[FOM] 461: Reflections on Vienna Meeting

[Andre.Rodin at ens.fr]

Dear Harvey,

thank you so much indeed for this very interesting and very rich report about
the past meeting in Vienna - and also for posting the link to slides of your
talk! I comment here only on some of your most general points.


As you can see, I began by emphasizing that there are distinct forms of
intellectual lives, where I briefly discuss

The Mathematical Life
The Philosophical Life
The Foundational Life

These have very different value systems and aims. The overlapping concerns
between the Mathematical Life and the Foundational Life, and between the
Philosophical Life and the Foundational Life, dwarf the overlapping concerns
between the Mathematical Life and the Philosophical Life, at least at the
present time and for considerable time past.


I'm not convinced that what you call the Foundational Life is indeed on equal
footing with what you call Mathematical and Philosophical Lives. In my
understanding FOM is a boarder area between maths and philosophy that itself
has no precise boundaries neither with philosophy nor with maths.  I talk now
about FOM  in the sense, in which this expression (I mean the full expression
rather than the abbreviation) is understood in the general intellectual
discussion - but not about FOM in your specific sense. FOM in your sense is one
among a number of other research programs that use the same title. I doubt, in
particular, that Voevodsky's recent proposal in (what he calls) FOM 
http://video.ias.edu/univalent/voevodsky  qualifies as FOM in your sense 
(please correct if needed). Lawvere's categorical foundations definitely don't
(as Lawvere himself makes it clear).   My own research in (what I call) FOM
doesn't qualify as FOM in your sense either, in particular, because it
essentially involves historical considerations, in a way which is incompatible
with your notion of FOM if I understand it correctly. This notwithstanding I
can see no reason to give up the title.

In my view this vagueness of the expression "FOM" in the general talk is not a
disaster but an advantage, because it facilitates a rational discussion between
different researchers working on close subjects; it goes without saying that a
preliminary for such a discussion is a clarification of what each researcher
means my FOM; then it may become clear that some of these people simply work on
close but different subjects or - which is the mostly important case - that
there are some genuine philosophical and/or mathematical controversies between
their approaches and arguments. Such controversies often turn to be both
mathematical and philosophical; they can be called FOM controversies without
any need to give a more precise definition what FOM is about.  A great thing
about the natural language is that no person and no group of people sharing a
common interest have a full control on it; the natural language stabilizes as a
result of multiple linguistic interactions. I'm convinced that this capacity of
natural languages is important also in science, maths and particularly
philosophy.

 FOM in your sense indisputably remains a strong continuing research program but
I'm not enthusiastic about it for reasons briefly sketched below. In your
slides you talk a bit about its history tracing it back to Gödel and then
further back  to the 19th century work on foundations of Calculus. As an
outside critic I agree with the first part of this description but have a
reservation about the second. The reservation is obvious: this part of the 19th
century maths may be equally claimed by people (by me for one) working on other
FOMs  than yours as a part of *their* legacy. Your claim that FOM *begins* only
in the 19th century is understandable given the narrow sense of FOM that you
make explicit - but anyway it strikes me as historically erroneous (I provide a
relevant argument below).

Thus delimiting the area of FOM in the way you suggest seems me a bad idea
because it prevents your research program from any external critique. Actually
the attention you pay to arguments of Angus and other people who think
differently (including myself) fully convinces me that to protect your research
from an external critique is not your real intention. So my critique here
concerns the terminological choice rather than the substance. However this
particular terminological choice seems me an important issue because among
other things it has a bearing on the power game between different group of
researchers.

Now briefly about the substance.  I agree that the set concept at the time of
its emergence appeared to be a powerful unifying concept in mathematics. I also
agree that this fact makes set theory relevant to FOM (so our different notions
of FOM coincide at this point). But I also observe that in the history of maths
there were a number of other concepts that played a similar role in different
times. In particular, in 17th century maths such a role was played by the
general concept of magnitude (that generalized on more traditional concepts of
geometrical magnitude and number). This is why (leaving more general
philosophical reasons aside) I think that FOM has its history dating back at
least to Euclid's time. Looking back in time one can observe that the history
of FOM involved drastic changes of fundamental concepts and that these changes
were essential for the mathematical progress. One consequence of these
historical consideration concerns the future but not the past. There are strong
evidences - and some leading mathematicians including Manin claim this
explicitly - that the set concept no longer affords to play the unifying role
in today's maths and is gradually replaced by the category concept. This is
still a relatively new development, so nobody knows yet how things are going to
be settled. What is obvious though is that there are rapid developments going
on here. And the history of the subject suggests that the ongoing
transformation of FOM is not something exceptional but something that happens
in maths all the time. Claims about the irrelevance of set theory (and hence of
FOM in your sense) made by working mathematicians (I refer now to you posting)
are further evidences that confirm my guess about the role of set theory in
today's maths. This suggest that a sound theory of FOM must take such
transformations of FOM into consideration. I cannot see that the FOM in your
sense does it.

Another critical point, that I would like to make, concerns the axiomatic method
used in ZFC and more generally in FOM in your sense. This method relies onto a
strong epistemic assumption according to which Logic is an ultimate foundation
of any mathematical theory. Historically speaking this is an old idea dating
back to Aristotle and then pushed forward by Schoolmen in their attempts to
make science, in particular make physics. The relative in-success of the
scholastic science and the great success of the modern science and the modern
maths that gave up the old aristotelian approach is a strong historical
evidence  that there is something fundamentally wrong in the aristotelian
approach. I don't believe that the replacement of the traditional aristotelian
by the modern FOL solves the problem. There are understandable historical
reasons why this rather old-fashioned epistemic approach has been retaken in
maths of the beginning of the 20 century but in my understanding today it
becomes more and more clear that it was a wrong strategic decision. The
relative in-success to extend FOM to science seems me to be an evidence for it.
This issue, of course, requires a more systematic philosophical argument, which
cannot be given here.

My final remark concerns the idea formalization. In my understanding a formal
mathematical theory is a mathematical model of a mathematical theory rather
than mathematical theory proper. The impressive development in logic and in
particularly in FOM in your sense is due mainly to the idea to apply maths to
logic rather than to the idea to apply logic (let alone Logic) to maths. In
particular it allowed to give precise mathematical answers to questions about
(in)completeness that you stress in your slides. However these mathematical
answers are all given by means of the "usual" maths (please correct if I'm
wrong); what is unusual in such a mathematical study is rather its object (a
formal theory) rather than its method. This leaves the following questions
wholly open: what is the evidence that formal theories are adequate models of
informal theories; what are criteria of such an adequacy?

It is surprising that in spite of so many fundamental disagreements with your
approach to FOM I readily agree with more specific things that you say in your
slides, in particular about the usefulness of proof assistants.

Thanks so much again for your arguments that make me react so extensively!
Unfortunately I didn't manage so far to get access to Angus's slides, otherwise
I could comment more precisely on your exchange.

Andrei