Formal Axiomatic Systems

Every mathematical proof can be formalized; otherwise, it is not a proof. The mathematical assumptions on which the proof stands and the logical rules by which each step of the proof follows from previous steps can be made explicit and precise. Mathematicians don’t usually do this; instead, they take shortcuts—quoting previous theorems, using English, skipping derivation steps, and often making the unfortunate use of the dreaded “clearly”. Nevertheless, every valid proof can be expanded and turned into a fully formal one.