[FOM] Comments on comments on Corfield's book

[Harvey Friedman]

Reply to Urquhart.

On 10/13/03 10:26 AM, "Alasdair Urquhart" <urquhart at cs.toronto.edu> wrote:

> I was surprised by the fact
> that a few people engaged in heated
> discussions on FOM of David Corfield's recent
> book "Towards a Philosophy of Real Mathematics"
> without having read it.

You can't include me in your list. But in their defense, I think they were
writing about accounts of Corfield's book, whether or not those accounts
were accurate. That is still an appropriate thing to do, as long as it is
clear that they are writing about accounts of the book, rather than about
the book. Do you think that any of these people were not clear about that?

In any case, I am now joining the club, in that I have not read Corfield's
book, but I am explicitly commenting on your comments on Corfield's book.

Is this a legitimate thing to do?

I say yes, because your very interesting email account of some aspects of
Corfield's book assumes a kind of life of its own as an independent
document.

So I will be explicit.

COMMENTS ON URQUHART'S COMMENTS ON CORFIELD'S BOOK.
 
> Anyway, I have now read the book cover to cover,
> and I found it interesting and stimulating.

Nice to hear that. However, you are probably in no need of additional
stimulation, already being overstimulated!
 
> I recommend it to FOM subscribers as an
> enjoyable and provocative book that presents
> a somewhat heterodox viewpoint on the philosophy
> of mathematics.  

If only I had the time... The FOM list itself keeps me too busy (together
with f.o.m. research, etc.), so I appreciate your posting.

>I don't always agree with it,
> but I think I may have more points of agreement
> than disagreement with the author.

>From reading your posting, I wasn't clear just where you disagree, except
with regard to Lakatos.
> 
> Corfield's book has two main aspects.  It
> is a polemical argument for a re-orientation in
> the philosophy of mathematics.

In a sense, I have been suggesting an expansion of activity in philosophy in
mathematics in the direction of closer ties with the foundations of
mathematics. However, I have never suggested that philosophers of
mathematics make any serious effort to do foundations of mathematics - but
rather to incorporate findings in f.o.m., make comments about f.o.m., make
philosophical interpretations of f.o.m., ask f.o.m. questions that may
affect f.o.m. research, etc.

I am in fairly steady contact with several philosophers of mathematics who
do this kind of work (as well as other work of a different character).

So I do not regard myself as particularly critical of the philosophy of
mathematics community.

> There are chapters on computer-assisted proofs,
> Bayesianism in mathematics, Lakatos's philosophy
> of mathematics, and a final chapter on higher-dimensional
> algebra.

Does the book suggest an interesting thread for the FOM regarding
computer-assisted proofs? There already has been some discussion about such
issues as "what level of certainty should be associated with
computer-assisted proofs".

Does the book suggest an interesting thread for the FOM regarding
Bayesianism in mathematics?

Does the book suggest an interesting thread for the FOM regarding Lakatos's
philosophy of mathematics?

Does the book suggest an interesting thread for the FOM regarding
higher-dimensional algebra?

> His main complaint about contemporary philosophy
> of mathematics is its narrowness of focus and lack of
> engagement with current mathematics*.

Foundations of mathematics is not to be identified with (the closely
related, symbiotic) philosophy of mathematics, and I have already given my
views on this issue of engagement with current mathematics. I will elaborate
somewhat on this. 

I say that the history of f.o.m. is a steady but deliberate expansion of the
features of mathematical practice that are incorporated into f.o.m. This
expansion is entirely orderly and entirely appropriate. As the power, depth,
sophistication, and experience in f.o.m. grows, expansion can be properly
supported.

By properly supported, I mean

*without compromise to the traditional f.o.m. standards of decisive results
of general intellectual interest*

> He is a sworn enemy
> of "neo-logicism" and the "foundationalist filter."
> By the latter, he means a lack of interest in anything
> beyond the ideas of the period from 1880 to 1930,
> as the following quote illustrates:
> 
> By far the larger part of activity in what
> goes by the name "philosophy of mathematics"
> is dead to what mathematicians think and
> have thought, aside from an unbalanced
> interest in the 'foundational' ideas of
> the 1880-1930 period, yielding too often
> a distorted picture of that time (p. 5).

I have some sympathy with this point of view. However, it has to be tempered
by some features of mathematics and mathematicians.

First of all, mathematicians are not generally concerned with "general
intellectual interest", "wider conceptual issues", "foundational issues",
"philosophical understanding", etc. The vast preponderance of them do not
see the value for their subject of such things, and develop some sense of
"furthering mathematics" that is left unanalyzed. The occasional attempts to
discuss such things are not impressive and not valuable. There is rarely any
attempt to start anything from first principles, even though it is
definitely possible to do so.

So for philosophers of mathematics to "incorporate what mathematicians
think" into their work, is quite difficult, at least if we are talking about
reasonably modern mathematicians.

In fact, it will take a great deal of imagination and insight to even
formulate the right questions for mathematicians in order to elicit any
informative response.

I say this, presuming that this new kind of philosophy of mathematics is
going to be done at a level beyond commentary. Let alone, in the hard nosed
tradition of f.o.m.

The work that I see as promising along these lines, but which needs to be
done FIRST, is a foundationally perceptive re-exposition of mathematics at
the advanced undergraduate and early graduate levels.

This material is incorporated in some major widely used textbooks that are
quite well fleshed out from the point of view of preparing math students to
move along the road towards a mathematical career. They are good "shop
books". 

However, they are intellectually bewildering, in the sense of providing no
clearly developed overarching themes of obvious general intellectual
interest. It's more like: people need to know this, people need to know
that, trust the author that this is important, trust the author that that is
important, isn't this beautiful, etcetera. (Of course, all this is
absolutely true!).

This is not an issue of foundations in the focused sense of laying down
axioms or deductive systems. It is foundations in the sense of conceptual
analysis, of delineation of motivating themes and purposes, etcetera. This
is certainly f.o.m.

The mathematicians have proved to, ordinarily, be not interested in this -
and/or not good at it. Also, not to see the value in it.

I have had definite ideas of such a re-exposition of mathematics at this
level, which I think is absolutely necessary before even attempting to
incorporate contemporary professional mathematics into work in philosophy of
mathematics. But where is the time?...

> Corfield advocates a "practice-oriented" philosophy of
> mathematics, rather in the style of recent philosophy of
> physics.

Let me comment on this from the viewpoint of foundations of mathematics, to
be distinguished from philosophy of mathematics.

One should not forget that foundations of physics is perhaps roughly at the
level of the foundations of mathematics in 1800. I am not in favor of any
hard push towards 1800 level foundations of modern mathematics. This would
be at odds with the deliberate and steady expansion I discussed above.

Of course, sometimes some narrow aspect of very contemporary mathematics can
be effectively addressed in appropriate f.o.m. terms. For the time being,
that is expected to be a rather isolated event. For major progress, we first
need the foundational re-exposition of curriculum mathematics I discussed
above. 

> I had occasion myself a few years ago to complain in a review
> of how little attention philosophers of mathematics pay
> to what mathematicians actually do, so I think Corfield
> is definitely on to something here.

Whether or not one is talking about philosophy of mathematics or foundations
of mathematics, one should be careful to distinguish various forms of this
view. 

1. There is the taking into account of mathematics at the level before 8-9th
grade algebra.

2. There is the taking into account of mathematics at the high school level.

3. There is the taking into account of mathematics at the college level
before rigor (epsilon/deltas).

4. There is the taking into account of mathematics at the level of calculus
done with epsilon/deltas.

5. There is the taking into account of mathematics at the standard
undergraduate level (math majors).

6. There is the taking into account of mathematics at the level of required
early graduate material.

7. There is the taking into account of mathematics at the basic classical
level, subject by subject - i.e., first graduate course in algebraic
topology, first graduate course in numerical analysis, first graduate course
in differential geometry, etc.

8. There is the taking into account of mathematics at the advanced graduate
level, which may be quite sophisticated, but which is well known and well
worked out, and, on average, at least several decades old.

9. There is the taking into account of mathematics at the contemporary
research level, which makes sense to the typical professional mathematician
whose is outside the subarea, where PhD topics abound, and which form the
basis of grant proposals, etcetera.

10. There is the taking into account of mathematics at the contemporary
research level, which is not penetrable to the typical professional
mathematician, due to unusual novelty and/or complexity, and which form the
basis of grant proposals, etcetera.

So what exactly are we talking about?

As I said earlier, 5 and 6 seem most appropriate for foundational treatment
(earlier levels 1-4 being less critical, but still interesting and
worthwhile, at this juncture).

> There are quite a lot of passages in the book that could
> be read as showing hostility to logic and formal research
> in the foundations of mathematics, but I believe that
> they are rather to be interpreted as directed against
> what Corfield takes to be a narrowness in the current philosophy
> of mathematics community.

Does Corfield have a solution? I have proposed a thoroughly foundational
re-exposition of 5,6. Philosophers can participate in such a project in
conjunction with appropriate people from math and f.o.m.

>This passage (contrasting
> philosophy of physics with philosophy of mathematics)
> illustrates this very clearly:
> 
> The prospective philosopher of mathematics
> quickly gathers that some arithmetic, logic
> and a smattering of set theory is enough to
> allow her to ply her trade, and will take
> some convincing that investing the time in
> non-commutative geometry or higher-dimensional
> algebra is worthwhile.  One of the main purposes
> of the book has been to argue against this (p. 235).

My view is a bit more complicated, as I have indicated above. Let me restate
it again.

1. "arithmetic, logic, and a smattering of set theory" is enough for a
philosopher of mathematics, who is gifted enough philosophically, PROVIDED
they manage to work properly with appropriate people in math/f.o.m.

2. I have serious doubts as to whether there is any point to philosophers
investing time in non-commutative geometry or higher-dimensional algebra, at
least not without locating and working with appropriately highly
mathematical collaborators. They will not be easy to find and work with, in
my opinion.

3. If the standard is lowered considerably to *informed commentary*, then I
am not sure what is or is not a valuable use of time.

4. A much better use of time is to start much lower in that list of 10
above, and trying to give it sophisticated and novel
foundational/philosophical structure.

> 
> * He makes an exception of writers like Penelope Maddy,
> for example.

The context of this footnote may have been forgotten by the reader of this
posting:

> His main complaint about contemporary philosophy
> of mathematics is its narrowness of focus and lack of
> engagement with current mathematics*.

Maddy sought out and had extensive discussions with California set
theorists. Many results of these discussions were incorporated into her two
books Realism in Mathematics, Naturalism in Mathematics.

I suggested to her that her Naturalism program be extended to core
mathematical contexts. I gave my opinion that there were a few
mathematicians worth contacting in this vein - not many - and that one had
to be rather imaginative about just how to approach them in order to get
truly useful information. This was several years ago.

I close by suggesting a principle of symmetry.

In the present culture, one can expect mathematicians to be about as good
at, or interested in, philosophy, as philosophers are good at, or interested
in, mathematics. 

Harvey Friedman