FOM: F.O.M./PureMath/GII

[Harvey Friedman]

This is a response to the latest from Lou, 3:31AM 12/28/97.

>I would have reacted earlier to some recent messages of Harvey,
>but I broke my right arm a week ago in a stupid accident, and had to
>take it easy for a few days. Let me get back into the ring.

Welcome back! I, and probably everybody else missed you! By the way, if you
appreciated FOM more, of course you wouldn't have broken your arm; or at
least it would have healed instantly. Such is the power of high level FOM.

>I won't descend to the level of Harvey's message to Thayer, but will
>otherwise not mince words, following the example of the "new rules" message.

That's because I didn't write anything like Thayer. Take a good look at it
again. If I wrote like that, you would have responded like I did to Thayer
- or at least you would have thought of responding like I did to Thayer.

**Lou uses GII for "general intellectual interest."**

>   Previously I called Harvey's notion of GII suspect.
>In an earlier round when Harvey and Steve appealed to GII,
>their ultimate recourse was: "you just don't get it", and "you are
>congenitally
>incapable of grasping certain basic facts". (Addressing Anand and me, I
>believe.)

"congenitally" is Steve's word, not mine.

>They, Harvey and Steve, claimed to be experts on the matter of GII,
>and we, as "pure mathematicians", had lost or had not developed these
>instincts.

GII plays little or no role in the choice of research topics of most pure
mathematicians, and they would admit this explicitly. However, it plays the
utmost role in the choice of research topics in high level FOM. And this
role is not forced. It comes completely naturally. Such is the fundamental
nature of high level FOM.

>   My *experience* in the course of 30 years has indeed made me suspicious of
>certain wide spread instincts that Harvey and Steve may have in mind here,
>and which I share:

I had trouble reading this at first. I think you mean that you are
suspicious, and you wish to share why you are suspicious. What I don't know
how to interpret is what you mean by "wide spread."

>these instincts, covered by a thin veneer of questionable
>philosophy, are often used to justify ignorance of major developments in
>mathematical thought of the last 200 years that are outside of the
>*relatively minor*
>and *exceedingly familiar* FOM-line:  Cantor, Frege, Goedel, ...

This extreme statement has very convenient language in it for me as I now
work you over in the ring with body punches. You say "mathematical thought
of the last 200 years." Well, there is a book conveniently titled
"Mathematical Thought from Ancient to Modern Times" by Morris Kline,
Professor of Mathematics, Courant Institute of Mathematical Sciences, New
York University, Oxford University Press, 1972. This book is 1,238 pages
long, consisting of 51 Chapters. These Chapters are:

1. Mathematics in Mesopotania
2. Egyptian Mathematics
3. The Creation of Classical Greek Mathematics
4. Euclid and Apollonius
5. The Alexandrian Greek Period: Geometry and Trigonometry
6. The Alexandrian Period: The Reemergence of Arithmetic and Algebra
7. The Greek Rationalization of Nature
8. The Demise of the Greek World
9. The Mathematics of the Hindus and Arabs
10. The Medieval Period in Europe
11. The Renaissance
12. Mathematical Contributions in the Renaissance
13. Arithmetic and Algebra in the Sixteenth and Seventeenth Centuries
14. The Beginnings of Projective Geometry
15. Coordinate Geometry
16. The Mathematization of Science
17. The Creation of the Calculus
18. Mathematics as of 1700
19. Calculus in the Eighteenth Century
20. Infinite Series
21. Ordinary Differential Euqations in the Eighteenth Century
22. Partial Differential Equations in the Eighteenth Century
23. Analytic and Differential Geometry in the Eighteenth Century
24. The Caluclus of Variations in the Eighteenth Century
25. Algebra in the Eighteenth Century
26. Mathematics as of 1800
27. Functions of a Complex Variable
28. Partial Differential Equations in the Nineteenth Century
29. Ordinary Differential Equations in the Nineteenth Century
30. The Calculus of Variations in the Nineteenth Century
31. Galois Theory
32. Quaternions, Vectors, and Linear Associative Algebras
33. Determinants and Matrices
34. The Theory of Numbers in the NIneteenth Century
35. The Revival of Projective Geometry
36. Non-Euclidean Geometry
37. The Differential Geometry of Gauss and Riemann
38. Projective and Metric Geometry
39. Algebraic Geometry
40. The Instillation of Rigor in Analysis
41. The Foundations of the Real and Transfinite Numbers
42. The Foundations of Geometry
43. Mathematics as of 1900
44. The Theory of Functions of Real Variables
45. Integral Equations
46. Functional Analysis
47. Divergent Series
48. Tensor Analysis and Differential Geometry
49. The Emergence of Abstract Algebra
50. The Beginnings of Topology
51. The Foundations of Mathematics

Now, do you think that Morris Kline was expressing "ignorance of major
developments in
>mathematical thought of the last 200 years that are outside of the
*relatively minor* and *exceedingly familiar* FOM-line:  Cantor, Frege,
Goedel, ..." when he wrote the following from Chapter 51?

A. "By far the most profound activity of twentieth-century mathematics has
been the research on the foundations."
B. "Hilbert, moved by the desire to answer the intuitionists' criticism of
classical analysis, took up problems of the foundations and continued to
work on them for the rest of his scientific career [from 1919 to 1943]."
C. From Chapter 45: "David Hilbert, the leading mathematician of this
century,..."

Not even you would made the absurd claim that this FOM-line is relatively
minor in the history of mathematical ideas. You can only mean that you
judge it as relatively minor within mathematics. I could agree with this,
provided that we agree that we mean mathematics in a sense that is
restricted to counting, and shapes, and matters directly arising thereof.
But if you mean mathematical thought - or mathematical science - then of
course it's completely absurd. My experience has been that the majority of
pure mathematicians agree with me on this point - as long as the proper
distinctions are made.

This is a perfect illustration of where you find it very difficult to look
at things from a general intellectual perspective. Pure mathematics is only
a small part of the picture.

If FOM is lumped together with mathematics, then of course Kline is
correct. Now he wrote this in 1972, and you can ask whether or not he would
or could still say this. I say yes, given what has happened in mathematics
over the whole century, and given what has happened in high level FOM over
the whole century. Yes, by far the most profound activity of 20th century
mathematics has been FOM, for the 20th century as a whole.

What on earth are you going to put up to compare with the following FOM,
from a mathematical thought point of view. Let's hear specifics. Let's see
what anybody other than pure mathematicians have to say.

1. Completeness theorem.
2. Incompleteness theorems 1 and 2.
3. Consistency of AC and CH.
4. Independence of AC and CH.

At the very least, you must concede that as far as mathematical thought is
concerned, it is at the very top of the century. A great many pure
mathematicians even agree with me completely. One thing you are very right
about, and that is when you say "exceedingly familiar."

>Agreed, the basic ideas of this last line are of fairly general
>*mathematical* interest,
>and easy to grasp (since one starts from scratch).

The word "fairly" is a gratuitous insult to FOM. It is trivial to grasp at
a conceptual level. However, at the detailed level, it is extremely
difficult for even the best pure mathematicians in the world today to grasp
it. It is much more difficult for them to understand technically than
almost anything else they experience. This is also true among the best
graduate students in mathematics. I have heard reports that the very best
graduate students at the very best Departments are demonstrably in logic in
disproportionate numbers. Maybe some of the fom subscribers can confirm
this.

Now some of this technical difficulty for the best mathematicians may stem
from the fact that at the elementary level, the concepts are so different
in style and character than those in core mathematics. Witness their
extreme difficulty in understanding the definition of predicate calculus,
models, satisfaction, etcetera.

>but I consider it
>preposterous to
>claim that the more recent development in the FOM-line (reverse math, for
>example)
>has this "general interest" character with respect to mathematics and
>computer science,
>let alone science in general.

It may be preposterous to you, but it's completely true. This year alone, I
gave about 30 talks at places including Cal Tech, UCLA, Princeton,
Stanford, Berkeley, Urbana, Harvard, MIT, San Diego. In many cases, the
attendance was abnormally large, and sometimes the largest of the entire
year at their Colloquia. This is simply due to the fact that the abstracts
I prepared and that were circulated in advance were of unusual "general
interest." Now that some major new developments have occurred, I expect the
attendances to get substantially bigger both in computer science and in
mathematics departments over the next couple of years. And these attendance
reports were compared to other talks in mathematics, not compared to other
talks in logic. Such is the general interest in FOM!

And I predict that you will regret having said "let alone science in general."

>This is not to deny certain virtues of FOM as
>pursued
>by Harvey and Steve: I do value the connections to certain parts of
>combinatorics,
>but why exaggerate the (future) impact of this on the rest of mathematics?

You're confusing some claims. First of all, the main emphasis I make of the
work that has been completed in FOM is not on "the rest of mathematics" but
on the general mathematical and intellectual culture - including the
culture outside pure mathematics.

I did claim a gigantic impact on all of mathematics when a major line of
work gets to the point that there are necessary uses of large cardinals in
order to prove new kinds of striking theorems that are highly valued among
practitioners in diverse areas of mathematics. Note that I now say "when"
instead of "if." That is because of what is happening over the last month.
But that is a story for another time.

I continue to say that "demonstrably necessary uses of large cardinals in
order to obtain highly valued concrete results in diverse areas of
mathematics" would be far more than simply the greatest mathematical
acheivment of the 20th century. But I won't argue this in this posting.
(This hasn't happened yet.)

>By the way, I do consider P=NP as of general mathematical interest, and
>recognize that the origin of this problem is partly in FOM (and partly in
>number
>theory and computer science), but present research around P=NP can hardly
>be claimed
>under the FOM flag (and probably Harvey doesn't.)

P=NP is much more than of general mathematical interest. It's of general
intellectual interest. There is much greater intellectual interest in it
than in any problem in pure mathematics right now. By the way, how do you
claim that the *origin* of P=NP is partly from number theory? Just asking;
not confrontationally. I see  possible *origins* in graph theory more than
in number theory. Are you thinking of factoring integers?

In any case, we can ask Steve Cook to tell us about the origins of P=NP.

>In one of his last messages Harvey puts forward an astonishingly
>ambitious goal for his intellectual activities: to make it possible that
>those of us with
>intellectual gifts can become once again as Aristotle and Leibniz were in
>their time,
>rising far above the level of being only an expert in a *relatively* small
>area.

Absolutely.

>Fantastic!
>This would surely be of revolutionary significance. But from what i know of
>Harvey's
>work in FOM, there is *nothing* there to make this ambition remotely
>plausible even
>with respect to, say, the major accomplishments of mathematics as of 1900.

Now here I don't know what "with respect to" means. I think this is an
incorrect interpretation of what I have in mind. Perhaps the following
might make this clearer.

I do not think that the actual development of mathematics - as of 1900, or
whatever - is canonical. Some of it surely is - e.g., are R^n and R^m
homeomorphic? This is inevitable. But there are lots and lots of
mathematical subjects that make just as good sense as what has been done -
perhaps in many cases more sense than what has been done - which were never
conceived of. And when they are conceived of and done, they will get a
growing constituency - at various rates of growh of course.

The FOM style of thinking is very very good at the development of these
sensible - or more sensible - mathematical subjects. Also the FOM style of
thinking is very very good at much more effective expositions of
mathematical subjects because it emphasizes a different organizational
structure. And this structure is particularly effective for nonexperts.
Even particularly effective for nonexperts who in fact mathematicians from
very different areas.

No, Lou, I don't mean that if you think in the FOM style, you will get a
simple proof of FLT. But if you rethink geometry in the FOM style, you will
undoutedly do new innovative things that are unusally sensible and have a
lot of constituency and are in new directions and are of competitive
interest with what's been done. To repeat: the organizational method is
different - and, in my opinion, much more powerful and effective
intellectually.

>On the contrary, FOM-preoccupations tend to make it harder to catch the
>spirit in which math is and was done, and is best understood.

It certainly makes it harder to catch the spriit of uninteresting technical
mathematics. In any case, I am convinced that pure math "preoccupations
tend to make it hard to catch the spirit in which" every subject other than
pure math - including applied math - "is and was done, and is best
understood."

> I have been thoroughly
>exposed to FOM throughout my career, and to other parts of mathematics as
>well, and
>I have experience in relating the two, as to their various claims and ways of
>doing things.)

Yes, but you have a color blindness about the way high level FOM is done,
and especially its purposes. So your comparisons are invalid. The FOM way
is much more like the way applied math is done, as well as just about every
subject other than pure mathematics. Core mathematics is done differently -
like art/sport - not as a subject, and with a wholly different value
system.

>And mathematics is only a small part of the big picture Harvey conjures up!

Right. There is a huge amouont of uniformity in the world. To a very large
extent, it's all the same. I see a lot of sameness in mathematics,
philosophy, computer science, and music. But none of these disciplines is
on the right path, and they all have serious strengths and weaknesses. FOM
style thought reveals how to redirect these fields - and unify them - as
well as many more.

>  Actually, I think it would be an interesting project to equip the average
>research mathematician of today with the mathematical knowledge, and more
>importantly, the mathematical flair and instincts of, say, Hilbert and
>Poincare, to mention
>the most prominent mathematicians of 1900. This would perhaps be a doable
>project,
>but of a much more modest scope than Harvey's. FOM would play a role, but
>in my
>opinion not a dominant one, in this limited enterprise.

Again, it's not FOM itself. Its the approach to intellectual life that FOM
represents. Its an FOM style of thinking.

>   Harvey's reactions to Franzen and Thayer (followed by the promulgation
>of the "new rules") have confirmed my other suspicions on Harvey's GII, which
>Thayer made explicit: clearly, Harvey's appeals to GII are often mere
>bluff and bluster,
>a stick to hit the opposition with.

They are a good stick to hit you with, since it is the only stick I know of
that might make you listen and perform these thought experiments: What are
you trying to accomplish in simple terms? What information are you seeking
in simple terms? Why do you care in simple terms? Why would anyone else
care in simple terms? How would you get anyone else to care in simple
terms? Who could you get to care in simple terms? Let me know when you have
completed these thought assignments. And you still haven't given me
anything in contemporary pure math whose GII compares with the highest
levels of FOM. It's you that are bluffing and blustering.

Tell you what: I'll match my answers to these thought experiments with
yours. We'll go through them line by line, and compare them on the fom.

By the way, do you really want to hold up what Thayer wrote as a good
example? Why don't you write something like that? You wouldn't dare. You
have a reputation to uphold.

>Nothing wrong with blustering on your own FOM list where you are the boss
>(as someone reminded us a while ago), though I wonder if this is how you sway
>serious intellectual opinion to your side.

Nothing wrong with blustering among pure mathmaticians and selected
logicians that share your biases, rather than to step out in the real world
and see if anybody else agrees with you, or can even make sense of what you
value.

>But why claim that the other guy
>(me in this case) must be the only one on the fom-list to doubt that certain
>FOM matters are GII-stuff, when you surely know better? The messages of
>Thayer and Franzen immediately refuted that particular claim. Does that
>explain the
>savage reaction?

Just drawing out as much silliness as I can find, trying to convince
everybody. Incidentally, Thayer and Franzen are not mathematicians, and so
there still is a question of what other mathematicians "on the fom-list
doubts that certain FOM matters are GII-stuff." Acutally, I am not aware
that you explicitly doubted that certain FOM matters are GII-stuff, have
you? If I am wrong, which ones do you doubt?

Of course, you are competent, and don't write like Thayer, and to a much
lesser extent, Franzen. Another difference: you're still on the fom, and
they're not. And that is a big difference. Thanks for staying on. Give me
your best shots. Make your best defense of pure math vs. FOM. Good luck.
You'll need it.