FOM: ReplyToMattes

[Harvey Friedman]

I wrote:

 "I do expect that for specialists in the
 overwhelming majority of disciplines outside mathematics, there will be
 more interest in and understanding of FOM than in/of core mathematics. "

You wrote:

>It seems to me the list of disciplines for which core mathematics is at
>least as  relevant than FOM will include:
>Physics, chemistry, biology, geology, astronomy, economics, engineering,
>history, any field that uses probability/statistics (sociology,
>psychology, ...), etc.etc.
>
>As a rough measure, I checked today on MathSci...

Notice that you use the word "relevant" which has nothing to do with my
statement. Was my statement short enough for you to be able to read it
carefully?

Upon your careful reading of my choice of words, it might have been clear that

1. My comment concerned the overwhelming number of people in these
disciplines who do not use any kind of modern mathematics or any (serious)
mathematics at all. I was not referring to the use of mathematics or f.o.m.
in actual publications.

2. Also, I was not referring to classical elementary mathematics and
statistics, but rather topics of current interest to professional core
mathematicians versus topics of current interest to professional fom people.

	The vast overwhelming number of people in these disciplines do not
understand modern core mathematics, and as a consequence of the nature of
that material, are therefore not (able to be) interested in modern core
mathematics. Furthermore, it is difficult to package modern core
mathematics in such a way that it is understandable, and can generate
interest. That's the nature of the subject as it is practiced.
Consequently, core mathematicians usually try to sell the subject on the
basis of applications - which is not easy to do and which rarely does
justice to what the core mathematicians would like to emphasize.

I mention, for example, the recent film about FLT. It was cleverly and
effectively done, but not a single core mathematical idea was conveyed -
they really didn't even try. People are of course quite interested in the
sociology of it. Compare this to a movie about Godel and f.o.m.!

Isn't the special fascination with and accessibility of modern f.o.m. for
specialists accross all of the disciplines you mentioned obvious to you?
Godel is truly a legend. And a lot of von Neumann's work is either in
f.o.m. or very f.o.m. friendly. Who are the comparable figures this Century
in core mathematics? When you ponder that question, you may begin to "get
it." In recent documentaries about Albert Einstein's approach to
intellectual life, it is clear that his approach was very f.o.m. friendly.
Very different in style from modern core mathematics.

I wrote:

I would like your reaction to the Morris Kline quote from Mathematics from
Ancient to Modern Times, Chapter 51, pp. 1182:

"By far the most profound activity of twentieth-century mathematics has
been the research on the foundations."

As a comment to both Mattes and McLarty - note that the work that Kline
describes in Chapter 51 starting at page 1182 is exclusively concerning
f.o.m. in the normal customary sense that I use the phrase - and definitely
not in any provacatively indiscriminate manner.

***I still think you (and others) are having trouble dealing with this
quote of Kline.***

Mattes wrote:

>You complained that some people supposedly "seek to minimize the special
>importance and status of Foundations of Mathematics" and quote Morris
>Kline (see above).
>First let us make clear that of course no one says Goedel's (or your) work
>is unimportant. But you seem to claim much more than just that it is
>important:
> "I do not expect you to concede that the special
> importance and status of FOM is "greater than" that of fundamental
> mathematics. You may well be much more interested in core mathematics
> than you are in FOM; . . ."
>So obviously you think that the importance of FOM is greater than that of
>other mathematics.

I have chosen my words very carefully (I hope!) in this regard on several
occassions in the FOM. What is special about the importance and staus of
f.o.m. is (at least) its extreme and unusual general intellectual interest,
and its special place in the broader area of Foundational Studies. These
particular features are not (normally) shared by other branches of
mathematics.

Mattes wrote:

>Does he say that foundations of mathematics are the most important part of
>mathematics? No: "... the calculus, which, next to Euclidean geometry, is
>the greatest creation in all of mathematics" [p.342].
>Does this book cover all of 20th century mathematics? No: "This book
>covers the major mathematical creations and developments from ancient
>times through the first few decades of the twentieth century." [p.vii]
>What was important about foundational work? "the development of these
>several philosophies [logicism, intuitionism, formalism] was the major
>undertaking in the foundations of mathematics; its outcome was to open up
>the entire question of the nature of mathematics." [p.1192] (see also
>below)
>
>I don't see how any of this supports your views.

As I said much earlier on the FOM, f.o.m. is not primarily a branch of
mathematics, although it is customary for many people to view it that way.
Evaluating it only as a branch of mathematics is absurd - as I have
discussed earlier. It is just as absurd, say, to judge statistics as a
branch of mathematics. So if Kline thinks that Euclidean geometry is the
greatest creation in all of mathematics, that in no way contradicts any
view of mine. Also, Kline said that it is too difficult to place very
modern (then 1964) developments in proper historical perspective. However,
he does mention a very very very modern development - Cohen from the
1960's!! That's pretty recent for a 1964 historical work. So he must have
thought that this had special status and importance. And "opening up the
entire question of the nature of mathematics" is something that is
obviously of special importance and status, and of great general
intellectual interest.

What modern developments would you quote as having "opened up the entire
question of the nature of mathematics?" Please apply some high standards
when you answer this question.

Mattes quotes Kline who quotes Godel as follows:

> "Perhaps more surprising is Goedel's statement of 1950 that
>    the role of the alleged "foundations" is rather comparable to the
>    function discharged, in physical theory, by explanatory hypotheses...
>    The so-called logical or set-theoretical foundation for number theory
>    or of any other well established mathematical theory is explanatory,
>    rather than foundational, exactly as in physics where the actual
>    function of axioms is to explain the phenomena described by the
>    theorems of this system rather than to provide a genuine foundation
>    for such theorems.
> What these leaders is acknowledging is that the attempt to establish a
> universally acceptable, logically sound body of mathematics has failed."

Mattes goes on to say:

>Again, this seems to me to be quite the opposite of what you are calling
>the 'genuine foundations'.
>As to your assertion that the quote from Goedel is out of context: Since
>Kline does not give a reference he presumably considered it unambiguous.

Again, I assert that you and I cannot make real sense of this quote out of
context. It is too easy to look at something like this and attempt to read
things into it for one's own special purposes. I have three volumes of
Collected Works of Kurt Godel, and I think I have the idea that Godel was
interested in classical f.o.m. - do you agree?

Sol Feferman is clearly one of the leading interpreters of Kurt Godel and
perhaps might be willing to comment on where this quote appears, what it
means, and can place it in context.

You keep coming back to this quote of Godel in order to avoid dealing with
the original Kline quote:

"By far the most profound activity of twentieth-century mathematics has
been the research on the foundations."

which may be puzzling to you. The context of this quote is his whole 1000
page plus book.

Mattes writes:

>This quite clearly says that Goedel thinks the so-called logical or
>set-theoretical foundations are not really foundational, doesn't it?  As
>you said, sometimes one finds telling quotes that are much
>stronger than anything one is saying - from people you would least
>expect.

No. E.g., it might well be that Godel's use of the word "explanatory" here
is totally compatible with my use of the word "foundational." I am shocked
that you can determine this interpretive issue so quickly without even
knowing where Godel wrote this. I am impressed.

I wrote:

But the main point I wish to emphasize is that we need to maintain high
standards for conceptually clear foundational expositions - otherwise
we really don't have anything which is penetrable by the general
intellectual community.

and Mattes wrote:

> "I doubt that the problems of the intellectual community with
> mathematics stem from a lack of clarity of the concepts in mathematics. I
> rather think they come about because a.) many of our important concepts
> are quite abstract and b.) even if they are not, we like to hide them
> behind lots of technicalities."
>Any comments?

Yes. I never said that the problems ... stem from a lack of clarity of the
concepts in mathematics. This is a far too simplistic view of the role of
f.o.m. - to make concepts clearer. Clearer relationships between concepts
is closer to the mark, but still is an oversimplification.

A conceptually clear foundational exposition of mathematics would bring out
the general intellectual interest of the material in a way that cannot be
done by traditional expositions. Old material would be recast in new terms
which would make the importance of the material apparent. New motivating
themes would evolve which would recast existing subjects in new terms, and
lead to new areas of mathematics.

A reason I didn't respond earlier to you about this is that although we use
different language here, we may well have a lot to agree with here.

Mattes writes:

>Goedel's theorem has been quoted as an example of being of interest
>to
>non-mathematicians. I just wonder, how much interest would there be if
>it
>were not a statement about mathematics? Without interest in
>mathematics/computation, who would be interested in Goedel?

I wrote:

analysis of proofs is far from a completed topic, but what has already
been achieved through Gottlob Frege, Godel, and others regarding
predicate calculus and formal set theory, stands tall with the greatest
intellectual achievments of all time. A similar kind of intellectual
activity later produced the first programming languages and their
implementation - and this stands tall with the greatest engineering
achievements of all time. Such is the power of the FOM outlook and
style of thinking. No other way of thinking about things is even
remotely as powerful.

Mattes writes:

> "This does not seem to answer my question. It goes without saying that I
> have no intention of denying
> the importance of Goedel's results. But it seems to me that they are
> considered important because they are statements about mathematics and
> computers, which in turn are considered important. In contrast, cosmology
> for example is important because it is about nature (if you want to
> appreciate it you just have to look at a clear night sky).
> ... How should the importance of Goedel's work have been explained to the
> nonexpert before the advent of computers? By explaining
> formalization of mathematics? Or was it not foundational then?"

1. Godel's completeness theorem is a profound result about the nature of
general reasoning (far beyond mathematics). That doesn't mean that there
may not be further profound results that take into account additional
aspects of general reasoning.
2. The notion of algorithm goes back to Greek times, and makes sense far
outside the context of mathematics - e.g., in cooking.
3. You say "cosmology ... is important because it is about nature - ...
just look at a clear night sky." Well, "f.o.m. is important (partly)
because it is about reasoning - ... just reason."
4. Godel's second theorem says that there is no complete system of
reasoning in a certain even weak sense. The sense of completeness is so
weak that one need only refer to, say, arithmetic with addition and
multiplication. The idea of arithmetic with addition and multiplication is
also ancient.
5. The idea that there are "normal" mathematical statements that cannot be
answered by the accepted axioms and rules for mathematics, and that these
accepted axioms and rules are very robust, is something that can be
understood via a very general understanding of the mathematical enterprise.
Current f.o.m. is partly concerned with sharper concepts of "normal" that
take more aspects of current mathematical practice into account.