FOM: ReplyToLou

[Harvey Friedman]

>The rumors of my "selling out" have been greatly exaggerated.
>Anyway, what is of "general intellectual interest" is I think
>a very difficult issue to discuss sensibly.

I use the phrases "foundational" or "general intellectual interest" in
connection with an implied "hierarchy of concepts" that cuts across all
intellectual activity. Now of course, no one I know is yet prepared to be
very formal about this: this way of looking at things is, I believe,
implicit in the approach of the truly great intellects over history, which
allows them to see the forest so clearly over the trees, and found the
great new subjects. We simply do not yet have a suitable treatise which
reconstructs a significant portion of human knowledge and understanding in
terms of an appropriate "hierarchy of concepts." This has not yet been done
appropriately even for classical mathematics. Until this is done, such
discussions are necessarily subject to the difficulties as indicated in the
next paragraph.

My experience in talking in these terms with scholars is that they either
naturally "get it" or they don't; i.e., they either have or do not have
natural instincts in this direction. As a function of the current
educational process, culture, academic reward system, etcetera, most people
today don't. Certainly most mathematicians don't get it. So when I talk in
these terms to most people, I find that I must ***jar*** them in some way,
and the most effective way I know of how to jar them is to grossly simplify
matters and talk about "how many people can understand it?" or "how many
people can appreciate it?" This is only an approximation that I use when I
talk to people who "don't get it."

So I can certainly agree with Lou when he says "Anyway, what is of "general
intellectual interest" is I think a very difficult issue to discuss
sensibly."

>I now realize that
>these theorems [Seigel's and Falting's] themselves do have equivalent
>forms that can be
>stated on that level. (And any number theorist could have
>probably told us this right away.) But I do have to look this up
>a little so I don't make some stupid mistake.

I did ask you to give your best formulation in terms of "foundational" or
"general intellectual interest" of Falting's theorem so what we can
quietly, slowly, deliberately, and leisurely compare it with my formulation
in these terms of MRDP (Hilbert's 10th Problem). Of course, you chose to
replace "foundational" and "general intellectual interest" with "high
school level" as a result of my necessarily having to jar you into
understanding something about what I have in mind.

>By the way, I do like
>this kind of exercise to some extent, and I am happy to oblige,
>Steve and Harvey, who'd like to see things stated on that [high school] level.
>But I 'd be careful to draw any kind of "dramatic conclusions"
>from such things, as foundationalists seem so eager and willing
>to do.

I agree that "dramatic conclusions" are not in order for simply reducing
something to high school level. You're saying this is the price that I pay
for jarring you into understanding what I am talking about.

 (By the way, the general notion of algorithm does not seem to
>me on the high school math level, so any result that uses this notion
>essentially seems to me not "of general intellectual interest"
>according to some of the definitions I have heard bandied about,
>like "you can tell it to your neighbour", etc. And if you are
>willing to make a case that Turing's or Church's analysis can be
>made accesssible on the high school level, you could make just as
>good a case (probably a better case) that complex numbers,
>and algebraic curves and their genus can be made accessible on
>that level.)

But here is where things diverge. The conceptual analysis of the notion of
algorithm is clearly not ordinarily high school material - not high school
level in the noraml sense of the word. However, it is unarguably of
foundational or general intellectual interest. One way this conflict can be
resolved for you might be for me to say that the *concept* (not the
analysis) of algorithm is of high school or even any level. The mere
existence of a UNIQUE analysis - regardless of how involved it is - is
sufficient for foundational or general intellectual interest. The
complexity or difficulty of that UNIQUE analysis is not an issue here.

When you finish doing your best with Falting's theorem, we can compare the
situations carefully. And as I have said - I am open minded.
********
I want to close with a comment on: what on earth are we arguing about??

Well, there is much much more under the surface than the actual focused
argument. I will take the liberty of stating my version of a portion of
your viewpoint. You obviously will want to correct me in certain places.
And then I will state my version of a portion of my viewpoint. This is the
real argument. And it is interesting and important. We certainly don't
disagree on everything.

MY VERSION OF LOU.

1. Lou seeks to minimize the importance of or special status of work in
Foundations of Mathematics by asserting that the special features and
audiences that the supporters of FOM claim for it are already features and
audiences for lots of important mathematics. In fact, Lou seeks to deny
that there is anything more Foundational about Foundations of Mathematics
than other important mathematics.

2. Lou does concede that there is something distinct and coherent called
Foundations of Mathematics. But having, in his view, stripped away any
special status of the field, Lou inevitably regards it as yet another
branch of mathematical thought that is to be judged like other branches of
mathematics. In particular, Lou regards it as only playing a minor role for
mathematics. It is therefore less interesting and important than the part
of mathematical logic that apply directly to core mathematics, and
certainly less important than core mathematics. Lou regards it as being
consistently too general to be "of any use."

MY VERSION OF ME.

1. Harvey seeks to maximize the importance of and special status of work in
Foundations of Mathematics by asserating that it has special features not
shared by the various areas of mathematical logic and also not shared by
core mathematics. Harvey claims that even fundamental mathematics is not
normally foundational. Harvey claims that much of classical mathematics can
be reconstructed in a foundational manner in the style of FOM, and that
this is very fruitful.

2. Harvey sees Foundations of Mathematics not as a branch of mathematics,
but instead a branch of what he calls Foundational Studies. As such, it is
the currently preeminent branch, and is very highly developed and
successful. Harvey believes that there are spectacular surprises to come
from FOM for the practice of mathematics - necessary use of new axioms for
mathematics in concrete problems. Harvey thinks that such surprises would
shake up the whole mathematical enterprise in ways that normal applications
of mathematical logic and ordinary important core mathematics will not.
Harvey regards FOM as being of a level of generality that is driven by the
genuine intellectual issues being addressed.