[FOM] mathematics as phenomena

[Harvey Friedman]

On 1/28/06 10:47 AM, "Timothy Y. Chow" <tchow at alum.mit.edu> wrote:

> Neil Tennant wrote:
>> So, that was "thumbing of the nose" #1 on the part of CMs to
>> metamathematicians and foundationalists (MFs).
>> "Thumbing of the nose" #2 concerned set theory.
> 
> Joe Shipman wrote:
>> What Friedman is criticizing is the determination of most
>> mathematicians to regard Godel's Incompleteness phenomenon as a
>> curiosity that is not relevant to mathematics as a whole, rather than
>> as a challenge to get involved in the METAmathematical pursuit if
>> identifying new axioms to be accepted as true.
> 
> While I certainly find Friedman's theorems in this area extremely
> interesting, I am also somewhat troubled by the extent to which some of
> these efforts (I do not speak specifically of Friedman here) seem to be
> motivated by annoyance at the fact that "f.o.m. don't get no respect."

It is true that I wear two quite different hats about this.

1. Scientific Hat. I treat not only mathematics itself but mathematical
practice as a phenomenon which is deep and rich and obviously worthy of
intense study for its own sake and other reasons. Clearly there is an
informal notion, with considerable objectivity, of what is mathematically
important, or natural, or beautiful, or interesting. Of course, there is not
anything like universal agreement in many cases. But there is definitely an
important objectivity surrounding these notions as used by practicing
mathematicians. As I have said on the FOM before, I continually check my
intuition about these things with practicing mathematicians. There are also
the usual subfields and subsubfields of mathematics, for which there is
considerable objectivity as to what falls in what category or categories. It
is also true that mathematicians generally are not engaged in serious
research as to what these various terms mean. I would guess that only very
few mathematicians would have the ability to say something illuminating
about what these notions mean, and that this would involve much more
extramathematical considerations than they are normally comfortable with.

2. Academic Profession Hat. I have no doubt that f.o.m. is in a very awkward
position in academia, falling into a special place between mathematics,
philosophy, and computer science. I also have no doubt that Godel's place at
the top of mathematical thinkers (notice I said thinkers, not
mathematicians) is permanent, and that f.o.m.'s place among mathematical
subjects (subjects with mathematical methodology) was at the top in the
1930's, and also at the top in the 2000's. Also I have no doubt that the
present perceived view of the importance, relevance, beauty, value, promise,
etcetera, of f.o.m. is profoundly wrong and reflects poorly on the
intellectual judgment of the people involved in those determinations. But
still there are objective reasons that allow this attitude to persist, that
are worthy of intense study, and so this feeds into 1 above.
> 
> Honestly, what do I care if someone thumbs his nose at me?  There seems to
> be something psychologically unhealthy about channeling enormous amounts
> of effort into winning someone else's acceptance or changing someone
> else's behavior.

You are not talking about me, because of 1 above. Also with regard to 2
above, there is present danger that f.o.m. as we now know it, will die as a
viable profession, at least in the most visible parts of the academic
community - and perhaps elsewhere. People have, in the past, turned the
other cheek, while they are systematically slaughtered. I am not known as a
pacifist. 
> 
> For example, it seems to me that one likely outcome of Friedman's results
> is that large cardinal axioms will join the axiom of choice in the
> category of axioms that one learns may be relevant sometimes and which one
> goes ahead and assumes if necessary.  However, "core mathematicians" will
> *still* exhibit no interest in engaging directly in f.o.m. and the "search
> for new axioms." 

I remember conversations I had with a well known analyst at MIT many years
ago - whose name I forget, but would like to remember - who was Chairman
there and also represented the AMS in Washington for a while. I think he is
no longer with us. 

I was talking to him in the 70's and early 80's when I really clearly
crystallized the program of getting incompleteness for "real, good, concrete
math". 

I remember vividly him telling me that most of the very top mathematicians
he knows over the years were attracted to the subject substantially because
truth and proof were not an issue - and he named some preeminent names. He
graphically asserted that these people would be visibly shaken by any
suitable good concrete independence result, and would very strongly want to
try to return math to the good old days where the rules of the game were not
an issue, and where everybody thought there were no logical difficulties
with real concrete interesting math. The only satisfactory way to do that is
for them to study the new axioms.

On the other hand, I have been in contact with a Fields Medalist who would
like to say something more along the lines that you are suggesting. That "we
will simply use more hypotheses and state them in the theorems."

But this was before I got into Pi01 and Pi00 (not yet posted), where it
might be more awkward to take this tack. (This particular famous
mathematician probably would not change his/her mind.) After all, the idea
of absolute truth for Pi01 and especially Pi00 sentences where the number of
objects under discussion is something like 8!! or 8!!!, is more ingrained in
mathematicians. 

Obviously the best guess is that an appropriate expansion of what I am doing
will have a spectrum of reaction, but with a considerable number of the
leading mathematicians far far more concerned about f.o.m. than they are
today. For example, many of them would readily participate in Symposia about
"WHAT POSITION SHOULD BE TAKE ON THE NEW USES OF THE NEW AXIOMS?", with
perhaps, even one of the focal points being: should we just neutrally add
hypotheses and not fret about any absolute truth of Pi01 and Pi00, or should
we try to recover the lost paradise that we had, illusory as it might have
been? Other critical issues might be: is there STILL some way of siphoning
off these new results as not good mathematics, that can be distinguished
from real mathematics?

Experiments will be conjured up. E.g., I might draw up a list of 10
mathematical statements, only some of which are provable in ZFC and others
are not. Ask mathematicians in such a meeting to pick out which ones are
provable and which are not.

The drama of such an experiment would of course crucially depend on how far
I get with this development. I am not there yet. But I can imagine
experiments like this that just might convince a very wide range of people
that they can't tell the difference in terms of "naturalness", "beauty",
etc. 

>If new axioms turn up in the course of studying "core
> mathematics," then one will pay due attention to them, but studying f.o.m.
> for its own sake will still be regarded as deviant behavior.  In other
> words, large cardinal axioms will be welcomed into the mainstream, but
> f.o.m. won't.

I doubt this for the following reason. I think history shows that there are
major swings over long periods of time, and that ideas that were undervalued
sometimes get overvalued. This happens all the time in the stock market.

The mathematicians will have to face just how this state of affairs came
about. In a vacuum? No. By studying f.o.m. for its own sake. Of course,
future reactions depend a lot on how far the program goes.

> I see little point in the Sisyphean task of trying to get f.o.m. respected
> in the same way that the more glamorous areas of mathematics are.
> Indeed, there is some danger that this "evangelical" motivation will draw
> effort away from the more meaningful task of formulating and pursuing
> productive agendas within f.o.m. itself.  (Though I don't think this
> potential problem has actually materialized yet in practice.)

But on the other hand, if no attention is paid to the f.o.m. respect issue,
then there may not be an environment in which someone can be paid to work
single mindedly on such things for 40 years straight...

Harvey Friedman