firstname.lastname@example.org wrote: > 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) >
We reflect on the axioms and decide whether we believe in them or not.