FOM: Re: FOM tone

[Harvey Friedman]

White wrote: 9:41PM 8/14/98:

>I want to make a few remarks about the tone of discussion in FOM, which
>I've been finding pretty damn awful. I've been considering formulating
>an academic argument against you guys, but I think it's probably more
>direct and accessible if I simply let off steam in this manner.

In your posting, you raise a number of interesting issues in passing, but
restrain from making points throroughly because to want to "let off steam
in this manner," which is unfortunate. Is this an example of the kind of
posting you are complaining about?

In my view, what you say about Simpson's postings shouldn't make you
hesitate to engage in the sort of issues you are concerned with. I find it
trivial to take the edge out of Simpson's postings, while preserving the
essential content. So I don't see what people are exercised about. That's
the main point I wish to make in this posting.

>Now philosophically
>I'm not a foundationalist; I *don't* believe, that is, that the question
>of foundations is *necessarily* the first, or the most important, in
>a particular area of philosophy. That doesn't mean that I'm a post-modernist,
>or an anti-realist, or a relativist, or any of those positions; it just
>means that I disagree with the foundationalists on that particular point
>of philosophical methodology. (And, in fact, I'm a realist, I'm also
>not a relativist,
>and I think that post-modernism is to philosophy as Madonna is to music; but
>none of that stuff is relevant to what I'll be saying.)

It would be interesting if you could say more about this issue "of
philosophical methodology," with examples.

>Given that, I also think that some things in category theory are of
>considerable philosophical interest (but not because they're foundational).
>Namely, I think that category theory gives an interesting formal account of
>mathematical practice; it tells us extremely interesting things about
>identity;
>and it may also give us an interesting formal account of intensionality.
>Now all of this is philosophically interesting, beyond the bounds of
>philosophy of mathematics; philosophers in general would like to know
>about practice, about indentity, and about intensionality. It seems
>to be an interesting philosophical furrow to plough.

It would be interesting if you could say more about what this "interesting
formal account of mathematical practice" is, and what "it tells us about
practice,  identity, and intensionality." I think that conventional wisdom
in f.o.m. is that it doesn't do these things, but rather is a surprisingly
convenient way to present some mathematics in some contexts, and
inconvenient in others. And conventional wisdom is that to the extent that
it tells us about practice, identity, and intensionality, the same things
can be told even more clearly by conventional f.o.m.

However, I have no doubt that if people in f.o.m. holding these
conventional views were to come to see the kind of things you have in mind,
that they would be formulating and proving theorems about this aspect of
category theory. So you potentially have a lot to gain by backing up your
views on this in a clear and convincing manner.

>However, I've not got the least interest in converting you guys; it seems
>to me that our projects ought, ideally, not to clash. I've subscribed to
>this list because I like to keep up with what's going on in reverse
>mathematics.

I would summarize the arguments between foundational category theorists and
traditionalists had on the FOM in this way. The foundational category
theorists emphasize what they view as gross inadequacies of traditional
f.o.m., and propose categorical foundations as a (the) cure. The
traditionalists claim that these categorical foundations are not
intellectually coherent in the precious sense that traditional foundations
are. Furthermore, that traditional foundations can be fairly easily
adjusted and recast in order to address these "gross inadequacies" in a way
that seems to do about as good a job as is claimed to be done by
categorical foundations. Put harshly, the traditionalists consider the
present state of categorical foundations to be a cure that is worse than
the disease.

You will obviously have to do a lot of work in order to convince people to
change their minds, and understandably you don't want to try to do this.
However, many traditionalists would be quite interested in reworking
categorical foundations from the ground up in order to make it really work,
foundationally and philosophically. In so doing, it may no longer look
exactly like the traditional category theory that is so useful in so mnay
mathematical contexts, however.

Matters get particularly contentious when categorical foundations people
reject as an illusion, or at least unimportant, the special intellecual
coherence of traditional foundations, thereby releasing them from the
requirement to make categorical foundations intellectually coherent in any
analogous way. Then there is the flat out claim by categorical
foundationalists that it is already obviously intellectually coherent.

>If someone could come up with a good logical account of
>foundations, that would, of course, be interesting, and the stuff that one
>discovers in an attempt could also be interesting (and maybe not even
>interesting for the reasons that you try to do it; an awful lot of things
>in mathematics have turned out to be interesting for different reasons than
>those for which they were first attempted).

What do you mean by "a good logical account of foundations"?

>Under the circumstances, then, I find the tone of this list rather
>distressing. Here are some examples:

Do you mean that if "someone could come up with a good logical account of
foundations" then you wouldn't "find the tone of this list rather distressing"?

>I) From the exchange between
>Thomas Forster and Stephen Simpson:
>>[Forster]
>> > My feeling is that an important source of hostility to set theory
>> > arises from mathematicians interpreting the foundational claims of
>> > set theory as somehow deconstructing their activity, and nobody
>> > likes being deconstructed!
>>[Steven]
>>Please define what you mean by "deconstruct".  This vague term
>>(borrowed from modern literary theory of the politically correct
>>variety) blurs a lot of important distinctions.
>>
>>Do you mean that many mathematicians don't like outsiders to analyze
>>what they do in terms of general intellectual interest?
>

Would you complain about Simpson's tone if Simpson wrote the following?:

I think it would be valuable if you could define "deconstruct" in this
context. How is this related to the apparent fact that mathematicians do
not like outsiders to analyze what they do in terms of gii?

White comments:

>This seems to beg the question: it assumes that the only way of
>"analysing [mathematics] in terms of general intellectual interest"
>is a *foundational* analysis. Maybe one could argue that the connection
>between
>mathematics and matters of general intellectual interest is fairly
>innocuous, but that foundationalism is perceived as being too pre-emptive?

It would be interesting for you to clairfy what you mean by "foundational
analysis" in this passage.

>II) Stephen again:
>>it's the job of f.o.m. to explain the meaning of
>>mathematics.  So I prefer to confront these hostile mathematicians
>>head on, rather than meekly surrender to them or try to get along with
>>them, as Forster seems willing to do.

White comments:

>Seems unnecessarily monopolistic; *only* fom "explain[s] the meaning of
>mathematics", and anyone who disagrees is "hostile".

Would you complain about Simpson's tone if Simpson wrote the following?:

An important goal for f.o.m. is to explain the meaning of mathematical
statements. We agree that the mathematicians are hostile, and I think that
we should join the issues with them rather than hide our disagreements as
Forster is suggesting.

>III) From the same email:
>>From [one] point of view, set theory is a respectable
>>but narrow niche; it is probably not of much interest to most
>>mathematicians, and it certainly does not have much in the way of
>>general intellectual interest.
>>...
>>On the other hand, there is the viewpoint that set theory is *not*
>>just another branch of mathematics.  Rather, set theory has a special
>>significance as a very successful foundation for *all* of mathematics;
>>this is the view of G"odel, Cohen, Friedman, ....  From this point of
>>view, set theory has tremendous general intellectual interest and is
>>potentially of great significance to all mathematicians.  This is
>>obviously a much loftier and more ambitious view of set theory.

White continues:

>Again, a rather arbitrary dichotomy: either set theory is a narrow
>mathematical niche, or it is of "general intellectual interest" precisely
>because it's foundational.

Would you complain about Simpson's tone if Simpson wrote the following?:

One aspect of set theory is a theory of sets, like: theory of groups,
theory of manifolds, etcetera. As such, there is not enough structure to
interest most mathematicians. And beyond the basics, purely as a theory of
sets, it certainly does not have much in the way of
general intellectual interest.
...
On the other hand, there is another aspect of set theory that is *not*
just another branch of mathematics.  Rather, set theory has a special
significance as a very successful foundation for *all* of mathematics;
this is the view of G"odel, Cohen, Friedman, ....  From this point of
view, set theory has tremendous general intellectual interest and is
potentially of great significance to all mathematicians.  This is
"set theory as f.o.m." and is of immediate gii.

>III) Stephen:
>>In fact, most of the recursion
>>theorists are boycotting FOM.  Only a few of them are subscribers, and
>>they have posted hardly anything.  I wonder why.  Do they explicitly,
>>consciously regard foundations of mathematics as irrelevant to what
>>they are doing?

White comments:

>"Boycotting"? Is this something the recursion theorists have *organised*
>among themselves? Or do they just individually find this an unpleasant list
>to be on?
>
>There seems to be a false dichotomy here: either you think of fom as
>completely irrelevant to your activity, or you subscribe to the fom list.
>But there are lots of positions in between; you might, for example, regard
>fom as, in principle, relevant, but you might not want to refer to fom at
>every turn.

Would you complain about Simpson's tone if Simpson wrote the following?:

There is an unfortunate and noticeable abscence of recursion
theorists on the FOM, who could add much to the discussion.  Only a few of
them are subscribers, and
they have posted very little. I wonder if they question the relevance of
foundations of mathematics to what they are doing, and to what extent this
plays a role.

>IV) Stephen (replying to Thomas Forster):
>>I like Adrian's article "The Ignorance of Bourbaki", which has been
>>discussed here on FOM.  In particular, my posting of 17 Nov 1997
>>16:20:11 contains a review from Mathematical Reviews.  But I don't
>>think Bourbaki is the only source of hostility to f.o.m.  The full
>>explanation of the hostility is unclear, but my best analysis is that
>>it's part of a general trend toward anti-foundationalism and
>>compartmentalization in academia.

White comments:

>OK. Two points here.
>First, I just don't believe that Bourbaki is part of a general trend to
>"anti-foundationalism
>and compartmentalisation"  Andr'e Weil a compartmentaliser? Dieudonn'e a
>compartmentaliser? Serre a compartmentaliser?
>This is so daft I can't take it seriously, and maybe you didn't mean it
>in that sense.
>
>The second point is that there's again this dichotomy: either you're a
>foundationalist, or you're a compartmentaliser.

Would you complain about Simpson's tone if Simpson wrote the following?:

I like Adrian's article "The Ignorance of Bourbaki", which has been
discussed here on FOM.  In particular, my posting of 17 Nov 1997
16:20:11 contains a review from Mathematical Reviews.  But I don't
think Bourbaki is the only source of hostility to f.o.m.  The full
explanation of the hostility is unclear. There does seem to be a general
trend toward anti-foundationalism and
compartmentalization in academia. What is the relation between these trends
and the hostility?

>V) Stephen:
>>My problem with the category theorists, Bourbakians, and other
>>structuralists is precisely that they fail to recognize this.  They
>>delude themselves into thinking that structure alone is important.
>>This has resulted in further fragmentation and isolation of
>>mathematics from the rest of human knowledge.  Jeremy, you are trying
>>to make friends with the structuralists, but is it worth it?

White comments:

>Do we? I don't. This seems to be a confusion of a mathematical area
>(category theory) with an ontological position (structuralism). Well, you
>might
>take the view (as I do) that category theory is philosophically interesting
>without necessarily being a structuralist. (You might also be a
>structuralist without also believing that "structure alone is
>important", just as you might believe that everything is material without
>also believing that quantum theory alone is important.) I also can't see
>why this results in "fragmentation and isolation".

Would you complain about Simpson's tone if Simpson wrote the following?:

My problem with some of the category theorists, Bourbakians, and others
adopting a strong
structuralist viewpoint is that they don't acknowledge this. Many of these
structuralists
have come to emphasize structure to the virtual exclusion of other aspects.
This view is alien to the way things are viewed in virtually every subject
outside of mathematics, and can result in isolation of
mathematics from the rest of human knowledge. Jeremy, are you a strong
structuralist?

>Well, enough of this. I'd really like to be a member of this list and enjoy
>the interesting things you people have to say about reverse mathematics,

That's no problem.

>and also I would not like to feel ranted at every time I get a message
>from this list; but the second seems not to be possible.

This is no problem, either; rewrite Simpson's messages along the lines I
have indicated.

>Ideally I would
>also like to have some reasoned, technically aware,
> *philosophical* discussion about the relation between mathematics and
>human culture in general; but the discussion on this list seems never to
>go into the philosophical arguments at all, merely to use philosophical
>slogans as a stick with which to beat imagined opponents with. Shame.

For this, just start a "*philosophical* discussion about the relation
between mathematics and human culture in general" and see what happens. In
addition to f.o.m. people, there are also professional philosophers on the
FOM.

>Still, it's sociologically interesting; only list I've ever been on when
>the list owner did more flaming than the rest of the list put together.

Gee, I thought I did more.