[FOM] Great Achievements of F.O.M. 1
Harvey Friedman
friedman at math.ohio-state.edu
Sun May 22 07:53:30 EDT 2011
This is intended to be a series in which great achievements in f.o.m.
are discussed.
My idea is to respond to all doubters of the greatest of these
achievements by refuting the objections. This often involves proving
new theorems, or making new conjectures, or setting up new research
programs.
As a primary example, when I was a Philosophy Professor at Stanford in
the 1960s, I had my office in the Math Dept. I would often go to tea
and talk to faculty there about f.o.m.. The lack of interest, and even
derision, expressed about formal systems was very striking to me.
I set up Reverse Mathematics as a kind of response to these objections
- "Formal systems have nothing to do with math. Formal systems are
artificially constructed. And so forth." Actually, my original forms
of RM were versions of SRM = strict reverse mathematics, which were
ahead of their time. But that is a rich story for later.
So if, in the course of dealing with doubters of the Great
Achievements of f.o.m., multiple subjects on the order of RM and SRM
get founded - as we hope - then this series will have been worthwhile.
In this first posting, I want to present some background for the later
discussions.
There is a considerable amount of skepticism about f.o.m. in the
mathematics community. The prevailing wisdom in the Mathematics
community, which I have run into many times in many forms and
temperatures, runs like this when most heateds:
"Since mathematics is done on an informal level, formalization is
pointless, meaningless, useless, misleading, countercultural, and
backward. Therefore any results in f.o.m. based on any formal system
is pointless, meaningless, useless, misleading, countercultural,
worthless, and naive. In fact, formalization is constraining of
mathematical activity, and in that sense, very dangerous. It is a
preoccupation that has no place in any self respecting Math
Department, because it is a preoccupation of third rate minds,
attracted to rigidity for its own sake. People who work with formal
systems and formalization know no mathematics, are devoid of any
understanding of mathematics, and have no talent for real mathematics.
Consequently, they hide in their inbred formal systems. This activity
and the people pursuing it must be purged from Mathematics Departments
in this age of limited resources. Given these truths, these people
properly belong only in Philosophy Departments. If they can at least
program a computer, then they might have some place in a Computer
Science department."
In my experience, this kind of view is predominate in the major
mathematics departments in the USA. I am measuring this in terms of
majorities. I know many individuals in these Departments, some of
which have iconic status in the Math community, who regard the above
point of view as utterly absurd. But these are just individuals -
famous and revered as they are.
How did this view come about? What circumstances perpetuate this view?
What is (can, should, will) be done about it, short term (medium term,
long term)?
Before getting into that, let me briefly state some highlights of my
own view.
There is the Foundational Life, and also the Mathematical Life and the
Philosophical Life. These are very different lives, with vastly
different intellectual value systems, perspectives, and expectations.
The two that are farthest apart among these three are the Mathematical
Life and the Philosophical Life. The Foundational LIfe is about
equally distant form the other two.
A very brief discussion of these three Intellectual Lives is presented
in http://www.math.osu.edu/~friedman/manuscripts.html Lecture Notes,
#55.
There are many other lives which are less relevant to our discussion.
These include the Scientific Life (which has major traditional
subdivisions, say into experimental and theoretical), and the Artistic
Life (which also has major traditional subdivisions, such as
performance and composition).
Of all of the major Intellectual Lives, it is the Foundational Life
that is
i. The highest form of Intellectual Life - the deepest, most profound,
and far reaching. Major full time and part time practitioners include
Aristotle, Leibniz, Newton, Frege, Russell, Einstein, Goedel.
ii. It requires an unusual combination of very strong technical and
conceptual powers.
iii. The number of serious practitioners is too small for it to have a
natural home in the current global academic environment.
iv. It is in a very early stage of development - with a tremendously
powerful, totally limitless, future.
v. Future developments in the Foundational Life will profoundly
revolutionize mathematics, science, engineering, politics, economics,
law, and art.
vi. Its most highly developed component of the Foundational Life, at
this time, is, by far, f.o.m.
We are of course going to focus on the f.o.m. component, as this is
the FOM email list.
However, let me just say in passing that I regard a number of my other
intellectual activities as part of my Foundational Life. These include
Concept Calculus (mutual interpretation of: i) groups of principles
about informal concepts; ii) abstract mathematical principles (via set
theory).
Software Verification (directed toward programming language design).
Interactive Educational Software (for teaching mathematical rigor).
Music Performance/Recording (includes both real time and non real time
piano performance, directed toward understanding musical
microstructure).
I have devoted my Intellectual Life to the Foundational Life. In the
earlier years focusing entirely on the f.o.m. component, and in later
years, branching out. However, there has been increasingly rapid
progress in the f.o.m. component of my efforts. The original intention
in graduate school was to spend about 5 years in f.o.m. before moving
on to a succession of areas within the Foundational Life. The 5 years
became 45 years (time flies!). I now look forward to substantial
expansion of my Foundational Life - better late than never.
This component, f.o.m., in turn grew out of the Philosophical Life. At
some point, it became rather evident that if f.o.m. is going to come
to grips with substantial mathematical issues of various kinds, then
it will need a rather substantial mathematical development on several
fronts. This spawned the development of mathematical logic - a spinoff
of f.o.m., where fundamental structures and notions arising from
f.o.m. are investigated for their own sake, and also for their
connections with various topics in mathematics proper.
So f.o.m. migrated from Philosophy Departments to Mathematics
Departments, where it morphed into mathematical logic. Currently,
there is very little f.o.m. being done in Mathematics Departments or
in Philosophy Departments - or anywhere. The remnants of f.o.m. are
Mathematical Logic in Mathematics Departments and Philosophical Logic
in Philosophy Departments. Yet there is a substantial interest in
f.o.m., as witnessed by the FOM email list which I founded as early as
1997 (with help from Steve Simpson, the first Moderator). There is a
hunger out there. Kurt Goedel continues to be revered.
The distinction between mathematical logic and f.o.m. is clear but
perhaps subtle; many people are not sensitive to this distinction,
particularly those in mathematical logic who grew up in an environment
where the f.o.m. perspective had long since virtually disappeared.
Yes, students generally have some sort of feeling that there is some
kind of huge difference between Goedel's work and current mathematical
logic. But this is rarely identified by students for what it is: a
major, even profound, shift in perspective. Away from profound
matters, and toward specialized matters.
So let us return to our questions: How did this view of mathematicians
come about? What circumstances perpetuate this view? What is (can,
should, will) be done about it, short term (medium term, long term)?
The (preponderance of) mathematician's repudiation of f.o.m. also
includes some "profound indifference". There is a set idea that
logical issues simply, on principle, cannot be of any importance, and
are a waste of time. This profound indifference may not be accompanied
by the kind of open hostility represented by the above diatribe in
quotes (starting with "Since"). However, when pressed to opine, in
some context or another, an at least partial expression of the above
diatribe surfaces. Often no verbalization is required among so many
like minded scholars. It is simply understood to be the case.
My theory as to what perpetuates this view goes as follows. Consider
how long mathematics flourished before even the serious beginnings of
the instillation of rigor - let alone the construction of ZFC. Clearly
there are great obstacles for most of even the greatest of
mathematicians to getting a sense of how formalization works, and just
how many ways there are of formalizing things. So the natural state of
affairs is to rely on one's intuition and not be concerned with
formalization - and never get schooled in just how surprisingly
flexible formalization really is.
Consequently, there is a widespread and mostly silent suspicion about
formalization among mathematicians. That formalization is somehow
inflexible and restrictive and artificial.
The truth of the matter is that it is extremely flexible. However, in
most ways of going about it, it is indeed artificial, since one is
generally forced to make a bunch of choices that make absolutely no
difference whatsoever, and never need be reconsidered. There are also
ways of formalizing which are trickier, but which largely remove these
artificialities. The artificialities, for the preponderance of
foundational purposes, are of course entirely harmless. Naturally, one
can investigate artificial-free formalization as a topic in itself (in
which I am interested), and that is a different story.
By far the most straightforward and well understood autonomous
formalization is through set theory. Here there is a great deal of
useful flexibility. One can also sugar it by adding a judiciously
chosen list of new primitives, with appropriate axioms, and then prove
appropriately formulated existence and uniqueness theorems. But for
the Great Achievements of F.O.M. that we have now, it is generally
simplest to avoid such sugar, and proceed in the well known
straightforward manner, accepting artificialities that don't cause any
problems. One proves that the artificialities don't cause any problems
by appropriate uniqueness and isomorphism theorems.
The usual foundation for mathematics via set theory - in particular
ZFC - employs a sharp demarcation between the so called logical part,
and the so called set theoretic part.
There are some serious delicate issues regarding just why things are
done in this way, and why the logical part is what it is - which is
first order predicate calculus with equality. In practice, it must be
heavily sugared. There is the problem of saying something interesting
about just what we mean by sugar. A major component of sugar is of
course abbreviation facilities. Without it, you can't really actually
formalize anything.
There is also the issue of just what is meant by "mathematics is
formalized in ZFC". There are several dimensions to this question.
First of all, it is false, if taken literally. People write papers,
which are not really formal at all. So one usually says "mathematics
can be formalized in ZFC". But what does "can" mean?
Let me insert a fact. When I interacted about a paper of mine with
Charles Fefferman, acting as Editor of the Annals of Mathematics, he
made a very clear and definite statement as follows.
"In order for a paper to be accepted by the Annals of Mathematics (as
far as he was involved), the proofs must be readily formalizable in
ZFC. Any further assumptions must be explicitly stated. Any assumption
that is part of ZFC does not need to be stated."
Let me move on to: What is (can, should, will) be done about it, short
term (medium term, long term)?
I classify what is happening now in logic in Math Depts in the
following rough categories.
1. Pursuit of Mathematical Logic for its own sake, as a spinoff of
f.o.m. Here connections with mathematics proper are besides the point.
Rather, one is interested in a kind of normal intense development of
the deep legacy of structures and concepts spawned by f.o.m. - in
which they are naturally extended and generalized - in a way
(allegedly) similar to what is going on in other highly specialized
areas of mathematics.
2. Pursuit of connections of Mathematical Logic with mathematics
proper. Here there is usually a rejection of any intrinsic interest of
Mathematical Logic development in 1, and the aim is toward connections
and applications (new proofs, simplified proofs, and new findings).
3. Foundations of Mathematics. Here the investigations are driven by
deep and systematically framed conceptual/philosophical considerations
of great general interest, following Goedel's modus operandi. However,
there is very little work generated in this way. Rarely is anything in
line 1 of general interest, even to mathematicians. Rarely is anything
in line 2 of general interest outside mathematics. F.O.M., at the top
level, is of great general interest - and, generally speaking, of
considerably higher general interest than major developments in either
Mathematics or Philosophy. This was clear in the 1930's with Kurt
Goedel.
Inevitably, things are not entirely black and white, and there is
blurring. Sometimes line 1 finds, perhaps accidentally, something in
line 2, which is opportunistically pursued somewhat before going back
to line 1. Or sometimes line 2 finds that it needs a period of line 1
development, with the expectation that it will eventually be in line
2. I mean this both at the people level and at the group level.
Line 3 may unexpectedly draw from line 1 and even sometimes line 2,
but only for a clearly stated higher purpose.
I am going to stop here. This will give you an idea of how I want to
talk about the Great Achievements of F.O.M., and how a careful
consideration of them, in detail, leads to an enormous number of new
(an old, and somewhat new) research directions in f.o.m.
Harvey Friedman
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