Feedback by Noga Alon and Yair Caro on Opinion 51

Feedback by Noga Alon

Of course I don't quite agree with the content of Opinion 51, but I can appreciate the reasoning; personally, I still find some value in the fact that I know a quick reason for the fact that 8765432187654321876543218765432187654321 is composite, and I will be less satisfied with a computer telling me that's the case.

Moreover, a proof of 4CT without computers will not change my conviction in the obvious fact that computers enable us to do lots of mathematics we cannot do without them. Hoping people will not find such a proof is essentially like hoping that people will not find a short argument for that question that someone asked John von Neumann, that can be solved either by computing an infinite sum (as von Neumann did quickly) or by multiplying the time by the speed. If we do not find the latter, shorter proof, that can serve as an evidence that we need a pencil and paper to do mathematics; do we really need evidence for that ? I don't.

Feedback by Yair Caro

I would like to oppose the opinion you stated in OPINION #51.
1) computers are the results of human efforts and human technology and 
as such they offer nothing deeper from the human thought that created  
them or that programmed them. It is just another way to do mathematics, 
where part of the classical reasoning and computing is replaced by a 
machine who was programmed by human to overcome our limited power to 
compute. I  am far from accepting the idea that the power of computing 
make the difference between shallow and deep.
There must be a much deeper criterion to make the distinction.
2)  you use the word shallow in an absolute form namely  everything we, 
human , can do  is shallow. This means that except of things created by God
or external power everything is shallow. This paradigm ( a sort of 
theology ) put everything that was proved in mathematics on the same 
level  of shallowness : everything is shallow. Isn't it the case that 
FLT has degree 9 ( say) in shallowness scale now and sqrt(2) is 
irrational has degree 3 ( say ) on the shallowness scale ??
3) that the computer can do things we can't do , needs no proof, it is 
obvious. In fact  there are several computations, which are trivial 
 from the mathematical point of view that are provably ( on the basis 
that we live 100 years X our ability to compute a multiplication of two 
digits in 1 second ) not within the human capability, but very easy for 
the computer power . I guess a reasonable example would be to compute 
exactly  10377! .
 So the fact that the computer can do something we can't and will never 
be able to is certainly not a certificate to be called "deep".
4) It is quite common in mathematics to have several proofs to the same 
theorem. The benefit is that we learn more about the issue in question 
by seeing various lines of attacking it and various reasoning that leads 
to a proof . A proof using computer and one that doesn't  are of 
interest if they suggest alternative or new ideas.
In my view Ideas are what make things deeper or leave them on the 
surface- shallow. By the way 4CT is in retrospect , in my view, less 
deeper than FLT just
because it is easy to understand the reduction ( as you said)  and the 
rest is a routine checking, and a routine checking is by no means deep.
So if we can't find a human proof will not make this 4CT deeper than it 
is now, it will only supply another proof that we are limited in our 
computational power with respect to computers and that we were very wise 
to notice that and to invent computers to help us in our weakness.
5) Time has a role : I am sure when Euclid first proved the infinitude 
of primes  that was a brilliant and deep idea ( still very nice today ). 
However as we make progress older results become less surprising and 
more understandable, and when we are less surprised we sometime might 
say "shallow".  The same will be the fate of proofs by computers as the 
power of computation will evolve. Evolution ( and time)  moves  the 
"deep and surprising" step by step into the "shallow and obvious" 
relatively to the new generation of thoughts, ideas and results.
6) The question whether the program that used by the computer belongs to 
the computer or to the programmer reminds me the same situation in the 
post modernistic debate if an article ( or paper, or book ) belongs to 
it writer or to the reader . While I am not sure about the philosophic 
attitude to this question it seems to me clear that if the program 
"belongs" to the programmer than "shallow" goes on to the results of 
computer finding.
7) I still prefer a more moderate and sensitive scale to be applied on 
ranking mathematics and proofs ( including FLT ) rather then say it is 
shallow because it was
proved by a man. Saying everything shallow leaves room for no other fine 
scale, and no way to make distinction between  : the sum of degrees in a 
is an even number ( elementary ) and FLT ( far from elementary ).

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