FOM: GII/clarification
Harvey Friedman
friedman at math.ohio-state.edu
Mon Jan 5 23:02:36 EST 1998
I have been intending to clarify some issues surrounding my use of the term
"general intellectual interest" for some time, but have delayed doing so
since in each case I have thought that putting the notion into action was
more urgent. Here I want to clarify and restate some points. There are many
further points about GII that need to be clarified, which I will get to at
a later time.
1. The general intellectual interest of contemporary high level FOM is very
high as compared to contemporary high level core mathematics. This is true
essentially regardless of how "general" the notion of general intellectual
interest being used. And there are several different "levels" of general
intellectual interest that one can consider. However, to take an extreme,
contemporary high level FOM and contemporary high level core mathematics
have the same general intellectual interest among the entire animal
kingdom. Namely: zero.
2. General intellectual interest plays a great role in the choice of high
level FOM research problems and projects, partly because of the nature of
high level FOM, and partly by explicit design. Since there are so many
topics of general intellectual interest for FOM that are wide open - yet
ripe for significant progress - it is practical to choose those of the most
general intellectual interest. In the context of contemporary high level
FOM, general intellectual interest and other kinds of criteria for the
choice of research problems and projects become intertwined and do not
conflict with each other. This is not, in general, true of contemporary
high level core mathematics.
3. In fact, this tension between general intellectual interest and current
development that is present in most contemporary high level core
mathematics can be documented by an analysis of the writings of the leading
contemporary core mathematicians and the oral presentations of those
leading contemporary core mathematicians. A good source are proceedings of
conferences in which the speakers are admonished to give talks of general
interest to the mathematical community. I mentioned that I would be
reviewing from this standpoint the entire Proceedings of the Centennial
Celebration of the AMS. Other good sources I would like to review from this
standpoint are the Proceedings of the International Congress of
Mathematicians - particularly the writeups of the plenary addresses. There
are also the statements made about the recipients of the Fields Medals that
would be interesting to review from this standpoint.
4. The general intellectual interest of contemporary core mathematics -
whatever one ultimately judges that it is - is not something that has been
made clear. Nor is it something that core mathematicians seem particularly
interested in or compelled to make clear. And this is at a time when core
mathematics is under considerable pressure from funding agencies to justify
itself. So by default, the general intellectual interest of contemporary
core mathematics is usually couched in terms of its applications. For
instance, there is an undeniable general intellectual interest to such
questions as how do fluids flow?, or how can we unify the various parts of
theoretical physics (unified field theory)?, etcetera. There is little
doubt that some aspects of contemporary pure mathematics are at least
connected with significant developments on such problems. By all accounts,
some of these connections with physics are quite tenuous, tentative,
controversial and speculative, and unconvincing to even rank and file
theoretical physicists. And in any case, such connections do not account
for the vast bulk of day to day concerns of high level contemporary core
mathematics.
5. The general intellectual interest of contemporary high level FOM,
although generally substantially clearer than that of contemporary high
level core mathematics, needs to be made even much clearer than it has
been. The fom list can serve as a vehicle for discussion of this interest.
Often one has to reflect carefully on high level FOM in order to state the
results in a way in which the general intellectual interest is most
apparent and compelling. Let me mention a primary example. Let us consider
Godel/Cohen to be the complex of results about the consistency and
independence of the axiom of choice and of the continuum hypothesis (in the
presence of the axiom of choice) from ZF (ZFC). One can on one extreme
a) state the axioms of ZFC, the axiom of choice, the continuum hypothesis,
the notion of formal proof; also give background about the notion of
cardinality in connection with the statement of the continuum hypothesis.
Then state Godel/Cohen.
Or at another extreme
b) assert that there is a commonly accepted system of axioms and rules that
is considered more than adequate to formalize all currently accepted
mathematical proofs. Assert that this commonly accepted system is based on
the single concept of set with membership, and is called standard axiomatic
set theory. That they form the usual axioms and rules of inference for
mathematics. Assert that the two most important problems in set theory -
indeed the two main problems emphasized by the founder of set theory - were
shown to be neither provable nor refutable within standard axiomatic set
theory. More specifically, the first was shown to be neither provable nor
refutable within standard axiomatic set theory, but has been regarded as
having the flavor of an additional axiom. And that when added as an
additional axiom - as is common today - the resulting system is not
sufficient to prove or refute the second of the most important problems in
set theory. This second of the most important problems in set theory is, in
contrast, not of the flavor of an axiom.
I can go on at considerable additional length with all kinds of points in
b) of general intellectual interest, including seminal open problems such
as - what is an axiom? where do they come from? specifically where do the
usual axioms and rules for mathematics come from? what is so different
about the axiom of choice versus the continuum hypothesis? etcetera,
etcetera.
But note that in b), ******one deliberately does not give any formal
definitions whatsoever.***** Of course, one is **invited** to get into
specific details. But the point I wish to emphasize is this: that b)
clearly states the general intellectual interest of Godel/Cohen in a far
more convincing and compelling manner than a). Specifically, the place of
Godel/Cohen in the history of ideas is made clear at the general
intellectual level.
I would like you to get the feeling that there is something rigorous going
on here, in the construction of b), and in the comparison of b) and a).
6. The general intellectual interest of high level contemporary FOM after
1962 is also very high, and the preponderance of this work can also be
couched in such terms. Furthermore, there is a massive amount of high level
FOM that will be done in the future with similar general intellectual
interest.
7. The whole of human knowledge will be recast in such terms, together with
an appropriate system of **invitations**. And these invitations lead to
further invitations, etcetera. This amounts to an hierarchical organization
of all ideas and knowledge that will realize the Leibnizian unification
vision. FOM will form an exemplary model of how this hierarchical
organization of ideas is acheived, as well as the virtually unlimited power
of such an hierarchical organization, not only for the construction and
development of research programs, but also for education.
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