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Before my recent editing, the second-to-last paragraph said:

mathematics is not immediately related to real things. Mathematics is a game where you set up arbitrary rules on an arbitrary playing board and combine them in a 'mindless' and playful way (any computer can do this, at least in principle).

As a mathematician who takes things seriously, I say: what a load of crap! It is true that the canons of symbolic logic treat mathematics as if that's what it is, and it is also true that philosophers who like to pretend mathematics is purely deductive like to describe mathematics that way. Here is a counterexample. Observe how the concept of group originated in the 19th century. It was observed that various concrete instances that would later come to be called "permutation groups" and "Lie groups" and the "additive group of integers", etc., shared certain things in common; they all had a binary operation obeying certain rules. It was found to be fruitful, in that it helped understand these various different groups, to isolate what they had in common into a new concept called a group. That process was induction, not deduction (not in the sense of "mathematical induction", but in the sense of reasoning from the particular to the general). It was not arbitrary; it was done for very good reasons. But those reasons are of a kind that the canons of deductive logic do not treat, so they look "arbitrary" if viewed only from the point of view of those canons. Anyone who thinks about mathematics from the point of view only of the canons of deductive logic would obviously be unable to do research in the field. Michael Hardy 00:10 May 2, 2003 (UTC)

I agree, The person who wrote this was obviously not a mathematician, I considered rewriting this but I'm not a mathematician myself -- just an undergradute student. Also check out what I did on Axiomatic system, it's (roughly) based on an undergraduate course, translated from hebrew.. (with terms googled from the net) Rotem Dan 00:28 May 2, 2003 (UTC)
I removed these philosophical comments. This article still needs a lot of work to become an accurate description of the process of axiomatization in mathematics. Historical timeline? axiomatization of set theory? related fields? Rotem Dan 13:04 May 3, 2003 (UTC)
  • Note: on Peano_axioms, it says that 0 is a natural number. Here, it says no. (That page references this one.)