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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.