Objectivity in Mathematics
One of the many things about the practice of mathematics
that makes the philosophy of mathematics so difficult, in fact
maybe the leader in that troublesome company, 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; if we
try to treat mathematics simply as a matter of what follows from
what, we capture the claim that the Peano axioms logically imply
2+2=4, that some set theoretic axioms imply the fundamental
theorem of calculus, but we miss 2+2=4 and the fundamental
1
See [2007], Part III, for more on the status of logical truth.
2
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.
But by what?
One perennially popular answer is that 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 the set theorist, Yiannis 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) 2
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.
2
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 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 and
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’.
3
Now it’s well-known that so-called Platonistic positions of
this sort are beset by a range of familiar philosophical
problems; 3 for myself, I’m more troubled by purely methodological
concerns, 4 but I won’t go into those here as I want to focus
instead on one prominent line of reaction to these difficulties.
This springs from a sentiment famously expressed by Kreisel -- or
perhaps I should say ‘apparently expressed’, as no clear
published source is known to me. 5
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 truth3
The canonical reference is Benacerraf [1973].
4
See [2007], pp. 365-366.
5
Dummett [1978], p. xxviii, identifies the source as something
‘Kreisel remarked in a review of Wittgenstein’, but if the passage in
question in the one pinpointed by Linnebo [200?] -- 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’.)
4
value. … The traditional battles in the 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 mathematical objects, but over the
objectivity of mathematical discourse. (Shapiro [1997], p.
37) 6
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 example).
I bring this up because my hope today is to float an idea
that would do Kreisel one better:
an account of mathematical
objectivity that doesn’t depend on the existence of objects or on
the truth of mathematical claims.
To get at this in reasonable
compass, I’ll have to skate over many themes that demand more
detailed treatment, but I hope what amounts to an aerial overview
of a book-length argument might be of interest, nonetheless. 7
The goal, then, is to uncover the source of the perceived
objective constraints on the pursuit of pure mathematics.
test case here will be my long-time hobby horse:
justification of set-theoretic axioms.
The
the
What makes this axiom
candidate rather than that one into a proper fundamental
assumption of our theory?
6
7
See also Shapiro [2000], p. 29, and [2005], p. 6.
The book in question is Defending the Axioms ([2011]) This paper was
written first, and the two now overlap in various places. Interested
readers are encouraged to consult the book for more complete versions
of this material.
5
The plan is to approach this question from a broadly
naturalistic point of view, so let me quickly sketch in the
variety of naturalism I have in mind.
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.
Along the way this inquirer comes to use mathematics in her
investigations.
She begins with a narrowly applied view of the
subject, but gradually comes to recognize 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.
(Here she recapitulates
the mathematical developments from the 17th to the 21st
centuries.)
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.
raises questions of two general types.
This
First, as part of her
continual evaluation and assessment of her methods of
investigation, she will want an account of the methods of pure
6
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 it is?
of its subject matter?
What is the nature
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.
Philosophy
undertaken in isolation from science and common sense is often
called ‘First Philosophy’, so I call her a Second Philosopher. 8
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.
I.
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. 9
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
8
9
For more, see [2007].
See Dauben [1979], chapter 2, Ferreirós [1999], §§IV.4.3 and V.3.2,
for historical context and references.
7
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 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 point-set P the set
of its limit points is conceptually co-determined; this I
will denote P’ and call the first derived point-set of P.
(As translated and quoted in Ferreirós [1999], p. 143)
Once this new set, the first derived set, P’, is in place, the
same operation can be applied again:
with P’, the set of its
limit points is ‘conceptually co-determined’; this P’’ 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. 10
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.
From a
methodological point of view, what’s happened is that a new type
10
Ferreirós ([1999], p. 160) notes that uniqueness continues to hold
if the ∀-th derived set is empty for some transfinite ordinal ∀, but
Cantor apparently never made this extension.
8
of entity -- a set -- has been introduced as an effective means
toward an explicit and concrete mathematical goal:
extending our
understanding of trigonometric representations.
(ii)
Dedekind’s introduction of sets
Around the same time, Dedekind also made several early uses
of what we now recognize as sets.
The first came in algebra, in
his theory of ideals, where he elected
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.
By side-stepping the computational
algorithms central to Kummer’s treatment, Dedekind was able to
demonstrate that the theory could be developed nonconstructively, and to explain why the properties of ideal
numbers didn’t depend on the details of how they were
represented.
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:
he introduces a promising new style of reasoning whose
mathematical fruitfulness was dramatically demonstrated as
abstract algebra went on to thrive in the hands of Noether and
her successors. 11
11
See McLarty [2006].
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’, 12 and in particular, to remove the
‘geometric evidence … [that] can make no claim to being
scientific’.
Since the calculus deals with ‘continuous
quantities’, he reasons it should be founded on ‘an explanation
of this continuity’, and he sets out to ‘secure a real definition
of the essence of continuity’.
The result, of course, is his elegant definition of
continuity and construction of the real numbers.
Competing
theories of Weierstrass and Cantor begin from particular
convergent series or sequences of rationals, identifying many
equivalent such items with a single real; in contrast, Dedekind
appeals to a cut, simply an infinite set of rationals. 13
This
approach wipes out all detailed series or sequence structure,
yielding one cut for each real, and the abstract characterization
allows for broad generalization.
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).
12
13
All quotations in this paragraph come from Dedekind [1872], p. 767.
He had already defined the integers and rationals in terms of
natural numbers (see Ferreirós [1999], p. 219).
10
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 representation-free,
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. 14
More globally, he sees himself as contributing to ‘the logical
foundations of all arithmetic and analysis’ (Zermelo [1908b], p.
200).
He despairs of finding a compelling and fruitful
definition of ‘set’ on which to base the subject -- something
comparable, say, to Dedekind’s definition of ‘continuity’ and its
role in founding analysis -- and opts instead to analyze the
14
See Moore [1982], pp. 143-160.
11
practice of set theory and ‘seek out the principles required for
establishing the foundations of this mathematical discipline’
(Zermelo [1908b], p. 200).
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.
The first is intuitive
self-evidence, which we might now describe as being implicit in
the informal ‘concept of set’.
Zermelo argues that Choice must
enjoy this sort of subjective obviousness on the grounds that so
many set theorists have used it, often without noticing.
But, as
we’ve seen, he despairs of defining the set 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 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 …
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)
12
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 the 1960s 15 as part of a broader search for new
15
See, e.g., Kanamori [2003], §27.
13
principles that might settle the problems in analysis 16 and set
theory 17 left open by the now-standard descendent of Zermelo’s
system, Zermelo-Fraenkel with Choice (ZFC).
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 [determinacy] 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)
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. 18
First, it
generates a rich theory of projective sets of reals with many of
the virtues identified by Gödel. 19
Second, Moschovakis’s ‘web of
implications … relating [determinacy] to large cardinal
hypotheses’ has indeed ‘grown steadily’.
In the decade following
16
E.g., the Lebesgue measurability of projective sets.
17
E.g., of course, the Continuum Hypothesis.
18
See Steel [2000], Koellner [2006].
And ADL() is necessary for this theory: it’s actually implied by its
consequences for definable sets (see Koellner [2006], pp. 170, 174).
19
14
Moschovakis’s book, Martin, Steel and Woodin, building on work of
Foreman, Magidor and Shelah, showed that determinacy follows from
the existence of large cardinals; indeed it is now known to be
equivalent to the existence of certain inner models with large
cardinals. 20
Third, a striking phenomenon in terms of consistency
strength has emerged; in John Steel’s words, ‘any natural theory
of consistency strength at least that of [determinacy] actually
implies [determinacy]’ (Steel [2000], p. 428).
Given the long-
standing foundational goal of set theory and the open-endedness
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 determinacy, this constitutes
a strong argument in its favor.
Fourth, in the presence of large
cardinals, forcing cannot succeed in showing a question about
projective sets to be independent. 21
This means that if any
question about projective sets is left unresolved by determinacy,
this can’t be shown by forcing; the independence involved would
have to be a new and unfamiliar variety.
Given that we want our
theory of sets to be as decisive as possible, within the
limitations imposed by Gödel’s theorems, this so-called ‘generic
completeness’ would appear a welcome feature of determinacy
theory.
20
See Kanamori [2003], §32, Koellner [2006], for discussion and
references.
21
If there is a proper class of Woodin cardinals, then L() is
elementarily equivalent to the L() in any forcing extension. (See
Koellner [2006], p. 171. Cf. Steel [2000], p. 430.)
15
In short, the current case for determinacy has blossomed so
impressively that many would agree with Hugh Woodin’s assessment:
‘determinacy is the correct axiom for the projective sets’
(Woodin [2001], p. 575).
II.
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 sets in the first place
and for determining their extent and their properties thereafter.
In broad summary, these rest on the pursuit of various
mathematical goals, from relatively local problem-solving to
providing foundations to more open-ended pursuit of promising
mathematical avenues.
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.
At the same time, she begins to appreciate
the extent to which these methods differ from her familiar
observation, theory-formation and testing:
for example, she
isn’t accustomed to positing entities to increase her expressive
power (as in Cantor) or rejecting a theory because it produces
less interesting consequences (as with the alternative to
determinacy’s theory of projective sets that results from Gödel’s
Axiom of Constructibility).
She might reasonably wonder if her
more familiar, tried-and-true methods could be called upon to
supplement or even correct these new approaches.
16
On examination, though, she concludes that the answer here
is no.
Ordinary perceptual cognition is most likely involved in
our grasp of elementary arithmetic, 22 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. 23
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. 24
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 essential tool 25 -- 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 justificatory methods that appear to be both
rational and autonomous.
22
See [2007], IV.2.ii.
23
See [1997], §§II.6-II.7, [2007], pp. 314-317, for discussion.
24
See [2008].
25
See [2007], pp. 330-331, 347, for discussion and references.
17
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:
these methods reliable?
are
Do they successfully track the existence
of sets and their properties and relations?
familiar with questions of this form:
Of course she’s
she investigates how
ordinary perception gives her information about the medium-sized
objects in the world around her; 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.
set theory is the same:
The case of
she’s observed the methods of set
theorists and now wants to know whether or not they successfully
track the truth about its subject matter.
question:
This raises the prior
should set theory be understood as describing a
subject matter, as attempting to deliver truths about it?
Now, 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
18
this will presumably rest more loosely on the way it is
intertwined with empirical science.
For now, I’d like to leave a
bookmark at this point, to return to it later.
For now, let’s
assume that the Second Philosopher is justified in regarding set
theory as a body of truths, and since she has no reason to take
its existence claims at other than face-value, 26 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.
III.
Thin realism
To this point, then, the Second Philosopher has determined
that set-theoretic methods are rational, autonomous and generally
reliable.
To explain why this is so, she must now delve more
deeply into questions of the second type, about the nature of the
human practice of set theory -- she’s now faced with the
26
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.
19
challenge of explaining what makes these methods reliable, what
sets must be like in order for this is 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.
Various familiar conclusions fall out
of this bare suggestion.
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 they 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’. 27
Let me call this Thin Realism. 28
What’s happened here is that the second-philosophical Thin
Realist begins from her confidence in the authority of settheoretic methods when it comes to determining what’s true and
false about sets, and draws from this a metaphysical conclusion
about the nature of sets, about their thinness.
For this sort of
realism, there is no troubling epistemological problem:
27
28
sets
Personal communication, 24 April 2002, quoted with permission.
The intended contrast is with robust versions of realism, like
Gödel’s, that involve rich metaphysical and epistemological theories
going far beyond ‘the positive things asserted by set theory’.
20
just are the kind of thing we can find out about in these ways.
There’s also no confounding worry about the determinacy of the
Continuum Hypothesis:
set theory is describing the set-theoretic
universe V, and CH or not-CH is a theorem.
This is not a version
of neo-Kantianism -- set theory doesn’t tell us that sets are
constituted by our practices or any such thing -- nor is it a
version of Carnapianism -- a decision about a new axiom isn’t a
merely pragmatic choice of a new linguistic framework, it’s
guided by reliable set-theoretic methods, a new discovery about
V.
Now despite these attractive features of Thin Realism, 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.
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
21
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 about what sets are,
without going beyond what set theory tells us, and our hope is
that asking the question this way might help.
constrains our methods?
So, what
Part of the answer lies in the ground of
classical logic, 29 but our interest here is in the mathematical
features.
To get at these, let me draw a brief compare-and-
contrast with Kant on geometry.
According to Kant, the concept of a triangle is defined by
us, so 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 here are shaped by the structure of our underlying
spatial form of sensibility, either in pure intuition (when we
29
For discussion of the ground of logical truth, see [2007], Part III.
22
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, but I think it provides a helpful analogy for what I
want to suggest in the case of set theory.
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 of
the sort we’ve sketched into good reasons to believe what we
believe.
What takes us beyond mere logical connections and
allows us to track something more?
more’?
And what is this ‘something
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
23
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. 30
So what
guides our concept-formation, beyond the logical requirement of
consistency, is the way some logically possible concepts track
important 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 (see [2007], pp. 351-355), for example, that
settles the Continuum Hypothesis.
Still, there are over-arching similarities.
Set-theoretic
concepts are formed in response to set-theoretic goals just as
the concept ‘group’ was formed in response to algebraic goals.
30
See, e.g., Wussing [1969] or Stillwell [2002], chapter 19.
24
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, 31 or the gradual
migration of the concept of measurable cardinal from its origins
in measure theory to the mathematically rich context of
elementary embeddings. 32
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:
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 the 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
31
See Kanamori [2003], p. 461.
32
See Kanamori [2003], §§2 and 5.
25
into a hierarchy that serves as an effective measure of
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 to the phenomenon I’m after here:
mathematical depth,
mathematical fruitfulness, mathematical effectiveness,
mathematical importance, mathematical productivity, and so on.
I’m using 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. 33
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
33
This is why I spend so much time rehearsing these various cases, to
give the reader a feel for what ‘mathematical depth’ looks like.
26
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.
mathematical goals.
This is what anchors our various local
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
27
facts of mathematical depth, not the other way ‘round. 34
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. 35
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
set-theoretic 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 various ways’; rather, the fact of measurable cardinals being
mathematically fruitful in various ways is evidence for their
existence.
34
35
Why?
Because of what sets are:
repositories of
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.
28
mathematical depth.
They mark off a mathematically rich vein
within the indiscriminate network of logical possibilities.
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 devices 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.
This, I
suggest, is the core insight of Thin Realism.
Let me sum up the Second Philosopher’s journey so far:
she
comes to realize that contemporary pure mathematics is a vital
part of her investigation of the world and to regard it as a body
of truths; she recognizes that its methods are new and
distinctive, sees no opening for correction or defense from her
more familiar methods, and concludes, in particular, that settheoretic methods are rational, autonomous and reliable guides to
the truth about sets; to account for this striking fact, she
forms the simple hypothesis that sets are the sort of thing that
can be investigated in these ways; and finally she discovers the
source of this fact, namely, that sets simply are means for
producing certain mathematically fruitful outcomes, and that settheoretic methods are expressly designed to track just these deep
mathematical strains.
Thus Thin Realism presents itself as an
attractive answer to our second group of questions:
set-
theoretic activity in the investigation of an abstract realm of
29
sets; its methods are reliable simply because of what those sets
are; the whole enterprise answers to the objective topography of
mathematical depth; the pursuit of new set-theoretic axioms and
of a solution to the continuum problem are legitimate parts of
this inquiry.
IV.
Arealism
So we’ve achieved a kind of objectivity here, but despite
its non-traditional aspects, it still relies on the existence of
abstracta and the truth of our claims about them.
What I’d like
to do now is return to that point where we left the bookmark, the
point where the Second Philosopher concluded that set theory is a
body of truths but her grounds were left vague.
Eventually I
want to return to the question of what those grounds might be and
the extent to which they’re persuasive, but first let me sketch
in the position that results if we take the other fork in the
road at that point, if we 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
30
problem. 36
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. 37
Let’s call this position Arealism.
Now we’ve noted that 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 application of mathematics
without regarding it as true?
There’s a complex story to be told
here, 38 but examination of the historical and scientific record
leads the Second Philosopher to believe that contemporary pure
mathematics works in application by providing the empirical
36
And, lest we forget, much of pure mathematics is still consciously
aimed at the goal of providing tools for empirical science.
37
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.
38
See [2008].
31
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 well-
described, just as one might illuminate a given social situation
by comparing it to a imaginary or mythological one, marking the
similarities and dissimilarities.
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
32
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 of mathematics, this way of talking seems
to me to invite mis-understanding.
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 it is 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)
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.
33
I hope and trust it’s clear that this is not a portrait of
the second-philosophical Arealist.
She doesn’t come to her
investigations with an 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 considered
a version of nominalism, it certainly isn’t what Burgess and
Rosen call the ‘stereotypical’ variety (Burgess and Rosen [1997],
p. 29).
V.
Post-metaphysical objectivism
At this point, we have two apparently second-philosophical
positions in play.
And, though they disagree sharply over truth
and existence -- the Thin Realist holding that sets exist and set
theory is a body of truths, and the Arealist denying both -still they 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, those
34
distinctive strains of mathematical depth.
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 true 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.
Where they differ is in their second-philosophical
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 settheoretic language and practice from an empirical perspective and
ask second-philosophical versions of the traditional
35
philosophical questions, questions in the second group we’ve been
considering.
So how is the Second Philosopher to adjudicate between 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 way mathematics is intertwined 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 over-arching 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). 39
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,
39
See [1997], pp. 203-205, [2007], pp. 345-247.
36
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 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 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
37
more evidence of an unexpected sort, or are they the trademarks
of a different sort of activity altogether?
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 this case ‘true’, ‘exist’, ‘evidence’,
‘believe’, ‘know’ -- mark fully determinate features or
attributes, that there is a determinate fact of 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 massive case against this picture, Wilson
[2006], rests largely on a wealth of fascinating and down-toearth 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
38
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). 40
And so on.
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
40
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).
39
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. 41
And notice
that this isn’t a version of the well-known Kripkesteinian
challenge:
what makes 1002 rather than 1004 the right
continuation of +2 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.
41
This time, the islanders might reasonably
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’.
40
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 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 pre-established 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. 42
Could it be that a similar brand of complacency is at work
in the case of the Second Philosopher faced with pure
mathematics?
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?
Do
honorifics like ‘true’, ‘exist’, ‘evidence’, ‘confirm’ -indisputably at home in those other studies -- belong in pure
mathematics as well?
Pure mathematics arose out of our empirical
study of the world; it remains intensely important as a tool for
that study, even in parts that weren’t expressly developed for
that purpose; it continues to be inspired by the descriptive and
42
See Wilson [2006], pp. 34-37, for more on the islanders, or [2007],
pp 186-188, for a somewhat more complete summary.
41
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 our central questions -- is pure mathematics of-a-piece with
physics, astronomy, psychology and the rest?
truths?
is it a body of
do its methods confirm its claims? -- that 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 Wilson’s
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
often baseless, and also largely harmless.
42
Likewise, once we understand how pure mathematics
developed, how it now differs from empirical sciences, 43 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
43
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.
43
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 chose -- 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.
And here, at last, we have 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.
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 or any extra-mathematical epistemology; what presents
itself to them is the depth, the importance, the illumination
provided by a given mathematical concept or theorem.
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
44
conviction.
For this reason alone, a philosophical position that
puts this notion center stage might be worthy of our attention. 44
Penelope Maddy
44
This paper was delivered as the Sixth Annual Thomas and Yvonne
Williams Lecture for the Advancement of Logic and Philosophy at the
University of Pennsylvania and as a workshop presentation to the
Southern California History and Philosophy of Logic and Mathematics
Group. I’m grateful to audiences at both events, and especially to the
organizers, Scott Weinstein and Thomas Williams, and Erich Reck,
respectively.
45
References
Avigad, Jeremy
[2006]
‘Methodology and metaphysics in the development of
Dedekind’s theory of ideals’, in J. Ferreirós and J.
Gray [2006], pp. 159-186.
Benacerraf, Paul
[1973]
‘Mathematical truth’, reprinted in P. Benacerraf and
H. Putnam, eds., Philosophy of Mathematics,
(Cambridge: Cambridge University Press, 1983), pp.
403-420.
Burgess, John, and Rosen, Gideon
[1997]
A Subject with no Object, (Oxford:
Press).
Oxford University
Caro, Paul
[1992]
Water, P. Thickstun, trans., (New York:
1993).
McGraw Hill,
Dauben, Joseph
[1979]
Georg Cantor, (Cambridge, MA:
Press).
Harvard University
Dedekind, Richard
[1872]
‘Continuity and irrational numbers’, W. Beeman and W.
Ewald, trans., in Ewald [1996], pp. 766-779.
[1888]
‘Was sind und was sollen die Zahlen’, W. Beeman and W.
Ewald, trans., in Ewald [1996], pp. 790-833.
Dummett, Michael
[1978]
Truth and Other Enigmas, (Cambridge, MA:
University Press).
Harvard
[1981]
Frege: Philosophy of Language, second edition,
(Cambridge, MA: Harvard University Press).
Ferreirós, José
[1999]
Labyrinth of Thought: a History of Set Theory and its
Role in Modern Mathematics, (Basel: Birkhäuser).
46
Gödel, Kurt
[1944]
‘Russell’s mathematical logic,’ reprinted in his
[1990], pp. 119-141.
[1964]
‘What is Cantor’s Continuum Problem?’, reprinted in
his [1990], pp. 254-270.
[1990]
Collected Papers, volume II, S. Feferman et al, eds.,
(New York: Oxford University Press).
Goodman, Nelson, and Quine, W. V. O.
[1947]
‘Steps towards a constructive nominalism’, Journal of
Symbolic Logic 12, pp. 105-122.
Kanamori, Akihiro
[2003]
The Higher Infinite, second edition, (Berlin:
Springer).
Kant, Immanuel
[1781/7]
Critique of Pure Reason, P. Guyer and A. Wood, trans.
and eds., (Cambridge: Cambridge University Press,
1997).
Koellner, Peter
[2006]
‘On the question of absolute undecidability’,
Philosophia Mathematica 14, pp. 153-188.
Kreisel, Georg
[1958]
Review of Wittgenstein’s Remarks on the Foundations of
Mathematics, British Journal for the Philosophy of
Science 9, pp. 135-158.
Linnebo, Øystein
[200?]
‘Platonism about mathematics’, to appear in the
Stanford Encyclopedia of Philosophy.
Maddy, Penelope
[1997]
Naturalism in Mathematics, (Oxford:
Press).
[2007]
Second Philosophy, (Oxford:
Oxford University
Oxford University Press).
47
[2008]
‘How applied mathematics become pure’, Review of
Symbolic Logic 1, pp. 16-41.
[2011]
Defending the Axioms, (Oxford: Oxford University
Press).
Martin, D. A.
[1998]
‘Mathematical evidence’, in H. G. Dales and G.
Oliveri, eds., Truth in Mathematics, (Oxford: Oxford
University Press), pp. 215-231.
McLarty, Colin
[2006]
‘Emmy Noether’s “set theoretic” topology: from
Dedekind to the rise of functors’, in J. Ferreirós and
J. Gray [2006], pp. 187-208.
Moore, Gregory H.
[1982]
Zermelo’s Axiom of Choice, (New York:
Springer).
Moschovakis, Yiannis
[1980]
Descriptive Set Theory, (Amsterdam:
North Holland).
Putnam, Hilary
[1979]
Mathematics, Matter and Method: Philosophical Papers,
volume 1, second edition, (Cambridge: Cambridge
University Press), pp. 60-78.
Shapiro, Stewart
[1997]
Philosophy of Mathematics: Structure and Ontology,
(New York: Oxford University Press).
[2000]
Thinking about Mathematics, (Oxford:
University Press).
[2005]
‘Introduction’ to Shapiro, ed., The Oxford Handbook of
Philosophy of Mathematics and Logic, (Oxford: Oxford
University Press).
Oxford
Steel, John
[2000]
‘Mathematics needs new axioms’, Bulletin of Symbolic
Logic 6, pp. 422-433.
48
Stillwell, John
[2002]
Mathematics and its History, second edition, (New
York: Springer).
Wilson, Mark
[2006]
Wandering Significance, (Oxford:
Press).
Oxford University
Woodin, Hugh
[2001]
‘The continuum hypothesis, Parts I and II’, Notices of
the American Mathematical Society 48, pp. 567-576,
681-690.
Wright, Crispin
[1992]
Truth and Objectivity, (Cambridge, MA:
University Press).
Harvard
Wussing, Hans
[1969]
The Genesis of the Abstract Group Concept, A.
Shenitzer, trans., (Cambridge, MA: MIT Press, 1984).
Zermelo, Ernst
[1908a]
‘A new proof of the possibility of a well-ordering’,
reprinted in J. van Heijenoort [1967], pp. 183-198.
[1908b]
‘Investigations in the foundations of set theory I’,
reprinted in J. van Heijenoort [1967], pp. 200-215.