FOM: Tymoczko's book; "quasi-empiricism"; the gold standard

[Stephen G Simpson]

Joe Shipman 28 Jan 1998 15:12:42 writes:
 > Yesterday I bought an anthology "New Directions in the Philosophy
 > of Mathematics" (Revised and Expanded Edition, Ed. Thomas Tymoczko,
 > Princeton University Press 1998, ISBN 0-691-03498-2) ....  I think
 > contributors to the FOM list (especially Harvey and Steve) need to
 > read this book and answer its challenge.  ....

To make a start on this "homework assignment", I got the the original
1986 edition out of the library.  So far I have read all of the
Tymoczko material and essays by Hersh, Putnam, Goodman, Davis/Hersh,
and Chaitin, plus I have skimmed some of the other essays.

It's an interesting book.  Tymoczko has carefully selected, ordered,
and introduced the various essays in order to support his main points,
which are: (1) "foundationalism" has got to go; (2) there is need for
a new direction in philosophy of mathematics, called
"quasi-empirical".  The latter direction is characterized by Tymoczko
and Hersh as paying greater attention to "mathematical practice",
i.e. the normal activity of the "working mathematician".

[ I don't know about the rest of you people, but I'm getting fed up
with "working mathematician".  This term seems to imply that the
mathematician is or should be some sort of proletarian factory worker,
concerned only with a specialized task on the assembly line.  I much
prefer the image of the "professional mathematician", who proudly
takes responsibility for the larger perspective and exercises critical
judgment.  Here on the FOM list, we are professionals. ]

Among the aspects of mathematical practice that Tymoczko emphasizes
are: conjectures, heuristic arguments (Polya), non-rigorous discovery
techniques, dissemination of results, errors in published proofs,
computer-aided proofs, changing standards of rigor, etc.  Tymoczko's
gripe with "foundationalism" is that, according to Tymoczko, it
ignores these aspects.

There are several strange things about Tymoczko's book:

1. In the introductory essay "Challenging Foundations", Tymoczko
attacks "foundationalism", but he never defines that concept.  Ditto
for the lead-off essay, "Some Proposals for Reviving the Philosophy of
Mathematics", by Reuben Hersh.  I view this omission as truly
shocking.  Is this typical of the standards in academic philosophy
publications?

[ I suspect that Tymoczko's implicit notion of "foundationalism" is
very different from f.o.m. as I have defined it in this forum. ]

2. Tymoczko never acknowledges that rigorous proof is the "gold
standard" in mathematics, has been so since Aristotle, and is likely
to remain so for the foreseeable future.

[ For those unfamiliar with this reference: Throughout most of
recorded history, the gold standard was a social institution whereby
gold was recognized as an objective medium of exchange and standard of
value, i.e. objective money.  In competition with other currencies,
gold was universally regarded as the best, most stable, most reliable,
etc. and was used for all high-level transactions.  In the 20th
century, governments abolished the gold standard by forcing acceptance
of fiat money which can be manipulated via monetary and fiscal policy.
(As a probably irrelevant aside, I seem to remember being told that
one of Hugh Woodin's ancestors played a prominent role in this process
of currency debasement.) ]

3. Tymoczko never acknowledges that, in the present historical era,
the orthodox and almost universally accepted explication of
mathematical rigor is formal provability in ZFC, Zermelo-Frankel set
theory with the axiom of choice.

4. Tymoczko's term "quasi-empiricism" would seem to promise an
objective, reality-oriented approach to mathematics.  Unfortunately,
this promise is not fulfilled.  Indeed, many of the essays emphasize
the social or cultural aspect of mathematics.  The overall message
seems to be that mathematical truth resides in a social process rather
than the individual mathematician.  Hersh explicitly advocates
subjectivistic consensus (see also my posting of 11 Jan 1998 03:55:21)
and rails against the idea that mathematicians should insist on
objective certainty.  What about the lone genius who sees farther than
those around him?  No answer.

5. Despite references to "quasi-empiricism", Tymoczko's book says
nothing about applied mathematics.  For instance, there is no mention
of current interesting work in fluid mechanics, where rigorous
mathematical analysis interacts with laboratory experimentation.

Obviously I'm not very happy about the overall tone of Tymoczko's
book, and his distortions regarding the current state of f.o.m.

Nevertheless, Tymoczko's book contains several interesting essays,
e.g. Tymoczko's own essay about the 4-color theorem, suggesting that
we need to broaden the concept of "mathematical proof" to include
computer-aided proof.  I would frame this differently, as a question
of distinguishing among various kinds of mathematical proof.  Rigorous
conceptual proof is the gold standard, but computer-aided proof can
also be valuable.  This reminds me of Aristotle's discussion of
reasoning which is correct but not fully scientific.  For Aristotle,
scientific reasoning gives not only knowledge, but knowledge of the
cause.

A note for the next book on philosophy of mathematics: I would like to
see this set in a wider context where issues and programs in
foundations of mathematics can be compared and contrasted to issues
and programs in other foundational disciplines.

-- Steve