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

[Michael Thayer]

It is with some trepidation that I sned this note.  Steve has kindly allowed
me to join this group despite my being neither professional mathematician
nor philosopher, nor barber.

I  am a computer consultant who reads mathematics and philosophy for
pleasure, and am interesting in some of the (as yet undelivered) promises of
this list.

Steve has said:


>>I want to try to clear the air.  ...
 This is a good idea.



>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.



On the contrary, my view is that this DOES NOT  form an acceptible working
definition for many on this list.
For my self, I see nothing in point 1. of Steve's paper that is even
plausible.  It also seems that many others have difficulties with some of
the underlying ideas that inform Steve's definition.  In particular,
structuralists (and many category theorists, etc.) do not believe in a
closed hierarchy of ideas.  Their view is similar to Hilbert's
chair-table-mug comment about geometry.

>
>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").


This is possible true, but there is also a long history of the notion that
foundational/fundamental/radical study of any field is aimed at
clarification, not reduction.  It may be that reduction is possible, but the
hierarchy that Steve posits may look like some non-founded set.


>
>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";

Let me try to rephrase what I think Anand (and some others ) are complaining
about.;
As to (1):
The definition is 'cultish' because it smacks of Objectivism, and
Objectivism is thought by some to be a cult.
I think that the first part of this is true, but the second is overly harsh.
As to (2):
Not everyone can swallow the simple hierarchy of concepts that is required
for the definition to go thru.  There seems to be some historical evidence
that implicit definition is as useful as explicit definition.  The former is
typical of the sturcturalist view; the latter of the reductionist view.
As to (3):
My personal offering would be that something is of foundaitonal interst when
it shows that something which has always been thought of in one way, can
also be concieved in a totally different way.  This seems to square with
many of Anand's examples and may be what he is getting at.

After a further discussion, Steve closes with
>
>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.

I would suggest that a more modest goal is equally acceptable.  But before
doing so, let me digress by way of illustration.  While at Harvard in the
early 60's I had many opportunities to hear B.F. Skinner and some of his
followers debate with various cognitivist  critics.  These discussions were
characterized by a talking-past-each-other that matches much of what as
appeared on this list (I have read all the archives, and so should be fairly
up to date).  My view was that Skinner was claiming far more than he could
prove, but that his critics were ignoring some very clear and definitive
results he had achieved, and which needed understanding within their own
frameworks.

Now to my suggestion.  I would hope that even those who are not sure they
can agree with the metaphysical underpinnings that drive Steve's interest in
foundation could still accept his program as a useful attempt to see how far
the search for fundamental concepts can go.  Even is Steve is ultimately
unable to show that his fundamental concepts are not replaceable by others
which serve equally well, the attempt is highly useful as a reductio
argument.

In this spirit, I agree to accept Steve's "home truths" for the purposes of
discussion, and to move on to something else.

Michael