FOM: 10:Pathology

[Harvey Friedman]

This is the tenth in a series of positive self contained postings to fom
covering a wide range of topics in f.o.m. Previous ones are:

1:Foundational Completeness   11/3/97, 10:13AM, 10:26AM.
2:Axioms  11/6/97.
3:Simplicity  11/14/97 10:10AM.
4:Simplicity  11/14/97  4:25PM
5:Constructions  11/15/97  5:24PM
6:Undefinability/Nonstandard Models   11/16/97  12:04AM
7.Undefinability/Nonstandard Models   11/17/97  12:31AM
8.Schemes 11/17/97    12:30AM
9:Nonstandard Arithmetic 11/18/97  11:53AM

A complete archiving of fom, message by message, is available at
http://www.math.psu.edu/simpson/fom/index.html
Also, my series of positive postings (only) is archived at
http://www.math.ohio-state.edu/foundations/manuscripts.html

I want to discuss some critical beliefs and attitudes of mathematicians
which are deeply held at various levels of consciousness by the silent
majority. On the whole, I find these attitudes and beliefs natural and
understandable. Detailed consideration of them is crucial to an
understanding of the attitude of the mathemtical community towards
mathematical logic and the foundations of mathematics. The thoughtful
consideration of them is also a supreme motivating force for much of the
current work in the foundations of mathematics.

Let me try to give a detailed explication of these critical beliefs and
attitudes. I will put this presentation in brackets, so as to make it clear
that it does not represent my view. My view is at advanced meta-levels,
since I am a professional in f.o.m. I know enough to know the problems
inherent in all views, and I seek to formulate and discover those key
scientific findings that forever change the nature of all relevant
discussion. But the exact nature of my meta-views is for a later time.

[ 1. Mathematics starts with counting and measuring. These matters are
closely connected with modeling various physical phenomenon involving space
and time.

2. These elemental considerations, which are the principal driving forces
behind the development of important mathematics, quickly lead to a number
of auxiliary considerations which are needed for the informative
development of mathematics. These include the construction of the principal
number systems, and the study of equations, inequalities, and their
solutions in these number systems (real and complex algebraic geometry,
number theory). And also various limiting processes and equations involving
them (ode's and pde's). And of course much more, such as the theory of
shapes (algebraic and differential topology and geometry, etcetera).

3. Very often, only a few key facts about the mathematical situations at
hand are needed for the desired conclusion. Great efficiency and clarity is
obtained by isolating these key facts (group theory, ring theory, field
theory, vector spaces,...; algebra).

4. There also needs to be some ultimate rules of the game, which support
what is or is not a legitimate mathematical construction and a legitimate
mathematical proof. Once this is clarified, nothing is to be gained by
going back to it or referring to it - as long as no crippling problems
arise. This has now been clarified, and we can get on with real mathematics
without worrying about it.

5. Items 3 and 4 lead to deliberately general formulations. E.g., one
really cares most about the algebraic closedness of the field of complexes
(fundamental theorem of algebra). But in item 3, one proves that every
field has a unique algebraic closure. And the best way of doing 4 is
abstract set theory where anything conceivable is considered.

6. However, 3 and 4 are never a main point. One really cares about them
only to the extent that they provide useful insights into the important
stuff in 1 and 2. One finds, time and time again, that 3 and 4 naturally
and inevitably assume a life of their own, and it is natural to try to
optimize the results in 3 and 4. All kinds of delicate and difficult issues
come up when one pushes 3 and 4.

7. Often pushing 3 does substantially illuminate 1 and 2. Often it does
not. One gets a feeling for what kind of 3 is illuminating and what kind of
3 is not. E.g., finite groups, Lie groups, algebraic number fields,
etctera, are illuminating, whereas general countable groups are not. It
often has a separate life of its own. Often its interactions with 1 and 2
are often not very substantial. The methods and techniques are often not
useful in 1 and 2.

8. Pushing 4 beyond very elementary and classical lines rarely illuminates
1 and 2. Very much less often than 3.

9. So in light of 6-8, one normally should stop pursuing 3 and 4 when it
becomes more of a problem than a help. Creates more darkness than light.
After all, it was set up to illuminate and clarify the development of 1 and
2. It will distract mathematics from its main line motivations. When 3 and
4 are pursued on their own, the onus will be on 3 and 4 to show how it
illuminates 1 and 2. 3 and 4, pursued on their own, without regard to 1 and
2, is not to be viewed in the same category as 3 and 4 purused with regard
to 1 and 2, not to be viewed in the same category as 1 and 2, and will not
be valued in the same way. And 4 is very rarely pursued with regard to 1
and 2.

10. The question of what kind of mathematical investigations in 3 and 4
that are not currently relevant to 1 and 2 promise to become relevant to 1
and 2, is inherently speculative. However, one has to make the best
judgments one can, and certain patterns seem to emerge. In general, the
more general the concepts considered, the less likely there will be
relevance. This is to be expected, since the mathematical objects of
greatest interest in 1 and 2 are obviously very very very special compared
to mathematical objects in general.

11. More specifically, as one passes to more and more general contexts, one
loses more and more important regularities that are crucial components of
our understanding of the mathematical objects in question. These are the
crucial components that support the fundamental theorems concerning these
objects. For instance, an algebraic function of C has at most finitely many
zeros, but an analytic function of C may have infinitely many zeros.
However, at least these zeros are discrete. A C^infinity function may have
even uncountably many zeros. But at least it has tangent planes which vary
continuously. However a merely continuous function doesn't even have to
have tangent planes. And solutions to various differential equations may
not be algebraic, but they are often better behaved than general analytic
functions.

12. In general, we call a mathematical object pathological if it does not
share the normal expected regularity conditions that have been the historic
focus of mathematics. E.g., arbitrary subsets of Euclidean space and the
Banach-Tarski paradox. Naturally, this is a relative term. The more
pathological an object is, the more crucial it is to explain why one is
interested. Usually, one can't explain this. The focus of mathematics will
always be on the less pathological, given its roots in 1 and 2. This is
natural because mathematical objects arising out of external reality tend
to be less pathological. Set theory represents the most extreme of the
pathological. That there are foundational difficulties with what axioms of
set theory are appropriate for set theoretic problems is no longer
surprising (if it ever was) - and also not relevant or mathematically
interesting or rewarding. Go let them revel in how many angels can dance on
the head of a pin. As long as they can teach calculus, and do
administrative work, we can continue to pay them - modestly. ]

This is truly mathematical logic and foundations of mathematics attacked.
Now that I have set the stage, the next positive posting will be an
overview concerning some the following: what is the defense? what research
in f.o.m. does the attack suggest? how far can research in f.o.m. go? what
has been done along these lines? what is the future of f.o.m.? will f.o.m.
survive as a subject?