FOM: General intellectrual interest

[Charles Silver]

On Thu, 18 Dec 1997, Michael Thayer wrote:

> Lou van den Dries responds to Harvey as follows:

> >Otherwise I am not conceding anything. Sure, "suspect" is vague, so is
> >"general intellectual interest", and perhaps even more, the significance
> >of "general intellectual interest".

Michael Thayer:
> I will go farther: it is blatant nonsense as used by Harvey and Steve.  What
> "general intellectual interest" do most of Harvey's "interesting technical
> results"  (to quote Anand) actually have.  How many people even understand
> large cardinals, much less care about them?  Far fewer than are interested
> in number theory, be it abc or FLT.

Michael,
	I can't agree with you completely.  To my mind there are several
separable points in the above.  I take it that Harvey and Steve wish to
take a constructive approach to the foundations of mathematics, to
investigate and solve various difficult problems that might shed light on
mathematical foundations.  The problems that they both work on do involve
technicalities, but I don't think they work on these problems simply
because of the technical challenges these problems present (though I think
that's part of it).  I think they think that at least some of the problems
they select are of "general intellectual interest" in so far as the
solution to them may reveal something "interesting" about the foundations
of mathematics.  They could of course both be wrong.  Perhaps you think
that the whole reverse-mathematics project is misconceived from the start.
Do you think this?

	I don't understand much about the reverse-math project, so I can't
offer much of an opinion.  From my extremely ignorant perspective, it
looks "interesting" (I am aware that Pratt has distinguished different
senses of 'interesting', and I admit I cannot sort out what I mean in
terms of them).  I like the idea of determining how strong a theory must
be to prove various mathematical results.  To me, this seems a 
quintessentially "foundational" project. 

	The other issue is whether Harvey's work is of "foundational" 
interest.  I'll pull a chunk from what you said above and put it down
here:

> How many people even understand
> large cardinals, much less care about them?  Far fewer than are interested
> in number theory, be it abc or FLT.

	I agree that large cardinal problems are technical and not readily
graspable by people outside the field, which contrasts with the statement
of--but *not* the solution to--FLT. But, I think the deeper and more
interesting question is whether technical, set-theoretical questions are
really and truly "foundational," aside from their lack of accessibility. I
take it that you don't think they are foundational, even if the
accessibility issue has been separated off, but I can't tell.  I can't
tell whether you are simply anti-foundational in general, meaning that you
object to *any* foundational approach, or whether you just don't see
technical work in set theory as really being of foundational significance. 
If the latter, I think you should say a bit more why you believe this.  I
am not disagreeing with you about whether technical issues in set theory
are really and truly foundational.  I'd just like to see your reasons. 


> I suspect that the folks who partake of the botanicals that grow in Doctor
> Cantor's garden have OD on them: we know since Goedel that number theory is
> incomplete, indeed incompleteable.  So details as to what large cardinal
> hypotheses are required to prove the graph minor theorem, or
> Paris-Harrington are about as exciting as improving the exponent on
> logloglogloglog(n) in the error term for some number theoretic function.

	I guess the question here would be how to sort out the merely
technical results in set theory from the truly "interesting" foundational
ones.


> The real "general interest" stuff has been mostly discussed by Tennant and
> Pratt in their discussion about mathematics on Quasar 1036, and Shipman et
> al in the recent discussion of "convincing proof".

	I think *all* the discussions so far have been extremely valuable
and interesting (though several points have been too technical for me to
comprehend).
	

> The two biggest foundational questions in mathematics are:
> 1. What is the origin of the certainty that many (but NOT all) people feel
> when presented with a mathematical proof?

	This seems interesting.  Do you think the feeling of certainty is
merely psychological, maybe just due to similar brain wiring?  Or do you
think there might be some underlying reality to this feeling that could be
(at least partially) captured by some foundational-type formalism? 


> 2. When, and why is this feeling of certainty justified?

	I tend to see technical issues in set theory as being related
(somewhat distantly, I'll admit) to the "justified feeling of certainty," 
since set-theoretical principals are normally assumed to hold in the
metatheory of the object-theories in which the proofs are presented.  That
is, certain set-theoretical principles are taken to justify the feeling of
certainty (when a proof is correct).  But, I assume you would take issue
with this blatantly foundationalistic view of the matter.


> as Machover says, he can recognise a proof when he sees one, although he may
> not be able to describe what is required.  He also mentions evolution etc as
> a reasonable explanation for the justification.  (Neil T. made similar
> remarks on the necessity of arithmetical notions for sufficiently advance
> organisms.

> Next to this sort of discussion (which may be too vague for Harvey) details
> of which uninteresting set theory statement follows from the existence of
> which implausible infinite entity seems remarkably like medieval theology -
> which is probably not of much intellectual interest to most people.

	I guess one task for Harvey, then, would be for him to explain why
and how an "implausible infinite entity" could be of intellectual
interest.  Would you agree with that, Michael?

Charlie