[FOM] Certainty in mathematical proofs

Vladimir Sazonov Vladimir.Sazonov at liverpool.ac.uk
Sat Oct 20 11:14:44 EDT 2007

On 19 Oct 2007 at 12:45, Arnon Avron wrote:

(Sorry for the excessive citations which is however difficult to avoid.)

> If pushed with determination, than there will
> be no escape from the claim that certainty is
> at the end a personal matter, and that one can never be certain about 
> what is certain.
> If we leave the issue at that, then there is no
> point in discussing FOM. The only choice would
> be: everyone for himself/herself, and that's it.

Why so pessimistic? All of us have sufficiently similar or comparable 
mathematical intuitions and can discuss these similarities or possible 
subtle discrepancies using quite precise methods of mathematical and 
metamathematical formalisms.

> Well I, for one, do think that it makes sense to
> discuss and make research on FOM - and *for the
> original goal*: to provide  secure and certain
> foundations for large parts of mathematics,
> and to do so on the basis of *objective* criteria,
> which goes beyond someone's personal feelings and judgements.

OK! *Objective* is what I want.

> Yes, it might indeed be impossible to objectively draw
> the *exact* line between the absolutely certain mathematical 
> propositions, and those which are
> less-than-absolutely-certain. Indeed, I do not pretend that I know 
> where the exact line is. But even without exact line
> there are obvious cases  of the two sorts. Thus the fact that there 
> are infinitely many primes was absolutely
> certain at the time of Euclid, it is still absolutely certain today, 
> and so it will  remain forever.

Yes, the proof remains the same. It is formal and quite "rigid" and 
therefore objectively existing constructive object which cannot be 
physically destroyed (unless all mathematical libraries in the world 
will disappear in the flame of nuclear world war). Nowadays the degree 
of formality of this proof is even higher than at the time of Euclid. 
Its formal correctness can be checked by computer.

Moreover, what means "formal" is now understood virtually by everybody 
using computers (like simple secretary staff practically using formal 
tools such as Microsoft Word editor, e-mailing system, etc), even 
knowing nothing essential on mathematics.

> The only way to change
> the status of a proposition to "absolutely certain" is by
> *proving* it on the basis of absolutely certain axioms,
> using absolutely certain methods of proofs.

I do not know what are absolutely certain axioms and absolutely certain 
methods of proofs (of mathematical truths) or what means "absolutely 
secure foundations". But I know what is a formal proof in a formal 
system. As a constructive object it is something extremely 
certain/rigid (in physical sense of this word).  I understand what is 
mathematical intuition - the real but not absolutely reliable substance 
of our mental world - related with formal derivations (concerning the 
natural numbers, cumulative hierarchy for ZFC, and any other kind of 
imaginary, fantastic, even illusory mental mathematical worlds). I 
think that psychologists can confirm the mathematical intuition exists 
objectively. This confirmation is practically unnecessary, because we 
all feel sufficiently well our personal mental world. But I have no, 
even remote idea on what is the "objective" or "absolute" mathematical 
truth or anything like that. It looks for me as a mystic conjuration.

>  Once we recognize

in which objective way?

that there are some absolutely certain
> mathematical propositions and some that are not (Of course,
> there will always be strange people who deny it.
> The only thing one can do about it is to ignore them),
> it is important to find out (as far as possible)
> what is the extent and applicability of those who are.

Aha, now I understand that mathematicians or those people working on 
f.o.m. should be unified into a big sect (the bigger the better - more 
"objectively") who "know" or believe that they know what is absolutely 
certain (mathematical truth or a special class of mathematical proofs 
of absolutely certain truths) and what is not. Yes, they realize that 
their personal knowledge is not absolute, but they believe in the 
highest "divine" ideal. The typical questions of people from such a 
sect are "do you believe?" that such and such things in mathematics 
have an absolute certainty, or something like that of a "religious" 
character. All others doubting and not believing are called "strange 
people" (or heretics?) which just should be ignored (and be happy that 
being not subject to inquisition!).

Yes! This is quite objective scientific criteria and methodology in the 
foundations of mathematics.

Vladimir Sazonov

P.S. In general I have nothing against predicativist   or any other 
concrete approaches to foundations of mathematics as soon as they are 
based on really objective grounds (formal approaches appealing to our 
intuition or to the real world). I know that Arnon also suggests some 
interesting formalisms and have no problem with that. The problem is 
with pretensions on some "absolute".

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