# FOM: history, funding, guidance

Harvey Friedman friedman at math.ohio-state.edu
Wed Aug 4 07:05:25 EDT 1999

Reply to Davis 3:07PM 8/3/99:

>1. Harvey and I write from two fundamentally opposed views of some
general
>social issues, issues for which f.o.m. isn't really an appropriate
forum.
>...My view is that the current situation of scarce resources for
>mathematics is artificial and anti-social.

The resources for applied mathematics in its various guises - so called
"applied math", statistics, mathematical computer science, mathematical
engineering, etcetera - is much greater than for pure mathematics. So I
assume that you are talking of scarce resources for pure mathematics.

Pure mathematics is additionally vulnerable because so few people outside
the field understand what it is about and why anybody cares about it. There
seems to be much greater understanding - or perception of understanding -
of other areas of science/engineering than pure mathematics. Pure
mathematics has traditionally shown very little concern for general
intellectual interest - something I am always talking about. As a
consequence, important and influential scholars and administrators don't
understand the point of it, don't feel its excitement, and start evaluating
it in terms of likely applicability - which is always iffy. So from this
point of view, it is very natural that resources are comparatively scarce.

>Mathematical talent and interest
>are rare enough and valuable enough (even from the most crass
utilitarian
>viewpoint) and the US is blessedly rich enough so there is need
for the
>perceived scarcity. ...

Mathematical talent is not generally perceived in the US to be valuable
unless it is working on applications. And the argument that the
applications will come unexpectedly is not proving to be all that
convincing. The applied mathematicians contend that it is far more cost
effective to support people committed to think about and look for
applications than to support people who may only accidentally run into
applications.

I'm not making any analysis of this issue of cost effectiveness. I'm just
stating my perceptions of people's concerns. But in any case, I definitely
think that the ignoring of general intellectual interest is making matters
much worse. And as you well know, general intellectual interest is not just
a code word for applications.

>I also believe that in conditions of scarcity people will fight
ferociously
>to defend turf. The kind of contemplation of what is best from a
purely
>intellectual
>perspective that Harvey (and I) would like to see (with very rare
>exceptions) can only be expected to arise when there's plenty to
go around.

My view is that if people see better ideas and get involved in them, then
the turf that they defend changes. I would like to see them defend some
more defendable and better looking turf.

>Of course the top places and top prizes will always be scarce; but
getting
>them for oneself and one's students would feel much less urgent under
>conditions of pleanty.

But if those top places and top prizes are filled with second rate ideas,
then second rate ideas become the standard, and with, as you say scarce
resources, the whole enterprise becomes infected with second rate ideas,
and first rate people will either conform, or be under pressure to go
elsewhere. First rate people don't like to in environments where they are

>2. Harvey has presented a schematic picture of the rise and fall of a
>mathematical subject. Lots of very smart people (e.g., Marx,
Toynbee) have
>tried to force history into well-defined schemes, and these work -
up to a
>point. But human history is so complex and intertwined, that these
schemes
>turn out to be, in the last analysis, simplistic.

The same can be said of any model of phenomena. They are all simplistic
since real nature is too complicated. But that doesn't stop the models from
being more or less useful and important. I think that the model I presented
is useful and important, and flexible, and explains a lot of phenomena.

You go on to talk about invariant theory and the Hilbert basis theorem. By
the way, I assume you know that if you ask for estimates, then Hilbert's
basis theorem is demonstrably useless. So:  many of those results that are
proved trivially using Hilbert's basis theorem have to be (and have been)
revisited as open questions if you want estimates. Aspects of this whole
matter are tied up with formal systems and reverse math.

>I certainly don't want to discourage critical thought especially
of the
>constructive variety. I object to attacks on a field from the outside.

Do you object to *criticisms* of a field from the outside of the field?
This is going on every day when somebody mentions that they want to hire
somebody in field X, and the hiring committee lets out an audible groan.

You almost never hear criticisms of a field from the inside of the field.
So I guess you object to criticisms of any field from anybody?

>Constructive suggestions for productive moves should always be
welcomed,
>whether or not they are called "renewal". But in the nature of
things, you
>are not going to convince mature mathematicians whose main stock
>virtuosity in a particular technique to abandon that technique.

How about suggesting that they look at new kinds of problems for which at
least some of their virtuosity will likely be useful?

>Who could be against "guidance." But it is as likely as not to be
misguided.

Do you object to guidance on principle, or object to *my* guidance? I claim
to see a pervasive fundamental flaw in the approach to intellectual life in
many intellectual communities. I see it clearly, I see the damage it is
doing to intellectual life, and I have recommendations to correct it. The
problems are at a very deep level. Not at some surface level like: reverse
mathematics is more interesting than new details about the lattice of r.e.
sets.

>Why do you suppose that guidance from you will
>be listened to more than from the powers that be.

I never said that it would be. I just said, implicitly, that mine would be
very effective if listened to. That it will be carefully argued. That wise
people might well see the value of it and recommend it to others. Since the
problems are so deep, pervasive, and destructive, the guidance has to be at
the right fundamental level, and be very well thought out.

But if wise men/women like Davis go around saying that it shouldn't be
listened to, simply because it is, after all, *guidance* and that
*guidance* shouldn't be listened to on principle, well -- that's going to
be very counterproductive and make it more difficult for me to get people
to listen -- to get the crucial message across.

Stevenson 1:26PM 7/29/99 provided the following interesting quote from von
Neumann:

>>"As mathematics travels far from its empirical source, or still
>>more, if it is a second and third generation only indirectly inspired
>>by ideas coming from reality,'' it is beset with very grave
>>dangers. It becomes more and more purely aestheticizing, more and
more
>>purely {\em l'art pour l'art}. .... In other words, at a great
>>distance from its empirical source, or after much abstract''
>>inbreeding, a mathematical subject is in danger of
>>degeneration. At the inception the style is usually classical;
when it
>>shows signs of becoming baroque, then the danger signal is up.''.
>>
>>   J. von Neuman (1943. The Mathematician.'' In *In the Works
of the
>>			   Mind.* Chicago, IL: University of Chicago.)

Friedman wrote:

>>When I first read this, I thought that it did not apply to f.o.m. On
>>further examination, it does, in the following sense. F.o.m. is
>>fundamentally about mathematical thought, and that can be viewed as a
>>"reality" or "empirical source." It is particularly evident that
in, say,
>>reverse mathematics, one is using mathematics itself as an empirical
source
>>that need to be tested empirically: e.g., the linear ordering under
>>interpretation of actual mathematical statements.
>>
Davis writes:

>I like "l'art pour l'art"; cf. Hardy for counter opinions. I
understand what
>von Neumann means, but what was he really fretting about? At the
time the
>major controversies were over what came to be called soft
analysis, Banach
>spaces, etc. In fact, these turned out to be very valuable.

Too simplistic. I am skeptical that von Neumann had the wrong idea; I
suspect he had a variety of things in mind. And Banach spaces, etc. - well
they were or were not valuable depending on what you do with them. It's
like saying that formal systems proved to be important. Well, it all
depends on what you do with them. Some of it is horrible, and some of it is
among the greatest things ever done.

>And as your
>example demonstrates, people applying the quote to their own specialty, will
>always find underlying empirical sources. Certainly people working in
>computability/recursion will have no problem doing so.

Not so fast. The reverse math empirical source being math itself is
extremely air tight -- and not flaky. I made an explicit empirical
conjecture *right at the inception of the field*, which is being verified
empirically as we speak! It would be interesting to see a proposed airtight
case like that!!