FOM: Evolution of Many Mathematical Subjects

Harvey Friedman friedman at
Mon Aug 2 06:14:00 EDT 1999

Simpson's posting "Friedman on defending specialized subjects" 4:34PM
7/28/99 gives a good account of a conversation I had with Simpson. In this
posting, I want to concentrate on how many mathematical subjects evolve.


History is very cruel. By this I mean that

**intellectual history is very selective**.

Many of us began our careers with the ambition - consciously or
unconsiously - of making permanently important contributions to
intellectual history, and had models of such contributions in mind. For
many people who became mathematical logicians, Godel's work figured
prominently as
a model. And of course, there are other historic figures in mathematics and
science whose work
also served as models for many people, including those who became
mathematical logicians.

There have been hundreds of thousands of substantial contributions
to mathematical subjects, even in recent years. But only a handful serve as
the kind of
inspirational model that we are talking about. And how are these handful
distinguished from the hundreds of thousands? We are talking here of
contributions that are inspirational to a very wide audience of
nonspecialists. The kind of contributions that strike a very wide spectrum
of scholars as obviously important and interesting. The kind of
contributions that deeply influence gifted person's choices of career. We
are talking about the highest forms of intellectual acheivement.

I like to use the phrase "general intellectual interest" to describe a key
way that history selects among those hundreds of thousands of substantial
contributions. There must be a selection process if only to enable young
scholars to meaningfully select an area in which to concentrate their
efforts. Accounts of subjects, or aspects of subjects from the point of
view of general intellectual interest appear at various levels of depth and
sophistication in the media, in the published literature, and in textbooks,
as well as in the classroom.

There are a number of important differences between such contributions at
this highest level. But a principal distinguishing feature is that the
contribution can be cast in general, informal, nontechnical, gripping terms
of general intellectual interest. The details - which may include proofs,
data, formal definitions, etcetera - may not be castable in such terms. But
what has been accomplished - what question has been answered, or what
phenomena have been explained, or what concept has been scientifically
treated, or what connection has been established - is naturally cast in
those terms.


Foundations of mathematics started in the late 1800's and had a spectacular
development in the 1930's. Since then, processes took place whereby
mathematical logic emerged as a name that is more faithful to what the
experts work on, which no longer draws on foundations of mathematics as its
principal motivation. Mathematical logic itself went through a process
where experts generally concentrate on four subfields which have more or
less autonomous standing and separate meetings/conferences:

1. Model theory.
2. Recursion theory.
3. Proof theory/intuitionism.
4. Set theory.

There are frequent meetings focused on 1,2,4 in the United States. Meetings
focused on 3 take place more in Europe. At this time, foundations of
mathematics itself is not generally regarded as a subfield of mathematical
logic, at least in the United States. There are few meetings in f.o.m. in
the United States. I regard myself as, primarily, an expert in f.o.m., and
am not regarded as a member of any of these four communities. E.g., I am
infrequently invited to meetings in any of these four subfields. At this
time, the interest in f.o.m. is quite low among these groups, at least in
the U.S., with the exception of 3 where it is definitely higher. The use of
f.o.m. as even a motivating force in 1-4 is pretty much only a distant
memory, especially in the U.S.

I have often thought about how this state of affairs came about. In
thinking this through, I have noticed great similarities between the
developments of these four subfields - which I will think of as subjects in
this posting. It is also useful to think of the whole of mathematical logic
as a subject for the purposes of this posting.

After some inquiries about the development of other mathematical subjects,
I came to the conclusion that there really is a kind of natural evolution
of many mathematical subjects. And that the inevitable patterns of
criticism and defense are essentially the same. In this posting, I will not
get into much detail about the patterns of criticism and defense. This will
be discussed more in other postings.


Many mathematical subjects gain their identity through a period of striking
achievements which are widely recognized and admired. They then evolve
according to certain patterns where the original motivation is no longer a
driving force, and where there is no adequate replacement for that original
motivation. The subject becomes less and less understandable to outsiders,
who can no longer relate to it. This pattern is typically left unchecked,
often because the experts do not recognize the problem, or minimize the
crucial role of general intellectual interest in intellectual life. When
left unchecked, this inevitably leads to the disintegration of the status
of the subject, which is evident from the availability and quality of jobs
for scholars concentrating in the subject.

This process of deterioration - where the general interest in the subject
drops, as well as the viability of careers based in the subject - can be
interrupted and even reversed by renewal. Such renewal is based on a
reconsideration of the basic thrust of the subject, often involving a
rethinking and reworking of the  motivations behind the original striking
achievements that put the subject on the map in the first place.

In a subsequent posting I will discuss subjects 1-4 in the terms laid out
in this posting. Let me just make two points without discussion right here.

1. Foundations of mathematics can be used once again as the principal
motivating force behind the development of subjects 1-4. This can result in
a major renewal of these subjects, which will draw upon the built up
expertise of the experts in 1-4. A wide variety of mathematicians,
philosophers, and computer scientists would become much more aware and
interested in these subjects than they are now.

2. Significant but limited progress along these lines is occurring, since
f.o.m. sometimes leads to prolific problem generators. But the force behind
these prolific problem generators - f.o.m. - is not generally acknowledged.

3. On the other hand, we are steadily reaching a point where the survival
of foundations of mathematics itself is in jeopardy. Experts in 1-4 are
concerned with their own survival, and their agenda does not include
foundations of mathematics.

We now discuss six stages of evolution of many mathematical subjects,
assuming that there is no renewal; i.e., no "reinventing" of itself.

As far as mathematical logic itself is concerned, or these four subfields,
it can be debated exactly where they are in this evolution of subjects.
Certainly none of them are in stage six - yet. And they are not all at the
same stage. I leave further discussion of this to later postings.


The kind of mathematical subjects we are talking about here didn't start
off as organized disciplines with a well defined identity. They evolved
until at some point a critical stage was reached where something or some
things particularly striking of general intellectual interest were done.
This work can take several forms. Examples: It may be a crucial definition.
It may be a surprising fact. It may be a method to deal with a body of
problems of wide interest. It may be any or all of these.

But no matter what particular form this critical work takes, the necessary
ingredient is that the work be of substantial interest outside the subject.
This is, in a way, obvious, since at the time this critical work is done,
there really is no organized subject yet.

Naturally, the greater the general intellectual interest of this critical
work, the more generally striking it is, the greater the chance that an
organized subject will emerge. It is at this critical stage that the
standard of general intellectual interest is most likely to be determinate.

In most cases, the founders will formulate a number of further research
projects whose interest is tied to the original striking developments. The
founders and their followers - who share their zeal - will make progress on
these further projects. Generally, people excited about the original
developments will, for the same reasons, remain excited about these
further, closely related developments.

At this early stage, criticism is rare. A general atmosphere of excitement
and purpose pervades the subject.


In this stage, it is still possible to motivate the further work in at
least one of several ways:

i) that it strengthens the original founding work by e.g., answering
objections as to the conclusiveness of that original work, or extending its
range, or adding precision or clarity to the original work;
ii) that it addresses a closely related issue of comparable interest to the
founding work;
iii) that it lays the technical groundwork for accomplishing i) and ii).
E.g., we need to understand such and such structures better in order to
make progress on something coming under i) or ii).

The researchers involved are generally the founders and their followers -
close colleagues who share their zeal - and also students of the founders
and followers. At this relatively early stage of development, these
students will naturally be intensely exposed to

a) the founding developments;
b) the general intellectual interest of these founding developments;
c) how these founding developments came about;
d) the exact relationship between their graduate work and the founding
e) the founders and their followers vision of the further development of
the subject.

The reason that these students are naturally exposed to this is

1) there is not all that much crucial material to learn since the subject
is so new;
2) almost all colleagues will be outside the subject - since it is so new -
and students will normally be required to give an explanation of what it is
and how it came about as they seek employment.

At this relatively early stage, criticisms are generally easily answered
with reference to the founding developments. During this stage, the subject
gains a sufficiently autonomous identity so that people can make a career
working entirely within that subject. The preponderance of research is
still motivated by the original purposes of the founding developments.


In this stage, a gradual process takes place where, by the end of this
stage, it is no longer possible to motivate the preponderance of work by
any of 1)-iii) above. Exceptions become rare, and are not the focus of much
attention by experts in the subject. However, there are still some experts
in the subject who do use i)-iii) to motivate their work. And there also
are experts who do not use i)-iii) to motivate their work, but are at least
concerned about general intellectual interest, and what motivation is to
replace i)-iii). The bulk of the experts, however, are unconcerned about
the issue of motivation and general intellectual interest. E.g., they take
one or more of these positions:

a) there is the indirect motivation that their work is in a long chain
going back to the founding developments;
b) that their work requires no motivation since it is intricate and/or
c) that talk of motivation is irrelevant to the progress of subjects;
d) that talk of motivation is bad for morale;
e) that talk of motivation is so subjective that it is beyond the scope of
consideration of experts in the subject;
f) that talk of motivation is irrelevant since no one can predict what will
become important later.

In graduate school, the motivation for the founding developments may only
be mentioned in passing, if at all. The impression on the students is that
the founding developments were from a long gone era, and the subject is
concerned with other matters. The motivation for these other matters is of
little direct concern. The passive message is: if you are fascinated by
what we do, then stay with us; otherwise, find something else you like

Criticisms start coming in. E.g.,

1) what is the point of this work?
2) how does it relate to the founding developments?
3) can it be used in other subjects?

Some of this criticism is from outside the subject, and some of it is from
inside. The inside criticism is by those experts who still use the founding
developments to motivate their own work, or those experts who talk to
people outside the subject and are concerned about lack of motivation of
the bulk of the work and are concerned with general intellectual interest.
By the end of this stage, tensions between the experts in the subject grow.
These tensions get resolved one way or another in the next stage.


At this stage, tensions that appear in the third stage heighten. By this
stage, there is a recognized market for people specializing in the subject.
The bulk of the experts are not concerned with motivation or general
intellectual interest, and are in control of the hiring process and Ph.D.
production. They strongly differentiate between the "hard" and "soft"
people doing "hard" and "soft" work. The "hard" stuff is more and more
intricate and more and more difficult to motivate. The "soft" stuff is an
attempt to remotivate the subject, and is comparatively less intricate and
more easily motivated. The "hard" people choose not to hire the "soft"
people, and choose to have "hard" students. They defend this by insisting
that they are hiring the "best" people regardless of other considerations.
The "soft" people complain that they are not being given due credit for
imaginative connections that serve to reinvent the subject. The "hard"
people judge the "soft" work on the basis of the technical content. The
"soft" people say that dangerously few people care about this "hard" work
except for the people doing it.

If the "soft" people win the battle and their ideas are sufficiently good,
there will be a renewal. But usually the "soft" people lose. We will
henceforth assume that they lose. [NOTE: There is the case where the "soft"
people win and their ideas are not sufficiently good. I won't discuss this
interesting case, particularly since I am not familiar enough with examples
of it.]

At this point, the "soft" people either

i) finish out their careers in the subject as second class citizens, with
little or no influece in the subject, rarely invited to meetings, cut off
from grant support, with few students, which generally become
underemployed; or
ii) switch to another subject where the motivation appears stronger to them;
iii) gather together to form another related subject with the idea of
getting to stage one.


In this stage, there is a steady breakdown in the status of the subject
within the hiring community (I am usually thinking of the mathematics
community). A set of interlocking and reinforcing phenomena appear.

1. A breakdown in the senior market for experts in the subject. When senior
experts vacate their positions (retirement, death, relocation), departments
push to convert the position to another subject, or at most refill the
position with temporary or junior people. This is particularly damaging
when the senior expert is one of the big "stars." This is one main way that
hiring units (departments) indicate that they do not view the subject as a
major component of their hiring unit.

2. An unwillingness of the most highly regarded hiring units to have
significant representation in the subject. This indicates that the subject
is regarded as a subnormal priority for hiring.

3. Relatively low visibility of the stars of the subject compared to stars
in other subjects. This is a consequence of 1 and 2. This includes the
infrequent awarding of major prizes and awards to stars in the subject
compared to stars in other subjects.

4. Relative weakness of letter writers for jobs. This is a consequence of
all of the above. Typically, candidates for jobs in, say, the top 40
mathematics departments these days have references who are primarily fields
medalists, recipients of major AMS awards, Harvard/Princeton/IAS/IHES...
senior mathematicians, NAS members, etcetera.

5. There is a drop in the level of research departments making tenure track
hires in the subject. This is partly a consequence of 4. Mathematicians
often deny that they discriminate against subjects by citing the relative
lack of eminence of letter writers.

6. Ph.D. production drops. This is a consequence of so many of the hires
being made at marginal research institutions without healthy Ph.D. programs.

7. Loss of high level talent to other fields. All of these factors lead to
a reluctance of high level talent to put a life long committment into a
subject experiencing such a breakdown in its status. Also, lack of clear
motivation adds to the loss of high level talent. Subjects with clearer
motivation and greater general intellectual interest tend to be more
attractive to high level talent.

8. A lowering of the status of the subject in the graduate student curriculum.

Yet not all news is bad news. There may be one top mathematics department
that wants to maintain a top tradition. Yet another top mathematics
department won't even consider replacing people even after all of them
left. And another lesser department may not even consider hiring anybody.
And yet another lesser department may be loaded up with many hires.

One measure is less variable: the level of intellectual committment to the
subject. In most cases where there is substantial support for hiring in the
subject, that support can be traced to purely political factors. E.g.,

i) people in the field are "owed" appointments because of their past
service on behalf of the hiring unit;
ii) the opportunity to be number 1 or close to number 1 in the subject with
little difficulty in light of market conditions.

The political factors tend to hide the persistently downward trend because
people in the subject can point to positive developments from time to time.


At this stage, departments with healthy Ph.D. programs will reject job
candidates in the subject outright - sight unseen - simply on the basis of
the subject, without considering credentials.

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