Stephen Montgomery-Smith wrote: > email@example.com wrote: > > Stephen Montgomery-Smith wrote: > > > >>[snip] > >>The responses have led me to think that some people here believe in the > >>modern axiomatic system like a religion. > > > > > > An axiomatic system is just a method of singling out a set of > > propositions (those that follow from the axioms). For example, > > the Peano axioms for natural numbers select a very definite > > subset of all the posible propositions (those that follow from > > the axioms). > > > > Axiomatic systems are thus just tools. In particular, the idea > > of "believing in the modern axiomatic system" makes as > > much sense as that of "believing in hammers". > > I disagree.
I honestly do not see what are you disagreeing with.
I cannot imagine you are disagreeing with my statement that axiomatic systems are descriptions of sets of propositions; this statement is completely equivalent to the statement that Chomski grammars are descriptions of sets of words.
I could understand if your point was that axiomatic systems make lousy descriptions of sets of propositions and/or that what the sets of propositions described by the "usually acepted" axiomatic systems is not the set one would be most interested in describing (be it because they include propositions which "should" be false, be it because they do not include propositions which "should" be true, and so on)
To the first point I would probably respond that in the whole history of mankind the number of effective alternatives to the axiomatic method that we have devised is zero.
The second point is not a problem with the axiomatic method, but with the fact that the usually used axiom systems may do not model what you (or I) believe that should be true or false.
Yet most people end up agreeing that Banach-Tarski's paradox is a fair price to pay for the existence of maximal ideals in arbitrary commutative rings, say. And no one lost any sleep when Shelah proved that the answer to Whitehead's conjecture depends on the set theory chosen: my guess for this is that essentially no one who has thought deeply about the issue has any real intuition as to what the answer should be, so it is not a problem if changing the set theory used changes the answer.
And that puts to the fore what it is that people believe in: their idea of what things should be, the set of propositions they believe/feel/intuit should hold. Now that set of propositons is very hard to communicate and it will certainly not work to base mathematics on "what Mr. X believes sets are" because eventually Mr. X will die and the decision procedure which consists in "asking Mr. X on the truth of propositions relative to sets" will stop working.
Axiom systems are a tool used when trying to describe the set of propositions people believe should hold. What people believe in precedes the axiom system. If some one came up with an alternative way of describing the kind of sets of propositions that people have in mind when they think about the integers, say, then people will try to describe the set of propositions about the integers that they believe should hold using that new method.
In short, what I am saying is: what people believe in precedes axiom systems (which are just an imperfect way of conveying it)