FOM: miniaturization
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Fri Sep 24 15:48:29 EDT 1999
Dear Steve and dear FOM readers,
This is an answer to some fragments of Steve's letter of Th. 16
Sep 1999.
I wrote that the difference between those who know logic and
those who do not know is unlikely to motivate interesting new mathematics
and I think that you agreed. And, I believe also that H. Friedman's
miniaturizations are not motivated by the task of popularizing
f.o.m., but by their mathematical interest (beauty) [importance].
Now, let me add that when we have a general method to obtain
certain things the relative importance of individual examples diminishes,
at least in the minds of mathematicians. [A famous example was Hilbert's
basis theorem which contributed to a diminished interest in the theory of
invariants. G.-C. Rota often criticised this, and explained eloquently why
this was an unfortunate outcome of the work of Hilbert, and why the theory
of invariants remains a fascinating area of mathematics.]
Returning to my topic, if we set to find a statement S(A) of
finite combinatorics equivalent (or stronger than) the consistency of some
set A of first-order axioms, there is the following general solution:
S(A) = (every finite subset of FIN(A) has a finite model).
Thus I asked what is the importance of Harvey's special miniaturisations.
I think that something like Rota's defense of invariants theory is needed.
You stressed that there is a large understandability gap between
those mathematicians who are educated in logic and those who are not. But
if you agree that this is not the reason for which Harvey works out his
miniaturizations this does not pertain to my question. The very statement
of the P - H theorem which you gave is the best answer, for we can see a
special value (beauty) in that statement or related statements of
Friedmann for stronger therories.
I think that our correspondence on this topic may well stop at
this point because complete understanding has been reached.
[Concerning the gap between mathematicians who know logic and
those who do not (which you mentioned), it is certainly unfortunate,
but it is not our responsibility as mathematicians to bridge it. This is
our responsibility as educators. The two tasks are very different.
For example Kolmogoroff told me in 1958 that first-order logic is
known by every reasonably educated mathematician. And Kolmogorov was
certainly entitled to define benchmarks in mathematics. The fact that
there are still many mathematicians who do not know f.o.l. is certainly a
matter of grave concern since it proves that they are not reasonably
educated. (And results on miniaturisation will not help in this state of
affairs.)
Let me add: the Greeks called the wild people Babarians and the
civilised Hellens. But the Greeks were subdued by the Romans (who did not
contribute to mathematics). However, this did not happen because the
Greeks failed to convince the Romans that they were doing interesting
things (in fact some Romans, e.g. Cicero, knew it). It is the Romans who
failed to learn from the Greeks, to educate themselves. Eventually they
induced stagnation (which destroyed their own empire). Will the US
also remain unwilling to educate itself in scientific matters, and rely
on narrrow specialists as it appears to be doing today?
Plato wrote in the Republic that it is good to KNOW-HOW to do
things, but it is a better to KNOW-WHY one should do them in this way.
A cult of know-how at the expense of knowledge is certainly dangerous.
If some mathematician is not interested in metamathematics, nothing can
be done for him. Fortunately young people are not like that. Almost every
person (every mathematician) has a flicker of interest in philosophy (in
metamathematics), and it is not hard to introduce a mathematician to
metamathematics (if he is not totally set in his know-how modes).
But we must not forget that mathematics is much much richer than
metamathematics. And, it is easier to do something interesting when one
knows many good problems. In metamathematics (like in science) important
problems which are not hopelessly difficult are not as easy to find as in
mathematics.]
Returning to your letter:
On Thu, 16 Sep 1999, Stephen G Simpson wrote:
> The problem is that, while we
> logicians appreciate the crucial importance and general intellectual
> interest of statements like Con(PA) and Con(ZFC), there is no obvious
> way to convey this appreciation to our colleagues in core math and
> other scientific disciplines.
JM: Steve you are some kind of pessimist! I do not believe this at all. In
fact it is very easy to explain Con(T) if one i willing to omitt boring
details. I have explained it to many people (not only to mathematicians)
who wanted to know what Godel has done.
And all mathematicians (who know what is an axiomatic theory)
understand it immediately. In fact Hilbert's problem about consistency is
fascinating to most people, and Godel's answer equally fascinating.
(However, it is my impression that mathematicians in the US work more and
discuss mathematics less than in Europe.)
> I believe your work on S(T) and the work of Paris-Harrington and
> Friedman on finite combinatorial independence results are both
> motivated in part by the twin problems of the ``understandability
> gap'' and the ``appreciation gap''.
JM: My work was not motivated in this way. [Frankly, it had to do with
my ignorance of the philosophical significance of Hilbert's epsilon
symbols, and was inspired by the work of Paris and Harrington. I wanted to
understand what is infinity and is it fully equivalent to potential
infinity (as H. Pioncare suggested). FIN(T) was my n-th step on this
path.
[At present I think that a formalisation of mathematics in
equational-two sorted logic with a sort for mathematical objects and a
sort for truth valued objects (i.e., formulas), explains mathematics
better than FINning it. (FIN is not so good since although it uses finite
universes those are too large, such that not all their elements can be
actually imagined. While the epsilon terms of the above mentioned
formalization represent things which are actually imagined)]
> By the way, when you say that S(PA) is 17 lines long, does that
> include the statement of the axioms of PA?
JM. No. But the P - H or Friedman's statements do not make sense outside
of some combinatorial theory. So the axioms of PA are lurking in their
background. Besides it would be a pity to convey somebody those statements
without informing him about their metamathematical significance. (Most
people are perfecly happy with the theorem of Ramsey and are not thrilled
by, say, P - H on its own.)
Regards
Jan Mycielski
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