[FOM] Haney and Tait on intuitive sources of mathematics

Gabriel Stolzenberg gstolzen at math.bu.edu
Wed Feb 8 00:02:28 EST 2006

   In this posting, I first respond to a question of Richard
Haney about the intuitionist's view that intuition is the
source of mathematics and then work my way into a mistaken
empirical claim about what constructive mathematicians do.
(One reason I'm prepared to call it mistaken is that I'm a
counterexample.)  It's in "The law of excluded middle and
the axiom of choice" by Bill Tait.

   In "Formal logic, constructivism and intuitionism" (Vol 38,
issue 7, January 17) Richard Haney asked,

>    Can anyone direct me to specific resources...where
>  intuitionists discuss their philosophical views "that
>  the unique source of mathematics is the intuition, and
>  the criterion of acceptability of mathematical concepts,
>  constructions, and inferences is intuitive clarity"?
> (The quote is by N. Shanin.)

   Intuitive clarity?  I don't think so.  E.g., I admire
Brouwer but if I had invented the concept of a free choice
sequence, I probably would apologize.

   Re "the unique source of mathematics is the intuition,"
I recommend looking carefully at the learning process, the
training.  Once the mathematics is codified, the training
creates "the intuition."  And different trainings sometimes
create different intuitions.

   But Haney may be thinking in terms of an intuitive source
that preceded the codification.  E.g., Errett Bishop cottoned
on to what he called "constructive distinctions in meaning"
while attempting, with little success, to sell his students
in a Math for Poets course on the truth table definition of
implication.  Eventually he and the class came up with, "If
a stick of dynamite is lit, then, depending on the relevant
physical facts, it will blow up by such and such a time" versus
the contrapositive, "If it never blew up, it never was lit."

   To Bishop, there was an intuitive distinction in meaning
between the two assertions.  He didn't know how to articulate
it but that didn't keep him from undertaking to redevelop
mathematics while maintaining this and related distinctions.
And as he did, there was never a sense that the mathematics
was responsible to the intuitive non-mathematical source that
got him started.  The mathematics had a life of its own.

   That was Bishop.  For those of us who came after, it is
not uncommon for us to view our mathematics as having an
intuitive "source," in the sense of it being about something
about which we have intuitions, something we call "a procedure"
or "a computation."

   But these intuitions are vague, their vagueness reflecting
the vagueness of the idea of 'can' (as in "My car can go from
0 to 60 in ten seconds") on which it crucially depends.  (More
precisely, the notion of "a computation" depends crucially on
the notion of "a step," which depends crucially on 'can.')

   Hence, this something (which we call "a procedure" or "a
computation") is not something that can serve as an arbiter
for what in the mathematics is true and what is not.  If it
were being asked to fulfill such a role, it could not.  But
it is not being asked to do that.

   At least, so say I.  However, on my reading, the following
statement from Tait's "The law of excluded middle and the axiom
of choice" implies that what I say above about the constructive
mathematical mindset (in particular, mine) is false.  He writes,

      "The circle is ineliminable in constructive mathematics,
       because whatever principles of logic are given, they
       must answer the challenge of whether they really yield
       'constructive objects.'  For the notion of being
       constructive is intended as a measure of correctness
       for any particular principle considered."

    Nevertheless, this claim about the intentions of people like
me is mistaken.  We break the apparent circle of which Tait speaks
(I call it a regress) the same way he does; indeed, I congratulate
him for having seen that it is the right thing to do.

    In the same vein (in the same paper), when Tait says,

   "I truly wish that the term 'constructive' had been reserved
    for just [constructing an object of a given type and a proof
    of a proposition] since it seems most appropriately applied
    to the view that the basic notion of mathematics is that of
    construction, without further specification of what kinds of
    construction are to be permitted,"

I can reply that, provided only that he and I are reading the
ambiguous last clause ("without further specification...") the
same way, in my constructive mathematics and, for all I know,
in everyone's, the term 'constructive' is, in fact, reserved
for just what Tait rightly wishes it to be.

   Gabriel Stolzenberg

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