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Are all scientific proofs ultimately based on a set of axioms?

First of all, science generates theories, not proofs. Theories are explanations; proofs are not.

A theory is a way of modeling observational data. To model a body of observational data is to posit an entity that is not itself observed but whose existence would generate the data in question. For one theory to be better than another is for it do a better job of eliminating causal anomalies and of diminishing the scope of that are not altogether eliminated.

This last property is one that can be had to varying degrees, and theories are therefore to be evaluated on a sliding-scale.

This does not hold of proofs. A proof either goes through or it doesn’t. This is because proofs identify fixed relations among statements and therefore aren’t answerable to observation; and because they aren’t answerable to observation, they cannot possibly account for the information borne by observations.

Sometimes proofs represent discoveries, but they are never explanations. Theories, by contrast, are of their very nature explanations.

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