FOM: Re: "Relativistic" mathematics?

[Charles Silver]

On Tue, 13 Oct 1998, Vladimir Sazonov wrote:

> If you will tell this story on Adam and Eve to somebody from the 
> street who really have NO mathematical experience and did not 
> learn at school some examples of mathematical proofs (and actually 
> of proof rules) he will be unable to understand even what are prime 
> numbers and any informal proof you will present. 

	You are probably right.

> 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). 


> 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.

> 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 that having *explicit* standards means having known 
> some rules of inference presented in any reasonable form. 
> Say, children learn at school how to use in geometry the rule 
> reductio ad absurdum. 

	Incidentally (I admit this is not relevant to the point), I don't
think we accepted Reductio proofs in elementary geometry class, not
because the proofs didn't establish what they purported to, but because
they seemed like cheating. 


> 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'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"?

Charlie Silver