Axiomatic Foundations

I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if mathematicians choose the axioms then we are inventing our own maths, surely that is what it is but as it has been practiced and built on for so long then it is too late to change all this so we have had to adapt and create new rules?? I'm not sure how clear what I am asking is or whether it is even understandable but would appreciate any answers or comments, thanks.

Mathematics does not exist in a vacuum. It is strongly related, via applications, to the world around us. Mathematicians choose axioms according to what works well when we try to use the insights and results flowing from these axioms to better understand problems (usually from science) that we care about.

To draw an analogy with painting, a painter can surely mix colours in endless combinations and spread paint on canvas in equally endless possibilities. But, artists don't just randomly spread paint on canvas. The reason is that their art does not exist in a vacuum. It is strongly related to human culture, the physical world around us, and the predispositions of the human brain. These dictate what is considered good art, and so guide the artist in the creation of a good painting.

There are (at least) four types of sources for axiomatic systems. Here are the scenarios that I have in mind:

(1) Some mathematical structure, like the plane in geometry or the system of natural numbers, has been recognized as useful for applications and has therefore been studied extensively. So many facts are known about it. In this situation, one might want to organize those facts in a logical system, showing which facts are consequences of which other facts. Of course, to avoid circularity, some facts have to be taken as basic, and then other facts are shown to be consequences of these. The basic facts are called axioms or postulates, and it is desirable to make them as simple and as few as possible, so that one is not assuming things that could rather be proved. Among the axiom systems that arose in this way are Euclid's axioms for geometry (and, in a more rigorous age, Hilbert's axioms for geometry) and Peano's axioms for arithmetic.

(2) Questions have arisen about the legitimacy of some arguments, so it becomes necessary to say exactly what the assumptions are that underlie those arguments. The clearest example of this is Zermelo's (1908) axiomatization of set theory. The immediate problem facing Zermelo was the axiom of choice. It had been used as if obvious, for example in the proof that the union of countably many countable sets is countable. But, when Zermelo pointed it out as an explicit statement and used it in his proof (1904) that all sets can be well-ordered, he got a lot of flak. There were also other points in need of clarification, such as Cantor's distinction between consistent multiplicities (sets) and inconsistent ones. So Zermelo set up a system of axioms on which to base not only the proof of his well-ordering theorem but also the other set-theoretic arguments of the time. (Nowadays, we can view Zermelo's axioms, as well as later extensions by Fraenkel and others, as falling under scenario (1) above, as systematizations of the known facts about the cumulative hierarchy of sets. But, as far as I know, the cumulative hierarchy is not mentioned in Zermelo's writings until 1930. So I regard their introduction in 1908 as a different scenario.)

(3) People notice that very similar ideas and proofs are occurring in different areas. The elementary arithmetic of addition of integers, or real numbers, or complex numbers is very similar to the behavior of the operation of composition of permutations of finite sets or of rotations of space. In this situation, it is worthwhile to isolate the basic features common to these different contexts and deduce other common features from the basic ones (axioms) once and for all, rather than treating each context individually. Thus, the examples I just mentioned are all subsumed by the axioms for groups. Notice that here the axioms are intended to apply to many different structures (numbers, permutations, etc.) whereas in (1) (and perhaps also (2)), the axioms are intended to describe one specific structure. In (1), the existence of different models of the axioms is an unintended feature or bug; in (3) it is the main reason for formulating the axioms.

(4) Just plain curiosity. For example, given Euclid's axioms for plane geometry, let's see what happens if we replace the parallel postulate by some contrary assumption. Nowadays, such non-Euclidean geometries are seen as descriptions of interesting structures (like the hyperbolic plane), but when such axioms were first considered, no such structures were known, and in fact these "strange" axioms were expected to be contradictory. In principle, anybody can make up and study whatever axioms (s)he wants. Whether anyone else will pay attention, though, is a more difficult question. Axiomatic systems that don't fit under (1), (2), or (3) above had better come with some serious motivation, or the person who introduces and uses them is likely to be ignored.