FOM: set/cat "foundations"

[Harvey Friedman]

This is a reply to Pratt, 11:15PM 2/25/98, and also Tragesser, 3:07PM 2/23/98.

Pratt writes:

>From: Stephen G Simpson <simpson at math.psu.edu>
>>My view is that "categorical
>>foundations" is a sham, false, misleading, etc etc.  When forced to
>>discuss it, I refer to it as categorical mis-foundations, categorical
>>dys-foundations, categorical non-foundations, etc.
>
>Steve is like a pianist who has played only Beethoven all his life.
>Confronted with a Scott Joplin score, his fingers tell him that this is
>not even music.

I would rather say: Steve is like a painist who has played only music with
clear rhythm and melody all his life. Confronted with very modern scores
with neither rhythm nor melody, his fingers tell him that this is not even
music.

>In that sense Steve's position is understandable, if not entirely
>reasonable.  For those whose world view makes every object a set, there
>can be no categorical foundations.

Neither Steve nor I has any commitment to "every object is a set," although
it is often very convenient to do so. The problem is that at the moment
there is no comprehensive categorical foundations, or any comprehensive non
set theoretic foundations.

>I have never met a category theorist who thought every object was a set.
>Category theorists simply don't think that way.

The more sensible ones accept the idea that category theory is grounded in
set theory, and go on from there, treating category theory as an important
organizational tool. I count MacLane as one of these sensible people.

>What distinguishes set theorists from category theorists is the order in
>which they add the ingredients.  Set theorists start with the points and
>will add glue if persuaded that it is worthwhile.  Category theorists
>start with the glue and will add points if you ask nicely.

But at present there is no theory of structures (glued points) that meets
the high standards that the theory of sets (no glue) meets. What is so
disturbing is the lack of recognition of the special intellectual standards
that are uniquely met by the theory of sets.

>Sand and glue are pretty basic mathematical ingredients.  Surely the
>prejudice that all foundations *must* start from sand can be overcome just
>as readily as can the various racist, sexist, and religious prejudices
>that cloud our thoughts (which may not be saying much).

But only if the appropriate intellectual breakthroughs are made. I have
thought hard in the past about an autonomous theory of structures, without
success. My efforts merely turned into a slavish translation of set theory.
But at least I recognize the difficulties, and what sort of intellectual
work is needed. This is why it is far more likely that people with my point
of view will make something serious out of something like category theory
as comprehensive foundations than people who mistakenly think that it is
already comprehensive foundations.

There is a way to view set theory so that the usual complaints about it
don't apply very well. I have made this point before. E.g., one can
criticize the identification in set theory of the real line with, say,
Dedekind cuts of rationals. However, the main point of this criticism is
that one should only care about structures up to isomorphism. Under that
criticism, one can view set theory has having provided one particular
isomorphic copy of the real line as a structure. And then, as I said
repeatedly on the fom, one simply adds a constant symbol c representing any
structure isomorphic to this particular set theoretic example of the real
line. Now one can no longer prove "silly things" like "any two real numbers
are comparable under inclusion."

The foundational advance that would support a theory of structures as
comprehensive foundations would be to somehow create a theory whereby one
can construct lots of structures, and perhaps mappings between them,
without first constructing the points. And do this in such a way that it is
conceptually coherent, and not a hodgepodge of specific items that look
like some sort of obvious generalization of set theory.

I think that something very good along these lines can and will be done,
but not by people who think that it either has been done, doesn't need to
be done, or don't understand what it means to do it.

Tragesser 3:07PM 2/23/98 writes:

>        Neither Maddy nor Tennant are facing up to the fundamental
>task of deeply understanding the reductive powers of
>SET and its limitations (what gets lost in translation).

Nothing whatsoever gets lost in the set theoretic foundations of
mathematics. For instance, in constructing the real line by Dedekind cuts,
one now has an existence proof of a structure with certain properties. Now
one can abstract possibly different desired properties for different
purposes, and prove that one has a structure with those properties,
introduce a constant symbol for a structure with those desired properties,
and move on from there.

In this scheme, one of course has to use legitimate set theoretic methods
to prove existence of structures satisfying certain properties. This is
usually trivial. However, one might want to directly construct structures
with desried properties without giving any set theoretic example. Coming up
with an interesting set of construction principles that aren't just really
set theory seems hard and elusive right now.

I don't see the mystery in the fact that the normal moves of mathematicians
are trivially set theoretic or trivially set theoretically understood or
trivially isormophic to set theoretic construction. After all, the most
basic of all mathematical concepts is, arguably, a set consisting of a few
objects. Before you count, you normally want to count something - and that
something is normally a set.

Is it a mystery that so many structures can be simulated as graphs (in
graph theory)?

>For example,
>it could be that set
>theory "works" because it is so rich in symmetry (so close to paradox)
>that most any bit of conceptual mathematics can break the symmetry (but
>not completely reduce to the asymmetry!).

Obviously, set theory is conceptual mathematics, which is one of many
reaons I don't have an adequate grasp of this sentence.

>Any other "system" as madly rich in symmetry could do the job of
>creating an impression of posessing like modelling power. (Indeed,
>dare I whisper: category theory?)

Set theory is special as f.o.m. because there is a clear powerful
conceptual understanding of the concept at least in somewhat limited
contexts. As I have discussed many times on the fom, one should ponder the
short initial segments of the cumulative hierarchy. V(0), V(1), V(2), V(3),
..., V(w), V(w+1), V(w+2),..., V(w+w), where w = omega, under epsilon, are
relatively clear, clean, and conceptually friendly mathematical
constructions. V(w+w) is far more than enough for the vast bulk of
mathematics. Of course, I am engaged in various projects showing where even
this is insufficient, but that is another story.

These small V's are the clearest, cleanest, foundationally potent objects
that we know. There are some others, such as HC. You should be awestruck by
the beauty, clarity, and power of these constructions.

>        (Perhaps a less fancy -- but less suggestive of workable analogy
>for
>exact development -- metaphor might be:  set theory is
>a clay which can assume mathematical shapes all too readily,
>and so therefore all too uninterestingly,  uninstructively.)

F.o.m. is not mathematics. I.e., the point of f.o.m. is not to do
mathematics, any more than the point of unified field theory is to do
biology.

>[1] In fact the reductions or translations from conceptual mathematics into
>set theory generally do not go over so nicely.-- Both mathematical concepts
>and proofs are truncated and distorted.  Here are a number of lite
>considerations, examples.
>        [i] as an exercise just try to so reduce the proofs of the
>first 20 theorems say of Hilbert's FoG as they occur there -- you are
>in serious trouble right off because a number of them -- or if one is very
>sensitive,  none of them -- go over into a predicate calculus derivation.

I have in my possession Foundations of Geometry, David Hilbert, as
tranlated by Open Court Classics, 1988. Not only is there no problem giving
standard set theoretic foundations for the entire book; it even was written
by Hilbert with set theoretic ideas in mind, and mentioned explicitly!!

E.g., I even quote from the first Definition on the first page of the first
chapter:

DEFINITION. Consider three distinct sets of objects. Let the objects of the
first set be called points and be denoted by A,B,C,...; let the objects of
the second set be called lines and be denoted by a,b,c,...; let the objects
of the third set be called planes and be denoted by alpha,beta,gamma,... .

I see nothing unusual about this work of Hilbert as opposed to the vast
bulk of mathematics. It lends itself just as well to formal set theoretic
foundations as other mathematics.

>[2] Instead of making google eyes at set theories reducing power (like an
>18th century creationist at the wonders of nature),  shouldn't one try to
>understand why,  make a serious problem of,  the reduction of so much of
>mathematics to set theory.

No one has managed to uncover any accepted fundamental principles of
construction that aren't trivially set theoretic. They become trivially set
theoretic as soon as one tries to "get rigorous." What would be interesting
is if one could point to a context where, when one tries to "get rigorous,"
one does something that is distinctly not set theoretic. And I mean:
distinctly not set theoretic in the eyes of an expert in standard f.o.m.
who knows how powerful and flexible the usual set theoretic foundations is.


>        THIS IS WHAT I DO NOT UNDERSTAND:  it's an old trick to dismiss
>what you can't examine by the tools of which you are a master to call
>that illusion or psychology.  But the fact that NT finds something
>to be amazed about suggests also that live,  practiced,  unreduced
>mathematics
>has about it a TRACTABLE content that is lost in set theory.

As I said earlier in this posting, nothing is lost when mathematics is
given a standard set theoretic treatment. In general, the set theory is
used to construct desired structures, and constant symbols are introduced
representing unspecified copies up to isomorphism. What is gained is a
level of perfect rigor that otherwise one doesn't have.

>[3]  The Bourbaki had the sense that set theory could be replaced by some
>other
>"foundational theory".-- That set theory is _not entailed_ by the
>mathematical
>structures they set out.

I have continually addressed this point on the fom, even in this posting.
Namely an autonomous theory of structures. So far this has proved elusive.
The most likely people to make the needed breakthroughs are experts in set
theoretic f.o.m. They know what genuine f.o.m. looks like, being steeped in
the relevant works of Aristotle, Boole, Frege, Cauchy, Cantor, Zermelo,
Godel, etcetera.