FOM: "home truths": what is foundational and what is not

[Stephen G Simpson]

I want to try to clear the air.  There has been so much obfuscation
that some "home truths" are now needed.  In this posting, I won't try
to make any novel points.  Instead I will try to lead the way to some
common ground where we can engage in productive discussion of exciting
issues and programs in the foundations of mathematics.

Terminological clarification:

  FOM = this mailing list

  fom = the e-mail address (minus @math.psu.edu) for posting to FOM

  f.o.m. = abbreviation for "foundations of mathematics"

  foundations of mathematics = ?; see below for my definition.

Dave Marker writes:
 > I find "what can be explained to one's barber" a very poor
 > benchmark for testing "general intellectual interest".

I also find barbers a very poor benchmark.  As Harvey and I have
explained, barbers and similar benchmarks are invoked only out of
desperate necessity, as crude surrogates, when dealing with people who
"just don't get it," i.e., are unable to grasp foundational issues on
any other level.

 > I would offer the test "could be discussed in a reasonable length
 > of time in a way that an intellectually curious layman would find
 > interesting".

This is a slightly better benchmark than the barber, but it's still
pretty poor.  If you try to use this benchmark, you will be reduced to
sharing pointless anecdotes about your experiences when dealing with
various segments of the population, e.g. 90th percentile highschool
students, 40th percentile college students, readers of Scientific
American, readers of Popular Mechanics, readers of UFO Gazette, et
cetera ad nauseam.

The only correct way to distinguish what is foundational from what is
not is to understand the concept "foundational" and apply that concept
in an intelligent, consistent way.  This of course presupposes that we
have a good working definition of "foundational" or "foundations of X"
on the table.  My view is that we do have a good working definition on
the table, namely my definition in

    http://www.math.psu.edu/simpson/Hierarchy.html

in terms of the hierarchy of concepts.  I don't claim that my
definition of "foundational" is the ultimate or optimal one; I'm open
to suggestions for improving the formulation; in fact, I have already
committed myself publicly, here on FOM, to one such improvement.
Nevertheless, I think my definition is pretty good as it stands.

To quote from one of my previous postings:

  My test for whether something is foundational is, how much does it
  focus on the most basic concepts (in terms of the hierarchy of
  concepts).  General scientific interest and intelligibility is a
  significant byproduct of this, but not the essence of it.

There is an obvious need for a concept of "foundational" such as this
one.  Moreover, I submit that this concept of "foundational" is
entirely consistent with the way "foundational" and "foundations of X"
have been used for the last 150 or 200 years in the general
academic/scientific community.  For example, it's entirely consistent
with Hilbert's use of the term in his two-volume work on f.o.m.,
"Grundlagen der Mathematik" ("Foundations of Mathematics").

The exchange here on the FOM list has many positive aspects, but one
of the annoying aspects is the behavior of people such as Anand, who
just don't get it.  Anand (1) exhibits hostility to my definition of
"foundational" (he finds it "cultish"); (2) offers no explanation of
what's wrong with my definition of "foundational"; (3) offers no
alternative definition of "foundational"; (4) makes wild claims to the
effect that any and all abstract unifying concepts in pure mathematics
are "foundational," when they are obviously not so; (5) attacks and
minimizes the interest and special status of genuine contributions to
f.o.m.; (6) exhibits personal hostility toward me and Harvey.  I don't
claim to understand Anand's motivation or mental state.  In
particular, I don't know whether Anand is deliberately trying to
prevent discussion of genuine foundational issues.  But I must point
out that Anand's behavior makes no sense, regardless of motivation.
If I don't point this out, the discussion of genuine f.o.m. is going
to bog down.

Let's try to put the nonsense aside and focus on the question at hand:
what is foundational.  My definition of "foundations of X" is, in a
nutshell: the systematic study of the most basic concepts of and
logical structure of subject X, with an eye to the unity of human
knowledge.  In order to discuss f.o.m., we have to understand which
concepts of mathematics are basic (with respect to the hierarchy of
concepts), and how those basic mathematical concepts serve to connect
mathematics to the rest of human knowledge.

We need to focus on the most basic mathematical concepts.  Here is a
tentative list of them:

  1. number
  2. shape
  3. function
  4. set
  5. algorithm
  6. mathematical proof
  7. mathematical axiom
  8. mathematical definition

By calling these concepts "basic", I am not trying to imply that they
are central concerns of every working mathematician.  Rather, the
point I am trying to make is that concepts such as these are
fundamental for all of mathematics (in terms of the hierarchy of
mathematical concepts) and serve to tie mathematics to the rest of
human knowledge.

For example, numbers and shapes underly most mathematical subjects and
are the key to virtually all applications of mathematics.  Algorithms
are methods of calculation that are used in applications.  Functions
are important in describing change and in many other contexts.  Sets
occur in many contexts and are somewhat analogous to the general
logical/scientific notion of "species" or "concept" or "class".
Proofs, axioms, and definitions are key logical/scientific notions,
here specialized to mathematics.

The above list of basic mathematical concepts is subject to revision.
However, I want to stress that higher-level unifying mathematical
concepts such as cohomology, Lie groups, categories, projective
varieties, etc etc, although of great interest and importance for the
working mathematician, are not basic and do not belong on this list.
If you think they belong on this list, you just don't get it.

Now that we have some idea of which mathematical concepts are basic,
we can begin to talk seriously about issues, programs, and results in
f.o.m.

Some familiar examples of foundational results:

  1. G"odel's completeness and incompleteness theorems, because they
  say something striking and basic about mathematical proof.
  
  2. The MDRP theorem, because it says something striking and basic
  about algorithms and numbers.
  
  3. The G"odel/Cohen independence results, because they say something
  striking and basic about sets and their role in mathematics, or
  perhaps about set theory and its role (depending on your
  philosophical view of the matter).

Some important theorems of pure mathematics which are not
foundational:

  1. Faltings' theorem.
  
  2. Wiles' theorem.
  
  3. The ergodic theorem.
  
  4. The Lefschetz fixed point theorem.
  
  5. etc etc etc 

There are hundreds of them.

A concluding note:

I hope that most people here on the FOM list can agree with the "home
truths" or fundamental points about f.o.m. that I have made above.  If
we can agree on these "home truths", then we can go on from there to
discuss more exciting matters, e.g. current issues and programs in
f.o.m.  If we can agree on these "home truths", then we will not have
to spend all our time defending the entire subject of f.o.m. against
attacks from people who just don't get it.

Sincerely,
-- Steve