FOM: sharp boundaries/tameness

[Harvey Friedman]

I refer the reader to my comment at the end of my posting 8:25PM 
2/14/02, where I wrote:

>GENERAL COMMENT: It is a lot of work to polish philosophical remarks so as
>to be immune to criticism of the sort in the recent postings of Frank and
>Pruss, etcetera. Generally, one is pushed into weakening the remarks so
>much as to be unproductive and not useful for progress in foundations of
>mathematics. So I invariably avoid trying to do this kind of work unless I
>think that such polishing would lead to something productive for
>foundations of mathematics. And then I will only do such polishing as long
>as it is or is likely to become productive. Obviously, I would find it an
>easy exercise to write such responses to my own postings (as Frank and
>Pruss have done)! I even thought of all these points while I was writing my
>postings!! However, I don't mind if FOM people wish to respond in this way
>- especially if they try to direct this into productive channels.

I'll play along a little while longer, but probably not too much longer.

>1. Certainly there are interesting theories of objects in general.  Take,
>say, Aristotle's _Metaphysics_ Z and _Categories_, Aquinas's _De Ente et
>Essentia_, Leibniz's _Monadology_, Kant's _First Critique_, and tons of 20th
>century stuff.  The theories make claims that are indubitably significant,
>if true.  They offer thought-out arguments for the claims.  Now, you may say
>that the arguments in all of these cases are fallacious and the claims
>false, but that is a substantial claim that would evidently require rather a
>lot of work to substantiate.  And of course even if there were nothing
>significant to be said about objects in general, that fact would itself be
>something significant--hence in any case, there is something significant to
>be said about objects in general.


This is not relevant to the discussion of the validity of, or the 
interest of, or the relevance of, a new notion of "arbitrary object" 
for foundations of mathematics or for philosophy. I.e., the 
discussion concerned a possible notion of "arbitrary object" that is 
supposed to be quite different than the usual notion of "object".

>
>2. It is of course true that the vast majority of mathematicians do not work
>on well-orderings of the reals--there isn't, I assume, all that much to be
>said about them as such, and the topic is rather narrow.  But there is a
>whole slew of topics on which the vast majority of mathematicians does not
>work.


I am providing a unified explanation for the general reaction of the 
mathematical community to the prospect of working on certain families 
of mathematical structures. A great deal of mathematics can be viewed 
as attempts to classify or otherwise understand families of 
mathematical structures - e.g., finite groups, manifolds, algebraic 
number fields, etcetera. Certain features stand out which seem to be 
necessary for a sustained effort of the mathematics community.

>  (There are even tons of topics on which the total number of
>mathematicians currently seriously working is under 50--indeed, I suspect
>that quite a significant number of mathematicians work in well-defined
>subfields in which no more than about 50 mathematicians seriously work.)
>But to cite another non-constructive example, there no doubt are significant
>numbers of mathematicians using Hahn-Banach mappings (MathSci find 1180
>papers containing "Hahn-Banach" somewhere in the MR entry).  Are they using
>them ineliminably?  Maybe not--but it is not obvious that they don't.


I am under the impression that the preponderance of applications of 
Hahn-Banach are to separable Banach spaces where one has 
constructivity, or at least pretty close to constructivity. E.g., 
this is far far below where any axiom of choice is needed.

Here is an interesting exercise: find mathematical theorems which are 
cited as the basis of prize winning work which are not provable in ZF 
(i.e., ZFC without the axiom of choice). Give dates and see how the 
recent compares with the past.

Here is another one: find mathematical theorems and conjectures which 
are cited in various important mathematical conference volumes (ICM, 
Math into the 21st century, etcetera), which could conceivably be 
connected to the axiom of choice. Again, trace this over time.

>
>3. Occasionally one nit-picks at some thesis, as I do in this case, because
>of a conviction that the thesis _cannot_ be given a satisfactory rendition,
>and this will come out when one starts giving obvious counterexamples.
>
I have no doubt that I can give a perfectly "satisfactory rendition" 
of such things, and many more related things. If the issue comes 
partly down to what mathematicians value and what they publish, one 
can run statistics and surveys on this, and also take into account 
the relative stature of mathematicians cited by following a recursive 
algorithm, citing the material that is prize winning, the material 
that is presented in books like "mathematics into the 21st century", 
etcetera. If the findings are overwhelming, then the burden of proof 
shifts to the doubter to argue that there is something defective 
about the outlook of mathematicians that is being cited.

I should mention that I *do* think that there are major defects in 
the outlook of mathematicians, generally, although at least some of 
the most highly celebrated ones seem to be generally devoid of such 
major defects. However, these defective outlooks are not directly 
relevant to the discussion at hand.

Of course, the most interesting justifications of the kind of remarks 
I am making are in various theorems that prove that examples cannot 
be given of certain things, or that examples cannot be given of 
certain things obeying certain conditions. These are very satisfying 
theorems, and although many of them are well known, they are 
certainly deep facts. They are an example of what I call productive 
foundational activity.

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