Jon Lawhead originally shared this post:
I've had a paper about the foundations of mathematics in my head for a while now. I had a long conversation with a good friend who does quantum field theory today, and found him unexpectedly sympathetic to the view. Now I'm thinking seriously about writing it. This is partly scratch paper for recording my thoughts, and partially an RFC. If anyone out there has any thoughts about this, please chime in.
I've always been sympathetic to a kind of formalism, and I think that the big objections that get raised to the formalist program aren't necessarily fatal. The spirit of the program can (I think) be decoupled from Hilbert's personal project of providing a complete and consistent foundation for arithmetic (which Godel torpedoed), and from the formulation that requires all of mathematics to computerized. The spirit of formalism just requires that mathematics be thought of as kind of symbol manipulation game in which we play around with constructed formal systems, deducing as many consequences as we can from a set of axioms. I think it's possible to give a concrete version of formalism that satisfies this spirit, but which doesn't run afoul of either the Turing or Godel-based objections.
In particular, I'd like to target the "unreasonable effectiveness of mathematics" argument (which sort of parallels the "no miracles" argument in the philosophy of science. The quick and dirty version of that argument is that if mathematics isn't "discovering" genuine truths about real objects in the world, it seems incredibly miraculous that so much of contemporary mathematics has turned out to be so useful for doing science. It seems to me that this is something like being astonished by the fact that so many words in the English language actually represent objects in the real world; if you start from the assumption that mathematics evolved in tandem with science–indeed, as a tool for doing science–then this shouldn't be surprising at all. Math is a good tool for describing the world because we made it to be that way.
My friend pointed out that much of mathematics is directly pursued not by people looking to advance the cause of science, but by mathematicians who are simply "playing around;" indeed, many "pure" mathematicians seem to look down on applied mathematicians. This is certainly true enough (though more true today than it was historically, I think), but isn't necessarily a problem. The program as a whole is shaped by a whole bunch of different concerns, so the objectives of individual mathematicians aren't really the important thing to attend to. The fact that mathematicians as people aren't always motivated by a desire to advance science doesn't torpedo the mathematical project as a branch of science any more than the fact that individual scientists might be motivated by a desire for fame (rather than a desire for truth) as a truth-seeking enterprise. Moreover, it makes sense that if mathematics is about exploring systems we've constructed, then it should be at least partially motivated by aesthetic considerations. Our notions of what counts as beautiful are shaped by quirks of our brains–quirks that are a result of biological, environmental, and chemical accidents in our development. If mathematics isn't something that's created by us, it would be very strange for it to correspond so frequently with our notions of beauty.
Finally, it seems to me that much of the "unreasonable" effectiveness of mathematics appears as a result of selective attention to cases of mathematical objects that turn out to be useful. Even setting aside cases where new mathematical concepts have been straightforwardly developed to move science forward (Newton's development of calculus, say), there are still a tremendous number of cases in which whole swaths of math don't have much relevance to the natural world, but are just other developments of the same formal systems that do contain useful structures. Number theory, for example, is not terribly important to any branch of the physical sciences. The illusion of unreasonable effectiveness comes from attending only to the cases where mathematicians seem prescient, while ignoring the cases where their developments have't found concrete application; it's the illusion of "synchronicity" for smart people. If you attend only to cases where (say) your dream of a friend from your past is followed by an unexpected phone call from the friend, then you might well think your dreams are portentous; the illusion is dispelled if you consider the number of dreams you have that aren't followed by contact with the characters in the dream.
The mix of applicable and inapplicable (but consistent) mathematical structures reflects the fact that the mathematical project is a hodgepodge of direct attempts to advance science, mathematicians pursuing individual aesthetically-motivated projects, and mathematicians just generally playing around within very robust formal systems. Given the number of tools mathematicians have produced over the years, it hardly seems miraculous that many (but by no means all) of those tools can be applied to describe the world. Science is concerned with the structure of the world, and mathematics is a study of many different formal systems that contain very rich structures.
Anyway, let me know if you have any thoughts.
Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one. This is to say, that if you interpret the strings in s…