FOM: General intellectrual interest

[Michael Thayer]

Charlie Silver writes:


>
>Michael,
...
>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 think that reverse-mathmatics is of some foundational interest, but of
almost no "general intellectual interest".
The reason for drawing this distinction is: the "general intellectual
interest" is found in the notion that not everything about natural numbers
can be settled in as formal system of the appropriate type.  The details of
this are of little or no "general intellectual interest", and the genreal
result has been aroung long enough to have started filtering out of
mahtmatics.

> 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.


Actually, HArvey put his finger on the general interest in most of these
areas: the sociological one.  The "general intellectual interest" of FLT,
the four color problem, the SImple Group Classification, etc is usually in
the human details, not either the statement of the result (except to show
that the question is easy to ask and hard to answer), or its proof.


>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.

A little bit of neither, I am afraid.  I think that foundational work is
interesting and useful EVEN though it will NEVER lead to a final settling of
the question of why we think mathematical reasoning is certain.

I think that set theory is OK as a foundational study,  but I would be
inclined to say that it is more foundations of analysis, than foundations of
mathematics, and I strongly disagree with those set theorists on this list
and elsewhere who refuse to give any other foundational approach a hearing.
Categories, as a continuation Schonfinkel's idea of using functions without
sets as a foundational approach is quite reasonable to me, quite possibly
because I am biased towards compouting.  I dont think that categories will
make the simplest and clearest foundation for analysis, but I think they do
a better job in other areas than set theory does, e.g. finite goups.

>
>
>> 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).

I agree they are interestign, but the "general interest" factor is usually
low, if it is too technical for you with your background and interests, how
can it possibly be of general intellectual interest ???

I guess a better way to state my objections is: Harvey (and Steve to a
lesser extent) try to bash people who disagree with their foundational
interests by using this "general intellectual interest " club.  I say they
have no more right to use than club than their opponents do, after all the
only example they have come up with is not even FOM in Harvey's eyes.  I
often feel like an Indian Buddhist watching Hindus and Moslems bash each
other: if these guys would practice instead of prosyletise we would all be
better off.  And in this instance Harvey, and not Lou or the others, is the
main instigater.

>
>
>> 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?


I tend to bristle at the phrase "merely psychological", but I can't imaging
what a feeling could be other than a psychological state.  The question is
how does this state arise: from normal or abnormal functioning of the
organism?  Is it delusional or base in some perceptual reality?  Neil
Tennant has suggested that there are some simple mathematical ideas that any
functioning organism of sufficient complexity will form.  I would like to
see this sketched out in more detail, but I am sure that there are probably
some which fit this description, since we seem to be able to use some
mathematics to manipulate the world.  (Here is one place reverse-mathamatics
does help a foundational enterprise, but the results we are talking about
are a lot LOWER on the hieararchy than most RM's seem to work.)

>
>
>> 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

I don't buy this at all: there are no sets in the perceptual world, they are
constructs based on nominalization of divided reference.  We use the
linguistic device and then suppose that the device actually points at
something real.

>
> 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?
>
YES, if you stick the much abuse word "general" in front of "intellectual",
then I would say he does need to show that.  At least I am willing to give
him a Scottish verdict at the moment.

Michael