[FOM] General Purpose/Special Purpose

[Harvey Friedman]

Some discussions concerning Alternative Foundations do not seem to take
into account the sharp different between General Purpose and Special
Purpose.

This distinction is already greatly exploited in computational work. We
often see general purpose programming languages gain wide acceptance, but
frequently one wants intense focused applications, sometimes involving
efficiency issues, and one designs special purpose tools, particularly for
large projects, especially when they are open ended.

The same kind of situation can be recognized in foundations, but with some
key differences.

The usual foundations of mathematics was intended to be completely general
purpose, in the sense of meeting certain reasonably clearly formulated
goals for all of mathematics. This makes sense because there does seem to
be a general notion of mathematics that transcends any particular kind of
mathematics, or any particular fashion in mathematics. There appears to be
a common thread.

As a very crude example that any mathematician or math student can
understand, if you are doing topology, one can envision a foundation where
"continuous function on a space" is taken as primitive, although a lot of
issues have to be resolved if one wishes to avoid bringing set theory into
the picture in a general form.

A more subtle special purpose, when looking at mathematics generally, is
computation. Henk and I are talking past each other, as my point only was a
large computation cannot be intellectually digestible, period. It would
seem that Henk does not think that this obvious point is relevant to the
discussion as he sees is, and he might tell us why.

Another subtle purpose is that a nice idea is to arrange for, crudely
speaking, "isomorphic structures are identical". The issue seems to be

1. Is there a way of systematically altering the way we treat identity that
is philosophically coherent?
2. If so, should it be completely obviously "safe" or should we rely on a
metatheorem that it is "safe"?
3. How "safe" is it in practice if one ignores the "safety" issue and
simply proceeds as one would like?
4. Is it worth redoing foundations for the purpose of, crudely speaking,
having isomorphic structures identical?
5. Using some nice finesse, can we simply adhere to the usual foundations
of mathematics, and still have the full effect of "isomorphic structures
are identical" by simply making some key definitions and conventions?
6. If so, then does this have the additional advantage that there is no
real alternative foundations, and that having "isomorphic structures are
identical" completely nonproblematic? I.e., can we have a powerful general
factoring process completely within the usual foundations of mathematics?
7. Should we consider alternative foundations based on ideas like
"isomorphic structures are identical" a legitimate form of special purpose
foundations, designed to facilitate the formalization of certain special
kinds of mathematical proofs? Or is there a deeper philosophical purpose?
8. For any alternative foundations, are issues of coherence, complication,
nonobviousness, safety, etcetera, that arise compensated by what is gained?
9. Should the commonly accepted great advantages of general purpose
foundations of mathematics be reassessed?

Harvey Friedman
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