[FOM] Re: pom/fom. directions

[Harvey Friedman]

NOTE: I write 

DON¹T QUOTE ME OUT OF CONTEXT

to indicate that if I am inappropriately quoted, people could think that I
am attacking FLT and/or the person(s) credited for proving it.

***********************************

Reply to Williamson.

On 10/24/03 7:33 AM, "Jon Williamson" <jon.williamson at kcl.ac.uk> wrote:

>> 1. 1951, 1959. A. Tarski. Decision procedure and completeness of axioms
>> for
>> elementary Euclidean geometry, real algebra, and complex algebra.

>> 2. 1963. PJ Cohen. The unprovability of the axiom of choice from ZF, and
>> the
>> unprovability of the continuum hypothesis from ZFC.

>> 3. 1970. Y. Matiyasevich, J. Robinson, M. Davis, H. Putnam. No algorithm
>> for
>> solving all Diophantine equations.
> 
> I certainly believe that these developments were immensely interesting and
> important for FOM, as you do and as I'm sure most members of this forum do.

Of course.

> But I fail to see why they count as being of general intellectual interest
> (gii). 

Are you trying to make a joke? It is funny.

Even the earlier result about exponential Diophantine equations that led up
to 3, was written up in the Herald Tribune at the time.

The 1970 result created quite a sensation even within mathematics. I am in
close contact with one of the most famous number theorists in the world who
is deeply interested in it to this day, and has asked me a number of
questions about it over the years.

The coverage of 2 was very extensive worldwide, and certainly had, e.g.,
Scientific American coverage, and NY Times coverage.

I have discussed all of the above very successfully with generally educated
people on many occasions, mathematicians, physicists, finance professors,
lawyers, psychologists, musicians, etc., where the people were very
impressed and receptive. Smooth as silk.

If you need it, I can write a "NY Times article" on all three of things
right here on the FOM.

DON¹T QUOTE OUT OF CONTEXT: I repeat that FLT is incomparably more
interesting to number theorists than 3, and also more interesting to most
mathematicians than 3.

However, it is completely obvious that the gii of 3 is not only great, but
clearly greater than that of FLT.

To someone outside math, the FLT story is only a human interest story. You
could substitute any neat little mathematical equation, and give the same
story, and the public would not know the difference:

Here is a conjecture of Erdos taken from
http://mathworld.wolfram.com/SquareNumber.html

"The only solutions to n! + 1 = m^2 in integers are (5,4), (11,5), (71,7)."

The other mathematics that FLT led to is also more interesting to
mathematicians than 3 (vastly more to number theorists), but does not have
gii, if only because there are no publicly understandable statements.

Just to make this point clearer, I am in close contact with a mathematician
(different than any mentioned by me recently) who talks to physicists a lot.
Some of the prominent physicists he talks to expressed a lot of curiosity
about the FLT. But as reported to me by the mathematician, the physicists
were entirely focused on the human interest side of the story - they had no
idea what the mathematics behind it was about, and furthermore they said
they had no idea why anybody would be excited about that particular
equation. 

In contrast, 3 is not a human interest story at all, and has an obvious
level of profundity about it - algorithms and impossibility results. There
is a major shortage back then - and also now - of decision problems known to
be algorithmically undecidable that are of general interest. (I am not
including any decision problems which themselves are close to the notion of
algorithm).  DON¹T QUOTE ME OUT OF CONTEXT.

>You don't mean as much as `of interest to the general educated
> population' by gii,

Yes, to the general educated population, in these three cases. Remember -
even with the exponential Diophantine pre-result, the Herald Tribune covered
it.

>but surely you mean more than `of interest to FOMers'?

Far more than that. Computer scientists, and philosophers for openers. But a
huge number of people are fascinated by such things, just as they are with
the work from the 1930's.

> (If not then I think you're unlikely to persuade Corfield or indeed any
> nonFOMer.) By gii I mean something broader than `of interest to FOMers'.

Yes, and these three easily meet those tests.
> 
> Although these developments are closely related to pre-1940's FOM
> developments that, we agree, were of gii, I don't think that qualifies them
> as being of gii themselves.

They are of obvious gii in and of themselves, independently of earlier work.
Of course, the achievements themselves depend on earlier work, as expected.

>The general interest of the earlier work in FOM
> arose from general interest in computability and in logicism and formalism
> in the philosophy of maths.

The gii of the work in the 1930's is so obvious that it completely
transcends any philosophical fashions at the time, then, before, and now.
Period. 

>By the 1940s logicism and formalism were
> generally viewed as having been refuted,

There are no refutations of isms, even today. There may be some refutations
of certain arguments for and against isms, but that's all.

>and by the 1950s the computer had
> taken something like its current form and there was more general awareness
> of the limits of computing

You are fishing for artificial reasons for *claiming* not to see the gii in
these three items - as it is undeniable and obvious. What you are talking
about is completely irrelevant to gii.

As far as the limits of computing are concerned, we are in the dark about it
at the most fundamental levels, even today.

>(and by 1970s there was a backlash against
> earlier bold claims about the achievements computers would yield).

Again, grasping for straws to deny the obvious. This has nothing to do with
the obvious gii of these three items.

>I think
> that explains why the developments you cite, though for FOM perhaps as
> important as earlier work, were of less interest outside FOM.

Less than what? And when? The work in the 1930's itself went through a
period before it really took hold. Besides, you are struggling to deny the
obvious gii of these three items, and now you talk of "less interest"?!
 
> I would say the proof of FLT achieved gii because:

DO NOT QUOTE OUT OF CONTEXT. The FLT event only achieved a human interest
story, in the general population of educated people. They wouldn't even
begin to see why anybody would care about that particular equation - along
with the top physicists I referred to above. This is very likely to remain
the case. 

Whereas, it is child's play to impress educated people with the profundity
behind 3 - both the fact that there are robust models of computation in the
first place (Turing), and the long term odyssey of Diophantine equations,
are involved. They make for a very powerful combination. DO NOT QUOTE OUT OF
CONTEXT. 

>I take it you agree with Corfield then that
> questions like
> "how do concepts get formed and adopted in maths?"
> "can Bayesianism be applied to mathematical beliefs?"
> "what is the nature of mathematical research programmes?"
> all make clear philosophical sense and are worthy of addressing, whether or
> not that involves work in FOM.
> 

The FOM email list uses FOM for the name of the email list, and f.o.m. for
"foundations of mathematics". This convention is due to Simpson in the
Rusted Age (inside joke).

These are great questions. I am not aware of any breakthroughs on these
items that have been made. Breakthroughs on these items are very likely to
be of gii. 

I don't think that substantial progress can be made on any of these items
without *at least* using the vast experience and tools of f.o.m.

If some initial progress can be made on these questions using little or no
f.o.m. tools/experience, then I have no doubt that that can be highly
leveraged into much more progress by invoking f.o.m. tools/experience.

I hope to be near the first in line if serious initial progress is made. Can
you present some serious initial progress on these items?

Harvey Friedman