FOM: constructive mathematics

Matthew Frank mfrank at
Fri May 26 00:48:07 EDT 2000

Those of us who work in constructive mathematics often feel misunderstood.
Let me take the opportunity of some recent comments by Simpson
(representing commonly held views) to state things as I understand them.

Most (if not all) of us who work in constructive mathematics also work in
classical mathematics.  Keep this in mind with the below.

Simpson says:  "We need a term to describe a person who endorses a
constructivist philosophy of mathematics".

I do not think we need any such term, since I know of only one person who
endorses beliefs like those usually referred to as constructivism (Michael
Dummett).  I think most discussions of constructivists miss the point, and
that people would say more reasonable and more interesting things if they
restricted themselves to discussion of constructive mathematics and the
people who work on it.

Simpson asks:  "If subjectivism is not the essence of the constructivist
philosophy, what is?".

I refuse to answer, since I would be explicating a position that almost no
one (again, except perhaps Michael Dummett) believes.

Simpson says:  "there are also full-fledged constructivists like Bishop,
Bridges, Richman, etc.", where by constructivism he means "a subjectivist
position, according to which mathematics consists of constructions in the
mind of the mathematician."

I doubt that that any of Bishop, Bridges, or Richman believe this (Richman
has been very explicit to the contrary); if you can find me quotes showing
that they do or did, I will be surprised.

Simpson paraphrases me as saying:  "there are people who `work on'
constructivism (e.g., by proving metatheorems about
constructivistically-inspired formal systems) without endorsing the
underlying constructivist philosophy."

I agree, but would like to emphasize that some people work on constructive
mathematics by proving theorems of (and not necessarily metatheorems
about) constructive formal systems.

There is still the question of why someone might do this--for instance,
why I do this--and so I refer again to the paper "Constructive
Mathematics:  Why and How" on my web page,


Appendix:  a brief bibliography of the above-mentioned people.

Michael Dummett, Truth and Other Enigmas.  (This is a book, reprinting
several papers of his; the one on "The Philosophical Basis of Intuitionist
Logic" is also reprinted in Benacerraf and Putnam's anthology on the
Philosophy of Math.)

Errett Bishop, Prolog and Chapter 1 of Foundations of Constructive
Analysis; these are reprinted verbatim in Errett Bishop and Douglas
Bridges, Constructive Analysis.  Some of Bishop's later papers are also
listed in the bibliography of the later book.

Douglas Bridges, article in the book Truth in Mathematics (also with
references to several earlier publications, many of which can be found via

Fred Richman's home page,

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