What Finite Minds Can Know: Epistemic Limitations and the
Existence of Infinite Mathematical Objects
Abstract
The classical British empiricists were finitists. They argued that we
lack an adequate idea of any infinite object on the grounds that we
cannot perceive an infinite object. The empiricist approach lingers in
contemporary constructivist philosophy of mathematics, such as the
views of A.W. Moore (1989), (1990), and (2007). Mainstream
contemporary mathematics is undeniably infinitist: one needs infinite
sets to do real analysis and set theory. This suggests that the main
philosophical motivations for finitism are flawed.
We reconstruct what we take to be the main argument for
finitism: the argument from epistemic limitations (‘EPLIM’). We
identify several key epistemological assumptions—verificationism,
idealism, and the finitude of perceptual experience—needed to make
the argument valid. We indicate why we hold that all these
assumptions are false. We consider in detail some arguments against
idealism about mathematical objects. We conclude by developing a
realist response to EPLIM and showing how epistemic limitations
are fully compatible with the reality of infinite mathematical objects.
1.
Introduction
Several branches of mathematics—such as number theory, geometry, topology,
real analysis, and set theory—express truths that can only be satisfied by infinite
domains. From a realist perspective, mathematical knowledge thereby requires a
vast ontology. How can this vastness be reconciled with the apparently limited
epistemic capacities of human agents?
The human mind has finite
computational and perceptual resources. The infinite objects of mathematics
threaten to outstrip the mind’s ability to know them. This problem, dubbed ‘the
problem of remoteness’ by Lavine [1994], is essentially an epistemological
problem.1 Like most epistemological problems, the problem is the result of bad
epistemology, an artefact of the way the problem is formulated. Or so we shall
argue.
Mathematical knowledge provides a special test case for empiricism.
Roughly speaking, empiricists hold that the source and basis of all meaningful
concepts is sensory experience. Mathematics appears to provide knowledge of
infinite objects (or at least of the concept of infinite objects). The concept of
infinite object does not seem to be derivable purely on the basis of sensory
experience. All that sensory experience presents is a finite object (a finite line
segment, a finite collection of objects) together with the open ended possibility
of repeating operations (such as dividing and counting) indefinitely. Thus the
limits of our experience together with the infinite character of some
mathematical knowledge put severe pressure on empiricist epistemology.
2
Moore [1990] suggests that the existence of the infinite is ultimately
empiricism’s undoing:
There can be no doubt that empiricism was one of the great
philosophical movements, of deep and lasting significance: but
this is partly because of lessons we can learn from its ultimate
failure. It is a very important feature of the infinite that it helped
signal that failure (Moore [1990]: 83).
Before we can evaluate Moore’s insightful comment, some terminological
preliminaries are in order. We can distinguish between three general varieties of
finitism: metaphysical, semantic, and epistemological.2 The tripartite division
between kinds of finitism is as follows:
(i)
Metaphysical finitism—the belief that there are no infinite objects.
(ia) Mathematical finitism—the belief that there are no infinite
sets or infinite numbers.
(ib) Physical finitism—the belief that the universe (or space, time,
is finite in size, or contains finitely many members.
(ii)
Semantic finitism—the belief that talk about the infinite is non-sense.
(iii) Epistemological finitism—the belief that only finitary reasoning is
epistemically secure; reasoning involving an infinite number of steps or
the use of quantifiers ranging over infinite domains is suspect.
With regard to (ia) mathematical finitism, we can further distinguish between
strict and liberal varieties. Strict mathematical finitists believe that there are
upper finite bounds to numbers and other mathematical entities.3 Liberal
mathematical finitists think that there may be a potential infinity of numbers (or
other mathematical entities), but deny that there is an actual completed infinity
of numbers (or other mathematical entities). In mainstream, classical
contemporary mathematics, strict mathematical finitism is assumed to be false.
Infinitism has won the day, as indicated by the widespread assumption of the
Axiom of Infinity.4 Moore’s comment suggests an argument from the truth of
infinitism to the falsity of empiricism.
This inference raises several
philosophical questions:
(iv)
Does empiricism compel finitism?
(v)
Are there any good reasons to be finitist? How about
epistemological considerations?
(vi)
Does the practice of classical mathematics provide an argument
against finitism?
Old fashioned empiricism is not a live option. However, several positions in the
contemporary philosophy of mathematics (e.g Dummett [1966], Moore [1989],
Wright [1982], [1985]) share empiricism’s view that mathematical knowledge
must be grounded in the appropriate experience of mathematical objects. Moore
[1989] argues that we cannot make sense of the idea of experiencing an infinite
object and therefore that statements about actually infinite objects are in fact
unintelligible. However, Moore allows that we can make sense of statements
about potential infinites. Moore thus argues for liberal semantic finitism.
The classic example of a liberal finitist position is Gauss’s interpretation
of the limit concept. Gauss stated:
I protest against the use of infinite magnitude as something
completed, which in mathematics is never permissible. Infinity is
merely a facon de parler, the real meaning being a limit which
3
certain ratios approach indefinitely near, while others are
permitted to increase without restriction (Gauss [1831]:216)
Liberal finitism appears to present the best of all worlds. On the one hand, it
allows for the use of the infinite in the calculus and elsewhere. On the other
hand, the actual infinite—which is epistemologically suspect—is banned from
mathematics. However, even liberal finitism is in tension with the mathematical
practice of quantifying over infinite sets. Using Quinean methodology, such
quantification carries an ontological commitment to completed, infinite objects.5
We argue that finitism (both strict and liberal mathematical varieties) is
not justified by traditional epistemological considerations. Section §2 of this
essay offers a reconstruction of the argument from human epistemic limitations
used by finitists. (This argument is referred to as ‘EPLIM’). Section 2.1 shows
that Hume and Berkeley embraced a version of the argument from epistemic
limitations. Section 2.2 excavates some of the bridging principles needed to
make the argument valid. Section 2.3 attempts to refute one such bridging
principle, the alleged datum (D) that human beings cannot encounter an infinite
object. Section §3 argues against bridging principle (I) which depends on a
consequence of idealism. Section §4 presents the realist response to the EPLIM
argument. We conclude in section §5 that the vastness of mathematical ontology
coheres well with due acknowledgement of the mind’s epistemic limitations.
To be sure, there may be better arguments for finitism than the argument
from epistemic limitations presented here. Nonetheless, as we show, the
argument has been tremendously influential. Finitists from Hume to such
‘Wittgensteinian’ finitists as A.W. Moore have embraced some version of
EPLIM. If we are right, then the single most influential philosophical path to
finitism has been closed off. Infinitists can have their cake and eat it too: they
can maintain the existence of the actual infinite in mathematics and respect
human epistemic limitations.
2. The argument from epistemic limitations.
2.1 Reconstruction of the argument
What is striking about the classical empiricists’s arguments for finitism is the
way they move from consideration of the mind’s capacities to a conclusion
about the nature and existence of mathematical objects. This kind of inference
(from mind to world) lends the argument an idealist character. The idealism is
explicit in Berkeley’s argument for finitism:
Every particular finite extension which may possibly be an
object of our thought is an idea existing only in the mind, and
consequently each part thereof must be perceived. If therefore, I
cannot perceive innumerable parts in any finite extension that I
consider, it is certain that they are not contained in it…
(Berkeley [1710], section 124: 77-8).
Berkeley’s conclusion is that any finite extension does not, indeed cannot,
contain infinitely many parts. Berkeley is a metaphysical finitist and atomist.
Only extreme idealism (such as Berkeley’s) would legitimate the leap from the
fact of our inability to perceive an infinity of parts to the conclusion that such
parts do not exist in spatial extension.
4
Like Berkeley’s argument, the main thread of Hume’s argument for
finitism moves from an epistemological premise to a metaphysical conclusion.
Hume’s epistemological premise is that the human mind has finite powers of
discrimination and can only form an idea of an interval with a finite number of
parts. Hume’s metaphysical conclusion is that space and time themselves are
only made up of a finite number of parts (‘minimal perceptibles’). As
commonly noted, the inference from epistemological premise to metaphysical
conclusion appears to depend on a kind of phenomenalism. Phenomenalism is
itself a species of idealism.6 Only if space and time themselves were the
product of human sense-impressions would it be legitimate to draw inferences
about them based on the character of human sense-impressions and ideas.7
Idealists hold that the nature of objects (space, time, and reality itself) is
itself constrained by the interpretative activity of the human mind. This reversal
of priorities is the point of the ‘Copernican revolution’ in philosophy that Kant
introduced, whereby ‘we try the hypothesis that objects conform to the structure
of our knowledge’ (Kant ([1781]/[1911]:Bxvi-xvii, 22). Consequently, idealists
should expect that they can infer the structure and properties of objects from
their knowledge of the structure of human concepts.8
Idealism interacts strongly with the belief that the human mind has finite
capacities, with striking repercussions for the philosophy of mathematics.
Suppose that one takes the finitude and limitations of the human mind very
seriously. Like a computer, the human mind has a limited ‘storage capacity’, a
limited time during which it can function and apply operations, and can only
take in and process a limited amount of ‘data’. If the objects of mathematics are,
in some sense, (as idealists hold) constructions of the human mind, then these
artefacts will necessarily conform to the limitations of that mind.9 It should be
obvious where this line of thought goes: a finite mind can produce (it is said) at
most a finite output, and so the objects of mathematics—the ‘output’—must be
finite (and finitely many) as well.10
We can generalize the argument and arrive at (the schema of) an argument
for mathematical finitism called ‘the argument from epistemic limitations’
(‘EPLIM’ for short):
(1) Human minds, being finite, are limited in their capacity to
comprehend and represent objects to representing finite objects.
(2) The objects of mathematics must be comprehensible and
knowable (by the human mind).
(3) Therefore, there are no bona fide infinite mathematical
objects.
The conclusion (3) is a statement of metaphysical finitism, since it concerns
mathematical objects. Premise (1) is a statement of a certain kind of epistemic
finitism, as it concerns the human mind and its epistemic capacities. Premise (2)
is intended to serve as a bridge between the two positions. Now, as it stands,
the argument is incomplete. Supplementary premises are needed to derive the
metaphysical conclusion from the epistemological premises. Neither premise
(1) nor premise (2) is innocuous. Both premises stand in need of disambiguation
and could be false. In particular, we claim that (1) is false if interpreted
properly.11 More surprisingly, (2) is highly contentious, and from a realist
5
perspective, false. As we will show, mathematical realists should tolerate the
possibility of a certain amount of ignorance of the properties of mathematical
objects as a consequence of their position.
To be sure, few philosophers nowadays want to be strict finitists as
recommended by (3). Most would prefer to be liberal finitists. EPLIM can be
recast as an argument for liberal finitism. Premise (1) of the original argument
can be relaxed to allow that the human mind can comprehend and represent
potential infinities. The revised argument then runs:
(EPLIM-revised)
(1-R) The human mind can comprehend and represent only
potential infinities, not actual infinities.
(2-R) The objects of mathematics must be comprehensible and
knowable by the human mind.
(3-R) The objects of mathematics are at most potentially infinite,
not actually infinite.
The revised argument, with its weaker conclusion, is still suspect. Like the
original version of the argument, the revised argument tries to derive a
metaphysical conclusion from epistemological premises. 12 From a realist
perspective, such a move is illegitimate. Unless there are a priori reasons to
think that objects must reflect the structure and limitations of the minds that
represent them, there can be no reason to conclude anything specific about the
nature of objects from our possibly inadequate conceptions of them.
It’s pretty clear that as it stands EPLIM is incomplete. It does not reflect
the pivotal assumptions of idealism and epistemic finitude central to the
empiricists’ arguments. The conclusion (3) simply does not follow directly from
the premises without the addition of a couple of bridge principles. Many
empiricists would carry these additional principles as part of their baggage and
so would tend to read them into EPLIM anyway. However, we argue that
EPLIM does not survive scrutiny of these bridge principles. Once these
assumptions are made explicit, realists can argue that we have good reason to
reject them (and the conclusion of EPLIM).
2.2 The Bridging Principles (V), (D), and (I)
One bridge principle must tie concepts to sense-impressions or experience.
After all it is the fact (if it is one) that we cannot experience there being
something infinite that leads empiricists to say that we lack a positive conception
of the infinite.13 However, the lack of an adequate concept of X does not license
the inference that X does not exist. To turn the foregoing observations into an
argument against the actual infinite, we need to conclude that our lack of an
adequate concept of the infinite somehow translates into a lack of ontological
reality. Such a move is verificationist. Here we agree with the suggestion of
Benardete (1966) that empiricist and idealist objections to the actual infinite are
based on a dubious verificationism.14 So our first bridge principle will be a
version of the verification principle of meaning (‘verificationism’ for short):
(V) A concept is meaningless and illegitimate unless it is a concept
of an object of possible experience.
6
In any case, the addition of (V) does not suffice to enable us to draw a
conclusion about mathematical objects based on premises about concepts in
mathematics. Another bridge principle is needed, linking concepts and objects
in such a way that concepts a priori impose their structure (and constraints) on
the corresponding objects. The simplest way to guarantee this feat is by
assuming a consequence of idealism:
(I) What concepts we have determine what mathematical objects
there are.
To derive the conclusion, we must add the bridge principles and the alleged
datum that
(D) We never experience anything actually infinite.
The additional premises (D), (I), and (V) function as ways of sharpening up the
original premises of EPLIM so as to yield the finitist conclusion.
(D), (I), and (V) are all highly contentious. However, they are all
necessary to derive the ontologically finitist conclusion. For example, if we
accept only (V) and (D), then all that follows is that we have no legitimate
concept of the (actual) infinite (in mathematics). It does not follow that there
can be no infinite mathematical objects. Acceptance of (V) and (I) with denial
of (D), however, also suffices to scupper the conclusion. If one thought that,
pace (D), one could have an experience of an infinite object, then it would
follow (even given (V)) that one could have a legitimate concept of an infinite
object and (given (I)) that such an infinite object exists. Of course denial of (D)
is out of keeping with the original spirit of EPLIM. In particular, the denial of
(D) contradicts premise (1) of the original EPLIM argument.
In full detail, with the additional premises inserted, the final revised
version of the argument runs:
(1*)
Our legitimate concepts of (basic) mathematical objects
must be derived from our experience, or at least possible
experience. [Verificationism (V)]
(2*)
We have no possible experience of anything truly infinite.
[Alleged datum (D)]
(3*)
We can have no legitimate concept of an infinite
mathematical object. [from 1,2]
(4*)
Our concepts determine what mathematical objects there
are. [Idealism (I)]
(5*)
Therefore, there can be no legitimate infinite mathematical
objects.
[from 4,5]
a.
The argument as it now stands is valid. Premises (1*), (2*) and (4*) –
Verificationism, the Datum, and Idealism, respectively— are all highly
contentious, and ultimately, we think, should be rejected as false. As these
premises form the crucial bridging principles needed to derive the conclusion,
we can conclude that the argument fails. In this paper, we shall focus our
discussion on reasons to reject (D) and (I). We think (V) is not really a live
option.15
We cannot refute verificationism here. However, a few brief remarks
7
may remind readers why it is objectionable. The verification principle of
meaning required for any meaningful statement S, that we have would have to
know, in principle, how we could verify S. Means of verification might include
observation as well as mathematical proof. The verification principle condemns
great swathes of mathematics to the realm of meaningless. For example, prior to
its proof, Fermat’s Last Theorem was, by verificationist lights, not meaningful.
Although we knew in general what was needed to prove Fermat’s Last Theorem
and we apparently understood the theorem, we do not know the specific proof.
Indeed, we did not even know that it could be proven. It seems far more likely
that the verification principle of meaning is too acidic rather than that
mathematicians do not possess the understanding that they take themselves to
have.16
2.3 Rejecting Datum (D): That we cannot experience
the infinite
Let us now consider datum (D) (premise (2*))— that we cannot and
could not in principle experience anything infinite.17 Although prima facie
plausible, (D) is in fact not conclusively indicated by any of the considerations
typically put forward in favour of it. The locus for discussion of (D) has been the
literature on the possibility of completing ‘supertasks’ (infinite tasks).18 It is
logically possible for someone to sub-divide a line according to the series
1+1/2+1/4+1/8 +…etc by performing the subdivisions at increasingly faster
rates, taking one minute for the first step, a half for the second, a quarter for the
third, and so on. Perhaps the laws of physics would have to be different in order
for someone to accomplish this feat, but there is no logical incoherence in
supposing this possibility to obtain.
Moore (1989) argues in a Wittgensteinian vein that although the idea of
completion of the supertask is not logically incoherent, we do not really
understand what it means to say that a supertask has been completed. Moore’s
reasoning is one of the best articulations of the case for finitism based on
epistemological (and semantic) considerations. We therefore examine and
criticize his argument in depth. Here is a representative passage from Moore’s
argument:
But might there not just be, say, infinitely many stars—or some
(natural) process whereby a particle, as a result of oscillating
through successively shorter distances, managed to complete
infinitely many oscillations in a finite time? Perhaps. But
again, this has to be understood in terms of the infinite
possibilities that might be written into what we could
encounter in experience. Let us suppose that space and time
themselves provide endless possibilities of movement and
reorientation. Then we can imagine empirical evidence to
suggest that, if we were to travel farther and farther away from
some point, or if we were to delve deeper and deeper into some
small region of space, we should always be able to find
specifiable phenomena (stars, the traces of the particle’s
movement, or whatever). But this is not to envisage an ‘infinite
reality’.
8
It may seem obscure what more could possibly be required.
But more is being presupposed when someone talks of
performing infinite many tasks , or more specifically, surveying
all the natural numbers, in a finite time; they are sanctioning
the possibility of a kind of direct encounter with the infinite.
We have yet to be presented with a situation, real or
imaginary, where it would be appropriate to talk in these
terms….It is an abuse of grammar to describe anything as
being, essentially the outcome of an infinite process, as it is to
describe any process as being infinitely old [italics ours].
(Moore, [1989], in D. Jacquette, ed. [2002]: 317).
Moore’s case for finitism consists of two steps. First, Moore suggests that we
do not understand the claim, because we cannot in principle have the requisite
experience corresponding to the claim.
Second, Moore tacitly takes
demonstration of a lack of understanding of a claim to suffice for showing the
claim to lack truth. We suggest that both of these steps are unwarranted.
With regard to the first step, why should the fact that we cannot clearly
delineate a situation, an experience that would count as ‘a direct encounter of the
infinite’ count against our understanding statements involving infinity, unless we
hold to a verificationist-empiricist of meaning? The position of the classical
rationalists in the debate over understanding infinity was that the idea of infinity
is not an idea of experience or imagination, but an idea of reason, intellect, and
understanding.19 Such ideas do not need to have a complete (or accessible)
empirical instantiation in order to be understood. Indeed, we want to suggest
that it is a mistake to understand knowledge of a mathematical truth (such as
‘The set of natural numbers N is infinite’) on the model of perceptual
acquaintance with an object. Rigid insistence on the model of acquaintance is
one of the empiricist’s great mistakes (and appears again in Moore (1989)).
Moore alleges that we cannot experience an infinite array of stars. But is
that claim true? We can experience a portion of an infinite array of stars. This
portion is compatible with the array being finite or infinite. Perceptual
experience itself cannot present an array as infinite, merely as finite and not
exhausted.
Moore argues that we do not understand statements involving actual
infinities on the grounds that we lack any corresponding (perceptual) experience.
This amounts to verificationism: the claims in question cannot be understood if
we cannot have possible experiences that would verify them. Moore might be
right that we cannot have an experience of an actual infinite, but this point is
irrelevant. Only on a crude empiricist-verificationist approach must we be able
to have an experience to ratify the meaningfulness of every statement. The
meaningfulness (and our capacity to understand) statements about actual
infinities might be ratified merely by the coherence of talk about such infinities
in ZFC set theory, for example. No sensory experience is required; this part of
mathematics is a priori discipline. Nonetheless, there might be experiences of
finite, but indefinitely large arrays, for example, that are relevant to explaining
how we acquire the concept of infinity. However, on our view experience does
not directly bequeath meaning to infinitary mathematical statements. Some
infinitary portions of mathematics may acquire their meaning indirectly, through
inferential relations with finitary portions of mathematics.20
Let’s now consider the second step of Moore’s argument. Moore moves
9
from our lack of understanding of a statement to the statement’s lack of truth.
Moore’s argument—like Wittgenstein’s observations on this topic—is entirely
about what statements we do and do not understand, and whether the concepts
involved are well-formed, and the statements thus ‘grammatical’ in
Wittgenstein’s philosophical sense.21
Statements that are completely
ungrammatical in the ordinary sense of being syntactically ill-formed cannot
express truths and may be gibberish. Wittgensteinians want to say that a whole
host of claims that pertain to metaphysics, including claims about infinity and
the claims of sceptics, are gibberish.22 But the problem is these statements are
not obviously ungrammatical and gibberish. The whole procedure of using
grammar to decide what is comprehensible and hence what statements could be
true is an idealist procedure that begs the question against realism from the start.
Why should the fact that we lack a completely adequate understanding of
statements about supertasks in empirical terms thereby mean that such
supertasks cannot be accomplished? We should not conflate statements, and our
understanding of them, with the reality that those statements purport to describe.
Moreover, given human fallibility and epistemic limitations, it is worrisome to
make the human imagination the arbiter of metaphysics. No defence of this
excessively restrictive verificationism about concepts is offered.
The
Wittgensteinian line, if it is not to collapse into sheer verificationism, must
maintain an agnostic position on the completion of supertasks.
3. Against (I): The problem with idealism.
So far we have discussed (V) and (D) as background assumptions of EPLIM.
Now we consider (I) (Idealism). We have seen that EPLIM is not valid without
a hidden idealist premise.23 On this view, mathematical objects are viewed as
constructions or ideas of the human mind, and have no existence independent of
the mind. In this way it becomes immediately explicable why mathematical
objects should conform to the limitations of the human mind. Each artificial
object bears the mark (and imperfections) of its maker. One of the imperfections
of the human mind is its inability to survey sequentially an infinite series.
Consequently on the idealist view the numbers must suffer a similar
imperfection: they must simply give out at the point from which surveying and
construction is not possible.24 These considerations may readily be taken modus
tollens as an argument for realism: there are infinite mathematical objects (such
as the class of natural numbers), where these objects exceed the mind’s capacity
to sequentially survey them, so such mathematical objects must exist
independently of the mind.
We are going to present two strategies for refuting idealism about
mathematical objects. The first strategy provides a wedge against idealism by
arguing that there are real items that are not conceivable in one sense (and hence
lack existence as ideas). The upshot of this discussion is that we should not insist
on knowledge by acquaintance of all mathematical objects. Insofar as premise
(2) of the epistemic limitations argument is a request for knowledge by
acquaintance, it must be rejected. The second strategy exploits a different tack,
arguing that the incompleteness of mathematical knowledge is something that
should be expected if realism is true. This style of argument for realism (and
against idealism) is quite indirect and less conclusive. Nonetheless, it does seem
that idealism naturally predicts that there should be few (if any) limits to our
knowledge in mathematics, a prediction at odds with mathematical experience.
Our wedge against idealism follows a strategy recommended by Thomas
10
Nagel in The View from Nowhere (1986). Idealists generally equate
conceivability of an item with its real existence.25 Idealists reason in accordance
with principle (Id):
If X is not conceivable by us, then this fact entails that X is not
real and does not exist.
In general, however, (Id) is not valid except for ideas, whose esse est concipi. It
is easy enough to generate counter-examples to (Id), both mathematical and
otherwise in content:
Example 1:The platypus was prima facie not conceivable by
European naturalists prior to its discovery by them, but was actual
and existent all along.
Example 2: The transfinite numbers were not positively conceivable
to Locke and Leibniz, but if numbers are real, they are real and yet
were not always conceivable.
Example 3: The infinite decimal expansion of  is still not entirely
conceivable to finite minds in full detail (they cannot imagine it and
survey it), but (according to realists), it enjoys full mathematical
existence; there is, for example, a fact of the matter as to whether the
n’th digit is 7, for all natural numbers n.
These examples are problematic. The first two examples are not obviously true
if the distinction between ideal conceivability and prima facie conceivability is
drawn.26 The third example begs the question against realism. In example 1, it
might be true that the platypus was always ideally conceivable, even if the
European naturalists failed in practice to conceive of it. Likewise, in example 2,
the transfinite numbers were ideally conceivable even if Locke and Leibniz were
not in a historical position to become aware of transfinite number theory.
Surely had such able minds as Locke and Leibniz lived in the 20th century they
would have been able to appreciate the truths of transfinite number theory.
However, appeals to ideal conceivability are problematic. It is unclear
what such ideal conceivability amounts to. We have no non-circular definition of
ideal conceivability. Ideal conceivability is supposed to allow us to abstract
away from ‘contingent limitations’. But it is not clear where we should stop in
our abstraction. Apparently everything is conceivable except that which is
contradictory!27
Example (3) would also be rejected by some idealists as a tendentious
example. Idealists would say that the lack of full conceivability in detail of π is
an artefact of the realist way of describing and interpreting the situation. Some
constructivists (intuitionists) hold that only so much of the number exists as can
be computed or specified by a construction.28 The statement that ‘there is a fact
of the matter as to whether the nth digit is a 7’ is an application of the law of the
excluded middle, which intuitionists reject.29 Though example (3) is
philosophically tendentious, it represents a mainstream view in the mathematical
community.
Such counter-examples are not conclusive. So another tack is needed.
Idealism implies a kind of arrogance about the scope of human understanding:
idealism implies that if x is real, then x will be (ideally) conceivable by the
human mind. To refute this principle, we need to find an x such that x is both
11
real and yet not conceivable by us. Of course it is a tricky business to specify
something that we cannot conceive.30 Is not to specify x to conceive x in a
certain manner? One way round this impasse is to argue by analogy. In general
we can see that there is most likely such an x for us, as there exist features of
reality that, although evident to us, cannot be conceived by minds less powerful
than our own. Nagel (1986) suggests that Gödel’s theorem is not understandable
by someone with a permanent mental age of a nine year old, but still true.31
Similarly, but by analogy with more powerful minds, we cannot survey the
infinitely many digits in π’s decimal expansion, but a more powerful artificial or
divine intelligence may be able to do so. Moreover, despite the fact that we
cannot clearly and distinctly perceive the whole expansion of π, it does not
follow that π does not have a determinate identity. 32 At least it does not follow
unless we have independent reasons to assume idealism.
Idealists will again appeal to the ideal conceivability of items in claiming
that the conceivability of an object is necessary (and sufficient) for its reality.
However, apart from the problems with ideal conceivability already noted, this
move sets the standards too high. Less than ideal conceivability should not
impugn the reality of an item. In fact we think there are many objects of
mathematics that are inscrutable and real. A prime example is the notion of a
real number. We have to allow that there are random real numbers. These real
numbers correspond to random infinite sequences of digits, which comprise their
decimal expansions. Random real numbers cannot be defined by rules laid down
a priori and cannot be surveyed in their entirety. Their legitimacy is due to
reasoning from inference to the best explanation. We know that the real number
line is uncountable by Cantor’s diagonal argument. We also know that we can
only compute countably many real numbers. So we infer that there are real
numbers that cannot be so computed and defined. These real numbers are the
`dark matter’ of analysis. They make our theory of real numbers come out right,
even though we lack an individual acquaintance with each of these real
numbers.33 Thus there must be mathematical objects that are not known by
acquaintance.
Premise (2) of the epistemic limitations argument does not elaborate on
the kind of knowledge required of the objects of mathematical knowledge. When
empiricists conclude that we have no positive idea of the infinite, they
presumably do so on the grounds that we lack acquaintance with the infinite.
This demand for knowledge by acquaintance is similarly present in Moore’s
assumption that to make sense of statements about infinite tasks, we require ‘a
kind of direct encounter with the infinite’. And it is dubious, or at least highly
contentious, that we do have knowledge by acquaintance of each mathematical
object; for example, it is hard to believe in recognisably distinct acts of
acquaintance with a 1000 sided polygon and an 1001 sided polygon. However,
this situation does not offer support for premise (1) of the epistemic limitations
argument, or for the finitist conclusion. For it may well be that we have other
ways of understanding infinitary mathematical statements that do not rely on
acquaintance.34
The realist has an advantage in allowing that our knowledge of
mathematical objects can be partial and incomplete at times.35 We do indeed
have epistemic limitations that prevent us from, for example, entirely
perceptually surveying an infinite decimal sequence. But such epistemic
limitations do not in themselves dictate that we should therefore assume that no
such sequence exists independently of our surveying and constructing it.
12
Intuitionists explicitly take such a stance: they hold that objects do not exist until
they are explicitly constructed, or at least until a recipe could be given for their
construction. The realist motto is: ‘completeness of objects, incomplete
knowledge’. The intuitionist (and idealist) tactic is the reverse: ‘complete
knowledge, incomplete objects’.
Our claim is that realism best explains incompleteness phenomena (both
of the informal sort I mentioned and the formal sort), and various forms of
idealism (intuitionism, constructivism) do not. However, we recognise that this
remark is not conclusive, since idealists will simply work with a different
ontology (a much truncated one) that does not exhibit the relevant
incompleteness phenomena. Nonetheless, classical mathematics does have
phenomena (eg random real numbers) which are unknown in the restricted sense
of not being objects of acquaintance (or particular construction). These
phenomena can be studied using indirect methods (knowledge by description,
inference to the best explanation etc.). Such incompleteness (or better, partial
ignorance) is something that we should expect from a realist point of view.
Realism precisely allows for facts that transcend our ability to verify (and in
some cases even conceive of) them. Thus realism about mathematics suggests
that premise (2) of EPLIM is false: it is not true that all the objects of
mathematics are knowable and comprehensible to the human mind. The respect
in which some mathematical objects are not knowable is that they are not
knowable by acquaintance (‘surveyable by perception’).
Idealists should not expect such partial ignorance of all the facts about
mathematical objects. In fact, in order to eliminate such ignorance, the
intuitionists have to tear down classical mathematics to eliminate it. This
observation about the philosophy of mathematics is actually a special case of a
general point. Nagel [1986] has observed that scepticism about our knowledge
of the world is to be expected (even encouraged and tolerated) on any robust
realist approach to the world. As Nagel puts it: ‘Realism makes scepticism
intelligible.’36 The same holds true, mutatis mutandis, for partial ignorance
(incomplete knowledge) and realism about mathematical objects. On the idealist
view, any partial ignorance that we display about mathematical objects must be
illusory: the objects themselves have fewer properties and less detail than we
thought. On the realist view, our partial ignorance is readily explained by the
richness of mathematical reality, combined with our epistemic limitations.37
4. The Realist Response.
In this section, we consider various broad realist responses to EPLIM. The
previous sections were mainly critical of the argument for finitism on the
grounds that it makes assumptions (eg idealism, verificationism, the alleged
datum) that cannot be defended. This section goes on the offensive, sketching
the realist alternatives that enable one to confidently reject the premises of the
epistemic limitations argument. As it turns out, none of the realist alternatives
depend on the detailed reconstruction of EPLIM (from section 2.2) The realist
response is best articulated against the original, simple two-step version of
EPLIM. As suggested at the outset, both premises (1) and (2) of the original
argument alone can and should be rejected by realists. The denial of these
premises results in interesting and plausibly true views that are not often
considered in the philosophy of mathematics.
The first realist alternative results from denying premise (1) of the
13
epistemic limitations argument, and seeing where that denial leads. We call this
solution ‘Cantor’s way’, since it accords with Cantor’s insistence that human
beings can represent infinite objects (such as the transfinite numbers and infinite
sets). In the Grundlagen (1883), he wrote:
What I assert and believe to have proved in this work as
well as in my earliest writings is this: that following the finite
there is a Transfinitum…that is, there is an unbounded stepladder of determinate modes which in their nature are not finite
but infinite, but which just like the finite, can be determined by
well-defined and distinguishable numbers. I am convinced that
the domain of definable quantities is not exhausted by the finite
quantities, and the bounds of our knowledge may accordingly be
extended without violence to our nature. ….In particular I
contend the human understanding has an unbounded capacity
for the stepwise formation of classes of integers. 38
Cantor’s point is consistent with the observation that we cannot survey,
sequentially, by a quasi-perceptual act, all of the infinitely many members of the
natural numbers. Our perceptual epistemic limitations are not denied. But our
powers of representation and comprehension go well beyond what we can
perceive. In other words, we could see Cantor’s affirmation of infinitary
knowledge as depending on a rejection of the empiricist perceptual model for
mathematical knowledge.
Cantor stands squarely in the rationalist tradition on which emphasis is
placed on an intellectual understanding of the infinite, rather than a sensory
perceptual experience of it. What differentiates Cantor’s views from the
rationalists is that he has the mathematical theory to back up his assertions.
Cantor provides a practical demonstration of the representability of such
infinities. In this connection, it is instructive to review briefly how Cantor
introduces the transfinite numbers, which he regards as actual infinities. The
natural numbers are producing by performing the successor operation (+1) on
an initial element 0, leading to the series N={0,1,2,3,4……..}. To get the idea of
the infinite number of natural numbers, another operation is needed: taking the
limit.  is the first ordinal number not reached by the successor operation. To
reach , one thinks of it as the first number to succeed all the natural numbers,
and simultaneously as the set consisting of the natural numbers in their canonical
order. The size of  can be gauged on a relative scale: it is 0, the first
transfinite cardinal number. To get the next transfinite cardinal number 1,
another operation is needed: forming the power set (set of all possible
subsets). Thus we take all different subsets of N to derive the power-set P(N)=
{Ø, {0}, {1}, {0,1}, {0,1,2}, {1,2,3}, {1,2,3,4}, ….}. For each element n of N
we can either put it in a subset or not, so P(N)=20. Cantor further proves that
2n>n as a general rule. So the power-set operation can enable us to keep
forming infinite sets larger than any transfinite number. These few operations
enable formation of Cantor’s original universe of transfinite cardinal numbers. 39
Let us consider the philosophical character of Cantor’s principles for
introducing infinite numbers. Mayberry [1977] perversely calls Cantor’s
position ‘finitism’. Mayberry intends to emphasize Cantor’s assumption that we
can treat and manipulate transfinite numbers in the same way as finite
14
numbers.40 His operations (successor, limit, power set) take us from finite
numbers to infinite numbers. Obviously, these operations would not be
acceptable to empiricists: they take us far beyond reflection on sensory
experience. Yet, these operations are part of a coherent theory, and we do seem
to form positive ideas of infinities through these operations. A purely negative
idea of the infinite-- all that is available according to empiricists such as Locke-is simply the idea that such infinity is ‘not finite’. But our knowledge of
infinities in set theory appears to go well beyond such restricted negative claims.
Cantor’s way of dismissing EPLIM, then, is simply to reject premise (1).
In this realist response we identify the different kinds of epistemic limitations
that we have, and then note that as regards mere abstract representation and
conception (as opposed to perception) we are not as limited as finitists think.
From the fact (however ill defined and understood it is) that the human mind is
finite, it does not follow that the human cannot have ideas of something infinite.
However, another kind of realist response is possible. This response
depends on disambiguating premise (2) of the original epistemic limitations
argument. Premise (2) states that the objects of mathematics must be
‘comprehensible and knowable’. Two interpretative questions arise. First, to
whom must the objects of mathematics be knowable? Second, in what way must
be the objects of mathematics be knowable?
Let us call the response that emphasizes the different kinds of minds
that might be said to know mathematics ‘the Augustinian way’. Although it is
natural to assume that what is at stake is comprehension by an ordinary, nonidealized human mind with finite capacities, there seems no principled reason
why this need be the case. Perhaps there is some unlimited divine or artificial
mind that can indeed survey infinite collections perceptually. Such a mind could
have an ‘intellectual intuition’ of all the natural numbers. Here the suggestion is
not to abandon the perceptual model of mathematical knowledge but to extend
the faculty of perception to include a non-empirical ‘intellectual intuition’.
Augustine suggests that God has such an infinite synoptic vision of infinite
collections that are too large for human minds to consecutively represent.
Augustine writes:
Although the infinite series of numbers cannot be numbered [by
us], this infinity of numbers is not outside the comprehension of
him ‘whose understanding cannot be numbered’. And so, if what
is comprehended in knowledge is bounded within the embrace
of that knowledge, and thus is finite, it must follow that every
infinity is, in a way that we cannot express, made finite to God,
because it cannot be beyond the embrace of his knowledge.
[Augustine, quoted in Hallett [1984]: 36] 41
Augustine suggests that the omniscience of God’s mind would require that there
can be infinite collections that are comprehended as complete and actual in the
same way that finite numbers are.42 If one were to accept Augustine’s view,
with all of its ontological baggage, then the epistemic limitations argument
would straightaway fall down. The argument would be invalid by equivocation.
Premise (1) concerns human epistemic limitations.
Premise (2)—for
Augustinians—concerns divine knowledge. The conclusion concerns
mathematical objects. Apart from the problems already identified concerning
the need to bridge the gap between epistemological claims in the premises and a
15
metaphysical conclusion, it is clear that if the argument is interpreted in
Augustine’s way, then the conclusion (3) cannot be drawn at all. Human
epistemic limitations are irrelevant to the ultimate content of mathematics. The
Augustinian position, however, is not such a happy resting place for realists.
Augustinian realism is, in the end, a form of idealism, whereby real existence
requires conceivability by an ideal mind. The essence of realism is to allow
objects to exist independently of any mind, human or divine.
We saw earlier that premise (2) of the epistemic limitations argument is
ambiguous in yet another way: it is unclear in what way the objects of
mathematics are comprehensible and knowable. We have suggested that the
empiricist insistence that all mathematical objects be knowable by acquaintance
is misplaced. In classical mathematics, there are more real numbers than we
can name, specify, or calculate. In addition, each real number can be
represented by an infinite sequence of digits in decimal expansion: we cannot
even imagine surveying all these digits. However, our inability to know the real
numbers in this way does not exclude our knowledge of them through other
means. We can now envisage an interpretation of the argument on which (1)
and (2) are true, but from which no finitist conclusion follows. According to this
interpretation, premise (1) asserts that human beings cannot perceptually survey
and represent infinite series. Premise (2) asserts that the objects of mathematics
have to be knowable in some general sense, but does not require knowledge by
acquaintance. Thus with a properly broad epistemology the existence of infinite
mathematical objects sits happily alongside human epistemic limitations.
There is yet another way to revise premise (2). As hinted earlier
(section 3) realists may wish to deny that the objects of mathematics are
knowable in the formal sense of being such that they can be characterized by a
complete axiomatic system. There is a connection between the infinity of
mathematical objects and formal incompleteness results. A detailed discussion
is beyond the scope of this essay. However, one should note the observation by
Torkel Franzén:
From a philosophical point of view, it is highly significant that
extensions of set theory by axioms asserting the existence of very
large infinite sets have logical consequences in the realm of
arithmetic that are not provable in the theory that they extend.43
Franzén’s observation suggests that epistemic limitations (at least as regards
what can be proven in a given axiomatic system) should be seen as an inevitable
result of trying to grapple with a deeply infinite, inexhaustibly rich ontology. In
other words, after Gödel’s results, we might see our epistemic limitations as a
reflection of an infinite ontology, not as a reason to deny the existence of such
an infinite ontology.
5. Conclusion
The argument against actual infinities in mathematics from human epistemic
limitations is a bad argument, yet it has been historically influential in various
guises. The premises of the argument were endorsed by the classical empiricist
philosophers, such as Hume and Berkeley. The premises have also informed
contemporary work sympathetic to finitism such as Moore [1989].
We have shown that although EPLIM is not valid in its initial form, it can
16
be made logically valid through the addition of controversial supplementary
assumptions such as (V) (verificationism), (I) (idealism), and (D) (the datum that
we do not experience anything infinite). These additional assumptions motivate
the finitism that we find in empiricism. We have suggested that EPLIM fails to
be sound because of the inclusion of these false premises. In particular, we
suggested that (I) involves the fallacy of mistaking our epistemic limitations for
constraints on reality. We further argued that even (D) is not obviously true, as
there is nothing logically incoherent in supposing that we might experience an
infinite object (cf. Oppy [2007]).44
We have seen that realists have a number of resources in responding to
EPLIM. First, they can and should reject the idealist supposition and the alleged
datum that were found to be hidden premises in the argument. Second, they
should disambiguate both major premises (1) and (2) of the argument. They can,
following the rationalist approach (endorsed by Cantor), suggest that we have
positive knowledge of infinities in mathematics where this knowledge is not
knowledge by acquaintance.
In summation, we have found ample grounds to agree with Moore’s
remark that the concept of the infinite signalled the failure of classical
empiricism as a philosophical movement. This does not yet demonstrate that
more contemporary empiricisms may not fare better. Contemporary empiricism,
such as Quine’s approach to mathematics and Maddy’s physicalism, allows for a
richer epistemology than the classical empiricist approach. Of pivotal
importance is the allowance for modes of knowledge that are indirect (inference
to the best explanation, knowledge by description, coherence as legitimation).45
In retrospect, we should not be surprised by the failure of an attempt to constrain
metaphysics by using an overly restrictive empiricist epistemology. Far from
being a cause to restrict the metaphysics underlying mathematics, epistemic
limitations should be taken as a reflection of the richness of that metaphysics.46
School of Mathematics and Statistics,
UNSW, Sydney Australia
17
In recent discussion ‘the problem of remoteness’ has been overshadowed by a
more ancient and basic problem: the problem of giving an account (preferably
causal) of the mind’s knowledge of abstract objects (see Benacerraf (1973]).
Yet for physicalists (such as Maddy (1990)) about mathematics the problem of
remoteness is more important than Benacerraf’s problem. If basic mathematical
knowledge concerns concrete objects, there is still the problem that some
mathematical truths require the existence of infinitely many such objects.
2 We find a similar set of distinctions in M. Giaquinto, The Search for Certainty
(Oxford: Oxford University Press, 2002).
3 The school of ‘strict finitism’ or ‘ultra-finitism’ is represented by
Van
Dantzig, ‘Is is 101010 a natural number?’, Dialectica, vol 10, 1955. One of the
originators is A. S. Yessenin-Volpin, ‘The ultra-intuitionistic criticism and the
antitraditional program for foundations of mathematics’, in Intuitionism and
Proof Theory, A. Kino, J. Myhill, and R. E. Vesley, eds., North-Holland, 1970,
pp. 1–45.
1
The Axiom of Infinity is a basic assumption of standard Zermelo-Frankel (ZF) set
theory. The Axiom of Infinity guarantees that there is at least one infinite set. In
symbols:
Here if the empty set belongs to the
inductive set I, and so does x, then the successor set of x also belongs.
4
We cannot address the opposition to Quine’s method here, though we note that
his criterion of ontological commitment (‘To be is to be the value of a (bound)
variable’) has come under increasing pressure from nominalists (see Azzouni
(2004)).
6 Part of what legitimates the inference, in Hume’s eyes, from idea to reality is
his infamous ‘copy principle’. According to the ‘copy principle’, every adequate
idea corresponds either to a sense-impression or can be decomposed into simple
ideas that do so correspond. Given the premise that the human mind has a finite
capacity to perceive, remember, and assemble sense-impressions, it follows that
human beings cannot construct or experience an infinite object. By Humean
empiricist lights, then, it follows that a finite mind simply cannot form an
adequate idea of an infinite object. This is exactly what Hume insists:
'Tis universally allowed, that the capacity of the mind is limited, and
can never attain a full and adequate conception of infinity: And tho'
it were not allow'd, twou'd be sufficiently evident from the plainest
observation and experience' (Hume (1740] Treatise I.II.1)
Consequently, even if an infinite object could exist, it could not exist as an
object of human knowledge in mathematics. Mathematical knowledge must
consist in knowledge of adequate ideas. So there is a swift empiricist argument
from our lack of an appropriate experiential basis for our idea of the infinite to
the conclusion that mathematics cannot treat of infinite objects.
7Cf. D. Hume, Treatise, Book I Part II section 1.
8 Realists of course reject this kind of inference, on the grounds that the objects
known are independent from the knowing subject.
9Proponents of strong AI may deny this claim on the grounds that human beings
can create computers that have capacities (such as the capacity to beat chess
champions) that most humans themselves lack. The computer metaphor is not
5
18
essential to the philosophical point.
10 Jacquette (2001] appears to endorse this line of reasoning in his defence of
Hume’s finitism as a relevant position in contemporary philosophy of
mathematics.
11 Even Kant seems to have been aware that assumption (1) is equivocally true at
best, true in one sense and false in another. For this reason, Kant rejects
conclusion (3) and maintains that the mind can represent space, for example, as
being potentially infinitely divisible. For the suggestion that Kant’s prohibition
on representing infinity by concepts resulted from his use of monadic logic, see
M. Friedman, Kant and the Exact Sciences (Cambridge, MA: Harvard
University Press, 1992). Furthermore, Kant’s straightforward acceptance of
Euclidean geometry as giving us knowledge of the structure of space commits
him towards viewing space as infinite (both in extension and division). So Kant,
at least as regards the realm of appearances, is no strict finitist.
12In contemporary terms, the issue is that premises (1) and (2) involve epistemic
operators and thus have an opaque context. In the conclusion (3), the epistemic
operators are ‘peeled away’ to derive existential conclusions about the objects
themselves. But this kind of move is suspect and related to the sin (according to
Quine and Kaplan) of ‘quantifying into opaque contexts’. However, we do not
think the Quinean diagnosis of what’s wrong with the argument is especially
helpful, in treating metaphysical arguments as though they were logical
mistakes.
13 Locke (1689] is typical: ‘So that what lies beyond our positive idea towards
infinity lies in obscurity, and has the undeterminate confusion of a negative idea;
wherein I know I neither do nor can comprehend all I would, it being too large
for a finite and narrow capacity….’ (Locke, (1689]/(1955) I.xvii., 151-152).
14 J. Benardete, Infinity (Oxford: Clarendon Press, 1966), chapter II.
15 Some argue that remnants of verificationism are found in Dummett’s
philosophy. For the charge that Dummett’s anti-realism is just verificationism
dressed up, see M. Devitt, Realism and Truth (Oxford: Blackwell, 1984); E.
Craig, The Mind of God and the Works of Man (Oxford: Clarendon Press, 1997):
“anti-realism is a very natural successor to positivism” (p.286). For a defence of
Dummett against the charge, see C. Wright, ‘Scientific Realism, Observation
and Verificationism’, in his Realism, Meaning, and Truth (Blackwell 1986,
second edition, 1993). For further argument that Dummett’s semantics is
unacceptably verificationist, see W. Alston, A Realist Conception of Truth
(Ithaca: Cornell University Press, 1995), chapter 4.
16 In any case, verificationism displays an arrogance about the human ability to
know reality that does not seem warranted. Craig (1997] singles out this passage
from the positivist Schlick (1936] as particularly awful:
…no meaningful problem can be insoluble in principle…This is one of
the most characteristic results of our empiricism. It means there are no
limits to human knowledge. The boundaries which must be
acknowledged are of an empirical nature and therefore never
ultimate…there is no unfathomable mystery in the world (Schlick
(1936) 156).
Without such arrogance about our ability to decide the truth of
propositions, verificationism has the consequence (unintended by its originators)
that we understand and know almost next to nothing, including in science. It is
not clear what the ground of such confidence is, but one guess is that it rests on
19
the phenomenalist and idealist view that objects are constructions of sense-data
undertaken by the human mind. If so, then verificationism’s plausibility seems
intimately linked with constructivism (Craig (1997) 286).
17 As we have seen, the empiricists endorsed this claim, and together with their
empiricist epistemology, it underpinned their finitism. Kant, as we might expect,
compromised between empiricism and rationalism on this topic. Kant asserted
that the infinite was not an object of perception while holding that it could be
intellectually cognized (Kant (1790)/(1952): 99-105).
18For a review of this literature, see G. Oppy, Philosophical Perspectives on
Infinity (Cambridge: Cambridge University Press, 2006). We agree with Oppy’s
claim that those sympathetic to finitism, such as Crispin Wright, have overstated
what can be concluded from the literature on supertasks.
19 Descartes’s Reply to Gassendi, Objections and Replies, ed. J. Cottingham, D.
Murdoch, and R. Sturgeon, The Philosophical Writings of Descartes, vol. II,
(Cambridge: Cambridge University Press, 1985), 252.
20This kind of picture is developed in detail by S. Lavine, Understanding the
Infinite, (Cambridge, Massachusetts: Harvard University Press, 1994). Lavine
views his program as Hilbertian.
21On grammar in Wittgenstein, see P.M.S. Hacker, Insight and Illusion (Bristol:
Thoemmes Press, 1989), 193-206.
22For Wittgenstein’s views on infinity, see his Philosophical Remarks (Oxford:
Basil Blackwell, 1975), including the Appendix 1: ‘The Concept of Infinity in
Mathematics’, 304-314. To be sure, there may be different meanings that attach
to the term ‘non-sense’ in Wittgenstein’s corpus. We have used the term
‘gibberish’ to single out the prejorative meaning of ‘non-sense’, on which there
is really no hidden transcendent meaning to the non-sense statement. There is
also a non-pejorative, mystical kind of ‘non-sense’ in Wittgenstein’s view,
which is connected with what can be shown, rather than said. For the mystical
kind of non-sense, see Wittgenstein’s Tractatus logico-philosophicus,
proposition 6.522 (London: Routledge, Kegan and Paul, 1922), p.187. However,
in the Philosophical Remarks, there are several passages (e.g. Appendix I, par.
128) where Wittgenstein seems to be saying that talk of the actual infinite is
non-sense in the pejorative sense. For more on Wittgenstein’s views on infinity,
see A.W.Moore, ‘Wittgenstein on Infinity’, forthcoming in Marie McGinn, ed.,
The Oxford Companion to Wittgenstein.
23 There seems to be more sympathy for idealism about mathematical
objects than for physical objects. One reason is that mathematical objects are
commonly supposed to be abstract, causally inert universals devoid of secondary
qualities such as smell, taste, and colour. So for those who find a full-blooded
Platonism about such objects unattractive, the thesis that mathematical objects
are created by human mental activity is a natural alternative. As Dummett
explains the motivation of intuitionism,
While, to a platonist, a mathematical theory relates to some
external realms of abstract objects, to an intuitionist it relates to our
own mental operations: mathematical objects themselves are mental
constructions, that is, objects of thought not merely in the sense that
they are thought about, but in the sense that, for them, esse est
concipi.23
Another reason for finding idealism attractive is the emphasis on
mathematics as a human activity and a failure to distinguish between the human
20
activity and its real subject-matter. However, as we shall see, as the general
reasoning underlying idealism is fallacious, the arguments against idealism
apply equally to mathematical objects and other objects.
24 One does in fact find this incredible thesis in B. Rotman, Ad Infinitum...the
Ghost in Turing’s Machine, Stanford University Press 1993.
25Nagel characterises idealism in these terms, as ‘the position that what there is
must possibly be conceivable by us, or possibly something for which we could
have evidence.’ (Nagel 1986: 93). His summary fits Berkeley’s views as
suggested by his remarks in Three Dialogues between Hylas and Philonous.
Berkeley has Philonous make Hylas concede that it is contradictory to speak of
(and conceive) of objects and qualities existing without the mind, hence without
conception. (See also Principles of Human Knowledge, section 23). To move to
conceivability rather than perception in ‘esse est percipi’ is, if anything, to make
the position more plausible.
26 Chalmers (2002] distinguishes between prima facie and ideal conceivability in
a bid to defend the principle linking conceivability and possibility (CP principle:
if x is conceivable, x is possible).
27 Notoriously, some philosophical positions—such as Graham Priest’s
dialetheism (Priest (1987])---would take issue with this conventional view.
28 See Christopher Ormell, ‘The Continuum: Russell’s Moment of Candour’,
Philosophy (2006), 81: 659-668. For a realist rejoinder, see A. Newstead and J.
Franklin, ‘On the Reality of the Continuum’, Philosophy 83 (2008), 117-27.
29 L.E.J. Brouwer, ‘Does Every Real Numbers have a Decimal Expansion?’, in
P. Mancosu, From Brouwer to Hilbert: The Debate on the Foundations of
Mathematics in the 1920s, (Oxford: Oxford University Press, 1998), 28-35.
30See Graham Priest’s reconstruction of Berkeley’s argument against ‘existence
unconceived’, in Beyond the Limits of Thought (Cambridge: Cambridge
University Press, 1995), pp.65-77.
31T. Nagel, The View from Nowhere (Oxford: Clarendon Press, 1986), p.95.
32 (Indeed to give up this principle would be to give up the comparability of the
real numbers, a tenet of classical analysis. Of course, that is just what Brouwer
and radical intuitionists do. But the consequences are pretty destructive of
conventional, mainstream mathematics. )
33A. Newstead and J. Franklin, ‘On the Reality of the Continuum’, Philosophy,
January 2008.
34One obvious consequence of this approach would be that knowledge of the
infinite in mathematics is not epistemologically basic. But this seems entirely
acceptable: perhaps our knowledge of the infinite is something we are less
committed to than our knowledge of finitary arithmetical truths.
35 By ‘incomplete knowledge’ we have in mind something more informal than
Gödel’s famous incompleteness results.
We simply mean incomplete
knowledge in the sense that not all of the object’s characteristics are known to us
in our current state.
36Nagel, The View from Nowhere, p.90.
37 To be sure, there may be other reasons to be finitist besides the
epistemological considerations. Marion (1998) suggests Wittgenstein maintained
finitism, but rejected the argument from epistemic limitations.
38Cantor, Grundlagen, in W. Ewald, (ed.) From Kant to Hilbert, II (Oxford:
Clarendon Press, 1996), 891.
39For details, see M. Potter, Set Theory and its Philosophy (Oxford: Oxford
21
University Press, 2004), 117.
40 Mayberry introduces this startling, counter-intuitive characterisation of
Cantor. It is counter-intuitive, of course, because Cantor is the defender of
infinities par excellence and no finitist in the usual sense. However, Mayberry’s
use of the term is meant to suggest that methodologically and epistemologically,
Cantor treats finite and infinite numbers as on a par. See J. Mayberry, ‘On a
consistency problem for set theory’, British Journal for the Philosophy of
Science, 1977, vol.28: 1-34, 137-170.
41Augustine, City of God, Book XII, chapter 19, as quoted in M. Hallett (1984)
op. cit, at 36 - denied by Wittgenstein, however, who thinks God should not
cheat by going beyond human mathematics: Remarks on the Foundations of
Mathematics VII-41 (3rd ed, Oxford: Blackwell, 1978), at 408.
42 The argument was revived by Constantin Gutberlet, Das Unendlich
metaphysisch und mathematisch betrachtet (Frankfurt am Mainz, 1878).
43 T. Franzén, Gödel’s Theorem: An Incomplete Guide to its Use and Abuse
(Wellesley, Massachusetts: AK Peters 2005), 152.
44 Graham Oppy, Philosophical Perspectives on Infinity, (Cambridge:
Cambridge University Press, 2006).
45 For a physicalist approach to mathematics, see P. Maddy, Realism in
Mathematics (Oxford: Clarendon Press, 1990). For a modified Quinean realism,
see P. Maddy, Naturalism in Mathematics, (Oxford: Clarendon Press, 1998). On
inference to the best explanation in mathematics, see C. Jenkins, ‘Knowledge of
Arithmetic’, British Journal for the Philosophy of Science 56 (4), December
2005, 727-47. For a moderate Aristotelian empiricism, D. Gillies, ‘An Empiricist
Philosophy of Mathematics and its Implications for the History of Mathematics’,
The Growth of Mathematical Knowledge, E. Grosholz and H. Breger (eds.)
(Boston: Kluwer, 2000), 46-51.
46 Work on this paper was carried out as part of a project funded by the
Australian Research Council (ARC) for 2007-2010, Discovery Project no.
DP0769997.
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