FOM: Re: "Relativistic" mathematics?

Charles Silver csilver at sophia.smith.edu
Mon Oct 12 20:00:19 EDT 1998

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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

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