[FOM] The gold standard and FLT and the meta-mathematical transversality theorem

[Josef Urban]

Thanks for this, I totally agree.

Vagueness seems pretty essential not just when expressing one's intuition,
but also for compressing communication in a context.

One could say things like "if you are close enough, ..." in particular
context and mean by that some "obvious" formal epsilon-delta argument.

Such compressed language will be often used by the human mathematicians, at
least when they explore the ideas. Sometimes the vagueness will cause
different understanding by somebody who is in a different context, which
may lead to conjectures, generalization, etc. And sometimes the vagueness
will start years of research.

I think/hope that a lot of the "more standard" cases can be reasonably
recovered (i.e. put into fully formal language) automatically by a
combination of statistical and semantic inference (trained on some math
corpora). There are no such systems yet, but I hope they will start to
appear soon.

I also think that building automated reasoning systems that in some parts
of their exploration use this human-style vagueness/compression/abstraction
natively will turn out very rewarding.

Josef



On Fri, Jan 12, 2018 at 9:35 AM, <katzmik at macs.biu.ac.il> wrote:

> Meta-mathematical transversality theorem and the FLT.
>
> The extremely interesting thread on FLT prompts me to look for a
> different angle on the debate involved that may explain the intensity
> thereof.  What I would like to propose is an explanation in terms of a
> meta-mathematical principle that I would call "meta-mathematical
> transversality theorem" (MMTT).  The *mathematical* prototype of this
> that I am referring to can be exemplified by a result providing
> sufficient conditions (say in terms of lowerbounds on f' and
> upperbounds on f'') for Newton's method converging to a zero of a
> function f if one starts with a point "close enough" to a perfect
> zero.
>
> Similarly, the seal of approval is affixed on a proof of a theorem in
> a given subfield as being "perfect" if it is "close enough" in the
> sense of having been understood by mathematicians specializing in the
> subfield and felt to be sufficiently convincing ("if the proof seems
> close enough to doing the job, there must be a true perfect proof
> nearby").  That's what I refer to as the MMTT.
>
> Obviously there is no "proof" of the MMTT.  The reliance of
> mathematicians on the MMTT is in the category of belief.  How far such
> belief extends of course varies from person to person.  Sometimes an
> argument proposed requires an unbearable stretch of belief in MMTT
> that cause some of the mathematical public to reject it while others
> accept it.
>
> What prompts this particular posting is the fact that some of the
> participants in the discussion of the status of FLT use terms such as
> "unbearable" rather than "unacceptable" in describing their reaction
> to the state of affairs with the proof of FLT.
>
> This is meant as a model intended to shed light on some of the
> workings behind the back-and-forth concerning the FLT.
>
> Mikhail Katz
>
> On Thu, January 11, 2018 00:12, Arnon Avron wrote:
> >
> > In reply to my words "standard, eternal criteria of mathematics"
> > (that meant, if it was not clear, the standard, eternal criteria of
> > a rigorous, acceptable proof in mathematics), David Brenton wrote:
> >
> >> I think you're getting this backwards. Math has not always been
> >> this "paradise of rigor", which has just stopped being so in recent
> times.
> >> On the contrary, for the longest time (with possibly the only
> >> exception of Euclid's elements), mathematics was an activity where
> >> the primary objective was to obtain results, rather than to be
> >> rigorous. Looking at most proofs from two or three centuries ago,
> >> they look much more like the kind of intuitive arguments that physicists
> >> do than like modern times proofs. The whole concern about foundational
> >> issues is relatively recent, its embryo probably first arose about
> >> halfway through the XIX century, and it only took full force in
> >> the early XX century. Considering that math has been around for
> >> millennia, I'd say that being careful about axiom systems, and
> >> what assumptions are being made, and about completely rigorous
> >> proofs, and the like, is more the exception, rather than the rule.
> >
> > And Timothy  Chow wrote:
> >
> >> Idealism is one thing, but historical inaccuracy is another.
> >> "Making explicit in a very precise way what are the assumptions that
> >> underlie the alleged proof" is almost never done in mathematics.
> >>
> >> For centuries, Euclid was the gold standard.  Today, we don't think
> >> that Euclid "made explicit in a very precise way" what all the
> >> assumptions were.
> >>
> >> If we look at mathematical practice across all fields and all of
> >> history, your "standard, eternal criteria" are far from being
> >> either.
> >
> > I was really amazed to read this kind of defense of the
> > current alleged proof, and especially of the attitude, of those who
> > claim to have proved FLT. I do wonder if those algebraic geometers
> > would be happy with the comparison made in these two messages of
> > their activity and proofs with those of  the mathematicians
> > in the 17th and 18th 0...
> >
> >  I also wonder, in view of what is described in these messages,
> > why we torture our students and ourselves by teaching
> > them what a rigorous proof should look like, and in
> > particular point out to them as severe mistakes  claims made
> > in their "proofs" that are not justified  by the collection of the
> > assumptions of the theorem they try to prove and
> > theorems that have already been proved before. What for? After all,
> > one needs not justify, or even make explicit, all the
> > assumptions one uses. Right?
> >
> >   Anyway, both Tim and David are confusing, in my opinion,
> > the norms of some area with the actual activity of the people
> > involved in that area. Thus many  religious people do sins.
> > Does this mean that they do not know what are the norms
> > in their religion and what is expected of them?
> >
> >   Despite what Tim and David note, mathematicians have had
> > a rather clear standard of what a rigorous proof is since
> > the time of Euclid. (And if they try to forget it,
> > people like Berkeley were there to remind them.)
> > Yes. There were periods in which the will to make rapid progress
> > in science caused mathematicians to neglect the old mathematical
> > norm of rigor. This does not mean that the mathematicians
> > at that time were not aware that they are not following
> > the standard eternal criteria. On the contrary: they felt
> > the need to justify this fact by claims like "for
> > rigor we have no time". (Sorry I do not remember now who
> > said this, and what were his exact words.) And of course,
> > At that glorious period really great mathematicians
> > were falling in traps like "proving" that 1-1+1-1+1...=1/2
> > Luckily, we have made some progress since those days. Are we
> > starting to go back? I am afraid that we are beginning to.
> >
> >   One more remark about Euclid. Tim emphasizes that
> > "Today, we don't think  that Euclid `made explicit in a
> > very precise way' what all the  assumptions were."
> > Absolutely true.  We have made indeed a great (temporary?) progress
> > since Euclid's time, and our  criteria
> > of rigor are (temporarily?) higher. But is there anybody who
> > infers from this, or thinks, that Euclid did not *try*
> > to make all his assumptions explicit? The crucial thing
> > is not whether Euclid succeeded in fully following
> > the general principles he forever set for mathematics,
> > but that he did set those principles, and was doing
> > his best to follow them. In contrast
> > (according to what I read here), the experts about the
> > alleged proof of FLT do not bother at all to make
> > their assumptions explicit!
> >
> >   Some years ago I predicted that all the great known
> > problems of mathematics will be solved in the next few
> > decades, because all is needed for this is to make "proofs"
> > very long and complicated on one hand (so that
> > nobody can really check them),  and not care
> > too much about rigor on the other... Isn't this wonderful?
> >
> > Arnon
> >
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