[FOM] Tait on constructive mathematics

Gabriel Stolzenberg gstolzen at math.bu.edu
Thu Feb 9 15:02:44 EST 2006

   This is an attempt to make clearer my criticism (in "Haney
and Tait on intuitive sources of mathematics" vol 38 issue 12)
of Bill Tait's criticism of constructive mathematics in "The
law of excluded middle and the axiom of choice."

   Here is a simple illustration of the "ineliminable circle"
I think he sees in alleged constructivist attempts to explain
the meaning of mathematical terms in pre-mathematical language.

    How would we explain what we mean, constructively, by a
function on a 1-point set?

   Classically, it is a trivial notion.  But constructively
it is not.  It is a procedure (whatever that may be) for
constructing (whatever that may mean) an element of the range
of the function and there is no limit to how complicated this
procedure can be.  Not even for a function from {0} to {0}.

   (Note. In constructive practice, most of what is described
as a proof of an existence theorem is really a construction of
a function on a one point set whose value is the object that is
asserted to exist.)

   Having said this, do we now have to explain what we mean by
"a procedure for constructing something"?  (If so, in terms of

   No, neither language nor reasoning works that way.  Instead,
at most what we do is to view such "explanations" as suggestive
remarks, not definitions.  (Fred Richman does not do even this.
For him, talk of "a procedure for constructing" merely points
to one model of the mathematics, one that doesn't interest him
very much.)  And in place of the latter, we work with the formal
ones given by the mathematics, which are not really definitions
at all.  But then what is?

   And it works!

   Gabriel Stolzenberg

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