[FOM] Scope questions

[Harvey Friedman]

Reply to Pratt 4/21/03 10:41PM.

I can only speak for myself, and guess about others.

>Having participated in this list some years ago, and having returned this
>year, largely in observer mode, I'm prompted to ask to what extent the
>subject of foundations of mathematics is defined by the interests and
>technical specialties of its contributors.

I try to have an overarching idea of foundations of mathematics that, 
at least in principle, allows for serious consideration of approaches 
that differ greatly from the conventional 
Frege/Russell/Zermelo/Frankel/Bernays/von 
Neumann/Gentzen/Hilbert/Turing/Godel approach.

I do not believe that any serious competitor to this standard 
approach to the foundations of mathematics has emerged, although some 
alternative approaches definitely give one the impression that they 
could become a viable alternative. However, in my opinion, none has 
been carried through with enough powerful ideas in order to do 
anywhere near the work that the standard approach does. This doesn't 
mean that it can't be done.

In particular, these remarks apply to categorical and/or algebraic 
approaches, such as the ones that Pratt is an expert in.

Experience shows clearly that it is not an easy task to convince 
adherents to alternative foundational schemes that their scheme does 
not do the job that the usual foundational scheme(s) do - that some 
essential matter is not handled in a convincing way.

In the FTGI series. I was planning to address the issue as to just 
what is accomplished by the conventional foundational scheme(s), and 
try to put it sufficiently strongly and comprehensively so that 
proponents of alternative foundational schemes would see just what 
additional challenging work they need to do.

I feel confident that

a) I am willing to consider alternative foundational schemes, and try 
to see just what is missing, or perhaps admit that nothing is 
missing, provided that the alternative schemes are presented in a 
sufficiently elemental and crystal clear way starting with no 
significant prerequisites - just as this can be done for the 
conventional scheme(s), and as will be done in the FTGI series.

b) There will be various spinoffs in the form of interesting f.o.m. 
(from all points of view) if the disagreements really get joined.

c) Proponents of alternative foundational schemes are unlikely to 
concede any substantial points in the process.
>
>A similar question can be asked of philosophy.  Religion, ethics, paradoxes,
>and logic are staples of philosophy, but literature, art, music, etc. are
>traditionally studied elsewhere, being viewed as simply outside the scope
>of philosophy.

Philosophy of art - aesthetics - is a recognized branch of 
philosophy. So is, by the way, Philosophy of Law.

>
>
>Returning to FOM, what is the place in the foundations of mathematics
>of each of: integers, rationals, reals, complex numbers, polynomials,
>relational structures, algebras, topological spaces, manifolds, buildings
>(in the sense of K-theory), coalgebras, categories, and toposes?  Are they
>all equally important foundationally, or are some of these out of scope?

I don't like the formulation of this question. For me, there are 
first and foremost

*foundational issues and foundational programs*.

There are certain foundational issues and programs in which some of 
these items play a major role.

I don't generally work from such concepts back to foundational issues 
and foundational programs. I work the other way.

However, you can ask me what foundational issues/programs surround 
integers? Rationals? Reals? Complex numbers? Polynomials? Relational 
structures? ... and I could be quite responsive, because I have 
thought about various foundational issues/programs in which these 
items have played a crucial role. But not all of the ones you 
mention. I would not be responsive about some of them, as they have 
not surfaced in various foundational issues/programs I have thought 
about.

>
>In particular (reflecting my own interests), what about coalgebras?
>Do these have equal foundational significance to algebras, more, less, ...?
>And to what extent does the answer depend on the number of people who both
>work with coalgebras and are interested in foundations?

Again, let me say that I for one operate in the other direction. I 
could imagine that Pratt could come up with some compelling take on 
foundations of mathematics for which it is clear that coalgebras play 
a major role. If I bought into it, I might be trying to prove 
foundationally relevant theorems about coalgebras.

However, this hasn't yet happened for me, and I have a huge amount of 
things to think about within the conventional foundational scheme(s).

>
>Coalgebras have been mentioned only twice on FOM, once in a posting by me
>dated 1/13/00 on defining the continuum from scratch as a final coalgebra
>(analogously to defining the natural numbers from scratch as an initial
>algebra), which I felt was a very foundationally significant observation
>about the continuum but which no one responded to; and once (tangentially)
>in an advertisement posted by a workshop organizer for a CONCUR'02 workshop
>mentioning coalgebras as being within its scope.

This is within the categorical approach to foundations of 
mathematics, which seems to be lacking in important ways.

An attractive idea is to replace the extremely clear and unified and 
natural existence axioms in the conventional f.o.m. with a different 
idea that certain kinds of algebraic/categorical objects exist with 
certain algebraic/categorical properties. The idea is to explicitly 
get away from the usual feature of conventional f.o.m. - that 
mathematical objects exist in one immutable absolute form, not up to 
some equivalence relation such as some sort of "isomorphism". E.g., 
there is exactly one empty set, and that is that. There are no copies.

I don't believe that anyone has made this kind of alternative 
foundations work in the sense of doing the job that the conventional 
foundations does so well.

>
>More recently I have worked on comonoids, which are to coalgebras as
>monoids are to algebras, modulo some nice twists.  In the course of this
>work it seemed to me once again that the material raised some foundationally
>significant issues.  This formed the topic of my invited talk at a coalgebra
>workshop in Warsaw two weeks ago.  Associated with this talk is an open
>problem I'd worked on without success some years ago that I was originally
>offering $500 for in January and which in the following months edged up
>to $2000.  Quite apart from any foundational significance, this problem
>would appear to be quite difficult (I won't embarrass those whom I know to
>be working on it).

I am waiting for you to do something that more clearly reflects the 
real Pratt - i.e., put four more zeros at the end of this reward. 
(smile).

>
>>From FOM's essentially complete neglect of coalgebras one could infer
>that coalgebras are of exactly zero relevance to FOM, my argument to the
>contrary notwithstanding.

Speaking for myself, since I never bought into this alternative 
approach to foundations, I never invested the time on such things. If 
it became clear just what is to be gained for f.o.m. through thinking 
about coalgebras along the lines you discuss, then I would have tried 
to add to the discussion.

>Or perhaps it is just that coalgebras have the
>same status for FOM as quantum mechanics has for philosophy: they will be
>studied for their foundational relevance after someone in foundations (this
>does not describe me, I know relatively little about the subject judged by
>the standards of this list) has spent some time with them, much as Popper
>felt equipped to write about philosophical aspects of quantum mechanics
>once he had worked with the subject for some time.
>
...after someone in foundations sees how it fits into an effective 
foundational scheme, or sees how to use it to make an alternative 
foundational scheme effective.