[FOM] Solution (?) to Mathematical Certainty Problem
Robbie Lindauer
robblin at thetip.org
Tue Jun 24 19:54:44 EDT 2003
Reply to Prof. Friedman:
It strikes me that you're trying to provide several things:
1) A reason to think that for some relatively long number that we
could be certain of the number of 1's in it.
2) A reason to think that we know (1) better than we know, say, that
yesterday the sun came up.
3) That some machine machine might be helpful in advancing this
project.
In response to this:
The naive notion of mathematical certainty, is given in a kind of
intuition "a feeling of certainty" which results in expressions like "I
am certain that 2 + 2 = 4, the whole world might be an illusion, but
I'm sure that 2 + 2 = 4".
Some mathematical questions are more complicated, but it's thought that
the same notion of certainty ought to apply:
"(some very large number of numbers ending in ....126785172389469117)
are prime" is true in the same sense that "2 + 2 = 4
THEREFORE
There should be some way of achieving certainty about them in the same
way.
Unfortunately, the proofs of some such statements are too complex for
us mortals to comprehend, in their entirety.
You propose to build a machine that ought to rid us of our feeling of
uncertainty by replacing all complicated steps in reasoning with
mechanical ones, reproducible "in principle" by anything that could
read the record of the proof.
But then there is the nagging question, how do we verify in any
particular very complicated proof, that it is accurate?
In the example given, we have a proof where there are a googleplex of
simple steps necessary to complete it. And so, us limited computing
machines, forget why and therefore whether our previous estimation of
the correctness of the initial 100,000 steps of the proof were correct.
This leaves us in the position of having to say "We know that this
computing method works for proofs less than 100,000 steps in length,
beyond that, we assume it still works, but can't be sure." We might
invoke a statistical argument "Well, it's been right for every problem
that we can evaluate ourselves, so it's probably right for every
problem that we can't." But then we've lost the certainty we were
looking for.
I draw these conclusions:
That the notion of mathematical certainty in the naive "2 + 2 = 4" is
not captured by reference to idealized computers, but rather by
reference to direct intuitions of the truth of primitive mathematical
statements. (You apparently would agree, yourself trying to reduce
complex proofs to less-complex ones.)
That the notion of mathematical truth is not as closely related to the
notion of mathematical certainty as we might have hoped, and that like
other epistemological issues, the question of whether or not a
mathematical statement is true is entirely different from whether or
not we could be convinced of it, much less certain of it, and that
machines may help guide us to have intuitions we ought to have while
understanding a proof, they can't have those intuitions for us, and we
can't accept their proofs and statements of their proofs just because
they are well made in the "mathematical sense" of certainty.
Finally the stipulation that this be an idealized computer makes your
position a little tenuous. On the one hand you want to say that
certainty is to be handed off to computers to deal with, but
simultaneously admit that any actual computer really couldn't be given
this job.
Best Regards,
Robbie Lindauer
More information about the FOM
mailing list