[FOM] The defence of well-founded set theory
Andrej Bauer
Andrej.Bauer at andrej.com
Wed Oct 5 10:31:41 EDT 2005
I would like to offer a criticism not of set-theory itself, but of the
idea that having a single powerful foundation of mathematics is a good
thing.
A reasonably powerful foundation of mathematics imposes a certain way of
thinking onto those who have been trained to use it. When such a
foundation is as wide-spread as classical set theory is among today's
mathematicians, this hinders development and acceptance of those new
mathematical ideas that can only be expressed in alternative foundations.
In my experience this is very much visible in theoretical computer
science. There are important links between computation and constructive
mathematics, as everyone on this list knows. I have met more than one
"computable" mathematician who has stated that he would like to, but is
unable to understand constructive mathematics because of his classical
training.
Another example that comes to mind is synthetic differential geometry,
as invented by Lawvere and Kock (I hope I am getting this bit right--if
I forgot to mention someone here, please correct me). It is a theory
which allows direct computation with nilpotent infinitesimals a la
Newton, Leibniz, Kepler, etc. It is precisely what physcists do in
practice. Yet, because synthetic differential geometry requires
intuitionistic logic, it has had little success in finding its way to
physicists' minds, primarily because ordinary mathematicians are
inclined to ignore it, as it is not a classical theory. So we keep
teaching physicsts epsilons and deltas, even though they are totally
useless to them, and physicsts keep using infinitesimals "secretly"
without proper mathematical foundation.
While logicians may be used to switching back and forth between
foundations, ordinary mathematicians are not. They are unwilling to
spend several years of "deprogramming" to be able to understand a
different foundation, even if this would have clear benefits for the
subject they are interested in.
The absolute domination of any one particular powerful foundation is
harmlful, as it makes choices that are not acceptable to everyone. We
should be conservative in how we teach and present foundations to
ordinary matehematicians. Mathematicians should be trained in a "base
theory" (perhaps something like intuitionistic Zermelo set theory).
Excursions into specific extensions of the base theory would be done on
a 'need-to-use' basis only.
There cannot be an absolute defense of classical set theory. It can only
be defended as a useful foundation for a particular kind of mathematics,
say for the study of the concept of iterative hierarchy of sets. There
are other kinds of mathematics for which classical set theory is largely
irrelevant (algebra, category theory) or even inappropriate as a
foundation (computer science, geometry). From the point of view of those
who do such mathematics--and have been appropriately
deprogrammed--classical set theory is clearly the wrong choice.
Andrej Bauer
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