FOM: fom as mathematics
Joseph Shoenfield
jrs at math.duke.edu
Tue Sep 1 11:55:47 EDT 1998
This is a reply to Harvey's comments on my posting with the above
title. We agre more than I thought we would, so I'll only comment on a
few points.
We disagree on the truth-value of the statement "fom is mathematics";
but this is less a disagreement over fom than a disagreement over the
criteria for saying something is mathematics. I find Harvey's criteria
interesting, and partly agree and partly disagree with what he says about
them; but I won't discuss this here, since it has little to do with my
original posting.
My original point was that the really important results in fom are
mathematical; that is, they consist of definitions, theorems, and proofs,
all meeting the established standards of rigor in mathematics. I said
this implied the best procedure when one encountered an interesting
intuitive idea in foundations was to replace it be a precise idea and try
to prove useful theorems about the idea. Harvey suggested changing
"replace" to "analyse formally", which is quite acceptable to me, but
otherwise seemed to agree with most of what I said. But he says:
>Some intuitive ideas may not yield to such analysis, but still may
be essential to consider. On doesn't simply pretend that the concepts
don't exist.
Sounds good; but what should one do? It is no use to just assert
very strongly that the concept is important and that those who do not
agree are obtuse. Perhaps one should just put the concept aside until
another day, as one usually does with problems one cannot solve.
Harvey disagrees with my statement that there are no significant
results on foundations in general, but I do not find his remarks on this
convincing. There are fields in which there has been substantial work on
foundations; e.g., physics and biology. Some good and not too
technical books on this are Sciama's "The Unity of the Universe"
and Crick's "What is Life". (The former, pointed out to me some years
ago by Robin Gandy, is wonderful.) I don't think the authors would
take kindly to what appears to be Harvey's suggestion that they use the
predicate calculus and other logical tools as a basis for their work.
The problem is that in each field, the substantial work on foundations is
done by researchers in that field; and this is natural, since one cannot
say anything substantial about the foundations of the field without a good
understanding of the work that has been done in that field. The result
is that the work on foundations will be done using the tools of the
particular subject. I think Harvey's dream of departments of foundations
appearing in academia and taking over thw work of foundations of
particular fields could only happen in an alternate universe.
Harvey seems to have no strong objections to my analysis of the
achievements of reverse mathematics. He says I should amplify my
statement that the next step should be to prove mathematical theorems
about these concepts (the five basic theories). Well, the theorm which
says that these theories are linear ordered has the right form, but has
two disadvantages: the proof is trivial and uniformative, and the
foundational significance of the theorem is not clear. I am sure Harvey
agrees that a theorem which people would agree explains this phenomena of
linear ordering would be a jewel of reverse mathematics.
I am sorry that I must disagree with Harvey's statement that it is
inconceivable that the idea of reverse mathematics is not a permanent part
of fom. Improbable perhaps; but not inconceivable. If the researchers
in reverse mathematics II do not obtain results at least as interesting as
those in reverse mathematics I, the subject will disappear from fom.
Researchers may continue their work, but they will be a lonely crowd. I
don't think that even Nostradamus could predict whether this will happen.
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