FOM: A 1997 message of Professor Pillay...His second ``overly polemical statement''

[Matt Insall]

Professor Pillay's second ``overly polemical statement'' on 25 Sept. 1997
dealt with the ``enormous abstract development of math in the past 60-70
years''.  Again, I agree with his assessment of what we should do.  He said
we should perhaps take a ``position'' on recent developments in mathematics.
Then we should see how much logic can be applied or is ``relevant''.  I feel
that this is what we should do, even when we are not expertly trained in a
particular area.  For instance, I originally worked in nonstandard methods
applied to lattices and universal algebraic structures.  Now, I did work for
a while only in that area, because I felt more comfortable with the
classical results that were already available, but I saw there are
connections to other subjects, that I think would be worth investigating, or
at least suggesting to students that they investigate them.  (Nonstandard
analysis itself is, in my opinion, an outgrowth of this type of thinking
about mathematics, its foundations, and the interactions between the two.
In fact, it was Robinson's insight that the compactness theorem could be
used in this way, and, with Zakon, the understanding that an algebraic
flavour could be lent the development of nonstandard analysis by appeal to
ultraproducts.)  After a while, I began some collaborative work with
computer scientists here.  Any collaboration seems to take some time to
develop, and although I was not trained in nonclassical logics (the
foundations of nonstandard analysis are decidedly classical in nature, with
an algebraic flavour introduced due to the ultraproduct construction), or
computer science, I undertook to work with them on applications of temporal
logic.  During this entire time, I have been willing to be wrong, and to
find myself struggling with concepts outside my own areas of past
mathematical experience, because I see the worlds of mathematics and their
applications, or of foundations and their applications to mathematics or
computer science as a whole, rather than as compartmentalized collections of
disembodied theorems.  In doing so, I have at times been perhaps overly
cautious about accepting some of the ideas of nonclassical logics (applying
some old-fashioned ``dogmaticism'' - see Professor Pillay's original post).
[I have had, at times, various technical reasons for being concerned about
the utility of nonclassical logics, but those are for another post.]  This
discussion about my opinions and experience has been intended to lead to the
point that I think it is necessary for us to be less shy about our
occasional (or even frequent) mistakes than we have been in the past, and
not say, as I have heard some mathematicians say ``I don't know much about
that.'' when the discussions turn to something outside our area, without
then at least suggesting that it would be interesting to find out more.  For
while it is (probably?) impossible to know everything, it is imperative that
we try to know more all the time about anything that may apply to our own
research or may be able to use our own research somehow.  One thing that may
be involved if one is to pay attention to some other area than one's own
specialized work is ability to ``take a position'' as Professor Pillay said,
even if the position one takes is that of a devil's advocate for a while, or
is ``conservative'' in some sense, or else is somewhat ``rebellious''.
Taking a position leads to a narrower sequence of thoughts about a
particular concept, and may thereby result in more, and more useful,
results.  For example, although there are certain foundational uncertainties
about statements like the continuum hypothesis, certain mathematicians have,
in the past, taken a position on them, and entire bodies of (useful)
mathematics have come from one position or another.  (Consider the climate
that has lasted for a while about the axiom of choice, in much of
mathematics.  While some mathematicians are concerned about it, others adopt
it, and use it freely, even though they do not know all of the foundational
implications of doing so.  This is, as I see it, wholly appropriate, even
for a student of foundations, for certain very useful and interesting
theorems can result.)  Now, there is a way in which Professor Pillay's two
first comments seem to me to be related.  It is this:  When we take a
particular position, we may be at odds with someone else who takes the
opposite position.  That is, a form of ``dogmaticism'' can appear in one
side or the other of a conversation.  I suggest we be careful about this,
but to some extent, it can be beneficial.  For instance, I have found myself
occasionally being somewhat dogmatic in this forum, and I do not think I
have been alone.  Sometimes I have been wrong.  If I had not shed the
shyness that comes from not wanting to get into a disagreement with someone
who is more adept and more well-read in one or more of the subjects in which
I am interested, I might not have had the appropriate shakeups that led me
to see my mistakes.  I think now that I have some information and new
motivation to check certain results in mathematics and foundations that I
had not previously seen or had not paid such close attention to before.  For
the profession as a whole, the dogmatic approaches of certain schools of
mathematicians or logicians can lead to controversy of course, but in some
cases, I feel this controversy itself will help us focus our attention on
certain problems and potential solutions.  That is, the conflict can, I
think, be beneficial, when ``properly'' dealt with.



Dr. Matt Insall
http://www.umr.edu/~insall