Are axioms chosen with the goal of "making things work" instead of some deep philosophies?

Question

Are axioms chosen with the goal of "making things work" instead of some deep philosophies?

If everything should be deducible, that is, provable from something else, then in this chain of deduction eventually you have to stop somewhere, because we can't work with infinity. So you have to accept some stuff as axioms, and you won't be able to find things from which you can deduce those axioms. You just accept them.

So the question then is, how do you choose where to stop? How do you choose the axioms? Is it done for the sake of making all known math work, or we have to do it based on some deep philosophies? But what if the philosophy is good, but still does not enable you to construct axioms from which you can build all the rest of math? Then it seems, you have to pick axioms, with the explicit goal of "making all the rest of math work".

Please give me some comments on this

Answer 1

How do you choose the axioms?

I think that "things go" from new ideas and results and "problems", to new axioms which "systemathize" and organize and unify and ... those ideas and concepts.

Set theory :

• first Cantor's "significative" results in early set theory : around 1870

• Zermelo's axiomatisation : 1908.

Geometry :

• Thales, credited with the first use of deductive reasoning applied to geometry : c. 624 – c. 546 BC

• Euclid's Elements : c. 300 BC.

Number theory :

• Pythagoras : 570 BC – c. 495 BC

• Dedekind-Peano axiomatization : Richard Dedekind - 1888 and Peano - 1889.

You can see these "ingredients" in play into Ernst Zermelo, Investigations in the foundations of set theory I (1908) [quotations from Jean van Heijenoort (editor), From Frege to Gödel : A Source Book in Mathematical Logic 1879-1931 (1967), page 200-on] :

Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions "number", "order", and "function", taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis; thus it constitutes an indispensable component of the science of mathematics. At present, however, the very existence of this discipline seems to be threatened by certain contradictions, or "antinomies", that can be derived from its principles - principles necessarily governing our thinking, it seems - and to which no entirely satisfactory solution has yet been found.

[...] starting from set theory as it is historically given [i.e.as originallyt formulated by Cantor], to seek out the principles required for establishing the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to exclude all contradictions and, on the other, take them sufficiently wide to retain all that is valuable in this theory [e.g.Cantor's theorem and the theorems on infinite sets].

In this article, we can see also the so-called regressive strategy in place, i.e.to regard an axiom as true if it is useful to generate as theorems those results we already believe on other grounds to be true. This is the approach followed by Zermelo in his new proofs of the well-ordering theorem, which needs the "tool" of the axiom of choice.

Answer 2

There are lots of stuff you just "have to accept". For example, you have to accept that Earth is a large ball of dirt and rock that we live on. There is no way to "prove" this from prior beliefs: there is simply the fact that we would not use the word "Earth" to refer to an object that is not a large ball of dirt and rock that we live on.

Many (arguably all) axiom systems can be viewed in this light. You have a mathematical structure that doesn't satisfy Hilbert's axioms for Euclidean plane geometry? Then we won't call it a Euclidean plane. You have another structure that does satisfy those axioms? Then we will call that one a Euclidean plane.