Checking proofs

Argumentative practice in mathematics evidently takes a number of shapes. An important part of understanding mathematical argumentation, putting aside its special subject matters (numbers, shapes, spaces, sets, functions, etc.), is that mathematical argument often tends toward formality, and it often has superlative epistemic goals: often the aim of a piece of mathematical argumentation is to prove that such-andsuch a student is logically true or logically valid consequence of some assumptions; a proved student thus seems to be indubitable, certain, or irrefutable. These aims generally do not depend on the subject matter of what is being argued about; whether one discusses functions, numbers, spaces, shapes, sets, arrangements, flows, figures, or fields, mathematical argumentation, in its final, published, form (and even in ordinary mathematical conversation) tends to be formal and self-consciously explicit about its own argumentative structure. The problem, then, is to better understand the notion of mathematical proof. We are interested in this paper in the phenomenon of mathematical proof considered as a species of argumentative practice in mathematics. For us in this paper the central feature of mathematical argumentation—specifically, mathematical arguments that are put forward with the intention of showing that a certain proposition is a true or valid–is its in-principle formalizability. By the in-principle formalizability of an argument we understand that there exists a formal derivation in some conventionally accepted formalism suited for mathematical reasoning of the proposition that commences from some conventional set of foundational axioms in a gap-free way all the way to the (formalized version of the) proposition.