Abstract:
For more than 2000 years, from Pythagoras and Euclid to
Hilbert and Bourbaki, mathematical proofs were essentially based on
axiomatic-deductive reasoning. In the last decades, the increasing length
and complexity of many mathematical proofs led to the expansion of
some empirical, experimental, psychological and social aspects, yesterday
only marginal, but now changing radically the very essence of proof.
In this paper, we try to organize this evolution, to distinguish its different
steps and aspects, and to evaluate its advantages and shortcomings.
Axiomatic-deductive proofs are not a posteriori work, a luxury we can
marginalize nor are computer-assisted proofs bad mathematics. There is
hope for integration!