FOM: the "normal" mathematician (reply to Friedman)

[Harvey Friedman]

Reply to Davis 11:54AM 5/12/00.

>Harvey invokes the figure of the "normal" mathematician as opposed to the
>set theorist and makes much of his views regarding such things as V=L as an
>axiom and how arbitrary mathematical objects should be regarded.

The word "should" is overstated. I am explicating a coherent view or views
about generality and pathology in mathematical thought. It is not the only
coherent view. But not only is it coherent. It appears to be, for all
practical purposes, universally held among the leadership in mathematics
worldwide - both pure and applied. Furthermore, the views are at least
implicitly held by the leadership in science and engineering.

Furthermore, given the ongoing goals of the mathematical scientific and
engineering communities, it is the most natural and compelling view towards
generality and pathology for them. Of course, it goes without saying, that
these are not natural or compelling views for the set theory community -
given its ongoing goals. This is obvious since such a major thrust of the
set theory community is to develop mathematical thought in its greatest
possible generality.

But it must be recognized that developing mathematical thought in its
greatest possible generality is a goal that is not shared by the
mathematical, scientific, or engineering communities. It is also a goal
that they often (not universally) regard as suspicious, repugnant,
repulsive, misguided, disgusting, and worthless. It is here that I,
personally, disagree with them, and am moving forward to refute
"worthless".

Moreover, these coherent views about generality and pathology are widely
held (certainly not universally held) even in the mathematical logic
community. The views are especially widely held - typically strongly - by
applied model theorists and proof theorists. The views are more weakly held
by recursion theorists. The views are virtually nonexistent in the set
theory community and rare among pure model theorists.

The coherence of these views - and their essentially universal acceptance -
does not contradict the fact that these views in their strongest forms are
only beginning to be subject to refutation.

>I regard the concept of the normal mathematician as quite like the
>proverbial man-in-the-street. It makes about as much sense to make serious
>mathematical judgements based on a poll (real or imagined) of typical
>mathematicians on such questions as it would to base scientific judgements
>in biology on a similar poll of ordinary Americans on the validity of
>Darwinian evolution. In both cases one is dealing with people, who however
>clever and well-informed about other matters, are utterly ignorant
>concerning the matter at hand.

I want to correct a common misconception of what I am doing when I cite the
mathematical leadership on these matters.

I regard their views on these matters of generality and pathology as
completely coherent on their face, and restrained forms of them as
permanent and natural features of mathematical and scientific thought. As I
said, these attitudes have driven mathematical and scientific thought for
something like 2500 years, and the exceptional period was an experimental
period - with people feeling their way with reagard to what mathematical
thought is legitimate or valid. Of course, this experimental period and
acceptance of legitimacy and validity laid the groundwork for the
acceptance of the anticiapted upcoming refutations of the strongest forms
of these coherent views.

Since I have gotten absolutely nowhere with set theorists and many
mathematical logicians concerning the crucial overriding urgent importance
of an intense consideration of these coherent views towards generality and
pathology, I invoke the fact that they are universally held by the
mathematical, scientific, and engineering leadership so as to persuade
intransient scholars to take them very seriously.

The fact is that a reconciliation of the ongoing thrust of mathematical,
scientific, and engineering research with the development of mathematical
logic is by far and away the most crucial, critical, profound, and urgent
matter for the mathematical logician today and for the forseeable future.
Other issues seem trite by comparison.

I now know of no other way to influence the development of mathematical
logic over the last few decades that I have been in the profession - other
than to emphaisize these universally held attitudes, views, and beliefs of
the leadership of the mathematical, scientific, and engineering
communities.

>V=L? Most mathematicians to whom I have
>spoken are utterly astonished to learn that everything they are doing can
>be regarded as internal to a countable structure.

Part of the problem is not their ignorance. It may be that "can be
regarded" can use some further thoughtful analysis, even for the logician.
The "can be regarded" requires acceptance of the importance of first order
predicate calculus - whose fundamental importance is only convincing
foundationally from a proof theoretic point of view, etcetera.

>They assume that
>Goedel-Cohen have settled CH and are truly surprised to learn that Cohen's
>forcing models are typically countable.

Of course, there is the Boolean valued model approach that is not
countable. But my own view of their views regarding CH is that they are
perfectly willing to accept the idea - rightly or wrongly - that CH and
related questions are absolutely undecidable (which may not be the case),
and that there is no need to settle them, since they have nothing even
remotely to do with normal mathematics. It is a "who cares if they are
independent" and "who cares if they are true or false" attitude.

I'm betting that the reaction is going to be quite different as the
independent statements move into their contexts, and are, at the same time,
pervasive throughout mathematics with a compelling theme, and the
mathematics is also completely accessible to them and rather stunning and
rather familiar in character. What is particularly vital here is that the
reason why the statements cannot be proved in ZFC has nothing to do with
any generality that is uncharacteristic of normal mathematics.

At first, only a significant minority are expected to be emotionally moved
- say 10%. This will increase over time, especially as the results move
into more and more mathematical contexts, and the results get much sharper
in other ways.

>It isn't exactly news that mathematicians tend to be mostly interested in
>well-behaved objects. The move to arbitrary sets, functions etc. developed
>historically out of the need for various kinds of completeness. ... A good
>student would quickly
>recover and point out that the Lebesgue integral has these terrific limit
>theorems. The full system of real numbers (of course set-theoretically
>equivalent to the power set of omega) is needed not because the "normal"
>mathematician has any interest in undefinable reals, but because the full
>continuum is needed for completeness.

This is a primarily example of where generality makes things simpler. There
is no doubt that nice canonical completions of essential structures are
very much appreciated by normal mathematicians. However, as soon as some
aspect of generality is developed that leads to great complications, they
take a knife to whatever generality they see that causes the complications,
and cut it out of consideration. They declare it to be excessive generality
whose complications have nothing to do with the underlying essential
structures and mathematical issues.

In particular, it is not the goal of a mathematician, scientist, or
engineer to ponder mathematical thought in anything like full generality -
looking for any complete theory of everything. The leadership in
mathematics is focused on arithmetic, algebra, and geometry, and whatever
has serious arithmetic, algebraic, or geometric meaning. Mathematics will
be evaluated by them in these terms.

This may appear rather narrow minded to the philosopher or the set theorist
- who, in their own ways, are looking for a complete theory of everything.
But that is a different goal. And more is not necessarily better. After
all, even if the philosopher or the set theorist succeeds in coming up with
a complete theory of everything, that doesn't mean that they will be able
to say anything new about arithmetic, algebra, or geometry. Or even say
anything interesting at all about arithmetic, algebra, or geometry

I, personally, am interested in all coherent points of view, and in
developing deep relationships and reconciliations that may exist between
them. I am also interested in establishing that there may be no deep
relationships and reconciliations in some cases - if that be the case.

There are serious issues of clear and profound significance with the
potential for a huge impact on intellectual history - and many of the
technical tools needed for major advances are already here. Let's get
serious about using them for dealing with the serious issues.