Opinion 102: Some Soul Reckoning on Yom Kippur: We Are NOTHING, and we Should Not Just SAY it, but truly BELIEVE It (and ACT accordingly)

By Doron Zeilberger

Conceived: Sept. 28, 2009 (during Yom Kippur 5770); Typed and posted: Sept. 30, 2009.

As I was sitting in shul, trying to repent, and praying for forgiveness, during the passages stating the triviality of humans, ashes to ashes etc., I was reminded of the ancient joke about the rabbi dramatically proclaiming: "Oh God, please forgive me, I am nothing!", that inspired the cantor to exclaim: "Oh God, please forgive me, I am nothing!", that in turn inspired the parnas to announce: "Oh God, please forgive me, I am nothing!". Deeply moved, Yenkel the shamas (janitor) repeated:

"Oh God, please forgive me, I am nothing!",

that lead the rabbi to whisper to the cantor: Look who thinks that he is nothing!.

We may laugh at the pompous and hypocritical rabbi of the joke, but we, human mathematicians, are not any better. We often say how little we know, and we know, on some level, that most mathematical open problems we would never solve, but we act as though we can potentially solve everything. We keep trying, very hard, to prove RH, P ≠ NP, 3x+1, Goldbach etc. with our limited human means, even though we know that the prior probability of success is infinitesimal.

But if we would realize that we are truly nothing, and that there is an intrinsic lower bound on the complexity of the proof that P ≠ NP, that far exceeds our limited human capacity, we wouldn't even bother to try and prove it (by ourselves)! If we want to raise our chances from ε2 to ε we should take full advantage of our beloved computers, and try to train them how to eventually try to prove P ≠ NP, RH, etc. etc.

Yet, Rome wasn't built in a day. A direct and frontal attack is hopeless. We should start out modestly, and try to have the computer, say, prove a lower bound of 3.1n, beating, the current (I believe) bound of 3n (for the full-circuit complexity), that was proved (in 1984) by Norbert Bloom. Now I said that you must use the computer, no credit for paper-and-pencil! Because, the skill that you would acquire in teaching the computer how to make a minor improvement, may, hopefully, one day, enable the computer to prove a super-polynomial lower bound on, (for example) CLIQUE.

Of course, we need, at present, all the human attributes that helped us prove conjectures and solve open problems, by hand, namely "cleverness", "creativity" and "thinking out of the box". But don't waste your (very) limited talent on doing things by hand. Remember that you are nothing by yourself, and your best bet is to get help from your friend the computer, as Appel-Haken and Hales did. This would require some investment of your time, and a sharp learning curve, teaching yourself how to program computers symbolically, and using meta-algorithms, and getting rid of the old hang-up that "computers can't think, they can just compute". Of course, they can't think, but neither can you!, both computers and us only compute. In other words, we think that we think, but we are really nothing but (lousy!) computers. Computers already far surpass us in numerics and routine symbolics, but soon they would also surpass us in concept- and idea- crunching.

Another piece of human vanity and superstition is the Krattenthalerian insistence on fully "rigorous" proofs, and "absolute" truth, and the all-or-nothing Boolean narrow-minded mentality. First, there is no such thing as "rigorous proof". All proofs are either done by us humans (and we, humans, are nothing!), or by our much more reliable computer brethren, that nevertheless are built and programmed by us humans (and we, humans, are nothing!). We should adopt a much more flexible attitude to "truth" (whatever it is), and encourage diversity. So it would be great if there would be a proposed proof-plan for, say, RH, with intricate lemmas and sublemmas, and subsublemmas, some of which would still wait for a fully "rigorous" proof, but they would have even a greater empirical plausibility, and empirical verification, than the statement of the parent statement, RH itself, that has only been checked for a few billion cases. Perhaps we can develop tools that would meta-determine (non-rigorously, but nevertheless reliably) lower bounds for the length of rigorous proof of each yet-unproved-piece, if it exists, and if the lower bound exceeds current resources, we should learn to live with it, and enjoy what we have! But of course, my dear freund Christian Krattenthaler would not accept this not-yet-full proof for his journal, since, according to him, and unfortunately according to ninety nine point nine nine nine per cent of currently living (human) mathematicians, you either have a proof, or you don't, and "almost doesn't count". Nonsense! Almost (or even a tiny bit) does count, since we are truly nothing, and we do what we can (if at all possible, taking full advantage of computers).

Opinions of Doron Zeilberger