[FOM] intuitionistic here, constructive there.

Gabriel Stolzenberg gstolzen at math.bu.edu
Tue Feb 14 23:55:19 EST 2006

  In comments (February 9) on a paper by Neil Tennant, Harvey Friedman,
notes that

> Tennant quotes Bridges...to the effect that "constructive mathematics
> is none other than mathematics carried out with intuitionistic logic."

   Harvey finds Bridges' statement "at best very controversial" and
explains why.  Here is part of it.

       This assertion of Bridges is at best very controversial....
       Z_2 has a well known and fairly well studied intuitionistic
       version, which is the same except that intuitionistic logic
       is used....However, most constructivists yesterday, today, and
       tomorrow, will not accept intuitionistic Z_2 as constructive.
       For that, they need to accept a rather mysterious notion of
       "species" that Brouwer proposed.

   This is a nice argument.  However, I think what is going on here
is that, although Harvey rightly distinguishes between 'intuitionistic'
and 'constructive' as names of different logics, in informal discourse,
others, Bridges included, sometimes say 'intuitionistic' when they mean

   Thus, although I've been extremely critical of some of Bridges'
statements about constructive mathematics, especially his revised
version of Bishop's "Foundations of Constructive Analysis," I
believe that, in the statement that Tennant quotes, when Bridges
says "intuitionistic," he means nothing more than "constructive."
And, as I note above, he is not alone in talking this way.

   Thus, in "Intuitionistic Logic," in the Stanford Encyclopedia
of Philosophy (2004), Joan Moschovakis says that

        "intuitionistic logic may be considered the logical basis
        of constructive mathematics."

   And in "Foundations of Constructive Mathematics" (pp. 191-2),
Michael Beeson has his one of his fictitious commentators say,

          IZF [Intuitionistic ZF] is intended to formalize the
          classically intelligible fragment of constructive
          mathematics....I don't know a single piece of
          constructive mathematics that is not classically

   But, intuionistic Z_2 is not classically intelligible--for the
same reason that, as Harvey pointed out, it is not constructively

    Finally, on my reading of in "Against intuitionism: constructive
mathematics is a part of classical mathematics," Bill Tait is not
talking about two different things, one of which is intuitionism,
(which, for some reason, he's against) and something else called
'constructive' mathematics, which, he reminds us, can be seen as a
restricted part of classical mathematics.  On the contrary, it seems
clear that, at least in the title, both words refer to constructive
mathematics, not to intuitionism.

   P.S.  Although it often is noted that constructive mathematics
is a restricted subsytem of classical mathematics, this is only
half the story.  The other half is that classical mathematics is a
restricted subsystem of constructive mathematics, the part devoted
to deriving consequences of the law of excluded middle (and any
other nonconstructive axiom).  Each is a restricted subsystem of
the other!

   I mention this here because, although it may seem like a trivial
point (and, of course, in a way, it is), if you wish to learn which
differences between these two frameworks matter for mathematical
practice (and how), this observation is a crucial place to start.

   Gabriel Stolzenberg


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