[FOM] for harvey

Gabriel Stolzenberg gstolzen at math.bu.edu
Sun Feb 12 14:23:15 EST 2006

   Harvey Friedman has invited me to comment on a remark that he
makes in his February 9 reply to a paper referenced in Tenant's
message of February 8.  If only for my own reasons, I'm glad to
do so.  But I'll have to do it in two stages.  This is the first.

   Harvey begins,

> 1. Tenant quotes Bridges on page 9 to the effect that "constructive
> mathematics is none other than mathematics carried out with
> intuitionistic logic."

   I need a little help with the terminology here.  If Bridges had
said "constructive" instead of "intuitionistic" logic, would there
still be a problem?  From what Harvey goes on to say, I assume not
but I need reassurance.

> 2. This assertion of Bridges is at best very controversial. Let
> us take an example. Normally Z_2 (not a great name) stands for the
> usual two sorted first order theory of natural numbers and sets of
> natural numbers with full comprehension in its language.

> 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.

   Indeed, in "Haney and Tait on intuitive sources of mathematics"
February 8, I said that if I had invented the notion of a free choice
sequence, I probably would apologize.  And there is Bishop's wonderful
line in "Foundations of Constructive Analysis" (page 6),

      In Brouwer's case, there seems to have been a nagging
      suspicion that unless he personally intervened to prevent
      it, the continuum would turn out to be discrete.  He
      therefore introduced the method of free-choice sequences
      for constructing the continuum, as a consequence of which
      the continuum cannot be discrete because it is not well
      enough defined.

   (Note. When Springer followed the advice of pre-publication
reviewers and said that the book would not be published unless
such "journalistic" remarks about Brouwer were removed, Bishop
published it instead with McGraw Hill.)

> I am not clear just how acceptable Brouwer found intuitionistic Z_2.
> But I think the overwhelming majority of his successors would not
> find it constructively acceptable. In particular, we know that Errett
> Bishop would NOT find intuitionistic Z_2 acceptable constructively.
> (Stolzenberg could comment on that).

   I assume that, by "successors," Harvey means something like "those
who came after."  But "successors" in the sense of followers is not
correct.  Bishop cottoned on to his "constructive distinctions in
meaning" on his own.  (See "Haney and Tait on intuitive sources of
mathematics" February 8 for how it happened.)

   Gabriel Stolzenberg


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