#
Opinion 141:
Biological Evolution Did not prepare us for reasoning logically and for rigorous proofs

and Analogously

Mathematical Cultural Evolution Did not prepare us to Use Computers Optimally

## By Doron Zeilberger

Written: Feb. 18, 2015

In a beautiful recent masterpiece, discussed in my previous
opinion 140,
Zvi Artstein claims, very convincingly, that the reason most people
(in fact, all of them, including us, professional mathematicians!) find math so hard, is that biological evolution
did not prepare us to the rigid discipline of formal mathematical and logical reasoning. In order to
survive in the jungle, we had to use informal, intuitive, `Bayesian' `logic', if you would call it logic at all.

But in spite of that, Mathematics blossomed, and has come a long way, both as *queen* and *servant*
of the physical sciences. For more than two millenia, the *cultural evolution* of mathematics,
and the notions of *axiomatic method* and *rigorous proof* ruled.
But neither Euclid, nor Gauss, not even Ramanujan, knew about the new *messiah*, the powerful
electronic computer, that would revolutionize **both** the *discovery* and the *justification*
of **mathematical knowledge**, and would (soon!) turn
mathematics into an *empirical *science, just like physics, chemistry, and biology, but dealing
with mathematical entities (like numbers, equations, groups etc.) rather than with electrons and stars, (or acids and bases, or cells and genes etc.),
and we would soon abandon our fanatical insistence on *rigorous proof*,
and very soon **semi-rigorous** proofs (see my manifesto)
would be fully acceptable, and soon after, completely non-rigorous proofs!

In a recent article,
Shalosh B. Ekhad and I present, as *case studies*,
a class of problems where fully rigorous proofs can be safely abandoned. I believe that this
would be the case, in at most fifty years, for the rest of mathematics. We will come to realize
that fully rigorous proofs are only possible for relatively trivial statements, for example
Fermat's Last Theorem, the Poincaré conjecture, and the Four Color Theorem.
But for really deep (and interesting!)
mathematical knowledge, we would have to be content, if lucky, with *semi-rigorous* proofs, where
we know that a proof exists, but it is too complicated for us, and even for our computers, to find, and
more often with fully non-rigorous (heuristic and empirical) proofs.

You are welcome to watch the lecture.

Opinions of Doron Zeilberger