FOM: Is fom mathematics?
Joseph Shoenfield
jrs at math.duke.edu
Mon Sep 7 11:16:15 EDT 1998
This is a reply to Harvey's reply of Sep 1. I think the series of
postings resulting from my criticism of Steve's evaluation of Harvey's
results has produced some interesting issues, but perhaps it is about
time to move on to something else.
Harvey quotes me as follows:
>We disagree on the truth-value of "fom is mathematics"; but this
is less a disagreement over fom then a disagreement over the criteria
for saying something is mathematics.
Since Harvey says nothing contradictory to this, I will assume he
accepts it. Now I said that my criterion for saying something is
mathematics (at least in this context) is that it consist of definitions,
theorems, and proofs given with mathematical rigor. Harvey correctly
points out that I omitted "definitions" in my original positing.
However, in that posting I gave several examples of accomplishments
of fom, including Harvey's favorite, the creation of the predicate
calculus. I said that it was not mathematical in the sense of my
definition, but that its accomplishment was to make possible the
proof of mathematical results such as the completeness theorem.
I later saw that I could say about the same thing more easily by
adding definition to theorem and proof. I don't think this is a
drastic change; formulating definitions has always been regarded as
a part of doing mathematics, just like formulating theorems and
finding proofs.
Harvey does not explicitly state his criteria for saying
something is mathematics, although some of them can be inferred
from his statements. I suspect that he, like most of us, uses
somewhat different criteria in different situations. This seems
like a sensible attitude, rather than trying to lock down the
meaning of "mathematics" once and for all.
Let me say a word about some of the criteria he appears to be
using. In discussing statistics and computer science, he uses as
a criteria the opinion of researchers in the field on the question
of whether their subject is mathematics. This is a kind of
criteria I like; it deals with how research is actually done rather
than with how we think it should be done. I don't know how this
would work for fom or logic (which were once the same field, but
are now considered separate by some). I think that most logicians
would classify logic as a field of mathematics.
Harvey says:
>The disadvantage of calling fom or mathematical logic a branch
of mathematics is that it looks quite weak as a branch of mathematics
compared to number theory, geometry, algebra, analysis, etc.,
especially in term of interactions with other branches of mathematics.
And interactions is perhaps the most highly valed criterion being used
by mathematicians to evaluate branches.
I don't think how good it makes fom look should be a criterion for
saying fom is or is not mathematics. Leaving that aside, I strongly
disagree with Harvey's evaluation of logic. It is a much younger
branch than the other branches mentioned, and during its lifetime it
has always has less researchers that these branches. Considering
this, I think it has provided a very respectable amount of mathmematics
which is outstanding by the usual standards of mathematicians. For
example, the work on forcing from Cohen to Shelah and his followers
is very good mathematics, as is the work on priority theory by many
researchers. I find the word interactions extremely vague, and I
hope Harvey will not feel I am misrepresenting him if I replace it
by applications. I think logic has outstanding applications to other
branches; the algorithmic unsolvability of the word problem for groups
and Hilbert's 10th problem, the Ax-Kochen proof of a modified Artin
conjecture, Hrushowski's recent work in algebraic geometry. I think
Cohen's proof of the independence of CH could be considered an
application of logic to set theory. All of these applications have
received great praise from mathematicians; Cohen received a Fields
medal. Moreover, I do not believe that mathematicians evaluating
mathematics value applications as much as Harvey thinks they do; see
the quote from Bourbaki in my recent posting.
I do not (as Harvey seems to think) advocate the elimination of
intuitive or vague concepts from fom. I do believe that if one
wants to prove mathematical results about such concepts, one must
first analyze them formally; and neither Harvey nor Steve denies
this. My main point is the following. If one thinks that an
informal concept is foundationally important, he is entitled to
say so; others are then entitled to express doubt. In this case,
one must either formally analyze the concept and prove mathematical
theorems which show that the concept is foundationally important,
or admit the the importance is still a matter of opinion. I see
not rational third course. As I said and was quoted by Harvey:
it is no use to just assert very strongly that the concept is
important and that those who do not agree are obtuse. I expected
that Harvey wouls agree to this; but his comment that "obtuse" has
an analogue "tone-deaf" in music suggests that he is not so sure.
I agree there is an analogue. I think that a music critic who
answered another critic's doubt of his statement on the value of
a piece of music by saying the other critic's opinion is worthless
because he is tone-deaf would be behaving in an abominal manner.
I will not be so foolish as to argue with Harvey about reverse
mathematics. During the beginning of reverse mathematics, I had
grave doubts that it would produce anything of interest to logicians
outside the field; but I was wrong. My remarks about "conceivable"
were not intended as a criticism of reverse mathematics; they apply
equally to any topic in logic. (I will disagree with one small
remark in Harvey's discussion on reverse mathematics. I don't think
that Godel was lonely (in the sense of having his work ignored) at
any time after his announcement of the Incompleteness Theorem.
As to foundations in general (by which I mean the part of
foundations independent of the particular field whose foundations
are being investigated), my alternate universe remark was a
picturesque way of stating my feeling that Harvey's vision of its
future could not possibly occur. There is little point in us
arguing about it; the future will tell us who is correct. I would
like to argue a little against what appears to be Harvey's feeling
that substantial foundational work has been done only in mathematics
and computer science, and with his statement that experts in fields
are notoriously incompetent and uninterested in doing foundational
work. I think physics is a counterexample. I think that there
has be much work on the foundation of physics, some of it by very
outstanding physicist, and that much of it has had a profound
influence on the everyday work of physicists. I don't have the
knowledge to document this in detail, but I will cite a few examples
I have come across in reading popular books on physics.
In the nineteenth century, Mach proposed as a principle that all
motion is relative and that absolute motion does not exist. This was
widely debated by physicists, and had a profound effect on Einstein.
Einstein's theory of relativity made big changes in physicists' notion
of time and space; according to some commentators, this is the
principle reason that it is so important in physics. Experts in
thermodynamics have long debated about whether the second law is really
a law of physics or simply a statement of high probability; according
to my teacher in graduate school, the question was still undecided at
that time. There has been much debate over the foundation of quantum
mechanics; for example, whether indeterminateness is really a general
property of the universe. This has involved physicists such as Bohr
and Schrodinger. In recent times, such physicists as Hawking and
Witten have explored other topics in the foundation of quantum mecahanics.
Harvey will likely say that these are not the true problems of
the foundations of physics. Well, they are certainly foundational
in some sense of that term, and they are regarded as key problems of
the foundation of physics by almost all physicist. A theory of
foundations which ignored these problems and concentrated, say, on
the formulation of physics in the predicate calculus would probably
be of very little interest to physicists.
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