FOM: Hersh &the Dumbed Down Mathematics Movement

[Robert Tragesser]

        Surely without intention on his p[art,  
but Reuben Hersh is widely regarded as 
the guru of what Saunders MacLane,  
perhaps pejoratively, has called the powerful,
deadly movement (in education) to 
"dumb down mathematics" [under
old materialist, multi-culturist, new historicist,
etc etc agendas which have already lowered beyond
belief the quality of thought allowed in the humanities
and are now at work on the sciences and
mathematics].

        I bring this up on FOM for two reasons,

[1] the (not quite coherent) philosophy of 
mathematics developed in Hersh's
What is Mathematics, Really? directly
feeds and further arms this movement, 

[2] it raises a deep issue of interest
on FOM: the function and nature of "rigorous
proof" in mathematics, and points to the
necessity of developing a General Proof
Theory of the sort variously envisioned
by Kreisel and MacLane.

        That [2] goes back 
to the cascade of papers and commentary
provoked by Jaffe & Quinn's "Theoretical
Mathematics: toward a cultural synthesis of
mathematics and theoretical physics" Bull
AMS(2) 29(1993),1-13. See Proof:A Many-Splendored
Thing by Kleiner and Hadar in Math.Intel. Vol.19 #3
1997 for an intermediate bibiography.  Jaffe-
Quinn was discussed in the early days of FOM
(in 1997).

        It should be noted that the issue
rather quickly widened from working out
institutional
strategies (of the sort proposed by Jaffe-Quinn)
to give speculative and rigorous mathematics
separate but equal status, to those a battle
among those who believe that rigrous proof is
the very sould of mathematics and those who
believe that rigorous math. might have
its place, but humanly understandable mathematics
human mathematics,  peoples mathematics
is intuitive, sensory,  pictorial, empirical. [Which,
I admit,  sounds more like chimpanzee math. to me]
We are familiar with this from Hersh.  But also see
William Thurston's "On Proof and Progress
in Mathematics" [Bull AMS 30, #2 April 1994],
which is more coherent and clear headed than
Hersh.  N.B.,  for comparison, notice how both
Hersh and Thurston speak elaborately about
how _people_ understand.
        In connection with "understanding",
Quinn/Jaffe,  Rota,  and MacLane have pointed
out how RIGOROUS PROOF IS A FUNDAMENTAL
TOOL OF UNDERSTANDING (and not "certainty") 
and that at the very least because
generally only rigorous proof gives
one the control of detail and subtlty necessary
to (mathematically) understand deep and complex
mathematical phenomena/problems...
        Both Kreisel and MacLane have in
different ways called for a General Theory of
Proof in which the function of rigorous proof
(beyond achieving transparence of validity)
is elaborated.  [MacLane, in Synthese Vol.111,
No.2, May 1997;  Kreisel, e.g., in Hintikka, ed.,
Essays on Mathematical Logic and Philosophical
Logic, Reidel,  1978,pp.3-23]
        In part,  of course,  the dumb-downers
(here I refer-- not to Hersh or Thurston but-- 
to that increasingly 
wide-spread movement of educational reform that
seeks to purge mathematics teaching of the
teaching of proof) have a leg up in advance because
mathematicians generally do not have that
self-reflectiveness about proof that enables
them to teach it well.  Recognizing that rigorous
proof does represent a barrier,  the pedagogical
move should have been to discover ways of being
step by step illuminating about what is going
on in each proof (for example). . .not an easy
teaching assignment --
simply because there is no tradition of thinking
about proof (save from the point of view of logical
validity).  Consequently,  the dumb-downers
can happily throw out rigor and proof altogether -- 
worse that no student is required to make all on 
their own more than a one step inference.

        I can speak from bitter experience that is
sending me fleeing from academia that the dumb
downers have tremendous power in even some of the
top 25 lib.arts colleges.  This movement really
began over there in the humanities,  and was
already there destroying the possibility of serious
thought.  Here is Harold Bloom: "the multiculturists,
the hordes of camp-follwers afflicted by the French
diseases,  the mock-feminists,  the commissars,
the gender and power freaks, THE HOSTS OF NEW
HISTORICISTS AND OLD MATERIALISTS...[have already made]
travesties of our universities.
        [Is it an accident that in Hersh one
finds New Historicist,  Old Materialist rhetoric
side by side?]
        The spirit here is moving into the
sciences and mathematics.  In the sciences
(even in good l.a. colleges) there is an increasing
emphasis of lab.technique and instrumentation,
replacing serious thought.  I go green and soft when
I think of some of the math. courses I've seen by
which students can satisfy their math. requirements.
Little dances to illustrate modular arithmetic.
Drawings of visual pyramids to show the important
of geometry for linear perspective. [Contrast
the latter with the quite accessible axiomatic
presentation of linear perspective by Brook Taylor 
[as in the Taylor series Taylor],  with
some nice theorems,  e.g., how a rectangular
mesh projected onto a smooth,  convex,  but
highly uneven surface will nevertheless look like
the retangular mesh when viewed from the point
of projection.--Euclid's Optics matured.  It
would be a beautiful setting in which to
appreciated the relation between the rigorous
and the unrigorous,  which is to say,  the
power of rigorous to make understandable
complex intuitive phenomena [perspective projections].
But how all too too Western male and immaterialist!

        The effect of dumb-downing is an extremely
 misleading, off-putting,  patronizing,  and dim-witted
mathematics. [It reminds one of Durant's embarrassingly
patronizing Story of Philosophy (written for blue-collar
workers who were supposed to have no capacity to
reason).]

        In short:  here a call for FOM thoughts
toward a General Theory of Proof.

Robert Tragesser
Professor in the history and Phil. of Science
and Mathemtics at [terribly sinking]
Connecticut College