[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 

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
There should be some way of achieving certainty about them in the same 

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

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