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

[Matt Insall]

Going through the archives, I saw a posting by professor Pillay dated 25
Sept. 1997.  One of his ``overly polemical'' statements about the
development of foundations of math ended with the statement that ``Current
work on Foundations should similarly be informed by the mathematics of
today, although not in a dogmatic fashion.''  I agree thoroughly.  As we see
from many experiences, dogmatism appears to occur in many parts of society.
I have not interviewed mathematicians or scientists about this, but I have
formed some of my own opinions, based mostly upon my experience of either
being dogmatic myself or facing some form of dogmatism.  If it is of
interest to this forum, I shall try to discuss it rationally, in the context
Professor Pillay's message establishes, namely in the culture of mathematics
and its foundations, and their interactions.  Much of what I shall say will
be speculative, but, if it is deemed pertinent, then some discussion may be
worthwhile, or some projects may be worth starting, in addition to this one,
that provide for a potential to improve the climate and reduce the dogmatism
that does appear to exist, and that does seem to hinder worthwhile
interactions.  Since these opinions are my own, and are based upon anecdotal
evidence and in many cases, personal experience and in some cases, upon
conclusions I infer, either ``correctly'' or ``intuitively'' from various
readings I have done by authors whose names I have in some cases forgotten,
I will provide few references, but in cases where I recall a specific
reference, I shall try to include it.

First, what causes a mathematician to become dogmatic?  I think that
culturally, we come into mathematics with certain pre-conceptions.  In some
cases, these must be overcome.  In some cases, they are correct.  In either
case, we then are educated by certain mathematicians who have a certain
``taste'' for a certain part of mathematics, who may or may not be dogmatic,
and in some cases may appear dogmatic because of the focus they have chosen
on a particular subject, or because of the fact that they want the answers
on their tests to be correct, according to some particular standard of
correctness.  I believe there are reasons for this concern for correctness
which are sociological-historical, scientific, cultural, mathematical,
philosophical, and foundational.  Some historical, or socio-historical
realities include the fact that much of mathematics originated in solving
problems for use in the physical world.  Anecdotally, it seems that in many
cases, accidents and disasters have resulted from some form of fault in the
design of some equipment, and the design always relies on someone with
experience to use some form of reasoning to produce a workable, safe design.
I expect this has been the case for much of history, and early
``mathematicians'', ``engineers'' and ``scientists'' recognized that what
were eventually perceived as errors in calculation or reasoning lead to
undesirable results.  Thus, a concept of ``right reasoning'' developed, and
entire subjects developed around what constitutes ``right reasoning''.
Already the concept of ``right reasoning'' was influenced by cultural dogma,
including religious preferences, but it seems that some of the most involved
work that was done was by theologians and philosophers for many years, and,
I guess, centuries.  The craftsmen and engineers learned from experience and
trial and error what worked and what did not.  The natural sciences
developed, and their needs drove various developments in the science of
reasoning that included attempts to quantify everything one used for the
purpose of solving problems.  Humans being the way they are, erroneous
attempts may have been looked upon with derision, and various forms of
intolerance, even in the sciences and the arts, developed.  On the other
hand, those who faced such derision, would I expect, respond with their own
form of intolerance, especially at times when they were in a position of
power from which to act.  Various ``schools'' developed, leading to multiple
forms of dogmatism, in and out of religious and philosophical and artistic
areas of study and human endeavour.  This situation may not have been
reflected very keenly for a long period of time in mathematics, but when
uncertainties about the foundations of mathematics and its applications
developed, and different perceptions of what constituted ``good'' or
``proper'' mathematical research became a more prominent factor, various
dogmatisms came more frequently and more bitterly into conflict.  (One
particular case I have been reminded of frequently is the case of Henri
Poincare.  According to a reference on page 3 of the book, titled
``Foundations of Set Theory'', by Fraenkel and Bar-Hilel and Levy on
Axiomatic Set Theory, which is one of my favourite old Set Theory texts,
Poincare was a strong supporter of Cantor's Set Theory (apparently he
``contributed to the propogation and application of set theory'') and upon
the discovery of the Russel-like paradoxes, made a full reversal.  This fact
about Poincare is probably no surprise to anyone in this forum, but is an
example of a rationalization by a prominent mathematician of the time that I
believe has the following characteristics (again, I am speculating about
something based upon various sources I cannot immediately put my hands on,
and no formal study of Poincare's addresses, letters, etc.):

1.  Poincare had much experience with mathematics that had not been based
upon set theoretic or logical foundations, and saw that it was quite
successful.
2.  Poincare had been involved in some of the foundational successes of
non-euclidean geometry. (See page 396 of Roger Cooke's ``The History of
Mathematics - A First Course'', for example.)
2.  Poincare watched with keen interest and clarity the development and use
of the most formal approach to mathematics he had seen up to that time,
something a prominent mathematician would do.
3.  Poincare either already had or came to have an incisive distaste for
contradictions.  (I do not recall reading anywhere that he disagreed with
the law of the excluded middle, like some mathematicians did or do.)
4.  Contradictions appeared in this new area of study, right at its
foundations, and in its most formal form of the time.

This is enough to turn anyone away, and Poincare was apparently not an
exception.  He seems to have believed that correct results could be obtained
in mathematics, but saw no use anymore for this new foundational approach to
mathematics, because he felt certain he knew that the mathematics he was
doing was worthwhile and correct.  Once disappointed with foundations,
Poincare became a staunch opponent.  (According to Fraenkel, Bar-Hilel and
Levy, he treated attempts to ``rehabilitate'' set-theoretic foundations with
``an air of mockery''.  Cooke just mentions that Poincare ``opposed it''.)
When such a strong and successful personality develops such an attitude,
rightly or wrongly, he will generally lead others to his way of thinking.  I
consider some of the reasons Poincare turned against foundations justifiale,
but I am not convinced he was justified, even if his actions were somehow
understandable.  It is my impression that entire schools of mathematics then
grew up, some specifically from people like Poincare, and some of which
distrusted foundational studies of mathematics.  Whether they knew it or
not, some new mathematicians were developing dogmatic attitudes toward areas
of mathematics outside their own, and especially against foundations.  I
would, of course, not say that all mathematicians became dogmatic, for I
think that would be a gross exaggeration.  However, these dogmatic
approaches seem to at times lead to various interesting successes, probably,
I expect, because the mathematicians who are dogmatic focus their attention
more on very specific subjects, topics and problems for longer periods of
time than some of those who show some interest outside their own world view
of what constitutes ``good mathematics''.  (Of course, there will always be
some who appear capable of doing almost everything.)  Thus, getting back to
Professor Pillay's comment, that Foundations should pay attention to today's
mathematics, without being dogmatic about it- as I said, I agree.  Various
forms of dogmatism about this would probably lead to certain interesting
discoveries, but I think it would stifle some worthwhile work by those who
do not see a need to be dogmatic.





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