory of transfinite cardinals without Choice—to
15
This principle is sometimes linked to the more familiar idea of reflection—the settheoretic universe V is too large to be fully specifiable, so any description true of V is already
true of some stage, VÆ (see Martin [1976], pp. 85–86, Solovay, Reinhardt, and Kanamori
[1978], p. 75)—but the connection is somewhat attenuated. Koellner [2009b] argues that no
true reflection principle can do this job.
16
See [1988], pp. 750–752, for a sketch of one such argument, with references. One might
also express supercompactness directly in terms of an elementary embedding, as Magidor has
shown (see Kanamori [2003], p. 302).
17
Cf. Martin [1976], p. 86: ‘as the axioms become stronger their link with the basic
principle becomes more and more tenuous’.
18
See the definitive Moore [1982].
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begin with, they needn’t be linearly ordered—for a taste of the kinds of
difficulties that arise in a Choice-less context. Choice in its many forms
is now fundamental in analysis, topology, algebra, and other branches
of the subject.
A more recent example comes from Martin’s experience in the early
days of determinacy hypotheses. In 1968, he proved a theorem of the
form: if a set of Turing degrees is determined, then either it or its
complement contains a cone. It’s not important for our purposes what
these terms mean; what matters is Martin’s reaction:
When I discovered the Cone Lemma, I became very excited. I was certain
that I was about to achieve some notoriety within set theory by deducing a
contradiction from AD. In fact I was pretty sure of refuting Borel determinacy. I had spent the preceding five years as a recursion theorist, and I knew
many sets of degrees. I started checking them out, confident that one of them
would . . . give me my contradiction. But this did not happen. For each set I
considered, it was not hard to prove, from the standard ZFC axioms, that it or
its complement contained a cone. (Martin [1998], p. 224)
Martin naturally sees this as a remarkable vindication:
I take it to be intuitively clear that we have here an example of prediction and
confirmation . . . The example seems fully analogous to striking instances of
prediction and confirmation in empirical sciences. (Martin [1998], pp. 224–225)
And the confirmation arising from the prediction is more compelling
for its initial implausibility.
As sketched in }II.2.iv, the fully-developed case for determinacy is
multifaceted and quite complex, and as we now note, largely extrinsic.
The first component is the rich theory of projective sets of reals that
arises from ADL(ℝ): the structure theory provable from ZFC for the
early projective levels can be generalized to the full projective hierarchy. For example, with determinacy, an important classical property of
co-analytic sets, first established by Luzin and Sierpiński in the 1920s,
can be generalized to the higher levels of the projective hierarchy. The
portion of the new determinacy-based proof that applies to co-analytic
sets alone depends only on the determinacy of open sets, and this much
determinacy is provable in ZFC, so we have here an example of what
Gödel calls a ‘verifiable’ consequence,
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demonstrable without the new axiom, whose [proof] with the help of the new
axiom, however, [is] considerably simpler and easier to discover, and [makes] it
possible to contract into one proof many different proofs. (Gödel [1964], p. 261)
Determinacy also yields ‘powerful methods for solving problems’
(op. cit.), for example, settling the questions Luzin thought would
never be answered: all projective sets are Lebesgue-measurable and no
uncountable projective set lacks a perfect subset.
Furthermore, the theory based on ADL(ℝ) is clearly preferable to the
alternative generated by V=L. As Steel notes, the latter lacks appeal
for an analyst, by which I take him to mean that it lacks what our
second-philosophical Objectivist would describe as the mathematical
depth of its rival. And, in any case, anything derivable from V=L can
be recovered in the determinacy context as part of the theory of L. In
sum, then, this determinacy-based structure theory would certainly
seem to qualify, to quote Gödel again, as
shedding so much light upon a whole field . . . that, no matter whether or not
[it is] intrinsically necessary, [it] would have to be accepted at least in the same
sense as any well-established physical theory. (op. cit.)
So this first component is paradigmatically extrinsic.
The work that led to the remarkable interconnections between determinacy theory and large cardinal theory that make up the second component was originally motivated by the hope of generating intrinsic evidence:
Because of the richness and coherence of its consequences, one would like to
derive PD itself from more fundamental principles . . . whose justification is
more direct. (Martin and Steel [1989], p. 72) The success of determinacy
axioms led to a revised program for doing descriptive set theory based on large
cardinal axioms: Show that large cardinal axioms imply determinacy axioms.
(Martin and Steel [1988], p. 6582)
Martin and Steel went on, of course, to do just that: building on work
of Foreman, Magidor, Shelah, and Woodin, they derived PD from the
existence of a supercompact cardinal; Woodin then improved this
result to the full ADL(ℝ).19 In this way, such intrinsic evidence as
19
The optimal hypothesis is weaker: countably many Woodin cardinals with a measurable above. See Kanamori [2003], }32.
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there is for supercompact cardinals transfers to determinacy, but as
noted a moment ago, some support for supercompacts comes in the
form of extrinsic evidence accruing from their implying determinacy!
I think it’s fair to say that the gain in intrinsic justification for
determinacy is significantly supplemented, and perhaps even somewhat overshadowed, by the extrinsic support accruing to both determinacy and large cardinals as a result of their interconnections. After
the dramatic inference from large cardinals to determinacy, Woodin
established a reverse implication: ADL(ℝ) implies the existence of inner
models with large cardinals.20 Considering that determinacy and large
cardinals arose in the course of such disparate, apparently unrelated
contexts of mathematical inquiry, this ultimate equivalence is quite
surprising and impressive:
This sort of convergence of conceptually distinct domains is striking and
unlikely to be an accident. (Koellner [2006], p. 174)
Our second-philosophical Objectivist understands the situation this
way: the fact that two apparently fruitful mathematical themes turn out
to coincide makes it all the more likely that they’re tracking a genuine
strain of mathematical depth.
The third and fourth components of the case for determinacy
resemble each other in that they’re both predicated on attractive
theoretical virtues. In the third component, the relevant theoretical
virtue is high consistency strength: assuming we want our theory of
sets to be as strong as possible, we will want theories of high consistency strength; all reasonable theories with consistency strength at least
that of ADL(ℝ) actually imply ADL(ℝ); therefore our best theory of sets
can be expected to include determinacy. This reasoning is cogent if we
have good reason to want our theory to be as strong as possible; I think
this can be traced to the foundational aspirations of set theory. Similarly, we want our set theory to be as decisive as possible, to settle as
many questions as possible, so generic completeness (that is, immunity
from independence arguments based on forcing) is a welcome feature;
this drives the fourth component. The two cases are structurally
20
Specifically, an inner model with countably many Woodin cardinals. See Koellner
[2006], p. 170.
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different—in one determinacy itself enjoys the relevant theoretical
virtue; in the other, determinacy is implied by virtuous theories—
but they both turn on the same sort of fulcrum: we want our theory of
sets to have certain features and we have good mathematical reasons for
these preferences.
At this point, even with this limited sampling, it should be clear
that a number of different kinds of justifications are being collected
together as ‘extrinsic’. We have some idea of what’s intended by
‘intrinsic’—here we’ve focused on ‘implicit in the concept of set’—
but ‘extrinsic’ is being applied willy-nilly to any compelling justification that isn’t clearly intrinsic. I won’t attempt a complete taxonomy here, but a few observations are in order. Notice, for example,
that a Robust Realist might find the considerations arising from
Martin’s Cone Lemma especially convincing; particularly a Robust
Realist impressed by an analogy between mathematics and natural
science will see here a clear set-theoretic counterpart to the undoubtedly legitimate confirmation of a scientific hypothesis by
successful predictions. To take the sharpest contrast, the purported
justifications based on welcome theoretical features will be problematic from this same point of view. As remarked in }II.4, the
fact that we prefer high consistency strength or generic completeness gives the Robust Realist no reason at all to suppose that
such theories are more likely to be true.21 So the metaphysical
perspective of Robust Realism requires careful distinctions between
various purported justifications we’ve counted here as extrinsic:
some are legitimate and some are not.
The matter looks quite different to our second-philosophical Objectivist. Martin’s results clearly support the consistency of determinacy hypotheses; indeed it was the hypothesis of inconsistency that
Martin actually entertained and disconfirmed. But there is more to
Martin’s accomplishment than this:
21
As noted in [2007], pp. 358–359, 365–366, the case against V=L presented in [1997],
}III.6, has a similar character: V=L is rejected because it’s restrictive, where restrictive
theories are avoided to further set theory’s foundational goal. Here the Robust Realist
should object that even if we don’t like restrictive theories, and for good reason, this doesn’t
show that they’re likely to be false.
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What was predicted, moreover, was not just individual assertions. Though
there had been much work on the structure of the degrees, no attention at all
had been paid to the notion of a cone . . . Afterwards cones and calculations of
‘vertices’ of cones became significant in degree theory. In determinacy theory,
the Cone Lemma became an important tool. What was predicted by the Cone
Lemma was thus a whole phenomenon, not merely isolated facts. (Martin
[1998], p. 224)
I think it’s fair to say that by this point the notion of prediction at work
has become somewhat strained. The thrust of this passage is that viewing
the theory of degrees from the perspective of determinacy hypotheses
helps to isolate a mathematically fruitful concept—the cone—and in
the process, sheds light on old questions of recursion theory, providing
new proofs and situating them in a broader context, much as these
same hypotheses illuminate classical descriptive set theory. This, too, our
Objectivist applauds as further evidence that determinacy hypotheses are
fruitful, are effective, and probably track something deep.
The upshot is that for the Objectivist, and likewise for the Thin
Realist and the Arealist, there is no call to rule against some classes of
extrinsic justification on principle. This isn’t to say that all purported
extrinsic justifications are sound or successful, but at least those we’ve
been examining do aim at the right targets: effective, productive,
important mathematics.
4. Intrinsic versus extrinsic
One unmistakable theme that runs through almost all discussions of
set-theoretic evidence is a strong preference for intrinsic over extrinsic
justifications. I’d like to conclude here with a brief exploration of this
preference, to see if our investigations might cast some light. Let me
begin with a look at the motivations of those who reject extrinsic
justifications altogether.
Consider for illustration Solomon Feferman’s discussions of CH and
the search for new axioms.22 Though he doesn’t explicitly rule out
22
See e.g. Feferman [1999], [2000].
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extrinsic considerations, Feferman opens with the Oxford English
Dictionary’s definition of
‘axiom’ as used in logic and mathematics . . . ‘a self-evident proposition requiring no formal demonstration to prove its truth, but received and assented to as
soon as mentioned’. (Feferman [2000], p. 402, [1999], p. 100)
He remarks that
I think it’s fair to say that something like this definition is the first thing we
have in mind when we speak of axioms for mathematics . . . It’s surprising how
far the meaning of axiom has become stretched from the ideal sense in
practice. (op. cit.)
So for example,
When the working mathematician speaks of axioms, he or she usually means
those for some particular part of mathematics such as groups, rings, vector
spaces, topological spaces, Hilbert spaces, and so on. (Feferman [2000], p. 403,
[1999], p. 100)
These Feferman calls ‘structural axioms’, noting that they ‘have
nothing to do with self-evident propositions’ (op. cit.).
Of course Feferman introduces this idea of structural axioms in
order to contrast them with something else:
Axioms for such fundamental concepts as number, set and function that
underlie all mathematical concepts. (Feferman [2000], p. 403, [1999], p. 100)
These he calls ‘foundational axioms’, whose role is ‘in the end to justify’
the ordinary practice of mathematics (Feferman [1999], p. 100). Here
we require axioms in the OED’s ideal sense: statements whose own
truth is self-evident, on which all other truths are based. Mary Tiles
makes this connection explicit: extrinsic justifications are inappropriate
to the conception of set theory as providing a logical foundation for mathematics. To claim this status for set theory it is necessary to claim an independent
and intrinsic justification for the assertion of set-theoretic axioms. It would be
circular indeed to justify the logical foundations by appeal to their logical
consequences, i.e., by appeal to the propositions for which they are going to
provide the foundations. (Tiles [1989], p. 208)
For this purpose, only intrinsic evidence will do.
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Whatever the attractions of this strong sort of foundational theory,
I think it’s clear that set theorists today are not in the business of trying
to provide one. We’ve seen that Zermelo didn’t insist that his axioms
provide an entirely independent justification for classical mathematics;
he appealed explicitly to their effectiveness for ‘productive science’,
as did Gödel and many others.23 Clearly no purely foundationalist
epistemology is intended. We’ve seen (in Chapter I) that by the late
19th century, there was considerable confusion about which mathematical structures could safely be assumed to exist; this uncertainty ran
through geometry with its newfound points at infinity and points with
complex coordinates, through analysis with its pathological functions,
through differential equations with its question of when they could
be trusted to have solutions,24 and of course, as we’ve seen, through
set theory itself. The goal of the axiomatization of set theory was to
remove these ontological uncertainties, to provide a single ontological
framework for classical mathematics. Obviously set theory doesn’t
provide a foundation in certain truths, nor does it provide, given
Gödel’s incompleteness theorems, what MacLane calls ‘a security
blanket’ (MacLane [1986], p. 406) against the risk of inconsistency,
but despite these epistemic shortcomings, it still plays a profound
unifying role, bringing all mathematical structures together in a single
arena and codifying the fundamental assumptions of mathematical
proof. There’s no doubting the mathematical value of a foundation
in this sense.25
So this blanket rejection of extrinsic justifications rests on an overly
strong understanding of the type of foundation set theory is intended
to provide, but I suspect that a different source of disapproval is lurking
nearby. As noted in passing in }III.5, set theory is sometimes regarded
as an essentially Platonistic theory: Feferman, for example, refers to
‘the Platonistic philosophy of mathematics that is currently claimed
23
Cf. Russell [1907] on his ‘regressive method’. Echoing Tiles, Potter [2004], p. 35,
writes, ‘the regressive method . . . seems powerless to justify a theory that aspires to be
epistemically foundational’.
24
See Kline [1972], pp. 699–707.
25
See [1997], }I.2, for discussion and references.
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to justify set theory’ (Feferman [1999], pp. 109–110). Given that
Gödel’s advocacy of extrinsic justifications sometimes intertwines
with his Robust Realism—for example, when he speaks of axioms
that ‘would have to be accepted . . . in the same sense as any wellestablished physical theory’ (Gödel [1964], p. 261)—one might come
to think that the viability of extrinsic justifications is grounded in some
such metaphysics: set theory aims to describe the objective features of
an independently existing world of mathematical abstracta; extrinsic
justifications in set theory are understood to function on analogy with
their counterparts in natural science. At that point, any and all objections to Platonism would apparently become objections to the use of
extrinsic justifications, and if one further holds, as Feferman does, that
‘Platonism . . . is thoroughly unsatisfactory’ (op. cit.), then extrinsic justifications would fall with it.26 But we’ve seen that this line of thought is
misguided: Robust Realism is not in fact well-suited to a defense of
many extrinsic justifications; for that purpose, Thin Realism, Arealism
and Objectivism are all more appropriate starting points.
Still, even if we agree that many wholesale rejections of extrinsic
justifications are ill-motivated (resting on an overly strong foundationalism or a mistaken association with Robust Realism) and ill-advised
(disallowing productive avenues to mathematical fruitfulness), there
remains a lingering sense that they are second-best, that intrinsic
justifications are the gold standard against which extrinsic justifications
are measured and often found wanting. In these final pages, I’d like to
float the heretical suggestion that in fact intrinsic justifications are
secondary to the extrinsic.
As a first pass at what I’m after here, let’s return to the case of groups.
We saw in }I.2 that despite Galois’s work around 1830 and Cayley’s
formal definition around 1850, the notion of group only caught on in
the 1870s, when there were finally enough known examples for it to
begin to do real mathematical work. The detailed histories of this
development show how the definition of a group evolved during
this period.27 To take one example, Cayley’s 1854 definition required
26
Feferman doesn’t explicitly link Platonism to extrinsic justifications in particular.
e.g., see Wussing [1969], Stillwell [2002], chapter 19, or the quick survey in [2007],
}IV.3.
27
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associativity, an identity, and cancellation laws (if xy = xz, then y=z).
As Cauchy had noted in his earlier work on the special case of finite
permutation groups, the existence of inverses follows (for some n and
m, xn = xm; if n<m, then by cancellation, 1=xm-n; so x(m-n)-1 is the
inverse of x). When infinite groups entered the picture, largely from
geometry, this inference broke down and inverses were no longer
guaranteed; an explicit axiom requiring them was eventually added by
Dyck in 1883. In this way, the concept of group was gradually shaped
so as to best serve a particular set of mathematical goals.
Now imagine that we’re back in the 1850s, with Cayley and the rest,
and our definition of group posits cancellation laws but not inverses.
Along come the geometers with their infinite groups, and Dyck
proposes a change in the definition to require inverses directly.
Given the goals of group theory, this is entirely reasonable; in other
words, there’s an impeccable means-ends extrinsic justification for the
change. Would it make any sense at this point to object that inverses
aren’t part of our concept of group, that this intrinsic consideration
trumps the extrinsic? I trust we’ll agree that it wouldn’t.
Now let’s try the analogous thought experiment for set theory.
Suppose someone proposes a new set-theoretic axiom that’s mathematically fruitful in the sorts of ways we’ve been considering, but
which doesn’t follow from, or perhaps even appears to conflict with,
our current concept of set. Would it be reasonable to reject the axiom
on intrinsic grounds? This scenario may sound far-fetched, but at least
one thread in the initial negative reaction to the Axiom of Choice had
this character: insofar as the logical conception of a set as an extension
was in the air, the existence of a choice set appeared problematic;
there’s no specifiable property that picks out all and only its members.
The Axiom was opposed on this score, but in the end, its extrinsic
merits carried the day.28 So again perhaps the case of sets isn’t so
different from that of groups after all.
28
See [1988], pp. 487–489, [1990], pp. 117–124, [1997], pp. 54–57, for discussion and
references, or better Moore [1982]. Now that the so-called logical concept of collection—the
extension of a property—has been replaced by the iterative concept of set, some regard
the Axiom of Choice as intrinsically justified (e.g., see Shoenfield [1977], pp. 335–336; for the
contrary view, see Boolos [1971], pp. 28–29).
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To push this one step further, consider for a moment why we value
intrinsic supports, what it is we gain from them. First I think we’d all
agree that it’s extremely useful to have a workable heuristic picture of
the sort of thing we’re investigating mathematically. At a minimum, it
can give us confidence in the consistency of our theory, especially if
the picture is sufficiently detailed; we feel, reasonably enough, that an
internal conflict would turn up there.29 More dramatically, a viable
concept so effectively guides our thinking that, given the choice, we’re
reluctant even to pursue a theory that lacks one (for example, Quine’s
NF30). And once we have a concept that’s mathematically fruitful, it’s
rational policy to exploit it further, to try to extend it in ways that seem
‘natural’ or harmonious with its leading intuitions, in hope of further
gains;31 thus we explore Large Cardinal Axioms as a way of exploiting
the ideas behind the iterative conception.
What’s striking is that all these perfectly reasonable ways of proceeding are in fact grounded in their promise of leading to the realization of more of our mathematical goals, to the discovery of more
fruitful concepts and theories, to the production of more deep mathematics. Ultimately we aim for consistent theories, for effective ways
of organizing and extending our mathematical thinking, for useful
heuristics for generating productive new hypotheses, and so on; intrinsic considerations are valuable, but only insofar as they correlate
with these extrinsic payoffs. This suggests that the importance of
intrinsic considerations is merely instrumental, that the fundamental
justificatory force is all extrinsic. This casts serious doubt on the
common opinion that intrinsic justifications are the grand aristocracy
and extrinsic justifications the poor cousins. The truth may well be the
reverse!
29
Cf. e.g. Kanamori [2003], p. 264: ‘The clear internal structure and striking global
coherence of inner models of measurability . . . provide a forceful argument for the consistency
of the theory: ZFC plus there is a measurable cardinal’.
30
See Fraenkel, Bar-Hillel and Levy [1973], p. 164: ‘there is no mental image of set theory
which leads to [NF] and lends it credibility’.
31
This might be called ‘heuristic rationality’: extending or generalizing as a way of
generating new concepts or hypotheses that could turn out to be justified in the stronger
sense of successfully tracking depth. But the fact that a new concept or hypothesis arises from
some rational heuristic isn’t enough to show that it’s justified; it’s just enough to make it
worth a try. See [2001], }I.
morals
137
In any case, this is what the Objectivist would tell us. For her, the
be-all and end-all of mathematics isn’t a remote metaphysics that we
access through some rational faculty, but the entirely palpable facts
of mathematical depth. She seeks concepts and assumptions that
illuminate previously intractable problems, that reveal surprising
interconnections, that open up new areas of mathematical understanding, and she does so using the familiar methods of mathematics
itself, all carefully honed for just this undertaking. From this point
of view, being part of our current concept only matters insofar as
that concept is well-chosen; presented with a fruitful avenue that
runs counter to current thinking, the Objectivist will happily throw
the old concept over and embrace the new without regret. Indeed for
her the often vexed distinction between finding out more about an
existing concept and changing to a new one matters not at all. What
does matter, all that really matters, is the fruitfulness and promise of
the mathematics itself.
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Index
abstracta 3, 8–9, 22n, 23, 27, 29,
31, 35–36, 59, 62, 69n, 76–77,
86, 90–92, 96–97, 99, 102, 104–105,
109–110, 114, 116, 118, 134
abstract algebra 6, 43–45, 52, 56, 73, 82,
85, 89, 127; see also, group
ADL(ℝ), see determinacy
applied mathematics Chapter I, 40–41,
53–55, 70, 85–86, 87n, }IV.2, 96,
101, 110–112, 113
Archimedes 15n
Arealism Chapter IV, 113, 116–117, 123,
124n, 131, 134
Arfken, George 25n
Aristotle 4n
arithmetic 22n, 24n, 28, 32, 34, 38, 39n,
45, 52–53, 114, 119
atomic theory 18–20, 22–24, 26n, 29–30,
38, 40
Austin, J. L. 107n
Avigad, Jeremy x, 43–44
Awodey, Steven 32n
axioms 14, 31–32, 81, 131–132, 135
Axiom of Choice 32–36, 45–48, 52,
80, 87, 94, 126–127, 135
Axiom of Constructibility (V=L) 49,
52–53, 85, 125n, 128, 130n
Axiom of Determinacy, see
determinacy
Axiom of Infinity 125
for logic 119–122
for set theory ix, 1, 22n, 32–34, 41,
45–47, 52, 56, 58–59, 69, 81, 83,
99–100, 113–114, 122–123,
}}V.3-V.4; see also ZFC and entries
for particular axioms
large cardinal axioms, see large cardinals
Pairing Axiom 125
Axiom of Power Set 125–126
Axiom of Replacement 125–126
Babbage, Charles 11n
Balaguer, Mark 91n
Banach, Stefan, see paradoxes,
Banach-Tarski
Bar-Hillel, Yeoshua 35n, 136n
Benacerraf, Paul 5n, 57, 59, 63, 116
Bernstein, Felix 46
Boltzmann, Ludwig 19, 20, 24, 25
Boolos, George 135n
Borel, Émile 35n
Borel sets 48n, 127
Boscovich, Roger 18n
Boyle, Robert 18n
Bremmer, H. 37n
Brown, James 118
Brown, Theodore 23n, 25n
Burge, Tyler 119–123
Burgess, John x, 39n, 54n, 57, 60n, 62,
67n, 73n, 76, 97–98, 99n, 103n,
104–105, 109–110
Calinger, Ronald 13n
Cantor, Georg 6, 41–46, 53, 56–58, 68,
72, 82, 85–87, 88–89, 98, 100–102,
104–105, 110, 122
Carnap, Rudolf 5, 66–69, 99n, 114n
Carnot, Sadi 19–20
Caro, Paul 106
Clausius, Rudolf 19
category theory 34n
Cauchy, Augustin-Louis 135
Cayley, Arthur 7, 134–135
CH, see continuum hypothesis
Clarke-Doane, Justin x
Cohen, Bernard 10n, 12, 18n
Cone Lemma 127, 130–131
consistency strength 50–51, 58, 80–81,
87, 129–130
constructivism 33, 43, 45, 86–87
continuum hypothesis (fluid
dynamics) 21–22, 29, 56n
continuum hypothesis (set theory,
CH) 21n, 37, 48n, 56, 59,
62–64, 66, 69n, 75, 77, 80,
81n, 89, 117, 131
D’Alembert, Jean-Baptiste 31n
Dalton, John 18
Dauben, Joseph 41n
148 index
Dedekind, Richard 7, 31–32, 43–46,
52–53, 58, 68–69, 72, 82, 85–87,
88–89, 98, 100–105, 110, 122
delta function 36, 37n, 73
depth see mathematical depth
Descartes, René 4, 6, 9–10, 11–12, 18,
40n, 118
determinacy }II.1.iv, 52–53, 58, 80–81, 85,
100–102, 105, 110, 122, 125–131
Dirac, Paul 36, 37n
Duhem, Pierre 19, 20n-21n, 26, 28
Dummett, Michael 115
Dyck, Walther von 135
general relativity 8, 22, 27–28, 90, 103
geometry 3, 7–9, 11–14, 27, 31–32, 44,
58, 65, 78–79, 81, 133
Gibbs, J. Willard 20
Glass, Matthew x, 82n
Gödel, Kurt ix, 5, 32n, 47, 49, 51, 56–57,
60, 62, 64, 72n, 114n-115n, 118, 122,
124–125, 127–128, 133–134
Goodman, Nelson 97
Grossman, Marcel 8
group 7, 28, 37, 53, 79–80, 89, 95, 116,
132, 134–135; see also abstract algebra
Guicciardini, Niccolo 11n
Ebbinghaus, Heinz-Dieter 45n
Eklof, Paul 37n
Einstein, Albert 8, 19n, 20
Emch, Gérard 22n, 24, 25n, 37n
Engel, Thomas 23n, 25n, 29n, 30
epistemology 1, 5, 57–59, 62–64, 66, 68,
69n, 70n, 71, }III.3, 77, 86–87, 97,
105, 109, }V.2, 116, 123, 133
Ernst, Michael x
Euler, Leonhard 6, 12–14, 15n, 19,
21–22, 30, 31n, 36, 53, 90n, 93–96
existence ix, 1, 54–56, 58, 60–62, 65n,
67–68, 70, 73–75, 77, 82–83, 86, 89,
97–98, }}IV.4-IV.5, 114–117, 134;
see also, ontology
extrinsic justification 47, 50, 56, 69, 79,
83, 89, 113, 119–123, }}V.3-V.4
Hale, Bob 117n
Hall, A. Rupert 10n
Halliday, David 25n
Hausdorff, Felix 35n
Heaviside, Oliver 37n
Heidegger, Martin 5
Heine, Eduard 44
Heis, Jeremy x
Hellman, Geoffrey 57, 116n, 117n
Hilbert, David 31, 45, 132
Huygens, Christiaan 4, 10n
Feferman, Solomon 87n, 131–134
Ferreirós, José 41n, 42, 43n, 44–45
Feynman, Richard 21–22
fictionalism 26n, 29, 91n, 98–99, 102, 103n
Field, Hartry 36n, 98
First Philosophy 40, 65–66
fluid dynamics 21–22, 29, 56n
Foreman, Matthew 50, 51, 128
formalism 99
foundations 13, 31–34, 44–47, 51–52, 56,
58, 63n, 73, 79, 89, 119, 129, 130n,
132–133
Fourier, Joseph 15–19, 26n
Fraenkel, Abraham 33, 35n, 44, 136n
Frege, Gottlob 119–123
function 6, 31–34, 37n, 45, 104, 132–133
Galileo Galilei 3–4, 9, 11, 14–15, 18–19, 30n
Galois, Évariste 7, 134
Gauss, Carl Friedrich 8, 11n, 15n
ice 106–107, 109, 111–112
idealism 65–66, 71
Idhe, Aaron 18n, 19
if-thenism 32n, 99, 114
indispensability argument ix, 90n
instrumentalism, see fictionalism
intrinsic justification 47, 49–50, 52n, 56,
69, 79, 89, 113, 123, }}V.3-V.4
islanders 107–109, 112
iterative conception of set 124–125, 135n,
136
Jech, Thomas 36n
Jenson, Ronald 50n
Kant, Immanuel 64–66, 68, 71–72,
75–76, 78–79, 81
Kanamori, Akihiro 36n, 48n, 50n, 80n,
126n, 128n, 136n
Katz, Jerrold 118
Kennedy, Juliette x
kinetic theory, see atomic theory
Kline, Morris 4, 6, 8, 9n, 10n, 11–14, 15n,
31, 133n
Koellner, Peter x, 49n, 50–1, 63n, 126n, 129
König, Julius 46
index 149
Kreisel, Georg 115–116
Kummer, Ernst 43
Lacroix, Sylvestre François 13
Laplace, Pierre-Simon 16–17, 18n, 21
large cardinals 36, 48–51, 80–81, 100, 122,
125–126, 128–129, 136
inaccessible cardinal 51n, 125–126
measurable cardinal 36n, 80, 82,
125–126, 128n, 136n
Shelah cardinal 80
supercompact cardinal 36n, 125–126,
128–129
superstrong cardinal 36n, 80
Woodin cardinal 51n, 80, 128n, 129n
Lavoisier, Antoine 18
Lebesgue measure 35–37, 48n, 128
Leibniz, Gottfried Wilhelm 10, 11n, 12–13
Levy, Azriel 35n, 136n
Lewis, David 102
Linnebo, ystein 115n
Liston, Michael x, 60n, 65n, 68n, 90–96
Liu, Chuang 22n, 24, 25n
logic 22n, 38, 39n, 63–64, 66, 69, 78–80,
83, 98–99, 101, 114, 117, 119–123
Luzin, Nikolai 127–128
Mach, Ernst 20n
Machamer, Peter 4, 15n
MacLane, Saunders 133
Malament, David x, 23n
Magidor, Menachem 50, 126n, 128
Marino, Patricia x, 98n
Martin, D. A. x, 48n, 50, 125n, 126n,
127–128, 130–131
mathematical depth 79–83, 86,
100, 112, 116–117, 128–129,
131, 134–137
mathematical nihilism 82n
Maxwell, James Clerk 19
McLarty, Colin x, 44n
McNulty, Bennett x
McQuarrie, Donald 23n, 25n, 29
miracle of applied mathematics 95–96
Moore, Gregory H. 32–33, 34n, 45n,
126n, 135n
Moschovakis, Yiannis 48–50, 114
Newton, Isaac 4, 6, 9–15, 18–19, 26n, 27,
30, 36, 53, 84n
naturalism ix, 39–40
nihilism, see mathematical nihilism
Noether, Emmy 44
nominalism 70n, 96–99, 102, 105, 109–110
Nye, Mary Jo 19, 20n, 21
objectivity 1, 46, 56, 58–59, 60, 62, 64,
66, }III.4, 100, 103–104, 107, 112,
113, }V.1, 119, 134
Objectivism (post-metaphysical)
}V.1, 123, 124n, 128–131,
134, 137
ontology 1, 36, 53, 66–68, 69n, 76, 86,
108, 113, 115–117; see also existence
Packman, A. J. x
Pais, Abraham 37n
paradoxes 32–33, 46
Banach-Tarski 35–36, 93–94
Parsons, Charles 32n, 64, 118
Partington, J. R. 18n
PD (projective determinacy),
see determinacy
Perrin, Jean 19–20
Plato 3, 5, 118
Platonism 56, 65n, 118, 133–134
Poincaré, Henri 20–21
Poisson, Siméon 16–17, 21, 26n
Potter, Michael 133n
predicativism, see constructivism
projective determinacy (PD),
see determinacy
projective sets 37, 48–53, 81, 85, 127–128
Putnam, Hilary ix, 115
quantum mechanics 23, 28–29, 37, 58,
90–91, 103
Quine, W. V. O. ix, 53, 55, 69n, 87n, 97,
101, 136
Rapalino, John x
realism ix, 60–61, 66, 114–118,
133–134; see also Platonism
Robust Realism ix, 57–59, 60, 62–64,
66, 72, 77, 86–87, 92n, 113–116,
}V.2, 130, 134
set-theoretic realism ix, 57, 72, 118
stereotypical anti-nominalist 104,
109–110
Thin Realism Chapter III, 88–89,
}}IV.4-IV.5, 113, 116–117, 123, 124n,
131, 134
real numbers 22n, 26, 31–32, 35–37, 44–45,
48–49, 52, 56n, 92–93, 95, 127
150
index
Reck, Erich x
reflection 126n
regressive method 133n
Reid, Philip 23n, 25n, 29n, 30
Reinhardt, William 126n
resemblance 126
Resnick, Robert 25n
Resnik, Michael 8, 90n
restrictiveness 130n
Rice, John 25n,
Riemann, Bernhard 8
Robust Realism see realism.
Rogers, Brian x
Roland, Jeffrey x
Rosen, Gideon 39n, 57, 73n, 76, 97–98,
104–105, 109–110
Russell, Bertrand 32, 133n
Schoenflies, Arthur 46
Schwartz, Laurent 37n
Second Philosophy }II.1, 103–111
self-evidence 1, 46–47, 119–123,
124, 132; see also intrinsic justification
Shapiro, Alan 10n, 18n
Shapiro, Stewart x, 32n, 56, 115–116,
117n, 118
Shelah, Saharon 50, 80, 128
Shoenfield, J. R. 135n
Sierpiński, Wacław 127
Silverman, Allan 3n
Simon, John 23n, 25n, 29
skepticism 73–76, 83, 107, 116, 117
Slowik, Edward 10n
Smith, Crosbie 11n, 15n, 16, 18n
Smith, George 10n, 12, 18n
Solovay, Robert 126n
Stalnaker, Robert 65n
statistical mechanics 22–25, 26n, 28
Steel, John x, 49–51, 60–62, 128
Stein, Howard 11n
stereotypical nominalist, see nominalism
Stirling’s approximation 25, 90, 92
Stillwell, John 7n, 79n, 134n
structuralism 32n, 99n
Tait, William 60n, 63n, 69n, 73n
Tappenden, Jamie x
Tarski, Alfred see paradoxes,
Banach-Tarski
thermodynamics 19–20, 22, 25
Thin Realism see realism
Thomson, William (Lord Kelvin) 11n
Tidwell, Scott x
Tiles, Mary 132, 133n
Tolley, Clinton x
transfinite numbers 6, 42n, 80, 126–127
trigonometric representations 41–42, 52,
57, 80, 82, 85, 89
Tritton, D. J. 21–22
tropospheric complacency 105–106, 108,
111
Truesdell, Clifford 10n, 12n, 13n, 14, 30n
truth ix, 1, 54–56, 60–62, 64, 66–68, 69n,
70–71, 77–78, 81n, 83, 86, 88–99,
}}IV.4-IV.5, 113, 115–117, 118, 123
Uffink, Jos 25n
Urquhart, Alasdair 36n
van der Pol, B. 37n
van Fraassen, Bas 40
V=L, see Axiom of Constructibility
Wagon, Stan 35n
Walker, Jearl 25n
Wedberg, Anders 3n
Weierstrass, Karl 42, 44–45
Wigner, Eugene 96
Wilson, Mark x, 20n-21n, 28n, 73n, 90n,
93, 105–109, 111
Wise, Norton 11n, 15n, 16–17, 18n
Witten, Edward 37n
Wittgenstein, Ludwig 115n
Kripkenstein 107
Woodin, Hugh 50–51, 80, 128–129
Wright, Crispin 55n, 70–71
Wussing, Hans 7, 79n, 134n
Zemanian, A. H. 37n
Zermelo, Ernst 32–34, 37, 45–48, 52,
85, 88–89, 98, 100–102, 105,
110, 120, 122, 124, 126, 133
ZFC 33–34, 37, 48–50, 62, 125,
127, 136n