To: Craig S.

From: Geoffrey Klempner

Subject: Mill's view that arithmetical knowledge is empirical

Date: 20th February 2009 11:59

Dear Craig,

Thank you for your email of 12 February, with your essay for the University of London Philosophy of Mathematics module, in response to the question, 'Explain and assess Mill's contention that arithmetical truths are known by induction from the evidence of our senses.'

I liked your essay on Mill. If there is room for improvement, it would be around the analysis of the phrase, 'are known by induction'.

Let's say that I am doing my accounts and I get a figure for annual turnover which looks a bit high. On the third and fourth attempt, I get the same figure. Finally, I ask a friend who is an accountant to try, and he gets the correct figure. From experience, I know that my friend is never wrong, especially when he is this confident. So now I *know* what the figure is.

Is this knowledge therefore empirical knowledge? No, because the fact about what the correct addition is (modulo Mill) is not an empirical fact. This is a remark about the concept of knowledge rather than about the philosophy of mathematics. (There is room for debate over this example. Some epistemologists would reject the 'reliabilist' model of knowledge.)

In this light, it would seem that questions about how we 'learn' or have 'evolved' to make accurate arithmetical judgments based on perception are irrelevant to the question raised by Mill concerning the epistemological status of arithmetic. (Further to your point about infants, I saw a program where very young children were taught to recognize the numbers of surprisingly large collections and perform the 'addition' intuitively. An typical example would be 43 plus 86, represented as two cards with 43 and 86 coloured dots. The film 'Rain Man' with Dustin Hoffmann, is a beautiful illustration of how autistic savants can perform amazing arithmetical feats without, apparently, 'counting' or 'performing calculations'.)

So there is a distinction to be made here between how human beings acquire mathematical concepts and knowledge, and the epistemological status of the knowledge thus acquired.

Mill's claim about the empirical basis of arithmetic is a metaphysical, as well as an epistemological thesis. This is the point of his opposition to Kant's 'false philosophy'. What he is saying, in effect, is that there is no a priori requirement that the world conform to Peano's axioms. Conversely, there is a possible world in which arithmetic (as we know it) fails. That is a very strong claim, inviting speculation about what kind of world this might be.

When Quine suggests that even the truths of logic might come into question (citing the example of quantum mechanics) he wisely avoids attempting to describe what such a world might be 'like'. He is arguing a posteriori: we have already encountered problems in applying logic, and who is to say that there might not be more. Quine's view would be that the 'onus of proof' is on the philosopher who claims that arithmetic is immune from revision, rather than on the one who claims there is no area of knowledge where revision is not conceivable.

So, one question one might raise is, Is the onus on Mill to describe such an alternative anti-arithmetical world? A relevant point to make here would be about geometry. Kant was wrong about geometry. The geometry of the world is not Euclidean. We know how to describe worlds in which Euclidean, or in which non-Euclidean geometry holds. And, in fact, all the current evidence points to its being non-Euclidean. What exactly is the difference in the case of arithmetic?

For a start, there are no mathematical results comparable to the proof of the consistency of non-Euclidean geometries. If Frege and Russell are right, there couldn't be, because the truths of arithmetic are ultimately truths of logic.

On a point of detail: you mention Frege's claim that numbers are abstract 'objects'. As you probably know, the attempt has been made to reconstruct large parts of arithmetic without this assumption, using first-order predicate logic with identity, as explained by Frege in 'Foundations of Arithmetic'. This in itself is very strong reason for thinking that there just couldn't be alternative arithmetics, because the concept of a countable unit (which is equivalent to the concept of identity) is all you need.

A world where arithmetic failed would have to be world without 'units', without 'names' and 'referents', 'sameness' or 'difference'. The only example I can think of is the 'One' of Parmenides (which isn't 'one' in the arithmetical sense because it is logically impossible that anything could be 'added' to it).

All the best,

Geoffrey