FOM: F.O.M. and foundations

[Charles Silver]

On Tue, 1 Sep 1998, Harvey Friedman wrote:

Shoenfield:
> >     We disagree on the truth-value of the statement "fom is mathematics";
> >but this is less a disagreement over fom than a disagreement over the
> >criteria for saying something is mathematics.

Friedman:
> You have not addressed my claim that practically noone in (core) statistics
> or computer science says or thinks that their subject is a branch of
> mathematics, either. Many classifications consider mathematics, statistics,
> and computer science "mathematical science."
> 
> You have also not addressed my claim that the extreme disadvantage to
> calling fom or mathematical logic a branch of mathematics is that it looks
> quite weak as a branch of mathematics compared to, say, number theory,
> geometry, analysis, algebra, etcetera, especially in terms of interactions
> with other branches of mathematics. And interactions is perhaps the most
> highly valued criterion being used by mathematicians to evaluate branches.

	It may then, as Harvey says, be politically advantageous to
consider mathematical logic as a "mathematical science" rather than a
branch of mathematics, but--politically advantageous or not--it is now, as
Shoenfield says, more widely thought of as a branch of math, and, as
Harvey says, it is (unfortunately) considered a weak one.  I wonder what
percentage of graduate math programs have a comprehensive examination in
foundations.  I'll bet it's low (< 15%). 

	I think Harvey's point is normative, rather than descriptive. 
Perhaps foundations *should* be considered a mathematical science.  I'm
wondering where Harvey would locate the subject.  Could it become a
separate department, like computer science and statistics?  Not likely.  I
guess if philosophy were done differently, it could be a branch of that
field. [?]

Shoenfield:
> >      My original point was that the really important results in fom are
> >mathematical; that is, they consist of definitions, theorems, and proofs,
> >all meeting the established standards of rigor in mathematics.

Friedman:
> In your original posting, you omitted "definitions." Also, f.o.m. at its
> highest level, seeks to understand mathematical proof and construction,
> etcetera, and so inevitably requires a very refined judgement as to what
> are the appropriate definitions. At the moment, not very much is known in
> the way of a theory of what constitutes a good or important or incisive or
> elegant definition. And in the case of f.o.m., the most common definitions
> are the introduction of formalisms, including especially formal systems.

	An analysis of what constitutes "good" or "important" or
"incisive" definitions is welcome, but there seems to be very-very-very
little of this kind of thinking.  As far as I know, Harvey is somewhat of
an exception in emphasizing this. 

Friedman:
> In fact, a case can be made that an unusually large number of the most
> important contributions to f.o.m. have been the correct definitions,
> starting with Frege. This makes f.o.m. a lot more like the rest of
> theoretical science and engineering than, say, number theory. This is a
> major reason why it is counterproductive to refer to f.o.m. as a branch of
> mathematics. Doing so also does not reflect on the special relationship
> that the foundations of a field has to the field itself.

	But, it seems to me that however welcome what Harvey's proposing
is, this is simply just *not* done in the field.  Look at JSL.  Does
anyone care any more (besides Harvey and a few others) that it's all
technical stuff, without much attempt being made to analyze the value of
the results?  I would guess that the main standard for publication in JSL
is just technical difficulty.  Am I wrong here?  Are Shelah's technical
gymnastics "interesting"?  (Maybe they are.  I'm not saying they aren't. 
I'd just like someone who understands what he does to offer an opinion.) 

> Shoenfield quotes Harvey:
> 
> >>     >Some intuitive ideas may not yield to such analysis, but still may
> >>be essential to consider.   On doesn't simply pretend that the concepts
> >>don't exist.

Shoenfield:
> >     Sounds good; but what should one do?   It is no use to just assert
> >very strongly that the concept is important and that those who do not
> >agree are obtuse.   Perhaps one should just put the concept aside until
> >another day, as one usually does with problems one cannot solve.

Harvey supports intuitiveness:
> Almost every mathematician every day - in fact, every intellectual - is
> using such intuitive ideas every day. This is because, e.g., every
> mathematician must decide what is worth working on, and what is worth
> publishing, which inevitably involves such criteria as "is this elegant?"
> "is this nontrivial?" "is this deep" "is this memorable" "is this a simpler
> proof" "does this explain the situation" "is this really new" "does this
> get to the heart of the matter" "is this interesting" "is this fundamental"
> "is this way of doing it understandable" "is this proof better than that
> one" "is this a direct proof"

	I personally like the above considerations and think they're
important, but Harvey is in my opinion (unfortunately) expressing a
minority view when he argues that such judgments should really be part and
parcel of f.o.m. 

Harvey:
> To a large extent, the importancce of one's work depends on how good one's
> intuition is regarding such matters. There is a severe limit to the level
> of acheivment of people who have poor intuitions along these lines, but who
> can construct complicated technical arguments.

	I think "importance" here goes back to earlier discussions of gii
(general intellectual interest).  Frankly, I think "complicated technical
arguments" now completely dominate (so-called) foundations.  "Foundations" 
is now a misnomer.  In the last, say, 20 years, how many papers in JSL
mention anything about the importance of something?  To anyone not
familiar with the topic of a standard paper in JSL, it is all technical
gobbledy-gook. 


Shoenfield:
> >      Harvey disagrees with my statement that there are no significant
> >results on foundations in general, but I do not find his remarks on this
> >convincing.

Some snippets from Harvey:
> Experts in fields are notoriously incompetent and notoriously uninterested
> in doing foundational work of the appropriate kind.

> I am convinced that
> the appropriate foundational work will be done by people who are either
> originally from other fields, or who are not working in the mainstream of
> that field.

> [F]oundational work involves an
> incomparably more penetrating understanding of the basic material. In fact,
> it should involve a major reworking of the basic material with new
> emphases.

> Today, computer science is the one area where foundational work of roughly
> the kind we are talking about goes on all over the map. 

	Summing up what I think Harvey's saying in the above (though
perhaps putting a sharper edge on it): mathematicians are "notoriously
incompetent and notoriously uninterested" in doing good foundational work. 
Good foundational work will be done "by people who are either originally
from other fields or are not working in the mainstream of that
[mathematical] field." Such important foundational work is being done in
computer science, but it is not being done in mathematics. 

	A Question:  Should someone create a "foundations" journal that
attends to reasons why a problem is "important," a definition is
"appropriate," a theorem is a "deep" one--besides including technical
results?  It's clear--isn't it?--that JSL is *not* such a journal.  Is
there a journal that's close to this?


Shoenfield:
> >I think Harvey's dream of departments of foundations
> >appearing in academia and taking over thw work of foundations of
> >particular fields could only happen in an alternate universe.

Harvey:
> It can be said that an intellectual revolution is the creation of an
> alternative universe. If you shared my vision and ideas for doing this, you
> would be working on it.


Charlie Silver