Mathematical Proofs

The aim I am pursuing here is to describe some general aspects of mathematical proofs. In my view, a mathematical proof is a warrant to assert a non-tautological statement which claims that certain objects (possibly a certain object) enjoy a certain property. Because it is proved, such a statement is a mathematical theorem. In my view, in order to understand the nature of a mathematical proof it is necessary to understand the nature of mathematical objects. If we understand them as external entities whose “existence” is independent of us and if we think that their enjoying certain properties is a fact, then we should argue that a theorem is a statement that claims that this fact occurs. If we also maintain that a mathematical proof is internal to a mathematical theory, then it becomes very difficult indeed to explain how a proof can be a warrant for such a statement. This is the essential content of a dilemma set forth by P. Benacerraf (cf. Benacerraf 1973). Such a dilemma, however, is dissolved if we understand mathematical objects as internal constructions of mathematical theories and think that they enjoy certain properties just because a mathematical theorem claims that they enjoy them. This is just my view. By speaking of general aspects of mathematical proofs, I do not mean their logical nature, as it could be expressed in one of the artificial languages actually used by modern formal logic, like the usual languages of predicative logic, or the lambda-calculus. I am not interested in suggesting a way of translating the arguments that mathematicians employ or have employed to prove their theorems in a suitable formal language, where these arguments are reduced to chains of formulas connected with each other by syntactical rules of inference, and to describe these chains in general. I am neither interested in suggesting a way of treating these arguments as the objects of a formal theory of proof, nor in considering the possibility of reducing them to executable programmes. Rather, I am interested in studying the internal structure of these arguments taken as such. The difference between my aim and the aim of the formal theories of proof should be clear enough, and I shall not insist on it. By contrast, I would like to say something more about the difference between my aim