"PARADOXES OF RANDOMNESS"
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.