# FOM: Re: constructive mathematics

Jeffrey Ketland ketland at ketland.fsnet.co.uk
Sat May 27 15:55:16 EDT 2000

Ayan wrote on 27 May (17:40 on my clock):

>One thing may be worth pointing out, that constructive mathematics
>(Bishop's style which I understand is the topic of current discussion) is
>currently seen as working with
>Intuitionistic logic. I guess this goes against any subjectivism in
>constructive mathematics.

Goes *against* subjectivism? I don't understand. The following is an extract
of what Arend Heyting wrote in his 1956: Intuitionism: An Introduction

(This is part of an excellent dialogue, between the characters called
"Class" (classical mathematician), "Form" (a formalist, seems to refer to a
mixture of Hilbert and Carnap), "Int" (an intuitionist, seems to refer to
Brouwer), "Letter" (a sort of finitist formalist), "Prag" (a pragmatist,
seems a bit like Quine) and a mysterious character called "Sign"):

-------------------------
Intuitionist mathematics consists
in mental constructions; a
mathematical theorem expresses a purely empirical fact, namely the
success of a certain construction. ‘2 + 2 = 3 + 1’ must be read as an
abbreviation for the statement: “I have effected the mental
constructions indicated by ‘2 + 2’ and “3 + 1” and I have found that
they lead to the same result”.

The characteristic of mathematical thought is, that it does not convey
truth about the external world, but is only concerned with mental
constructions

In fact, mathematics, from the intuitionistic point of view, is a study
of certain functions of the human mind.
(Heyting 1956) (quoted from Benacerraf & Putnam 1983, pp. 72-73).

------------------------------------------------------
Frege gave an example like the following in Foundations of Arithmetic (Die
Grundlagen der Arithmetik: 1884). If someone were to say that astronomy
(say) doesn't deal with bodies in the external world, but is "only concerned
with mental constructions", one would rightly judge that they were an
idealist or subjectivist of some sort. Frege was arguing that numbers are
not (subjective) ideas.
Surely the above quotation from Heyting makes it plain that intuitionism (or
at least the form of intuitionism advocated by Brouwer and Heyting) is based
on a mental (i.e., subjective) conception of mathematical entities.

Regards - Jeff Ketland

Dr Jeffrey Ketland
Department of Philosophy C15 Trent Building
University of Nottingham NG7 2RD
Tel: 0115 951 5843
E-mail: Jeffrey.Ketland at nottingham.ac.uk