FOM: f.o.m. as math?

[Harvey Friedman]

Here I want to respond to Shoenfield 12:55PM 8/21/98. In a separate
posting, I want to write on the issue of "combinatorial statements," and
try to get to the real issues without getting bogged down in terminology.

>   In a recent communication, I stated rather casually that fom is a
>branch of mathematics.   Here I would like to explain what I mean by that
>statement and what consequences I think it has for fom.
>     In a reply to my statement, Harvey asserted that fom is a
>mathematical subject but not a branch of mathematics.

Yes, like statistics and computer science.

> I do not
>understand the difference, and I do not see why fom and statistics are not
>branches of mathematics and geometry and algebra are.

Because the aims and goals of statistics are very very different from
geometry and algebra.

*Rather than get bogged down in terminology*, instead of saying that "fom
is not a branch of mathematics," I could say "the aims and goals and modes
of evaluation of work in f.o.m. are very different from that of core
mathematics. The differences are so great as to explain the very low level
of interaction between f.o.m. and branches of mathematics, and the people
working in them, as compared to, say, the level of interaction between the
various branches of  mathematics." Furthermore, this comparatively low
level of interaction is completely natural and does not reflect in any way
on the intellectual stature and importance of work in f.o.m. However, if
f.o.m. were a branch of mathematics, then this low a level of interaction
might well reflect very badly.

F.o.m. and mathematical logic are treated in the mathematics community as
branches of mathematics with comparatively very low levels of interaction
with other branches of mathematics. This leads to great misunderstandings,
and poor employment opportunities for f.o.m. and logic.

This is why it is so important to emphasize that f.o.m. is not properly
thought of as a branch of mathematics.

This same scenario occurred for statistics, computer science, and, to some
extent, for applied mathematics. They did not stand for this, and broke
into autonomous groups in many places. There is an underlying intellectual
significance to these groupings within Universities. Statisticians and
computer sciences do not like to call themselves mathematicians, and
certainly do not say things like "statistics and computer science are
branches of mathematics the same way as algebra and geometry are."

Of course, f.o.m. cannot really break off autonomously. However, f.o.m. as
a branch of a wider subject called foundational studies can, and if I have
my way, will break off autonomously to form the largest and most
influential autonomous group in Universities.

> He also says that
>fom is a branch of the subject of foundations.   This is a truism, but a
>useless one.

It is an essential point with lots of consequences. However, I grant that
foundations, generally, is not so well developed. However, I would like to
make a contribution to this. I have been thinking more seriously about,
e.g., foundations of probability and statisitics.

>There are no significant results on foundations which can
>be used in the various foundational studies.

The whole setup of propositional calculus and predicate calculus, with its
completeness theorems, provide essential background information, as well as
the fundamentals of recursion theory and complexity, and also model theory.
There will be a full blown development of formal systems for science and
engineering, where one establishes the independence of certain scientific
and engineering principles from others, as well as the existence or
nonexistence of decision procedures for problems in science and
engineering, and the definability and nondefinability of certain concepts
from others in science and engineering.

>By contrast, mathematicians
>have though much about mathematics and reach many agreements of how
>mathematics should be done by consensus.

I said above that foundational studies is not very well developed at this
point. But I expect the situation to be very different by 2050.

>     My statement was not intended as a truism, but as a statements that
>all of the significant results in fom are mathematical.  I challenge
>anyone to find a significant advance in fom in which the principle
>ingredient is not the formulation or proof (or both) of a clearly
>mathematical theorem.

Frege's setup of predicate calculus. I explicitly agree that f.o.m. is a
mathematical subject, but it is not properly viewed as mathematics. At
most, some sort of applied mathematics.

>     If this is correct, it has consequences for the study of fom.   I
>suggested one such consequence concerning the study of intuitive ideas
>which arise in the consideration of the nature of mathematics.   I said
>that the object in studying such concepts should be to replace them by
>precise concepts which we can agree capture the essential content of the
>intuitive notion.

Of course, some intuitive ideas may not yield to such replacement, but
still may be essential to consider. One doesn't simply pretend that the
concepts don't exist if one has no idea how to replace them. But rather
than "replace" I would use the word "formally analyze."

>When we have done this, we come to the most important
>part.   This is to formulate and prove mathematical theorems about the
>precise concepts which increase our understanding of the intuitive notion.
>Sometimes we discover properties of the intuitive notion which we would
>probably not have even thought about in an informal discussion.   (I am
>sorry that my earlier communication seemed to suggest that the above is
>all of fom; Harvey was quite right to say that there is much else.)

Fine.

>     Let me use the above to show what I consider to be the acievement of
>reverse mathematics.   The original object was to discover what axioms are
>needed to prove the theorems of core mathematics.   To make things
>manageable, researchers confined themselves to mathematics expressible in
>the language of analysis (= second order arithmetic); this is certainly a
>reasonable restriction.   The main result was that over a very weak system
>of analysis, all of the theorems which they considered were equivalent to
>one of a small number (I believe 5) theorems.

With some caveats that Simpson can discuss more fully; and see his
forthcoming book.

>An additional point
>(emphasized by Harvey) is that these 5 theorems are linearly ordered by
>provable (in the weak system) implication.   I take this to mean that the
>intuitive notion of "theorem of core mathematics" gives rise to five
>precise notions which are related in a nice way.

But this account doesn't take into account the fact that *reverse
mathematics II* will be developed with a somewhat weaker base theory,
incorporating more phenomena more sensitively, and where 5 is no longer the
appropriate number. The idea of reverse mathematics transcends the
particular way it is currently executed.

>The next step, I
>believe, should be to prove significant mathematical theorems abou these
>concepts.

This intriguing statement needs some amplification. Please elaborate.

>Thus I think that reverse mathematics has contributed
>significantly to fom, but its future progress will decide whether it
>becomes a permanent part of the theory of fom.

It is inconceivable that the idea of reverse mathematics is not a permanent
part of f.o.m., even though the base theory may change in future
realizations.