FOM: basic concepts; structuralism

[Stephen G Simpson]

Dear FOM people,

I'm delighted that our discussion of foundations of mathematics
(f.o.m.) is proceeding so vigorously.

Recent comments by Hilary Putnam, Vaughan Pratt, and John Baldwin
touch on some issues in f.o.m. which I regard as crucial.

Hilary Putnam writes:
 > What makes a concept of mathematics "basic"? I am inclined to think
 > that very few concepts are *essential* to mathematics: perhaps only
 > the (better: a certain) concept of *proof*. 

Hilary does not provide any context for his remark about proof.
Therefore, I find the remark very difficult to understand.  In the
context provided by my definition of f.o.m. at

  http://www.math.psu.edu/simpson/Hierarchy.html

my reaction to Hilary's remark would be: How can you talk about
mathematical proof in the absence of basic mathematical concepts?  In
the final analysis, doesn't *proof* entail *reduction to basic
concepts*?  Perhaps Hilary is distinguishing between "basic concepts"
and "essential concepts"?  If so, what's the distinction?

On the positive side, although I find Hilary's remark maddeningly
cryptic, nevertheless I will be happy if this evolves into a
discussion of certain issues in f.o.m. which I regard as important.
Among those issues are: what are the most basic mathematical concepts,
why are they basic, etc.

Vaughan Pratt replied to Hilary's remark by saying that sets are
probably essential to mathematics, but membership is not; at least I
think this was the gist of it.  If I'm not mistaken, Vaughan's remarks
are informed by a category-theoretic perspective, wherein "set" is
interpreted as "object in a topos", but "elements of sets" are murky.
I know a little about topos theory, but I have never really understood
the way category theorists seem to view the rest of mathematics, not
to mention the rest of human knowledge.  I know that some topos
theorists such as Peter Freyd have sometimes asserted that category
theory or topos theory is or can be turned into a good foundational
theory for all of mathematics or at least a good portion of it.  I
have never understood how this can be.  It seems wildly exaggerated.
Could somebody please try to explain it clearly?  (I.e., no
mumbo-jumbo.)

For example, we know how standard set-theoretic foundations explains
real analysis in set-theoretic terms.  This explanation is successful
in a certain sense.  It may not be very illuminating or satisfying,
but right now that's beside the point.  The question right now is, how
can category theory be used to explain real analysis?  And if it
can't, then what does category theory have to do with f.o.m.?

John Baldwin writes:
 > Vaughn Pratt's note reminded me of a fancy word I learned at
 > the meeting for Bill Tait's retirement that seems relevant to
 > this discussion.  The word was structuralism and I while I'll
 > yield to the philosophers for the proper technical usage my
 > interpretation was this.  One should understand mathematical
 > structures in terms of the relationships amongst the objects
 > of the structure with our regard to their internal properties.
 > Thus we study the reals as a complete ordered field, not
 > Cauchy sequences of equivalence classes of integers.
 > 
 > So I suppose the question to Steve  is whether any analysis
 > beginning at this level could be viewed as foundational?

John has asked me a question, so I'll try to answer it.

Yes, this kind of analysis might be viewed as foundational, because
the concept of number is probably basic in some sense.  But to make
this analysis succeed qua f.o.m., you have to show that this concept
of real number (i.e. a real number as an element of a complete ordered
field) is truly basic and does not call for any further explanation in
terms of more basic mathematical concepts.  And that is questionable,
because this particular explanation of real number appears to depend
on several other concepts: field, order, complete.  In particular,
it's natural to ask how completeness of the reals would be defined in
the absence of the concept of an arbitrary bounded set of reals.  I
think it was considerations such as this that led to set-theoretic
foundations, in the late 19th and early 20th century (Dedekind et al).

This probably doesn't answer what John intended, but I think it does
answer what he asked.

John also didn't ask me what I think of structuralism, but I'll
comment anyway!  The main point that I would like to make is that
structuralism as a general intellectual trend is ultimately
anti-foundational and anti-scientific.  The structuralist creed is
that each subject exists in its own internal world or framework and is
understandable only from that narrow viewpoint.  Thus structuralism
tends to isolate subjects and cut off all of their connections with
the rest of human knowledge.  It's the opposite of the foundational
approach, which regards the unity of human knowledge as paramount.
This critique of structuralism applies to John's example, as follows:
If you understand the real numbers "structurally" as a complete
ordered field and nothing else, then how can you connect real numbers
to the rest of human knowledge?  It seems you are left with no way to
make such connections.

If anyone else would care to comment on structuralism vis a vis
f.o.m., I'd be most interested.

John Baldwin continues:
 > Another point that I made in my essay and think bears repeating is
 > that it seems fruitful to look not only a single global FOM
 > but a foundationS of mathematics where different techniques and
 > viewpoints are appropriate for various areas of mathematics.

I'm not sure, but I think John is harking back to his earlier point
about "local foundations".  The point was stated obscurely and
elliptically.  Was this deliberate, in order to blur f.o.m.?  In any
case, my guess is that John is referring to what I would call
"foundations of particular branches of mathematics".
E.g. "foundations of topology", "foundations of algebraic geometry"
(Zariski etc), "foundations of analysis", "foundations of model
theory", etc.  To me it seems obvious that these specialized
"foundationS" (as John might call them) are interesting subjects, but
it's also obvious that their character is very different from that of
"foundations of mathematics as a whole", i.e. f.o.m.  And it is also
clear (to me at least) that these specialized foundational subjects
are interesting only to pure mathematicians, whereas f.o.m. is of
general intellectual interest to the educated public.  This distincion
is elaborated in my essay referred to above.

The pure mathematicians and their little brothers may want to
obliterate genuine f.o.m., and may therefore attempt to disagree with
the point I have just made, i.e. the distinction between f.o.m. and
Baldwin-style "local foundations".  But they will have a very hard
time explaining how results in sheaf theory, general topology, model
theory, etc should be or could be of interest to the general
scientific/philosophical public.  By contrast, a number of results in
genuine f.o.m. (e.g. G"odel's completeness and incompleteness
theorems) do have this quality of general scientific interest.

Sincerely,
-- Steve