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Axioms don't exist independently from our feelings and processes, we pick axioms we feel are good, and axioms defines mathematics.

Mathematicians even argue which axioms we should have, it isn't objective in the slightest, mathematics is therefore very closely linked to our feelings and intuition. Remove that and you just have formal logic, a very different field.

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Is formal (aka "mathematical") logic part of mathematics? Now that's a philosophical question.

From my perspective, I feel you restated what I said with the opposite conclusion. You say that axioms "defines" mathematics. If I were Claude, I'd say that the word "define" is doing a lot of work, is load bearing or something like that.

"Define" is where we turn these axioms into consequences - what I call "truth". As opposed to all the other stuff people could say that don't follow from these axioms. These are nonsense and, most certainly, un-mathematical.

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