Wittgenstein's Philosophy of Mathematics
I'm back from snowy Kent with a much clearer idea of Wittgenstein's views on mathematics. Some of the speakers there were the kind of hard-core Wittgensteinians who can recognise a quotation of the master from 100 yards and tell you its section number. An important point that was insisted upon is not to treat all the published writings equally. In the days before word-processors, Wittgenstein would jot aphoristic comments in his notebooks, type up the best of them, and then cut out and paste them into what he took to be the best order. It is very dangerous, then, to take writings from any of the stages of this process as having the same status. Would you be able to stand by everything you've written in notebooks, e-mails, or anything that a student has taken down from your lectures?
With this priviso in place, we come to the assessment of Wittgenstein's views. And they seemed to fall into two camps: a stricter reading, and a more generous reading. The stricter reading finds Wittgenstein realising he is in "a bit of a pickle" by the late 1930s. He has wanted to have us see mathematics through different lenses, so that the charm of certain parts of mathematics, e.g., transfinite set theory, dissipates. He wants to point to concrete demonstrations of arithmetic facts, such as showing 3 X 4 = 12 with pebbles, as paradigmatic examples of doing mathematics. The meaning of such propositions is precisely the proof act of manipulating and counting the pebbles. What then of universal arithmetic statements? Well, their meaning is constitued by their proof, typically a proof by mathematical induction. Don't get lured into believing in the completed infinity of natural numbers. Instead, make the conventional decision, if you wish to join this language game, of being able to assert f(n) for a natural number n when you have seen an inductive proof of for all n f (n).
But this idea of a statement taking its meaning from its proof meets with problems. It suggests that a proposition has no meaning before it has been proved, and that we can't say we have two different proofs of the same proposition. Can it really be that we don't understand Goldbach's conjecture, that any even number greater than 2 is the sum of two primes, since we don't yet have a proof? We might want to say that we don't fully understand this area of number theory, and that a proof, and further proofs, would augment this understanding. But it seems to me to be too generous a reading to say that this is what Wittgenstein was driving at.
Other generous moves are to suggest that the later Wittgenstein was pointing to new things to do in the philosophy of mathematics in an era dominated by logicism, formalism and intuitionism: to look at picture proofs, to look at applied mathematics, to wonder why pieces of mathematics are 'interesting' or 'surprising'. All good suggestions, I agree, but I'm not sure Wittgenstein has carried out much in the way of useful groundwork. The problem, it seems to me, for him and many others, is the tendency to restrict oneself to assessing mathematics purely at the level of propositions and proofs. I think it is necessary to think of individual statements as part of a greater system. Mathematics comes in larger sizes - projects, programmes, traditions.
My favourite programme - higher dimensional algebra (aka n-category theory) - is bubbling along nicely. Aside from being drawn to it aesthetically, my interest has been sustained over the years by the wonderful web-publishing of John Baez. He has been joined in this activity by another mathematical physicist Urs Schreiber who has largely made The String Coffee Table his own. What adds to the interest is that where Urs comes from a background in string theory, John favours its rival, loop quantum gravity, and yet they have worked together on developing a categorified gauge field theory. The last few entries and comments on the blog point you to the latest moves. If you're a beginner who wants to join the higher-dimensional algebra party, try papers 39, 49, 52, 53, 68 & 73 from here, or chapter 10 of my book.
One small contribution philosophers could make to mathematics would be to pay attention to the expository efforts of the exponents of research programmes to encourage this activity. A further obvious candidate for philosophical treat is Alain Connes' noncommutative geometry, another programme with a strong articulated sense of direction. This case has the additional intriguing feature that there are other rival takes on what noncommutative geometry should be. I'm a great believer that the study of rivalry and disputation can be very revealing.
With this priviso in place, we come to the assessment of Wittgenstein's views. And they seemed to fall into two camps: a stricter reading, and a more generous reading. The stricter reading finds Wittgenstein realising he is in "a bit of a pickle" by the late 1930s. He has wanted to have us see mathematics through different lenses, so that the charm of certain parts of mathematics, e.g., transfinite set theory, dissipates. He wants to point to concrete demonstrations of arithmetic facts, such as showing 3 X 4 = 12 with pebbles, as paradigmatic examples of doing mathematics. The meaning of such propositions is precisely the proof act of manipulating and counting the pebbles. What then of universal arithmetic statements? Well, their meaning is constitued by their proof, typically a proof by mathematical induction. Don't get lured into believing in the completed infinity of natural numbers. Instead, make the conventional decision, if you wish to join this language game, of being able to assert f(n) for a natural number n when you have seen an inductive proof of for all n f (n).
But this idea of a statement taking its meaning from its proof meets with problems. It suggests that a proposition has no meaning before it has been proved, and that we can't say we have two different proofs of the same proposition. Can it really be that we don't understand Goldbach's conjecture, that any even number greater than 2 is the sum of two primes, since we don't yet have a proof? We might want to say that we don't fully understand this area of number theory, and that a proof, and further proofs, would augment this understanding. But it seems to me to be too generous a reading to say that this is what Wittgenstein was driving at.
Other generous moves are to suggest that the later Wittgenstein was pointing to new things to do in the philosophy of mathematics in an era dominated by logicism, formalism and intuitionism: to look at picture proofs, to look at applied mathematics, to wonder why pieces of mathematics are 'interesting' or 'surprising'. All good suggestions, I agree, but I'm not sure Wittgenstein has carried out much in the way of useful groundwork. The problem, it seems to me, for him and many others, is the tendency to restrict oneself to assessing mathematics purely at the level of propositions and proofs. I think it is necessary to think of individual statements as part of a greater system. Mathematics comes in larger sizes - projects, programmes, traditions.
My favourite programme - higher dimensional algebra (aka n-category theory) - is bubbling along nicely. Aside from being drawn to it aesthetically, my interest has been sustained over the years by the wonderful web-publishing of John Baez. He has been joined in this activity by another mathematical physicist Urs Schreiber who has largely made The String Coffee Table his own. What adds to the interest is that where Urs comes from a background in string theory, John favours its rival, loop quantum gravity, and yet they have worked together on developing a categorified gauge field theory. The last few entries and comments on the blog point you to the latest moves. If you're a beginner who wants to join the higher-dimensional algebra party, try papers 39, 49, 52, 53, 68 & 73 from here, or chapter 10 of my book.
One small contribution philosophers could make to mathematics would be to pay attention to the expository efforts of the exponents of research programmes to encourage this activity. A further obvious candidate for philosophical treat is Alain Connes' noncommutative geometry, another programme with a strong articulated sense of direction. This case has the additional intriguing feature that there are other rival takes on what noncommutative geometry should be. I'm a great believer that the study of rivalry and disputation can be very revealing.
3 Comments:
Here are a couple of books for those interested in getting in a field.
The first pertains to Hilbert’s geometry (of his Foundations).
Hartshorne, R. (2000). Geometry: Euclid and Beyond. New York, Springer.
Hartshorne thoroughly investigates the relation between Euclidean and Hilbertian geometries, showing, in particular, how Hilbert improved on Euclid. Hilbert’s geometry is thoroughly presented and developed. Furthermore, reading the book does not require a great amount of mathematical background. The book is billed as a text for undergraduate mathematics majors.
Kendig, K. (1977). Elementary Algebraic Geometry. New York, Springer-Verlag.
Kendig’s leads the reader into this subject without overburdening the text with heavy-duty theory at the beginning. The first hundred pages, e.g., do not use ideal theory. Then it is introduced. Although its been over ten years since my last mathematics graduate class (I took only the basics of analysis, algebra, and topology plus a little more) I found it quite accessible when I started reading it recently.
I haven't read up on W's views on mathematics, but I have read a little on his later philosophy.
I was surprised with your summary description "meaning is proof", where the normal gloss over W's view on *linguistic* meaning is "meaning is use". I say "gloss" because it is my understanding that rather than advocating an examination of the use patterns of a word or proposition to arrive at the construction of its meaning as an abstract object, W is making the argument that there really is no well defined object (abstract or otherwise) corresponding to "the meaning of X", and that "meaning" is really just a shorthand way of refering to our knowledge of X's use patterns, so that to understand the meaning of X is simply to be able to use that word or proposition, and *nothing more*.
Looked at this way, the gnarly problems of denotation, sense, reference, etc., are ill-concieved, being based on the wrongheaded notion that somewhere meanings "exist" as abstract objects.
It is surprising, then, that W should equate the meaning of a mathematical proposition with its proof - since any mathematical proposition can more or less tediously be formulated verbally, and thus enter the linguistic domain, one would have expected the same gloss (and its underlying thought) to apply, and perform the same service to mathematical Platonism (in its many forms) as the purely linguistic formulation attempts for the various quasi-Platonic theories of linguistic meaning...
So I wonder if a view is possible which would regard understanding the possible uses of a mathematical proposition as what is meant by understanding its meaning?
In that case, knowing that "for all n, f(n)" gives one license to assert f(12), f(15), etc., as in your summary, but the problem of unproved statements is avoided: we can say that knowing the meaning of Goldbach's conjecture is really just understanding that (if true) it allows one to say that for any particular n greater than 2, n is the sum of two primes.
The collapse of meaning in the absence of proof, in the view you have described, seems eerily reminiscent of the difficulties some (quasi-Platonic) theories of linguistic meaning have with statements about unicorns, etc. - just the sort of theories that W was keen to undermine in his later work.
Pete
I agree the 'meaning is proof' idea looks odd in the context of W's later philosophy of language. One might try to make the case for a change between the middle W, when he was interested in constructive and verificationist themes, and the later W. However, Michael Potter (Cambridge), one of what I called the strict interpreters, made a good case for there being little change in his philosophy of mathematics between these periods.
There does seem a plausible W'ian account along the lines you suggest. Given W's interest in concrete applications, you'd think he'd have no problem saying that the Goldbach conjecture could be used, so partially constituting its meaning, to say that if you give you me an array of pebbles with 2 equal rows, I should be confident that there is a way for me to separate the pebbles into two groups in such a way that whichever group you chose you'd never be able to form a rectangular array, other than a line.
But then it's hard to see the need for his finitist scruples. Mathematics which refers to infinite collections finds use both inside and outside mathematics. The question that intrigues me is how constraints operate to limit the invention and elaboration of mathematical language games. Answers range across: the needs of science & mathematics itself; the wishes of the powerful; the way humans think; an objective scarcity of rich, formal games; and so on.
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