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WHEN IS IT RATIONAL TO BELIEVE A MATHEMATICAL STATEMENT 195
Yet the mathematical community seems to have accepted the non-existence of the plane of
order 10 as an established fact, just like the proposition that four colors suffice to color any
map.
4. Conclusion
So where does this leave us? Both arguments we encountered seem to share the same flaw—
the argument for the Riemann Hypothesis in section 2 was merely probabilistic (even though
the probability for RH being true was one), and the computer-aided proof for the nonexistence
of finite projective planes of order ten in section 3 is only accurate if no random
hardware errors have given us a false solution, which in itself is highly unlikely. Why, then, do
we accept the second, but reject the first argument?
One of the more common arguments here goes roughly as follows. Mathematicians operate
under the premiss that acceptable proofs are those that can in principle be transformed into a
formal chain of deductive, gapless arguments in some formal calculus. The conclusion of such
a chain of reasoning is the established theorem, which then holds immutably true, given that
the starting points of the deductive chain are true. Perhaps the most fervent supporters of this
view were the group of mathematicians who collectively worked under the nom de guerre
Nicholas Bourbaki:
In practice, the mathematician who wishes to satisfy himself of the perfect correctness
or ‘rigour’ of a proof or a theory hardly ever has recourse to one or another of the
complete formalizations available nowadays, nor even usually to the incomplete and
partial formalizations provided by algebraic and other calculi. In general he is content
to bring the exposition to a point where his experience and mathematical flair tell him
that translation into formal language would be no more than an exercise of patience
(though doubtless a very tedious one). (Bourbaki 1968: 8)
Under this view, rejected argumentations like the one by Denjoy would fail to satisfy
mathematicians because the impression is that they cannot be explicated into a formal chain
of reasoning. On the other hand, since the calculations done by the computers to establish the
non-existence of a finite projective plane of order ten are by their very nature Turingcomputable
and thus recursive, they can certainly be formalized, and hence the calculations
constitute a proof.
I think there are several problems with this view. First, there is the general issue that “proof
in a formal calculus” is, strictly speaking, a notion devoid of content. Formal calculi come in
various shades and forms, and even under the reasonable understanding that we want
formalizations that are in some form tractable, we are left with a huge variety of possibly
admissible rules, ranging from the mathematical puritanism of Russian-style recursive
intuitionism, through the cautiously liberal formalism in the vein of Gentzen to the hedonistic
irresponsibility of higher set theory. And this does not even begin to face the problems
stemming from the fact that we can derive any result we want by introducing suitable
premisses and axioms.
Much more pressing, however, is the fact that Denjoy’s reasoning can be brought into formal
form—at least as much as we assume that any purely mathematical argument within analytic
number theory can. We end up (presumably) with a gapless sequence of logical deductions
that lead, invariably, to the “absurd” conclusion that the Riemann Hypothesis might be false,
but with probability zero. On the other hand, the central theorem of section 3 can be brought
into formal form if and only if no computational errors occurred. In case some random
hardware errors influenced the calculations, the whole argument can no longer, even in
principle, be formalized. Note that this holds even if the hardware errors are negligible in the
sense that they do not lead to a false result. Even if the conclusion is factually correct and