In purely formal mathematics - meaning using inference rules to derive new theorems from axioms (like in Hilbert calculus) - there is explicitly told how to deduce new theorems. However in most of mathematics, we use not very obvious implicit "rules" for deducing new theorems (meaning like in informal proofs). Would you be able to somehow explicitly tell what "rules" we use in those types of deductions (I guess it is some kind of semantic meaning of the words by which we reason)? If there are no explicit rules for deducing new theorems in this type of deduction, how do we know that the theorem really follows from those axioms?
2026-03-26 09:16:27.1774516587
What are those implicit "rules" by which we deduce theorems from axioms in mathematics?
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