A proof only becomes a proof after the social act of "accepting it as a proof".

To put it simply, we first explain what we are talking about, and then explain why what we are saying is true (pace Bertrand Russell).

The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.

A good proof is one that makes us wiser.

Of the properties of mathematics, as a language, the most peculiar one is that by playing formal games with an input mathematical text, one can get an output text which seemingly carries new knowledge. The basic examples are furnished by scientific or technological calculations: general laws plus initial conditions produce predictions, often only after time-consuming and computer-aided work. One can say that the input contains an implicit knowledge which is thereby made explicit.

A proof only becomes a proof after the social act of "accepting it as a proof".

Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life.