Abstract:
I’ll discuss how Gödel’s paradox “This statement is false/unprovable” yields his
famous result on the limits of axiomatic reasoning. I’ll contrast that with my work,
which is based on the paradox of “The first uninteresting positive whole number,”
which is itself a rather interesting number, since it is precisely the first uninteresting
number. This leads to my first result on the limits of axiomatic reasoning, namely
that most numbers are uninteresting or random, but we can never be sure, we can
never prove it, in individual cases. And these ideas culminate in my discovery that
some mathematical facts are true for no reason, they are true by accident, or at
random. In other words, God not only plays dice in physics, but even in pure
mathematics, in logic, in the world of pure reason. Sometimes mathematical truth
is completely random and has no structure or pattern that we will ever be able
to understand. It is not the case that simple clear questions have simple clear
answers, not even in the world of pure ideas, and much less so in the messy real
world of everyday life.