FOM: Mattes, McLarty, Hilbert, Yang, Chow, Pratt, Chu, Sazonov

[Josef Mattes]

On Wed, 5 Nov 1997, Stephen G Simpson wrote:

>My perspective on what has been going on the last few days:
>
>Josef Mattes gave a list of 9 diverse mathematical subjects and asked
>me whether I would consider them foundational, i.e. within the scope
>of f.o.m.  I'm not sure what the point of Mattes' question is, 

To better understand your posting concerning what you called "home
truths". You wrote 

 "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.
 ...
 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
 ...
 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. "                                           

which basically looks quite reasonable to me. I think it does, however,
raise some questions: 

1.) I'm not completely sure what you mean by "shape". Can I
translate it as "basic object of geometry"? Does it contain the concept of
"symmetry"? Do you think that a concept like "algorithm" is obviously more
basic than symmetry? Or more widely applicable (symmetry is important in
mathematics, particle physics, art, chemistry, . . .)? Should one add
"symmetry" to your tentative list of basic concepts? 

2.) Should the concept of limit be added to your list?  Probability?

3.) You switch from "fundamental to all of mathematics" to "underly most
mathematical subjects" to "used in applications". Therefore I don't know
how general you require the concepts to be. 

4.) Your selection of examples is also raises some questions:

 "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 "

No question that the three examples you give are foundational. But why are
all three out of the area of logical/set-theoretic foundations (F-R-H-G)?
Do you really think there has been nothing foundational in, say, geometry?
 
And even if we assume that everything foundational is of the
Goedel-Cohen-MDRP type, is there no theorem in set theory that is not
foundational? (I assume you would have included such a theorem in the
list of examples for 'not foundational',  since you know that some of us
are bothered by what we see as an overemphasis on set theory.)

In short, the tentative definition you gave sounded much more openminded
than the list of examples; the questions I raised where intended to
help me understand what you really meant.


>since I
>don't know whether Mattes is interested in f.o.m. at all.  If he
>isn't, then I guess the only purpose of the question was to disrupt
>the discussion of f.o.m.  But let's give Mattes the benefit of the
>doubt.  In that case, my answer is that I wouldn't *automatically*
>consider these subjects foundational.  In each case, someone would
>first need to explain clearly and honestly, starting with concepts
>that are of obvious general intellectual interest, exactly how subject
>X is related to the most basic mathematical concepts, and exactly what
>is the general intellectual interest of subject X.  This is probably
>what Harvey means by a foundational exposition of subject X.  Once
>this is done, we would have a basis for deciding whether subject X
>itself is foundational.  

I'm curious to see your foundational exposition which showed that the
ergodic theorem is not foundational. (I should probably add that I suspect
that it indeed isn't.)


Josef Mattes