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WHEN IS IT RATIONAL TO BELIEVE A MATHEMATICAL STATEMENT 191
But we saw earlier that M(n) = O n 1 2 +ε is equivalent to RH itself, so by substituting one for
the other, we get:
The Riemann Hypothesis has probability one.
Have we proved our conjecture, and with it, the equivalent Riemann Hypothesis? Well,
technically, we haven’t. We assumed that it makes no difference whether we (a) pick n
numbers at random, or (b) choose a number n and ‘pick’ the numbers one to n. The latter is
what we need to prove our conjecture, the former is the correct starting point for our
probabilistic reasoning. But substituting the random pick for the ‘one-to-n’-pick is justified if
the distribution of +1's and -1's in our μ-evaluation of the numbers one to n is truly random. If
it is, the first n numbers are nothing special; they are just as good as any random sample of n
numbers.
What we have established, however, is that RH is true with probability one. We have not
established the hypothesis itself, but we have given ample reason to be completely convinced
by it. In fact, the probabilistic account leaves no option even to be carefully agnostic—it would
be irrational to doubt an event that has probability one.
But this is not a proof of RH, as evidenced by the fact that the community does not treat it as
such. Edwards, one of the authorities on all things zeta, calls Denjoy’s argumentation “quite
absurd”, and overall sentiment in the community seems to agree with him. 9 One might
suspect that the reason for this rejection is due to the ‘merely’ probabilistic nature of the
argument. I do not believe that this is the case. To argue for this, let's look at another example
from mathematical practice.
3. Finite Projective Planes
Just as Euclidean Geometry takes place in a Euclidean space, so projective geometry
presupposes a projective space. And just like the two-dimensional reduct of a Euclidean space
is a Euclidean plane, so we have projective planes. Like its Euclidean sibling, projective
geometry is most usually practiced within an infinitary such plane, but finite projective planes
do exist. 10
Definition: A finite projective plane of order n is a collection of n 2 +n+1 lines and
n 2 +n+1 points such that:
1. Every line contains n+1 points,
2. every point lies on n+1 lines,
3. any two distinct lines intersect in one point, and
4. any two distinct points lie on one line.
The simplest finite projective plane is of order one: it contains three lines and three points,
and looks like a triangle. The next smallest finite projective plane is of order two, contains
seven lines and points, and looks like this:
9
Edwards discusses Denjoy’s argument in his book Edwards 1974: 268.
10
Throughout this section we will follow the exposition set forth in Lam 1991.