FOM: {n: n notin f(n)}
Harvey Friedman
friedman at math.ohio-state.edu
Thu Aug 29 21:56:12 EDT 2002
As Buckner and Heck (and others) continue their back and forth, I
would like to make a few comments.
1. As far as foundations of mathematics is concerned, the set
theoretic interpretation of mathematical statements and mathematical
proof is the currently accepted standard with no peer. There are
other approaches, such as restrictions on the set theoretic
approach, or categorical foundations. The latter does not, as least
yet, seem to have status independent of set theory, but even if one
insists that it does, the current formulations are immediately
mutually interpretable with set theory.
2. This set theoretical interpretation of mathematics is extremely
coherent, and totally natural, especially if one restricts to the
level of set theory that is comfortably sufficient for the great
preponderance of mathematics. I am thinking of, say, Zermelo set
theory with the axiom of choice (ZC). Of course, it is of great
interest to see in what way one can or cannot get away with much
weaker set theoretic commitments than ZC.
3. In fact, set theory and its fragments appear to be the only fully
rigorous totally natural completely coherent wide ranging powerful
systematization that we have throughout the whole intellectual
landscape. Its rivals, e.g., Newtonian mechanics, Bayesian
statistics, Euclidean geometry, religious doctrine, etc., fall apart,
in comparison, on the fully rigorous side and/or on the wide
ranging/powerful side. There is nothing else like it, and I regard it
as a major challenge to develop something like it in a different
context. I think this can be done in an effective and impressive way
in many contexts, but it so far has proven too difficult for people
to accomplish. We are not even at the point where people realize just
how vitally important it is to accomplish this, and what is to be
gained by it.
4. The kind of attitudes and objections being raised in the FOM
discussion have not, at least yet, been shown to be any relevance or
usefulness for the foundations of mathematics - or even foundations
of science generally. In comparison with the great systematizations
for f.o.m., these attitudes and objections are merely incoherent,
useless, vague, and arbitrary - at least in their present form.
5. It is the burden of those who hold such attitudes and objections
to develop something substantial out of their attitudes and
objections. If these attitudes and objections are not intended to be
relevant to f.o.m., or the foundations of science, then that should
be stated at the outset.
6. Also relevant is something I have said earlier on the FOM:
The progress of mathematics and science has necessarily lead to the
consideration of concepts that do not coincide with any that are
commonly used in ordinary informal discourse. For instance,
physicists and chemists do not convey their discoveries in terms of
"fire" and "hot" and "wet" and "dry". It was only through great
progress that "water" was, to a substantial extent, preserved.
It is not clear whether mathematics (ultimately formulated in set
theoretic terms) has an interpretation in ordinary informal
discourse. I have conjectured that even set theory with large
cardinals has a convincing interpretation in
ordinary informal discourse, sufficient to prove its consistency
(freedom from contradiction) within ordinary informal commonsense
reasoning. However, it appears that we are some distance from
establishing that conjecture.
The work I presented in my postings #155-157 is intended as a step in
this direction.
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