FOM: Re: "Relativistic" mathematics?

[Vladimir Sazonov]

Charles Silver wrote:
> 
> On Tue, 13 Oct 1998, Vladimir Sazonov wrote:
> 
...
> > Our teachers
> > actually (usually implicitly, by examples) said us which are
> > "correct" rules of inference.
> 
>         What if a teacher taught us the following rule:
> 
>         P --> Q
>         Q
>         -------
>   Therefore: P
> 
> Would that make it a correct rule because the teacher taught it to us?
> No.  The teacher would be just plain wrong, which shows that it's not the
> mere inculcation of *some* formal system, but getting things right (based,
> I think, on prior principles that are intuitively acceptable).

Note that the last paragraph of my posting you are commenting now begins
as

> > And finally, I think it is unnecessary to discuss very much that
> > mathematics deals only with *meaningful* formalisms based on
> > some *intuition* ...

(Cf. below.) Probably I was non-right and it was necessary to discuss 
this not at the end! I think this note should be quite enough. 
The teacher will present not only a rule, but also its meaning. 
It was never my intention to say something contrary. 

> > So, if somebody have any
> > reasonable mathematical education and training, then he actually
> > knows something like first-order logic. (But, most probably, he
> > does not know that he knows this. But this does not matter.)
> > Then he will implicitly, without even knowing this, formalize
> > (some essential features of) your story.  Anyway, any proof
> > which will present or understand that person will be formal in
> > some essential respect. Mathematical proof is something which
> > can be *checked* on correctness mostly relative to its form,
> > rather than to its content.
> 
>         You think formal systems come first.  I think intuitions are
> epistemically prior.

I think that formal systems are creations of peoples to have formal 
approaches to various intuitions. But many formal systems arose 
before we were born. Thus, a teacher explains us these formal systems. 
Of course this is done *together* with corresponding intuitions. 
Both formal system and corresponding intuition should *coexist* 
in the process of education. Moreover, *mathematical* intuition even 
cannot be presented in a pure way, without (essential features of) 
corresponding formalism. Actually mathematical intuitions may be born 
only *simultaneously* with corresponding formalisms. In a sense 
formalism may exist alone, but it hardly can be understood and used 
by *people* without invoking some intuition. Thus, they actually may 
only coexist. 

Pure idea without any formalism is like amoeba without skeleton. 
Pure skeleton is something not alive. 

> 
> > Let me also recall what M. Randall Holmes"
> > <M.R.Holmes at dpmms.cam.ac.uk> wrote on Fri, 2 Oct 1998 11:16:41:
> >
> > > One cannot be more or less rigorous if there is no standard
> > > of perfect rigor to approximate.
> > >
> > > We may _not_ doubt that the conclusion of a valid argument follows
> > > from the premises.  We do have explicit standards, which we can spell
> > > out, as to what constitutes a valid argument.  This is the precise
> > > sense in which mathematics is indubitable.  No natural science is
> > > indubitable in this sense.
> 
>         Was mathematical induction correct prior to its being formalized?
> Or, did it become correct once it was formalized. In my viewd, it has been
> enshrined as a rule only because it was previously intuitively correct.
> (If you don't believe this, try gaining acceptance for an incorrect rule
> of inference, like the one presented earlier.)

I think I have alredy replied. 

> > ...
> ...
> > To sum, I mean by a formal system such a system of axioms and
> > proof rules to which the term 'formal' may be applicable in any
> > reasonable sense. Thus, even any semiformal proof is
> > mathematically rigorous.
> 
>         Maybe you are right, but then a proof can be rigorous but wrong,
> in the sense of it following rules that later turn out to be mistaken.

I think I have alredy replied. 
 
> I'll let you have the last word:
> > And finally, I think it is unnecessary to discuss very much that
> > mathematics deals only with *meaningful* formalisms based on
> > some *intuition* and that even formal proofs are mostly
> > understood by us intuitively, may be with the help of some
> > graphical and other images and that there is a very informal
> > process of discovering proofs so that on the intermediate steps
> > we have only some drafts of future formal proofs (and sometimes
> > of axioms and proof rules, as well). Anyway, in each historical
> > period (except the time of a scientific revolution) we usually
> > *know* what is the ideal of mathematical rigour to which we
> > try to approach in each concrete proof.
> 
>         Sorry, but I can't help asking one more question: Is the above an
> application of Kuhnian philosophy?  Are you interpreting formal rules as
> examples of his "paradigms"?

Unfortunately, I do not know about this. Could you give, please, any
comments? 


Vladimir Sazonov