It's been interesting to read the messages on the maths ontology and
teaching issue. A few more bits and pieces.
I did say that you could have deep understanding of mathematics without
having the kind of viewpoint on mathematics that a pure mathematician has,
so I agree with the messages that supported "deep understanding" as an
educational goal, but I don't accept the privileging of the specialized
view of mathematics by pure mathematicians that claims that their
understanding is the only deep one. I don't even grant them the right to
own and dictate what "mathematics" is or can mean to members of other
communities. I generally happen to like pure mathematicians personally; in
fact more often than I warm up to "applied" mathematicians (maybe the
appliers show some of the defensiveness of the disrespected; they are very
much patronized by their purer colleagues -- though this may change in
another generation, see below).
But just as I refuse the mystique of pure science, I think we also should
refuse it for maths (I use the British form here, it's easier to type than
the long word!). This is intellectually harder for many people to do
because (a) almost no one else except pure mathematicians has any idea what
pure math looks like to them, and so it's hard to deny the implications
they draw from their view; (b) pure mathematics is today the epitome of the
classical (read patriarchal, euro-canonical) ideals (and idealist
inclinations) of western scholarship -- where philosophy and theology have
fallen on hard times (their claims to Truth seem shaky to most
intellectuals today), and physics keeps having to change its mind when new
data or superseding theories come along, Maths remains the purest of the
pure, the truest of the true, the universal, the trans-cultural, the
a-historical .... and all that other glorious crap of yore. Even I feel
terrible about refusing to bask in the shining mystique of pure mathematics
(which, by the way, I do appreciate aesthetically, regard as largely
innocent of evil-doing, value for its unpredictable if rare moments of
applicability to the material world, value for its imaginative openness to
endless possible realities, and find fascinating as a problem in the
history of semantics and semiotics).
Precisely because pure mathematics is the last bastion of outright
Platonist idealism (whether it was also the first is an interesting
question, turning on just how "pure" the geometry of Plato's day actually
was), it is an excellent site to examine the dialectic of "pure" vs.
"applied". Pretty much everywhere else it is EASY to subvert this
dichotomy; only in the case of mathematics is it still plausible to anyone
that there is a qualitative and discontinuous jump from any applied
mathematics to the one pure mathematics.
Unfortunately, again, it is just really hard to characterize the difference
between applied maths and pure maths to anyone who has not been initiated
into the latter at a reasonably advanced level (say ring theory, algebraic
topology, etc. -- some area where there is NOTHING in the maths books
except definitions and theorems, and after about six pages you can no
longer remember half the definitions, much less the implications of the
second half-dozen theorems -- unless you are talented or trained in the
special cognitive manner(s) of good pure mathematicians). One earlier
message to the list gave a typical, but again easily misunderstood,
characterization: that the pure mathematicians look more at the syntax than
at the semantics, that is, more at the formal patterns than at any concrete
meaning they might take on in some real situation or example. There are
problems with this characterization. It is a suggestive metaphor only. Pure
mathematics deals just as much with semantics, perhaps mainly with
semantics; but it is the semantics OF the syntax, not a semantics that may
be attached to or be supported by the syntax. Pure Maths is meta-discursive
in the highest degree; it is a discourse that is wholly about itself; it is
what Chomsky's linguistics and pure classical logic aspire to, but which
are, for each of them, dead-ends insofar as they also aspire to say
something about what is said by people in the world, and not just about the
possibility of saying anything in any world.
But, still, there is a revolution that has been going on in mathematics now
for the last few decades, and it turns precisely on the breakdown of the
qualitative distinction between applied and pure specializations. It has
come, as many of you probably know, because of the computer. It has two
prongs. The less obvious one is the result of what happened when pure maths
finally tried to haul itself up by its own bootstraps to prove that it was
totally autonomous from reality -- this was the project of
meta-mathematics, begun by Russell and Whitehead, systematized by Hilbert,
and demolished by Goedel (sorry, I can't do an umlaut that will work around
the xmca world). As everyone knows, the outcome was more or less that a
system of formal proof can't prove itself, and so you can never be
absolutely sure that you have proven something (not quite right, but close)
in purely formal terms. You could say that syntax cannot be its own
semantics; there has to be a gap, a level-difference, a distinction of
logical types. Since Goedel there have been others who have generally made
things worse and worse for the ideals of pure mathematics (Gregory
Chaitin's work makes many pure mathematicians literally feel suicidal, or
at least seriously ill).
The way out? philosophers of mathematics, though as yet not too many pure
mathematicians, are starting to give up on Platonism, and trying to recast
mathematics as an "empirical" discipline, as a system of activity, and
proofs as not so much being discursive genres as elliptical forms of
action-recipes (injunctions, sometimes called, to link to discursive
categories), to be judged by whether they can actually be carried out
(finitely, i.e. by human mathematicians, or conceivable machines limited by
thermodynamics or quantum cosmology) ... and so pure mathematics is coming
back down to earth, or at least re-entering the physical universe.
One version of all this is the "algorithmic" approach, inspired by
computers (as computers were inspired by Hilbert, indirectly), which
Chaitin uses (you can download his "proofs" and run them on a PC), and it
forms a bridge to the second prong.
The methods of applied mathematics, such as computation, are now being used
to solve classical mathematical questions, and to create proofs that are so
long and complicated that no human being, only another computer, can
"check" them to see if they are syntactically well-formed (no silly errors
in the steps). And mathematical problems inspired by computation (in
theoretical computer science, and in nonlinear dynamics) are becoming part
of the cutting edge of what mathematicians are doing today that is of
interest to other mathematicians. Applied mathematics is becoming part of
pure maths at the same time that pure maths is finding it has to re-ground
itself in something empirical.
So a general distinction between pure and applied maths just won't hold up
to x-ray laser inspection. Instead, I suggest, what we have are a lot of
fuzzy-boundaried, overlapping, intersecting, and mutually relationally
arrayed mathematical specializations (i.e. subcommunities of practice,
systems of practices, activity systems, or what you will) ... SOME of which
have claims of varying degrees on presentation in the curriculum, based on
the usual criteria of known practical usefulness, potential practical
usefulness, and need for awareness of the existence of (and little tastes
of) a wide variety of intellectual practices. To me this is a VERY
different view from saying either (a) that "pure maths" represents THE deep
understanding of any mathematical practice, or (b) that "applied maths" is
JUST rote proceduralism for wage-slaves.
There is no intellectually serious USE of mathematics that does not require
some theoretical sophistication about HOW to use the mathematics, but that
sophistication is NOT usually identical to the specialized view of the
mathematics used that is taken by "pure" mathematicians. Theoretical
physics is mostly a form of applied mathematics (and mathematical physics
IS applied mathematics), but just as it is notorious that pure mathematics
sometimes turns out to be (surprisingly?) applicable to the physical world,
so it is also well known among theoretical physicists that delving into the
pure mathematician's view of the math most often is no help at all. And
critiques of the mathematical foundations of modern physics (of which there
have been many) have, so far as I know, never led to any breakthroughs in
the development of physics (yet; it could happen). Theoretical physicists
learn their mathematics from other theoretical physicists, and from pure
mathematicians only as a last resort. My point here is that deep
mathematical understanding can and should be an understanding which is
specific to the field of use, and that there is not one universal such
understanding monopolized by pure mathematics, which is itself just another
specialization.
JAY.
PS 1. Nonetheless, I think pure mathematicians can usefully wrench school
mathematics out of its narrow-minded rut and should be included in
curriculum planning, but never be given the last word on what should be
taught or how.
PS 2. I don't know where the brief note on my own empirical work came from,
but I've done work on: the teaching of calculus to physics students, the
role of mathematics in scientific discourse and classroom discourse, the
relationship of mathematics to natural language, and the history of the
symbolic notations and semantic concepts of basic mathematics. This area
has never been at the center of my research, but a frequent side-line or
contributing element. ... and besides, I take the position that there is no
intellectually defensible distinction between "theoretical" and "empirical"
research in ANY field ... "theory" that is not grounded in empirical
experience is just bullshit, and "empirical" research that does not also
advance theory is always superficial and often misleading. So there! :)
---------------------------
JAY L. LEMKE
PROFESSOR OF EDUCATION
CITY UNIVERSITY OF NEW YORK
JLLBC@CUNYVM.CUNY.EDU
<http://academic.brooklyn.cuny.edu/education/jlemke/index.htm>
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