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

[Matt Insall]

Peter Schuster wrote:


>> 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.?


     Please bear with me.  On this list, I am the little fish in the big
pond, but perhaps what I say here can contribute something worthwhile to the
discussion anyway, or perhaps if someone chooses to respond, I can learn
where I am mistaken and take actions to correct those misunderstadnings.
     I am not sure that foundations can be very well described in any such
short answer as you seem to desire.  Thus in the ``short'' answers I give
below, I conjecture that I have left out a.e. thing relative to a.e.
relevant measure on the space of all potential (short) answers to this kind
of question.
     There is an aspect of philosophical enquiry involved in foundations.  I
never studied formally very much philosophy or the concept of ``synthetic
knowledge'' as it relates to ``analytic knowledge''.  (I only took two logic
courses in the philosophy department as a senior in university, but my
sophomore literature course had already brought me face-to-face with several
extant philosophies that I have since pondered, but never really felt
completely adept at discussing.)  I studied mathematics, and tried to
occasionally incorporate whatever I learned that bordered on some sort of
philosophical enquiry, if and when it seemed to make sense to do so.  It
took me a while to understand why some of the discussants on this list refer
to ``a priori'' knowledge.  I am still learning the vocabulary and jargon
that are involved in these discussions.  Now, I will give you some ``in the
trenches'', near-layman interpretations of what I consider to be
foundations, intentionally avoiding much of that vocabulary and jargon.
     I doubt that I need to say the following, but I will anyway:  Please
let me know when I am wrong.  (In some cases, the attempt to say too much in
too small a space of both temporal and spacial extent results below in a
severely constrained sentence formation problem, so that perhaps my mode of
expression will fail grammatically, or in some other technical manner, or
even semantically, but a long enough explication can unwind all the details
hidden in those tortured sentences and fragments and run-ons.)

1.  The study of foundations involves finding very sophisicated ways to
pretend to yourself that you know essentially nothing, and then explain to
yourself everything, whether you know it or not, in the hope that this
results in an explanation of something that every sentient being with a
certain level of training can understand.  Foundations of Mathematics is
just an instantiation of the foregoing description, and has existed almost
since the dawn of time, having been investigated to various levels by both
philosophers and mathematicians.

2.  The study of foundations involves modeling existing or potential
processes of intellectual enquiry, deriving consequences of the model one
develops, and then communicating with the rest of the world the consequences
of that model, in the hope of establishing long-standing theories about the
structure and stability of various forms of knowledge possessed by or
possessable by human (or other) beings or their agents, and in hopes of
finding out which properties of the models seem to fail to live up to
expectations, so that they can be excised from the model and then either
they can be replaced by empirically determined aspects, or perhaps they can
be replaced by guesses that are passable for accurate aspects or properties,
of either existing or proposed processes of intellectual enquiry.  The
Foundations of Mathematics is an application of this to the modes of
intellectual enquiry involved in learning, discovering, creating,
communicating or in some other way, producing accurate Mathematical results
for either societal amusement or educational purposes or application to
other disciplines.

3. The study of foundations involves finding and removing errors, incorrect
or inaccurate or ill-posed hypotheses or inefficiencies about or in some
specific discipline, or in or about the concept of intellectual enquiry
altogether.  Foundations of Mathematics involves all this when in any way
involved with Mathematics and Mathematical Sciences.



[Disclaimer:  I do not intend any of the above to be pejorative, although
upon re-reading it, some of what I said above seems to be reminiscent of
certain pejorative approaches taken in and outside various intellectual
disciplines.  For example, in the first item, in the first sentence, I use
the phrase ``finding very sophisicated ways to pretend''.  Although in some
contexts, either non-intellectuals or anti-intellectuals would use similar
phraseology pejoratively toward all those who make a career of some
intellectual pursuit, that is certainly not my intent.  In fact, when I
teach students how to prove something, I try to get them to imagine that
they ``know nothing'', and then ``explain everything to themselves''.
Remember also that the term ``sophisticated'', while sometimes used
pejoratively by someone who claims to be ``from the trenches'', it is
sometimes used also as an expression of awe or as a statement of fact.  In
fact, relative to the following revised version of my first sentence in item
1, the use of the word ``sophisticated'', as well as other terms was, on my
part, merely a ``sophisticated'' way to state the following everyday fact to
people who undertand the more sophisticated language:  ``The study of
foundations involves finding very educated but explainable ways to imagine
that you are as dumb as a brick, and then explain to yourself everything in
such a way that when the lights are on it turns out to mean that someone
really is at home.''*]

Matt Insall


*Reference:
A common saying that appears in a colourful way in a song written not long
ago by a fairly well-known pop rock star named Robert Palmer who once was a
member of the musical group ``The Eagles'', in the following poignant
phrase:  ``Your lights are on, but you're not home.  Your life is not your
own.''  The lyrics of the song, ``Addicted to Love'', are avilable at the
following url:  http://www.su.okstate.edu/students/mcbane/addicted.html