"It seems to me that the great and unique contribution of classical Greek
mathematics is not specifically the axiomatic method but, more generally,
the idea of proof. To return to the starting point of this discussion--my
guess is that the affinity that Hardy and Littlewood felt with classical
Greek mathematics was not so much to do with axiomatization (their own
work was certainly quite remote from this!) but with the more general
notions of proof and rigour."
I would like to stress this aspect of our debate: the role of "proofs",
a role which has been strongly debated, as a foundational issue, in recent
years, since Lakatos.
From a historical point of view, formally rigorous proofs have been
thought absolutely necessary in two periods: in Greek classic geometry
and in the last two centuries mathematics. Even from Renaissance to
Enlightenment there was no urgence of formally rigorous proofs, and those
were surely not 'dark ages' for mathematics.
Then I set forth the following statement: "Formally rigorous proofs
are deemed necessary only when the truth is believed deeply hidden beyond
apperances and sharply separated, by an abyss, from plain and empirical
natural reality"
This seems reasonable, because if the truth is not 'visible' you need
a very special method to get it: the proof. Is this even historically
correct?
In Greek culture that opposition is a constant whose most clear
example is Platonic philosophy.
On the contrary, in the century of the triumph of modern mathematics
and physics, Newton believed his fluxions method directly derived from
nature, Descartes' (but even Leibniz') faith in 'distinct' ideas left no
room for rigorous proofs, british empirical trust in observations made
proofs useless. If something was wrong, a better understanding of nature
could fix it.
After Kant that abyss began again to increase, truth became again hidden,
and today our strongest weapons to discover the "natural" laws are those
huge particles accellerators which credibly are the least "natural" places
in the world.
If this is true, a first consequence is the crucial link that appears
to exist between the prevailing epistemological mood and the basic of the
mathematical sciences.
Yours sincerely
Luigi Borzacchini