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I called it an axiom for the sake of brevity. I felt that reciting the Peano axioms would only cloud my point.

Was I 100% precise? No. Did you add something meaningful to the discussion? That's for you to answer.

"axiom" is two letters shorter than "theorem", so no, brevity doesn't cut it as an excuse. The difference between an axiom and a theorem is, undoubtedly, germane to a thread about the different ways we can know things.

I would go further: is "Axioms exist" true? You don't find them lying around in nature. They exist only in people's minds. I would argue that this is an inherent metaphysical question.

The scientismist's (?) standpoint could be that it exists because it is manifested as a concept in human brains. But by the same argument, wouldn't they then need to acknowledge that God exists?

The crux of the matter is to do science, you need formalism to express your theories, to interpret measurement results etc. Those formalisms are what we usually call "math". But any mathematical system has axioms, i.e. something we need to just "believe" in in the first place, and those believes are not themselves provable by science.

Axioms can be shown to be useful or not. This is why some mathematicians reason with or without axiom of choice. Likewise, systems of axioms can be shown to be consistent or not.

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