[FOM] Great Achievements in F.O.M. 2
Harvey Friedman
friedman at math.ohio-state.edu
Mon May 30 11:50:22 EDT 2011
I now have a clear idea of how I want to develop this series. My
numbered postings 1-463 are very different - they are (almost always)
abstracts of recent results with a reasonably high level of
confidence. The latest numbered posting is http://www.cs.nyu.edu/pipermail/fom/2011-May/015464.html
, although #464 has been submitted.
In this series, I am going to systematically discuss all coherent
lines of research in Mathematical Logic and f.o.m.
I will focus on the extent to which these lines of research and their
results constitute substantial contributions to f.o.m. - why and in
what way. In most cases, these analyses will lead quickly to
reformulations of these lines of research that substantially upgrade
their f.o.m. content.
This process will not only result in a very large collection of
substantial f.o.m. projects, presented for the first time in an
immediately accessible way, but will also flesh out in much greater
detail than ever before just what is the essential major difference
between
i. f.o.m., with its great general intellectual interest; and
ii. comparatively narrow developments in mathematical logic,
mathematics, and philosophy.
I strongly encourage readers to offer their own substantial f.o.m.
projects. As time permits, I will discuss these offerings in terms of
whether they meet the criteria for being substantial f.o.m. projects.
I may be arguing for or against their inclusion as substantial f.o.m.
projects. As indicated above, I will strive to reformulate these
offerings in order to upgrade their f.o.m. content.
For practical reasons, in order to limit the scope of the discussion
to a manageable level, I will restrict the discussion to research
programs in Mathematical Logic and f.o.m., avoiding any detailed
discussion of projects relating to, e.g., Computer Science or General
Philosophy. Obviously, I regard it as also vitally important to
discuss such projects as well, and may well decide to expand the scope
much later - or in a different series.
I will now start the discussion with some preliminary remarks in order
set the stage for this series - and to give you a clearer idea of the
kind of thing I have in mind.
First of all, Mathematical Logic is a spinoff of f.o.m., which
comprises a detailed investigation of the fundamental intellectual
structures created by f.o.m.
Mathematical Logic can be usefully divided into two strategic
objectives. Of course there are cross connections, but still this
division is very clarifying:
a. Investigations of these f.o.m. created structures for their own
sake, positioning such investigations as legitimate subareas of
mathematics. These inevitably lead to refined structures and more
investigations, in some generally coherent way. E.g., we see r.e.
sets, degrees of unsolvability, transitive models of ZFC, with set or
proper class domains, arbitrary models of first order theories, etc.
b. Using, or augmenting or refining, these f.o.m. created structures
to connect up, or apply to, topics in core mathematics. E.g., first
order definability, o-minimality, model complete, stability,
amalgamation, imaginaries, Dialectica interpretation, etc.
These strategies a,b are in sharp contrast with the incomparably
grander vision of f.o.m., which is part of what I call the
Foundational Life.
In a blurb I recently created in connection with my Musical Life -
which I regard as an integral part of my Foundational Life - I wrote
as follows:
"Friedman follows The Foundational Life, which focuses on fundamental
investigations into the essence of diverse areas of thought. Friedman
is best known for his work in the Foundations of Mathematics."
The Foundational Life deals directly with clearly stated challenges
that are of immediate interest to the widest possible audience of
scholars and intellectuals - i.e., challenges of general intellectual
interest. Specialized issues whose audiences are limited to scholars
in a particular area are of secondary importance in the Foundational
Life.
In the Foundational Life there is special attention paid to the choice
of fundamental concepts, which, upon appropriate creative analysis,
are likely to lead to the creation of penetrating new forms of
systematic knowledge. But without the right choice of fundamental
concepts, either an appropriate creative analysis is impossible or out
of reach, or such an analysis will not lead to the creation of
penetrating new forms of systematic knowledge.
Generally speaking, the weakness of the Philosophical Life as compared
to the Foundational Life is that in the Philosophical Life, there is
generally a focus on concepts which may be fundamental in certain
senses, but are not likely - usually obviously not likely - to lead to
the creation of penetrating new forms of systematic knowledge.
Of course, by far the biggest exception to this general rule is the
lead up in the philosophical community, which, together with work
among leading figures in the mathematical community, led to f.o.m. I'm
thinking of, above all, Frege and Russell.
F.o.m. treats *mathematical thought* as an object of study. I don't
want to use *mathematical structures* here since that needlessly
carries an ontological commitment.
As far as f.o.m. is concerned, its fundamental structures - which
spawned mathematical logic as a spinoff as indicated above - were
created to serve a much higher purpose, which is the Foundational
Life. As f.o.m. develops, new fundamental structures are created,
again for higher purposes. All earlier fundamental structures created
by f.o.m. are continually examined and reexamined, evaluated and
reevaluated, as to their appropriateness for further developments.
There is no focus on development of the fundamental structures for
their own sake, except to the extent that such investigations are
deemed likely to be useful for much higher purposes.
In contrast, this constant reexamination and reevaluation, with
everything up for grabs every second of every day, week, year,
century, millennium, etcetera, is not in character with mathematical
logic, and the Mathematical Life. This also doesn't occur in anything
like the same way in the Philosophical Life, and the reason for this
is that in the Philosophical Life, the creation of penetrating new
forms of systematic knowledge is not the primary concern.
Since f.o.m. treats *mathematical thought* as an object of study,
there is no particular emphasis on whether developments in f.o.m.
actually affect mathematical practice. Indeed, developments in f.o.m.
have directly affected mathematical practice, usually in the form of
impossibility results. I.e., nobody tries to give a general algorithm
for certain important decision problems, and nobody tries to prove or
refute certain statements - at least without knowing full well that a
new axiom foreign to existing mathematical practice is required. Of
course, if you include mathematical logic as a spinoff of f.o.m., then
the indirect effect on mathematical practice includes b) above.
Imagine one has a new theory of star formation. One does not measure
the power of this theory on the basis of how it affects actual star
formation. Or a new theory of evolving life forms. One does not
evaluate this theory on the basis of how it affects evolving life forms.
Thus the focus of f.o.m. is quite far from "influencing mathematical
practice" - despite the fact that it has affected mathematical
practice in the limited way described above. F.o.m. is an integral
part of, and in fact the most highly developed part of, the
Foundational Life. The Greater Intellectual World has no particular
interest in the minutiae of mathematical practice - except for Grand
Structural Features. Typical results in core mathematics are generally
far beneath what is required for general intellectual interest. Grand
Structural Features of mathematical thought ARE appropriate material
for general intellectual interest.
Of course, one sees, on rare occasions, highly atypical results in
core mathematics, which may arguably rise to the level of general
intellectual interest. Obviously this is rare, as evidenced by such
measures as the TIME/LIFE list of 20 most influential thinkers of the
20th century. This lists 2-3 f.o.m. scholars, 5 physicists, and 0 core
mathematicians.
So strategy b) above is in very sharp contrast to that of f.o.m.
Furthermore, strategy b) is generally marked by an uncritical
acceptance of the intrinsic importance of virtually any topic in core
mathematics. This is a viewpoint and modus operandi that is entirely
foreign to f.o.m. and the Foundational Life, which applies a uniform
standard of general intellectual interest, even if some development
has the label "core mathematics", "core philosophy", "core physics",
"core computer science", "core legal theory", "core political theory",
"core music performance", etcetera.
I will stop here, as this should set the stage for further postings in
the series, as well as useful feedback from you.
Harvey Friedman
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