[FOM] Concerning proof, truth, and certainty in mathematics

[Harvey Friedman]

That mathematics is certain in a sense transcendentally beyond all  
other intellectual endeavors is very well established, both  
practically, and also theoretically through the classical work in the  
foundations of mathematics.

In practice, both the method of professional mathematical interaction  
and publication, as well as the laying down of the accepted axioms is  
already enough to prevent any dispute about correctness of proofs from  
lasting very long. This is exactly the opposite situation in all other  
intellectual endeavors.

It could be argued that this completely singular ability of  
mathematics to easily resolve its correctness disputes would be true  
even without the laying down of the axioms.

However, the thought that led to the axioms has a lot of overlap with  
the thought that led to the common viewpoint which so easily resolves  
all correctness disputes. It may be hard to separate these two forces.

Furthermore, the singular kind of certainty of mathematics is made  
even more apparent by the rather successful construction of actual  
formally impeccable proofs, via computer technology. I have previously  
written extensively about this development on the FOM.

For the largest compendium of actual formally perfect proofs, look up  
MIZAR.

Nevertheless, given any kind of certainty whatsoever, there might be a  
modified kind of perhaps yet greater certainty. Or a proof that no  
greater level of certainty can be achieved. This sometimes is  
reflected in the technical buzzword "accepting the kernel of a  
system". And of course, there is a kind of certainty that can never be  
achieved - at least in the sense of not allowing anyone to raise  
doubt. This cannot be achieved simply because anyone can read a script  
which says "I have doubts", without necessarily even thinking about  
the matter.

I have long planned some research on "certainty in mathematics" to  
flesh these matters out in a novel way.

On Jul 31, 2010, at 9:30 PM, ARF (Richard L. Epstein) wrote:

Concerning proof, truth, and certainty in mathematics

> On Jul 21, 2010, at 8:59 PM, Vaughan Pratt wrote:
>
> "The critical difference as I see it is that even a very strong  
> argument
> may nevertheless not be sufficient to establish the truth of a  
> proposition."
>
> It seems that Dr. Pratt believes that proofs in mathematics are  
> meant to lead to knowledge of
> truths, and hence that strong proofs are not acceptable in  
> mathematics.  This can only be if proofs
> in mathematics are meant to yield certain knowledge, as opposed to  
> proofs in other intellectual
> endeavors.

Pratt is completely correct in this matter.

> If mathematics is a body of truths, with certain knowledge of them,  
> then such knowledge is the same
> kind as theological knowledge.

This is false. Mathematical knowledge is vastly different than what  
transpires in theology.

>  It is not just that
> mathematicians leave lots to the reader.  It is far too hard to fill  
> in all the steps, as anyone who
> has tried to formalize a proof in mathematics knows.  It is even  
> harder to read such a proof, and
> hence less likely to lead to certain knowledge.  Hence, a proof in  
> mathematics purports or attempts
> to show that an inference is valid, but, except in rare instances,  
> is itself at best a strong
> argument for that.  Such proofs cannot yield certain knowledge.

This paragraph is not informed by the rather extensive work I cited  
above, concerning the actual construction of perfect formal proofs.

> It cannot be held that mathematics is aloof from all other human  
> endeavors in yielding certain
> knowledge unless one is a feist.  I have set out this view along  
> with an analysis of the nature of
> mathematical reasoning and mathematics in my essay "On mathematics"  
> which can be found in the 3rd
> edition of the book Computability by Walter Carnielli and me or  
> directly from me.

A large number of scholars hold this view, including me - in the  
opening paragraph of this posting.

I am curious as to how deeply your book has gone into the actual  
construction of perfect formal proofs.

I looked up "feist" and found http://www.google.com/search?hl=en&client=safari&rls=en-us&defl=en&q=define:feist&sa=X&ei=d_9VTLqELML-8AaxnayMBA&ved=0CBUQkAE
What this matter has to do with a "nervous belligerent little mongrel  
dog" escapes me.

Harvey Friedman