ABSTRACT
We do not learn to demonstrate from the manuals of logic, but from the books which are full of demonstrations, which
are the mathematical and not the logical. Galileo
SUMMARY
This essay is primarily concerned to make the following points:
1 that formal-logical structure is not essential to mathematical proof and, at times, can even serve to conceal the important role that understanding plays in it,
2 that a well-described possible proof can have the same epistemic benefits as an actual proof (thus showing that a clear line is to be drawn between mathematical and empirical warrant and providing a promising direction to pursue in trying to explain the a priori character of mathematical knowledge),
and
3 that a proof or chain of proofs cannot leave anything unproved, contrary to the common idea that proofs must begin with assumptions that are not themselves proven. These facts point to difficulties in regarding proofs either as themselves being formal-logical derivations or as being satisfactorily represented by them. The formal-logical model idealizes away aspects of proof that are vital to mathematical thought, particularly obscuring the complex role that understanding plays in it.
