FOM: Kline/GII

[Harvey Friedman]

A brief response to Martin Davis and his quote from Lou. First of all,
Davis writes:

>Morris Kline was my colleague at NYU for many years. He was a kind man,
>and I always got on very well with him. He had played an important role
>in the early days of the Courant Institute and was mourned by many when
>he died a few years ago.

Too bad we couldn't hear directly from him how he became so incredibly
impressed with the profundity of FOM.

>However, when it came to foundational issues he truly didn't
>have a clue.

This is also true of the vast majority of all pure mathematicians. Are they
interested in them when imaginatively, creatively, and powerfully
presented? Yes. Have a clue to the technical ins and outs? No.

>Every few months he'd come to me asking for an explanation of
>the apparent paradox that while he knew how to prove that Peano's
>axioms were categorical and that the same was true for the axioms for a
>complete ordered field, nevertheless there were these non-standard
>models. Each time I would explain as carefully and simply as I could,
>and each time it was as though the previous time had not occurred. I
>believe that he never grasped the concept of a formal logical system.

The same would be true of the vast majority of all pure mathematicians -
that they certainly have never grasped the concept of a formal logical
system sufficiently to see their way through such matters. However, that
doesn't mean that they can't easily relate and absorb the highest levels of
FOM.

Without appropriate exposure, the vast majority of mathematicians wouldn't
have a keen enough interest in these more specialized matters to keep
asking you about it. It's remarkable that Kline kept asking you about such
things. In any case, the technical issues are second nature to you and me
and Lou, but are notoriously difficult for even the best mathematicians of
our time.

>I don't think that his opinions on foundational questions should be >given
>very much weight.

Nor do I. What I give great weight is his obvious extreme interest in
foundational questions, and his extreme (and entirely appropriate) respect
for work done on them this Century, in the context of a grand survey of so
much mathematics and the fact that he was so involved in the mathematical
community. That is what I am emphasizing. And this is all in a context in
which he has no apparent personal stake in the matter, having apparently
not made any contribution himself to FOM.

Lou writes:

>I am familiar with Kline's book, and value parts of it. But while it's
>a good source of information on the 19th century it really only covers
>the 20th century till about 1910, except for the very last chapter,
>which I find the weakest of all.

There are a number of celebrated more modern developments discussed there,
excluding the last chapter on the foundations of mathematics.

Kline discusses work on Betti numbers by Alexander in 1919. Work on Banach
spaces in the 1920's up through 1929. Work of Noether on ideal theory in
1921. Work of Birkhoff and Kellogg on fixed point theorems in function
spaces in 1922. Work of Alexander on Betti numbers in 1922. Work of Weyl on
Lie algebras in 1925. Work of Urysohn on metrizability and on the Hilbert
cube in 1925. Shroedinger's equation in 1929/1930. Von Neumnn's work on
Hilbert space in 1929/30. Work of Poincare, Brouwer, Urysohn, Menger on
dimension theory in 1925 and in 1930. Work of Wilhelm Magus on the word
problem in 1932. Work of Schauder and Leray on applications of this to
differential equations in 1930 and 1935. Work of Novikov on the word
problem in 1955. Work of Michel Kervaire, 1958, and John Milnor, 1958,
using work of Raoul Bott, on the possible division algebras with real
coefficients. Work on Pioncare's conjecture by Smale, Stallings, and
Zeeman, 1960 and 1961. Work of John Milnor on the Hauptermutung of
Poincare, in 1961.

Kline says explicitly that

"We shall terefore limit our account of 20th century work to those fields
that first became prominent in this period [20th century]. Moreover, we
shall consider only the beginning of those fields. Developments of the
second and third quarters of this century are too recent to be properly
evaluated. We have noted many areas pursued vigorously and enthusiastically
in the past, which were taken by their advocates to be the essence of
mathematics, but which proved to be passing fancies or to have little
consequential impact on the course of mathematics. However confident
mathematicians of the last half-century may be that their work is of the
utmost importance, the place of their contributions in the history of
mathematics cannot be decided at the present time."

What is so striking about

"by far the most profound activity of twentieth-century mathematics has
been the work on the foundations"

is that Kline said this at the same time being obviously aware of the
extremely highly regarded work he cites above. And also this is undeniably
a very thoughtful and carefully written book, and he surely was fully aware
of how strong this statement is, and that it isn't buried somewhere in the
middle - it's in the last Chapter. And that he would be severely criticized
for saying this. He must have been unboundedly impressed by FOM. He also
had no personal stake in FOM having not made any contribution to my
knowledge to it.

Lou - I know you think that the Kervaire, Milnor, Bott, Smale, Stallings,
Zeeman, and Milnor stuff cited above is incomparably more impressive and
interesting and important than Godel's 2nd incompleteness theorem, but
don't you see how funny people think this is?

Kline also talks in the book about the usual attacks made against pure
mathematics, that started to be made around the turn of the century, and
have accelerated to this day. These attacks are usually formulated as a
pure vs. applied controversy.

But I would like to reformulate this controversy. The real controversy is:
the general intellectual interest of contemporary pure mathematics has yet
to be explained in a convincing manner. The general intellectual interest
of applied mathematics is, in general, much easier to explain, and often is
completely self evident. That is why attacks on pure mathematics are so
often couched in terms of pure vs. applied.

But as I have indicated on several occassions on the fom, that the FOM
style of thought can reexposit and reorganize significant portions of pure
mathematics so that its general intellectual interest is apparent without
forcing pure mathematics into the realm of standard applications.

>Anyway, it's a pretty routine
>account, with the usual inordinate attention to marginal matters
>like paradoxes, and indeed some statements that I strongly disagree
>with, like the one Harvey mentions.

You disagree with "by far the most profound activity of twentieth-century
mathematics has been the work on the foundations"? Gee, I didn't realize
this.

By the way, you have shown gross bias here, in that you refer to "marginal
matters like paradoxes." Lou, despite the fact that you told us on the fom
that you solved the paradoxes when you were a teenager, they remain the
greatest seminal puzzles of all time, and are nowhere near having been
resolved or mined appropriately. Godel felt that they were not properly
understood, and that a better understanding of them would lead to great
insights. We have the great mystery of how to approach the axioms of
mathematics now that we are beginning to see how expansion of the usual
axioms are essential to give proofs of concrete statements we would like to
know. How are we to think of the consistency and/or truth of these
necessary higher principles? Clearly this is likely to be tied up
inexorably with the paradoxes - you know, the ones you gave complete
solutions to when you were a teenager. When most of us were trying to get a
car and go on dates and get more allowance, you were resolving the great
paradoxes with the greatest of ease! Most remarkable. I want your
autograph.

>(But perhaps this is just an example of "familiarity breeds contempt".)

I was going to discuss "familiarity breed contempt" in connection with this
discussion. But you beat me too it, and I could not possibly have said it
better. You know, its human nature. Repeat after me:

"familiarity breeds contempt" "familiarity breeds contempt" "familiarity
breeds contempt" "familiarity breeds contempt" "familiarity breeds
contempt" "familiarity breeds contempt" "familiarity breeds contempt"
"familiarity breeds contempt" "familiarity breeds contempt" ...