Axioms can't be deduced because they're axiomatic. A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
Also, thinking again about subitizing, it is not only repeatable but also people can agree on the result. The qualia (what it feels like to subitize four as opposed to counting to four) is irrelevant to a large extent. It's just extremely interesting that we can subitize and it may even have linguistic implications for grammatical number, but just because your native language doesn't have counting numbers doesn't mean you can't learn them, and the possible connection between perception and grammar is not so surprising since both perception and grammar involve the same brain. A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
I'd be surprised if basic arithmetic - certainly addition, possibly subtraction, probably not anything more advanced - wasn't similarly rooted in experience.
But the basic point was that arithmetic is rooted in experience, and doesn't exist independently of it. Trying to prove it using formal logic makes for an interesting scenic trip, but eventually you end up standing over a hole which logic can't fill for you.
Axioms - to use the term strictly - establish the Rules of the Deductive Game you're playing, intellectually speaking. Nothing prevents anyone else from declaring other axioms and playing other games. This is exactly how non-Euclidean geometries is/are developed.
Deductive Truth arises from the manipulation of the Terms and Operations, within the Axioms, of A Deductive System - as guided by the particular Interpretation of that system. Even within the good old Categorical Logic there are two Interpretations: Aristotelean and Boolean.
Verification of the results - the Truthyness of the Truth Value ;-) of a particular Axiomatic System - is a whole 'nuther topic.
