# Does Arithmetic Require an Appeal to Intuition?

In his essay 'Mathematical Truth', Paul Benacerraf lays out a problem
which he feels affects all current theories of the Philosophy of
Mathematics. He argues that there are two central requirements of a
mathematical ontology and theory of truth - namely, that
firstly there should be a straightforward account of what it means
for some mathematical claim to be said to be 'true', and secondly
that there should be an account of mathematical *knowledge*
we should be able to explain how we come to know things about
mathematics. He claims a tension arises in that any theory of
mathematics that places mathematical truth in a similar position as
truths about other sciences, and that thus allows mathematical syntax
to mesh with the truth - and, indeed, allows mathematical claims the
same format as claims about other things - will stand in the
way of the possibility of mathematical knowledge.

To unpack this claim, we must consider the perspective from which
Benacerraf is coming to this debate. The theory which he argues is
capable of explaining the truth of mathematical propositions, and
placing them in the same sort of syntax as that of other truths, is a
simple form of mathematical realism. If to say that 2 + 2 = 4 is true
simply because it expresses some fact-of-the-matter, some
relationship between the entities *2* and *4*, this aligns
itself simply with common-sense realism - the belief that to
say that a claim we might make about medium-sized physical objects is
true is simply to say that the medium-sized physical objects hold
some property (or that there is a relation between more that one of
them) which makes the claim true. So (to use Benacerraf's example) to
say that '*There are at least three perfect numbers greater than
17*' is just the same sort of claim for the mathematical realist
as '*There are at least three large cities older than New York*'.
The nature of these mathematical entities is, to a large degree, a
matter of debate, but what we do know about their nature (if they
exist) is that they are abstract, inert (i.e. not playing a causal
rôle), and not physical objects located in the world.

Benacerraf believed the current best
theory of knowledge to be the *causal*-account.
He describes what he calls 'combinatorial' accounts of mathematical
(anti-)realism - whose proponents argue that mathematical
'facts' are simply anything that follows from the (potentially
arbitrary) axioms of our mathematical system. It is easy, he argues,
to see where this mathematical knowledge comes from, since we need
simply understand the (potentially arbitrary) rules for generating
'correct' mathematical sentences to see how we might go about
broadening our knowledge of the subject (even if we consider these
facts simply to be a product of our axioms, and not any actual
truths-in-the-world).

Since Plato first formulated it,
most discussions of what knowledge consists in began with the
tripartite definition of knowledge as justified true belief, and this
definition remained, on the whole unscathed, for a couple of thousand
years. Gettier, however, effectively sunk this account, by showing
how we can easily generate any number of cases of justified true
belief that do not amount to instances of knowledge. His short paper
relied on the fact that, if we could have justified false belief, as
we must be able to if the 'true' in 'justified true belief' is not to
be entirely superfluous, and as seems, intuitively, very likely to be
a possibility, it must be possible to have a justified belief that
just happens, coincidentally, to be true. To take Maddy's example,
consider a set-up whereby, through a series of mirrors, our guinea
pig is caused to believe that there is a tree in front of him (where,
in fact, he is just seeing the reflection of a tree to his side). Our
guinea pig has a justified belief that there is a tree in front of
him - justified by his simple, visual evidence - but it
is a false belief. Now consider the parallel case where, in fact,
there is a tree just in front of him - it could even be in the
exact 'position' of the reflected tree's image - the
guinea pig still believes that there is a tree in front of him, and
has just the same justification as he did before for believing there
to be a tree in front of him, and yet, this time, the belief is true,
and so he has a justified *true* belief that there is a tree in
front of him. Intuitively, however, we don't want to call this
knowledge, since it rests on a coincidence, and no better evidence
than the case where the belief just happened to be false. Various
attempts have been made to save the tripartite conception, frequently
fundamentally changing its nature, however, and the causal account is
one such attempt.

The causal account argues that what is wrong with the Gettier cases is that, whilst there is some sort of justification, it is not of the right sort to constitute knowledge. This has a great intuitive appeal - we are likely to feel that the problem Gettier has highlighted is one with such a vague idea of 'justification', and we may well hope that, in laying down better guidelines for what constitutes justification, we might save the conception of knowledge. The causal theory argues that, to have any sort of justification, the belief must in some way be caused by the truth of the fact being debated: since the actual tree in front of the subject plays no part in the 'justification' involved in our mirror case, it is not 'the right sort' of justification.

So will mathematical realism be incompatible with the causal account
of knowledge? It does seem very likely that it will be, since,
traditionally, abstract, inert objects can play no part in
something's causal history, and thus cannot be said to have caused a
particular belief. What really *isn't*
clear to my mind is that the causal account of knowledge does what
Benacerraf claims it does. Consider what one might call 'the
Gettier-Gettier case', where, rather than seeing a tree, the subject
believes he sees some moving object, let's say a cow. Now, rather
than a mirror in front of him, we place a sheet of glass, and behind
the mirror, we place a pressure plate, calibrated exactly to the
weight of a cow. Now, when there is a cow standing on the pressure
plate, it causes a bright floodlight to shine on the guinea pig's
side of the glass, rendering it (since the area behind is in
darkness) an effective mirror, in which can be seen the reflection of
a *different* cow. Now,
our subject will have a justified true belief that there is a cow in
front of him, and, crucially, this will be caused by the fact that
there is a cow in front of him. Yet, surely, we would still not call
this knowledge; if we do indeed refuse, then the causal account fails
- demanding that the belief be caused by the fact being debated
before calling the claim 'justified' *still*
doesn't ensure that the believer has knowledge.

It might well be possible for the
supporter of such an account to argue that the Gettier-Gettier case
simply shows that, whilst a causal relation is necessary, this can't
just be *any* causal
relation. My problem with this is that it really isn't clear where
this regress will end - we need a true belief with a
justification which is in some way causal in some appropriate(ly
relevant?) way: we need some explanation of in just what this
appropriateness lies. Those proposing the causal account play up its
internalist component, whilst relying on an externalist factor to
sort out their problems - and I would assert that, as it
stands, the causal account doesn't actually go very far in explaining
how we might have mathematical knowledge. We should not construe
Benacerraf analogy between the causal account and the combinatorial
account of mathematics too strongly, but I feel it fails for just the
same reason. Sure, we can see how '2 + 2 = 4' might follow from the
rules of the game of mathematics, but that certainly doesn't seem to
justify our* knowing*
that '2 + 2 = 4'. The problem seems to be that, bizarrely, Benacerraf
seems to be suggesting that the combinatorial account's proponents
still view their claims as being objectively true - and yet,
whilst he is quite right to point out that simply 'labelling' such
claims as 'true' just makes a pun of the word 'true', which is
being used in two different ways, he is happy still to label these
beliefs, simply the product of the (potentially arbitrary) axioms and
rules of the game of mathematics as knowledge. It seems a potentially
simple claim that the problem with a combinatorial account of
mathematics holding knowledge as 'true belief justified by its
following a set of axioms and rules' is that there is no way in which
the beliefs can be said to be true in an objective
(how-the-word-is-actually-used) sense - purely because the
combinatorial account's proponents would seem very unlikely to
construe mathematics as objectively true. Far from showing that
Gödelian intuitions are unnecessary to know things about
mathematical truth, Benacerraf reinforces this idea, since what he
counts as knowledge is certainly not about mathematical *truths*.

Maddy comes up with a similar, but interestingly different,
comparison to be drawn. Essentially, Maddy, rather than suggesting an
account of the *truth*
of a mathematical system is important, holds that we need an account
of how a mathematical system comes to be useful. The distinction,
therefore, is that, whilst we can (at least intuitively - it
isn't clear that such reasoning will automatically stand up to
serious debate) understand why mathematics would be useful, were
mathematical realism to hold, namely because it is *true*,
and about objective facts in the world. The primary question in
Maddy's mind is why, for a conventionalist account, mathematics
should be more useful in problem solving than, say, chess. Both are
games, with arbitrary rules, and these rules lead to terms which are
'pejorative', but only within the game - in mathematics we are
taught to like only sentences which are labelled 'true' according to
its rules; in chess we are taught to shrink from the 'illegal move'.
Why should, Maddy asks, mathematical 'truth' be more useful than
chess' 'legal moves' for problem-solving, if not in virtue of
mathematical 'truth' meshing with* actual*
truths?

Maddy's answer is to assert that we must have some sort of realism or
*Platonism* with respect to mathematics, though the particular
sort that she comes up with seems fairly bizarre. She argues that,
whilst there are such things as sets which (passing over problems
with the completeness of set-theory since the collapse of Hilbert's
programme) she takes as capable of underpinning numerical and
mathematical notions, these *can* indeed be perceived directly.
She claims that, when we see sets of three objects we are able to
abstract from them to an idea of the number three - that it is
fact a property of these objects - in much the same way that
when we see many lumps of gold we are able to abstract and understand
what 'gold' is, and how (she suggests followers of Kripke might
argue) we come to baptise them with names. I would hope that to most
people such an account would be dubious at best - it simply
isn't clear how we would go about performing such an abstraction.
With any set of objects there will be infinitely many 'common
properties' that we might abstract from them - it is an
instance of Wittgenstein's rule-following problem - and it
would never be possible to establish that that property which had
been isolated was, in fact, 'threeness'. She argues that it is a fact
of our perceptual capabilities that we are able to see number, in
virtue of the way that, presented with an apple, we generally see it
as one thing - the singleton of the apple - rather than a
collection of atoms, or two halves, or whatever. Even if such
perceptions are 'hard-wired' into our eyes and brain, and we invent
hard visual boundaries between objects and their backgrounds where
none really exist, it seems very bizarre to base upon *this*
fact any claims about what is *true* in the world. She takes
this to be an easy alternative to Gödelian intuitions, and yet
her account sounds just as implausible, if not more than Gödel's,
in virtue of her attempting to put an empirical spin on it, rather
than accepting, as Gödel did, that were we able to know anything
about mathematical facts, this would have to be due to some
mysterious, ineffable new faculty.