There are two differences. First, group theory was already a well developed and important subject before the axioms were written down. Lagrange, Galois, Jordan, Cauchy had all worked on group theory. The axioms simply allow us to talk about a “group” in the abstract, not just a transformation group (say).

The other thing is that a robust concept like a group or a matroid can be axiomatised in many different ways. I wrote several posts on this a year or two back. An independence algebra of rank 1 is the same thing as a group acting on a set, although the axioms look completely different. Similarly, a matroid is a structure whose independent sets satisfy the exchange property, or a structure in which the greedy algorithm always works, or the dual of an IBIS family of sets (a definition modelled on computational group theory). I might agree if you told me that the concepts are created by God, but I think the axioms are our attempts to create something reflecting these concepts as accurately as possible.

]]>There exists a number 1 such that 1 x a = a for all a is an element of the Complex numbers

There exists a number 0 such that 0 x a = 0 for all a is an element of the Complex numbers

1 != 0

– based on faith alone? If we firmly believe that these things are true then we get answers that comfort us, but if we do not then all numbers are equal. So it works very well but there is no way of actually proving these statements – they are axioms and so can only be used to prove other statements.

And an axiom is a basic tenet of faith, no?

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