Stephen Montgomery-Smith wrote: > I read what you wrote, and I admit I am not quite getting it. So let me > describe my position and hope that it is a worthy response to what you > wrote. I'm not quite sure whether I am disagreeing with you or not.
You asserted that "the responses have led [you] to think that some people here believe in the modern axiomatic system like a religion". I replied that "believing in the axiomatic system" does not make sense, because axiomatic systems are just a way to describe a set of propositions, and you said "[you] disagree". I assumed you meant that.
In any case, there is no obligation to agree or disagree (despite what one might think after reading certain newsgroups ;-) ) One can simply be interested in trying to understand what others think.
> There is a way in which people try to persuade other people of their > belief system. I think that our current scheme (axioms, tautologies, > modus ponens etc) we use in math is a very good description of what we > feel are really effective ways of persuading other people. (Of course > we know that using pure reason doesn't work with most people, but we > have this sense that there is a Platonic reality as to how an ideal > argument should work.) However, I believe that our ultimate sense of > what we think is true and not true comes down to simply "what do I > believe." And the methods of logical deduction are no more than one > other set of things that I simply "believe."
IME, it is not only that axiom systems and formal theories are good at persuading other people. And it is not only that they are good in modeling our intuitions (what we believe). They are actually good at modeling reality. Say, ZFC and its deduction means can be used to model the physics of elementary particles so well that at the moment we cannot carry on experiments fine enough to confirm the predicted mass of the electron with the precision we are able to compute.
I wouldn't say I believe in logical deduction. I'd be symtomatically more verbose: I believe in there is a "reality", external from myself and from by beliefs. I do not know if we can know that reality (I do not even know what that means, in fact). But I know that we can attempt
to build models of it. And so far, the models we have built, using modus ponens, and logical deduction in general, have been quite good.
> On a completely different tack, while methods of logical deduction seem > to be a good description of how to do mathematics, I don't think it is > how mathematicians really think. The example I would cite is Euler's > (rather brilliant) discovery that sum 1/n^2 = p^2/6 (which basically > involves pretending that the Taylor series of sin(x)/x is a polynomial). > His original proof certainly was not rigorous. Rather it was a flash > of insight, and certainly thinking out of the box.
No one who knows how a mathematician works, and least of all a working mathematician, would ever say that mathematicians work by logic deduction. Insights do not come in the form of chains of applications of modus ponens.
If you think about it, the fact that such a limited tool as the axiomatic system, and formal theories in general, with not much more than modus ponens, provides a good model for genial intuitions such as Euler's, is one of the things that validates it.