Math is an odd discipline. Up through nearly the end of the 19th century people didn't really pay much concern to what math was. They just did it. Then people started worrying about mathematical foundations and we had things arise like constructivism, logicism, and so forth. (Basically the idea that math is being able to manipulate symbols to construct a proof, or complicated logical operations -- that's the overly simple answer because they're not what I'm here to talk about) Well as philosophers and occasionally mathematicians focused in on mathematics they started finding all sorts of odd things. One problem was infinities and how to deal with them. In general, depending upon what school of foundations you come from, you'll reject certain kinds of proofs involving infinities. Of course there always were the mathematical platonists. They thought that mathematics was something "real" and independent of any human thinking about it. Actually the platonists were fairly common in math departments. But philosophers don't generally like platonists, so they don't get a lot of respect.
One of the first big "counter-attacks" in mathematics against all these more linguistic or logical views of mathematics was from Kurt Gödel. He had his famous incompleteness theorem. The aim of the theorem, as I understand it, was to argue for mathematical platonism. The basic conception is that there is some mathematical theorem we can understand but can't prove true or false. Once again I'll not get into the details, beyond saying that Gödel's theorem is probably the most abused theorem in mathematics. It's right up there with the abuses of physics notions like the second law of thermodynamics and the uncertainty principle. It keeps getting applied in places where it just doesn't belong and by people who clearly don't understand it. And sometimes by people who do. (cough Penrose cough)
Well an other interesting approach to mathematics which doesn't get the discussion time that the constructivists or logicists or even platonists get is quasi-empirical methods. Now coming from a physics background, I should add that quasi-empirical methods seem natural to me. I suspect most physicists latch onto them. If you've been in a math class where part of the class are actual mathematicians and then part of the class are physicists, you can quickly tell them apart - especially in terms of how they do their proofs. It's hard to explain. But basically both mathematicians and physicists skip steps and leave out explanations for things that are self-evident. Mathematicians like to pretend they don't do this. But they do. They just do it in different places from physicists. Which leads to no end of frustration. Not to mention the fact that the mathematicians love the proofs for their own sake while the physicists want to understand the idea of what is going on. As often as not this means physicists will introduce unproven (at least to the mathematician's satisfaction) conjectures that seem right and that bring about an understanding.
Hold with me, we're nearly getting to my point.
Now mathematicians have conjectures that are rather famous. One recent one that has been in the news was Poincare's Conjecture. (In the news because it was recently solved by a Russian mathematician, winning him a cool million dollars but more importantly opening up a lot of important implications in topology and probably string theory as well) The problem is that mathematicians don't like to use conjectures in proofs until they are proven, even though every thing points to them being "true." (I'll put that in quotes since it often isn't clear what true means relative to mathematics)
This takes us to the quasi-empirical methods. Now a physicist would, likely as not, take a solid conjecture and treat it "as if" it were true. He'd worry about the empirical implications. But he'd tend to judge the conjecture in terms of how successful it is in various other theorems. So long as one doesn't encounter contradictions down the line, where's the problem? If one is wrong, so what? One simply redoes the work where one was wrong. That's the spirit of science. But it is, admittedly, a spirit that mathematicians (and even some mathematically minded theoretical physicists) can't really feel good about.
Which takes us to Hilary Putnam. Now Putnam wrote a very provocative paper called "What Is Mathematical Truth?" (Sadly not available online) Putnam basically makes a call for these sorts of quasi-empirical conjectures in mathematics. He even appeals to Gödel to justify it. After all there will be statements we can't ever prove, but which might be true. Why not take them as true? Further he argues that quasi-empirical methods have always been part of mathematics. Consider the series expansion of the Sum( 1/n^2 )= Pi^2/6. (Sorry too lazy for graphics) One can prove this by the traditional means you learn in freshman mathematics. However it is also possible to know it by induction on observations. We calculate the terms for a ways and see where it appears to be converging. Now to a physicist that is natural and common. To a mathematician, not so much so.
Interestingly the second half of the paper is Putnam making a case for realism. (Thus, presumably the appeal to Gödel) Now it isn't realism of the sort that one traditionally found in mathematics. It is more akin to the realism one thinks of in science applied to mathematics. However what is common to both kinds of mathematical realism is the idea that sentences about mathematics are true or false independent of social processes, symbol manipulation or so forth. Basically the argument of the quasi-empiricists is that if there is a mathematical realism why not act like it. Approach it the way we do in say physics
What brings all this to mind? Well over at Prosthesis they have a similar post. This one, instead of being about Putnam though is about Chaitin. Now Chaitin is a mathematician who also argues for quasi-empirical methods. Chaitin is one of the big names in information theory and has a famous proof related to randomness. He proved there is a number (Chaitin's constant) which can be given a mathematical definition but which is random. But of course if it is random, that makes it very difficult for traditional mathematical methods to make use of it. That's because even in simple mathematics like arithmetic, there is randomness. (Arithmetic is where a lot of these proofs arise - it was the case with Gödel as well)
Why does Chaitin's proof pose problems for traditional mathematical methods? Because it entails that some things might be true for "no good reason." Yes that's vague. But basically if something is random, you can't derive it from basic axioms. So why not just say somethings are true but without foundations?
By this point you'll start to see the parallels with anti-foundationalism in epistemology. Which isn't surprising. After all Putnam is a pragmatist and has praised Peirce's anti-foundationalism. Can we have a mathematics without foundations? I think so. Whether the mathematicians will come around to the physics way of doing things is an other matter.
I discussed a related issue to Putnam's paper here.
Interesting post, but I think you made a bit of a misstatement about mathematical practice. From what I understand, mathematicians in general actually do try to prove things using unproven conjectures. Part of the reason most of the Clay prize problems (like the Poincare conjecture) are such big problems is that a lot of work has already gone into deriving consequences of these conjectures. Sometimes this is an attempt to prove the conjecture false by finding a contradictory consequence, and sometimes it's an attempt to help to prove the conjecture, by finding out deep connections between it and other areas of mathematics. But I think a lot of the time it really is just an attempt to continue mathematics in its normal sense - after all, just about everyone believes that the Riemann Hypothesis is true and P=NP is false, and so proving that something or other follows from one of these facts is almost as good as proving it from already established principles.
Hmm. Interesting. Thank you for that information. I thought it was only physicists who were using various theorems who did that. For instance the Poincare Conjecture is used in various ways in superstring theory to deal with some topological issues.
"(...) mathematicians love the proofs for their own sake while the physicists want to understand the idea of what is going on", you wrote. Surely, mathematicans are always looking for sweet, new and elegant proofs, but they don't do it just for esthetics. A main reason is also that by proving the theorem they can find out much about what's really going on - the proof is meant to be an answer to the question "Why is the theorem true?". [An example of a proof not answering this question very well is the Appel/Haken-proof of the four-colour theorem, but then mathematicans don't find this proof very nice either, but just acceptable.]
That line was meant a little tongue in cheek. But I do think the physicist tends to perhaps view "why" questions in math differently from"why" questions in physics. Not all do, of course. There obviously are some very mathematically inclined physicists just as there are some very philosophically inclined physicists.
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