Are axioms based on intuition?

Question

If axioms are assumed to be true (although I have read that nowadays they are used as logical premises and their self-evident nature is a 19th-century idea), then how do we know that they are true, other than through our intuition (i.e. subconscious judgements)? Axioms are assumptions and even though they may (I am not sure about this but I think someone verified some of Euclid's axioms) be verifiable, the people who first proposed them did not logically decide them but felt that they were obvious truths.

Any help would greatly be appreciated.

Answer 1

Axioms define your language. If you think of mathematics as a way to organize structure, or a language of reason, then your axioms define how this language works. You can in principle imagine all kinds of axioms -- but only the intuitive ones will yield an language that can be used in intuitive ways.

And I would claim that if you e.g. want to do physics with this mathematics, so have a connection to the real world, intuition will be necessary to some extent. Even more basic, things like the natural numbers and addition coinciding with our way of counting objects are not a coincidence, of course: The language is built this way by using the right practical, intuitive axioms.

Answer 2

The axioms of group theory are not assumed to be true. They should be taken as the definition of the term, "group". That makes them true, literally by definition, so long as you are doing group theory, but they may fail to be true when you find yourself working with semigroups (or metric spaces, or some other structure). They are not based on intuition, but on experience, as objects that satisfy them keep on coming up wherever Mathematics is done or applied.

The axioms of Euclidean geometry were assumed to be true. It was assumed, based on intuition, that they applied to the physical world, and that things could be no other way. Nowadays, we know things can be another way, both in abstract Mathematics and in (mathematical models of) physical space, and the situation is more like that in group theory: the axioms of Euclidean geometry are taken to define Euclidean geometry. Alternatively, the axioms of Euclidean geometry can be proved to hold by bringing in Cartesian co-ordinates and arguing via analytic geometry (but then you're bringing in a whole other troop of axioms...).

The axioms of set theory, I think, come closest to the model you propose. They can be taken to define the subject matter of set theory, but the reason the current axioms are chosen, and not some others, is (I think) that the current axioms seem to capture the intuitions that mathematicians have built up about sets. Their history has been a mix of logical deductions and "obvious truths". Occasionally, logical deduction has shown that some "obvious truth" actually leads to a contradiction (I'm thinking in particular of Russell's demolition of Frege's conception of sets), and that has led us to fall back on somewhat less obvious truths. Like I said in the comments, it's complicated.

Answer 3

I think of it this way: We always show structures in maths like $A\rightarrow B$ or that A is equivalent to B. Think about the different ways to define compactness. Or the different ways to define the axiom of choice. You either require there to be a set which only contains one element from every partition. Or you want there to be a choice function. Then you show that those axioms are equivalent.

Similarly you use those axioms to build all/a lot of the know theorems. But what you show in reality is: If those axioms hold, then...

Now Theorems already have premises. The theorem shows that $A\rightarrow B$ and $A$ are your premises. Now you could just add all the axioms to your premises and then you don't need to think of them as universal truths. Instead of writing:

"If you only consider the real numbers, then all norms are equivalent",

you write: "If the axioms of Set theory are true and you only consider the real numbers, then all norms are equivalent"

Mathematics is more the structure between statements. But for simplifying things, we just assume that these axioms are true. Because it is really a pain to write this every time you write a theorem. But that is similar to having the convention, that from now on all the vector spaces in this lecture will be finite.