FOM: RE: What is f.o.m., briefly?

[Matt Insall]

(I apologize in advance for the length of this post.)


Matthew Frank wrote:
``I use the term "foundations of math" to mean:  Axiomatics with related
conceptual and methodological investigation of mathematics.''

I like this one.  Brief and to the point.  In some cases, when speaking to
various people, some details may need clarification, but for those of us on
this list, I think this is quits understandable, and, while verbally
economical, packs much information in a small space, for this audience.
Now, I wouldlike to examine again professor Schuster's original question:

``When one is asked to give an explanation as brief as
possible, could one perhaps reduce foundations of
mathematics to the question whether synthetic knowledge
a priori is possible---and, if so, which, how, etc.?''

It is a different type of conversation I was imagining when I saw this
question from Professor Schuster.  What I mean is that when I read ``When
one is asked to give an explanation as brief as
possible'', I did not think he meant that we were asking each other what is
fom.  I thought of myself sitting in front of a board of curators or
legislators, trying to justify the start of a new department, or talking to
freshman students about what kind of research a mathematician or
mathematical logician can do.  I thought about the conversations I have with
my friends, relatives and acquaintances who would like to think I am doing
something worthwhile, but keep telling me that math was their worst subject
in high school and they do well to keep up with their checkbooks,and that
they don't see how any of that stuff is useful, except for balancing their
checkbooks.  But then I read the later part:  ``could one perhaps reduce
foundations of mathematics to the question whether synthetic knowledge a
priori is possible''.  I then thought about philosophers, who know more than
I do about ``a priori'' knowledge and ``synthetic'' knowledge.  I thought
about computer scientists, who frequently model human knowledge in order to
simultaneously model machine learning and machine knowledge and information.
I thought about psychologists and psychiatrists, who investigate the role
that knowledge, information, and the ways in which they are processed, and
their origins, play in human behaviour patterns, in order to treat mental
disease.  And then I thought of all the lawyers to whom the concept of
knowledge and a multitude of philosophical concerns that affect the way laws
have been, are or will be written and how laws should be written, which, I
expect, is again related to ethics, a branch of philosophy in which concepts
such as analytic knowledge and synthetic knowledge may make sense and may be
terms that are used every day.  All these possiblilities bring me back to
the following concern about either professor Schuster's approach to handling
the question or Professor Frank's approach, each of which packs its own
elegant charm in its completeness and brevity:  Each of the proposed
descriptions of fom is very limited in terms of who is likely to understand
it, or even who is likely to understand very much of it.  In some cases, the
majority of your given board of curators will be a collection of lawyers,
and in some cases they will remember something about axiomatics, or about
synthesis or analysis, in the sense in which you intend.  Perhaps I am
wrong, but I think that this will in most cases not communicate much to
them, just because the audience for which you constructed your description
(mathematicians and philosophers and others on fom) is not an audience of
which they are a part.
     Thus, perhaps before any of us answered, it would have been better to
ask professor Schuster to tell us who his audience actually is.


Matt Insall
www.umr.edu/~insall