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196 STELLING
there is no finite projective plane of order ten, an error in one of the computational branches
that ‘accidentally’ gives the correct result will still mean that at this point of the argument the
theoretical transformation into a rigorous formal deduction breaks down. And, as we have
seen, the probability that such an error occurred is actually fairly substantial. So the reason
we accept the computational but reject the probabilistic argument cannot lie in the
underlying assumption that one but not the other is a formal proof in essence, if not in
execution.
Note also that another obvious explanation doesn’t work: We might be tempted to point out
that the significant difference between the two cases is simply that, while Denjoy’s argument
is indeed a proof that RH has probability one, it is decidedly not a proof that RH is true. And
since the latter is what we’re looking for, the reason the argument isn’t accepted by the
community is that it proves the wrong thing. The proof by Lam, Thiel and Swiercz, on the
other hand, does prove what it sets out to prove, and therefore delivers the correct content.
This objection, however, presupposes that there is a significant difference between the
proposition that RH is true and the proposition that RH has probability one. And, while this
may be so, simply stating that it is won’t help us. Instead, we need an explanation as to why
the two propositions are different enough to allow for the explication that Denjoy’s argument
simply delivers the wrong content. Needless to say, the explanation will have to be applicable
to the order ten plane as well, if we want to use it to shed any light on the situation as a whole.
And any such explanation will presumably at some point run into the objection that the
possibility of random hardware errors having lead us to the wrong result can not logically be
ruled out. But at that point the nonexistence of the order ten plane is probabilistic in nature,
much like Denjoy’s argument, so we have come full circle: we are still in need of an
explanation as to why the two cases are different, even if they’re both probabilistic.
My proposed explanation, then, is this. There is, I want to argue, a feeling of due diligence
that forms part of the paradigm of the current state of mathematical research. Formulated as
a maxim, it can be read as saying something like, “get as close as possible to the strongest
solution conceivable for the problem in question.” This needs some explanation. I do not want
to say that mathematicians automatically strive towards general over particular solutions
(though they might). Neither am I suggesting that mathematicians search for solutions that
have a maximum impact across mathematics over solutions that influence only small portions
of some particular field (though they might). What I mean is this:
(a)
(b)
If Denjoy’s argument ends up being wrong, i.e., if RH turns out to be false, we as
mathematicians have made a mistake in the argument. There is nothing outside of
us, the mathematical reality, and the proof as written down on the papers on our
desk, and if RH turns out to be false, the buck stops with us.
However, in the case there actually is a finite projective plane of order ten, we are
not to blame in the same way. If it turns out that there really was an undetected
hardware error that lead us astray, then hardware degradation, solar flares,
compounding effects on the quantum level or any of a number of other unfortunate
circumstances are at fault. In that case, there’s us, the programs we wrote,
mathematical reality and the outside world, the latter of which foiled our plans.
Due diligence, in this sense, is the feeling that, even on the off-chance that circumstances
should have conspired to foil our plans, we have done as much to solve the problem as can
reasonably be expected. 18 Mathematicians do not have influence over the reliability of
18
It is tempting to construct an analogy to the way in which the axiom of choice, even though it leads to
patently absurd conclusions when interpreted as a statement about the physical world, nonetheless
finds acceptance within the mathematical community. It is almost as if there is a tacit agreement that, as
long as the mathematics is beautiful, if the external world doesn't want to play along, so much the worse
for the external world.