That doesn't make for a perfect field, or even a good field, for machine learning to thrive in; what we care about is finding useful results. Starting with arbitrary axioms is a good way to prevent that from happening.
Compare this discussion from an algebra textbook I've been reading recently:
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The possibility of combining two elements of A(S) to get yet another element of A(S) endows A(S) with an algebraic structure. We recall how this was done: If f, g ∈ A(S), then we combine them to form the mapping fg []. We called fg the product of f and g, and verified that fg ∈ A(S), and that this product obeyed certain rules.
From the myriad of possibilities we somehow selected four particular rules that govern the behavior of A(S) relative to this product.
[...]
To justify or motivate why these four specific attributes of A(S) were singled out, in contradistinction to some other set of properties, is not easy to do. In fact, in the history of the subject it took quite some time to recognize that these four properties played the key role. We have the advantage of historical hindsight, and with this hindsight we choose them not only to study A(S), but also as the chief guidelines for abstracting to a much wider context.
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It takes work, a lot of work, to determine what axioms you should use. Where do you think the information necessary to make that determination comes from?
