FOM: Quasi-empiricism and anti-foundationalism
rhersh at math.math.unm.edu
Fri Sep 11 15:22:13 EDT 1998
A few additions to Peccatte's interesting and useful description and
analysis of quasi-empiricism and related subjects.
An important omission is Philip Kitcher's The Nature of
Mathematical Knowledge, Oxford, 199?. Also the introduction , by
Kitcher and Asprey, to their interesting collection of papers from
a meeting on history and philosophy of mathematics in Minneapolis.
My "What is mathematics, really?" (Oxford, 1997) is half historical.
Ernest's new "Social Constructivism as a philosophy of mathematics"
(SUNY,Albany) has useful historical discussion.
The term quasi-empiricism is due to Lakatos. In my opinion,
it does not have his usual flair for nomenclature. The real issue is
neither empiricism not quasi-empiricism.
Another often used name is "social constructivism," which is
not inaccurate, but suffers from an irrelevant and damaging false
association with postmodernism.
I used the name "humanism" because I wanted to avoid the
misleading sound of quasi-empiricism and the irrelevant and odious
associations of social constructivism.
It is interesting to catalog the different names tbat have
been attached to this trend in the philosophy of mathematics.
There is naturalism (Kitcher) and fallibilism (P.Davis) and
still others that escape me at the moment.
In my opinion, the fundamental cleavage in philosophy of
mathematics is between those who think mathematics is a human
activity and a human creation, and those (the dominant view) who
think mathematics is in some way or other superhuman or inhuman.
This disagreement goes back long before Lakatos or Mill; at least
as far as Aristotle vs. Plato. In my book I traced the unbroken
line of descent of humanism vs. superhumanism (usually called Platonism)
down to the present.
The issue of foundationalism (one of Lakatos' neologisms)
was clearly stated by Lakatos, in the preface to Proofs and Refutations,
and in the articles collected in volume 2 of his collected papers.
The indubitability attributed to mathematics was a major supporter
of Christian theology (see Descartes, Leibnitz, St. Augustine, et al.)
Consequently, the difficulties raised by the adoption of set theory
as a foundation of mathematics, and the contradictions ("antinomies")
in set theory, were very disturbing to philosophically minded
mathematicians and philosophers of mathematics. "Restoring the
foundations" became an overriding concern of a few brilliant
logicians and mathematicians. Three principal solutions were proposed
(excuse me for repeating this too famililar story)--logicism,
formalism, and intuitionism. But they all encountered grave
difficulties, and a consensus emerged that the enterprise of
restoring the foundations (restoring indubitability) was hopeless.
But the foundationalist enterprise had developed enough
momentum that it continued to dominate philosophy of mathematics,
as a thing "in and for itself", isolated both from general philosophy
and from mathematics and mathematicians.
My intention was to revive the philosophy of mathematics
as a concern and activity of mathematicians, portraying and
explaining their work based on their own experience. Having
come across more than one patronizing or condescending reference
to mathematicians who had the audacity to engage in philosophy
of mathematics, I was disappointed but not surprised at the
response I received on this list.
It is possible to regard philosophy as a technical
activity open only to those who have earned the required credentials.
It is also possible to regard it as a careful, serious reflection
on living experiences. My goal was to deal with philosophical
issues in mathematics in a way that would be meaningful and
helpful to mathematicians.
In describing foundationalism in the sense of Lakatos,
(Frege, Russell, Hilbert, Brouwer) I did not consider or have
in mind the research in logic and set theory of Harvey Friedman
and his circle. I do not regard this as part of foundationalism,
because the restoration of indubitability is not its goal.
It seems to me to be part of logic and set theory, which are part of
mathematics. For whatever reasons, Friedman says it is not
part of mathematics. In any case, whatever I wrote about
foundationalism has nothing to do with Friedmans "foundations of
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