Tuesday, February 5, 2013

Significance of Russell's Paradox for the philosophy of mathematics

To: Craig S.
From: Geoffrey Klempner
Subject: Significance of Russell's Paradox for the philosophy of mathematics
Date: 21st March 2009 12:26

Dear Craig,

Thank you for your email of 12 March, with your essay for the University of London Philosophy of Mathematics module, in response to the question, 'What is Russell's Paradox and what is its significance to philosophy of mathematics?'

The two questions from the UoL Internal exam papers you refer to are:

'Explain the view that mathematics is really logic, in light of the fact that mathematics, unlike logic, is about specific kinds of entity such as numbers and functions. Why is Russell's paradox a threat to this view?'

'What is Russell's paradox? Why might it be thought to undermine the logicist view of mathematics?'

It is significant that unlike the question you have composed, both exam questions explicitly refer to the logicist program. This actually makes a big difference. You have produced a concise and very readable history of the philosophy of mathematics emphasizing the importance of Russell's Paradox. As you state, the paradox is not the only significant result; there is also Godel's Theorem, etc.

Another point about your question is that there is a world of difference between asking for the significance of Russell's Paradox *for* the history of mathematics and philosophy of mathematics -- its reception, reactions to the problem and proposed solutions or projects -- and asking about its significance *to* present-day philosophy of mathematics, that is to say its place in a foundational inquiry into the nature of mathematics.

Both these are interesting questions, but if I were writing an exam paper I would not choose them because they are too open ended. The historical inquiry is far too big a topic, and doesn't really get down to the philosophical nitty gritty. Accidental historical reasons play a part too. The question about foundations is profound, but there are too many issues to consider: for example, what is the value of a foundational inquiry? (you touch on this with your remark about 'fruits' and 'roots'). To what extent is a solution by stipulation acceptable? (for example, Russell's Theory of Types). Or there is the question about the similarity (is it merely surface similarity or does it point to something deep?) between the self-referential aspect of Russell's Paradox and the use of Godelization to create a self-referential loop, which is the motor for Godel's Theorem, as well as other paradoxes of a similar nature.

The impact on logicism, by contrast, is a precise question. Although there is the historical question to consider, the main thrust of your answer has to be along the lines of asking what, if anything, is worth saving in the logicist program.

For example, you talk about the admired rigour and certainty of mathematics. Frege's Begriffschrift was intended precisely to increase the rigour of mathematical proofs. If you have sufficient rigour, why even raise the question about whether mathematics can be reduced to logic? The answer is simply because we can. Once it became clear just how powerful modern Fregean logic is, there was every reason to think that it could be possible to provide a logical foundation.

From this perspective, Russell's Paradox (as you state) is one of a number of results which undermine the logicist program. However, it could well be argued that Frege's reaction, in the Appendix to his Grundgezetze, was unwarranted. Russell's Theory of Types is a pretty good response, as is ZF set theory. OK, this doesn't give the 'pure' foundation that Frege was looking for, but the result is impressive all the same. It is a valuable and illuminating exercise.

Frege shows in his 'Foundations of Arithmetic' how statements about numbers can be analysed in first-order predicate calculus with identity. This is something we take for granted not but it is surely an amazing result. You can see how someone might think that all the Kantian stuff about intuition and time is just so much irrelevant bunk. This sharpens the question why Frege is so keen to demonstate that numbers are *objects*, like sets -- indeed, that they are sets. If you can do arithmetic (some, anyway) without numbers, isn't that enough to show that arithmetic is, at its core, logic?

I accept that if we give up ambitious logicism, then a question inevitably arises of what alternative account one gives of the nature of mathematics. So mention of intuitionism, formalism as philosophies of mathematics would be relevant. However, there is also the wider question (which, in effect, you raise) about why we need a 'philosophy of mathematics' in this sense, that is to say, the interest or importance of a metaphysically inspired theory about what mathematics 'really is' or what numbers 'really are'. (Both intuitionists and formalists see themselves as 'anti' metaphysics, i.e. anti Platonists, but their views are in their own way just as 'metaphysical'.)

All the best,

Geoffrey