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Re: [xmca] Where is thinking

Yes, this does sound like a wonderful course. I'd be interested to know about the entering predispositions, and mathematical experience, of the students who do well in it. The culture of mathematicians, especially the "pure" or more Poincare-ish ones (and it can get a lot purer than that of course these days), seems to me to be rather of an exceptional taste. And I always wonder just how much of it, and its products, are appropriate for general education. As opposed to a more workaday, applied maths approach to using the tricks of the trade outside the professional discipline of mathematics.

This concern perhaps relates to another posting that worried about whether the abstraction-is-divine culture of modern mathematics does for most students ever get concrete enough to be meaningful, much less useful. Of course in one sense everything people do, even mathematicians and philosophers, is concrete, as practice, for them. But when we first come to enter into, or seek to borrow from, such a culture, then its habits of formulating ideas and practices in terms which build castle upon castle into the stratosphere and exosphere of "abstraction" do seem to present a challenge to being grasped in concreto for those of us who do not live high in those castles. (For a very thoughtfully explicit version of what castle-building means and its educational implications, see Anna Sfard's new book: Thinking as Communicating.)

My own take, for what it's worth, on the simplicity of Relativity, is that long after he first got it sorted out, Einstein became able to explain its key ideas in simpler, more concrete, less mathematized, and more widely accessible forms of discourse (with the occasional diagram), and others helped out in this dissemination project (Eddington was notable, if not always successful, and there is a now little-known book by Lillian Lieber, liberally illustrated, that is a marvel). But in the beginning, the original insights were not at all as they were later portrayed and canonized. It had nothing much at all to do with space and time, and only a bit with motion. That was a later interpretation provided by Minkowski and adopted by Einstein. So far as I can determine, the original idea was that something had to be wrong with the fact that Newton's laws and Maxwell's equations were not both invariant under the same set of mathematical transformations (transformations which represented relative motion at constant velocity with no turning). The ones that worked for Newton (Galilean transformations) did not work for Maxwell. The ones that worked for Maxwell had just been discovered (Lorentz transformations). So Einstein appears to have asked himself, what would Newton's equations look like if they were invariant under the Lorentz transformations?

This was of course nonsense because everyone assumed that Newton and Galileo had to be right and there was just something weird about electromagnetism. If you change Newton so his equations are Lorentz invariant, they look ugly and clearly wrong. But what if, nonetheless, they are correct and Newton wrong? The problem was to make sense of how these ugly neo-Newtonian equations could possibly be right. Why? How could they be derived from something more basic and intuitive? And this line of reasoning ultimately led to all the discussion about space, time, measurement, and simultaneity. In some ways the end result, the view we now have and teach about relativity, is simple and argues from "first principles", like how to tell whether two clocks are synchronized or not if there is some fastest possible speed of signaling between clocks in different places. But this simplicity comes far down the line from trying to figure out how to reconcile Newton and Maxwell, in a mess of complexity that builds on practically all the basic physics of the previous 250 years and a lot of its mathematics.

The "beauty" in relativity comes from Minkowski's clever mathematical trick of representing motion in 3 dimensional space as an imaginary rotation in 4 dimensional space-time. Once you do that, a mathematician can immediately tell you how to construct anything you want so that it is Lorentz-invariant: you make a 4-dimensional vector or tensor. Now the ugliness disappears and everything looks simple and neat, and as a bonus you need many fewer equations, indeed fewer concepts though we don't exploit that, not to mention the possibility of making really big bombs. But is it simpler? or is the complexity just hidden away in all the procedures for unpacking the tensor equations and getting a calculation of just how big that explosion is going to be? Beauty yes, apparent formal simplicity yes, but overall conceptual simplicity, no. IMHO.

A generation or two later, it's no longer just a nice way to conceal complexity behind apparent simplicity. It offers a new way of thinking about things, though so far as I know it is never taught that way. I happened to rather accidentally and precociously learn the tensor version before I'd even even heard of F=ma. I still find the concept of "force" totally ugly and confusing, and when people and textbooks say that mass increases with velocity, I think they're just nuts (or wrong). The effort to keep the old concepts while fitting them to the new mathematics leads to anything but simplicity or beauty. But people claim that the old ways are still more "intuitive" and easier to connect to concrete experience. I've always thought that human beings are capable of finding anything concrete and intuitive if they grow up learning to think that way.

Simplicity, like beauty, is indeed in the eye of the beholder.


Jay Lemke
Educational Studies
University of Michigan
Ann Arbor, MI 48109

On Apr 25, 2009, at 6:18 AM, David H Kirshner wrote:

How can you explain to someone the beauty of Goedel's proof of his Undecidability Theorem? ... How do you communicate the beauty of these symbolic practices?

I teach a course called The Culture of Mathematical Problem Solving in which I seek to enable math teachers to connect with the culture of their discipline. The mathematical dispositions I seek to inculcate include not only mathematical aesthetics, but also problem solving heuristics, appreciation of logical nesting and paradox, and the like. Dispositions, in my terms, are culturally specific forms of engagement (with problems and artifacts, with others, with oneself).

In my own typology of pedagogical methods, I rely on two that are relevant to inculcation of valued dispositions: "enculturationist pedagogy," and "acculturationist pedagogy." Enculturationist pedagogy involves the nurturing of the targeted dispositions within the classroom microculture. The teacher, using this method, needs to know the target dispositions, as well as have a sense of their developmental precursors. The teacher supports the gradual evolution of those dispositions by surreptitiously encouraging increasingly sophisticated forms of engagement. Students learn through their enmeshment in the classroom microculture, rather than through any kind of explicit learning process. As more sophisticated forms of engagement become normative in the classroom culture, the teacher moves on to the next level.

Acculturationist pedagogy is appropriate for students who are identified with the reference culture (in this case, mathematical culture) and seek to acculturate themselves to it. The teacher, in this case, needs to be--and to signify as!--a member of the reference culture. Instruction consists primarily of modeling the valued forms of engagement for students who are eager to emulate the teacher's cultural practices.

Combining these two instructional forms in my course, we spend a good part of the semester enmeshed in solving puzzle problems from my personal collection. These non-technical problems embody specific dispositional characteristics, which we reflect upon collectively and individually. At the same time we are reading from the philosophy and psychology of mathematics (e.g., Lakatos; Poincaré), interviewing mathematicians about their experiences, and the like. In this way, we have the opportunity to engage in mathematical cultural practices even as we reflect on our own developing cultural orientations with respect to mathematics.

David Kirshner

-----Original Message-----
From: xmca-bounces@weber.ucsd.edu [mailto:xmca- bounces@weber.ucsd.edu] On Behalf Of Andy Blunden
Sent: Friday, April 24, 2009 8:25 PM
To: eXtended Mind, Culture, Activity
Subject: Re: [xmca] Where is thinking

I've never heard this discussed before Ed. Very interesting.
How can you explain to someone the beauty of Goedel's proof
of his Undecidability Theorem? Like with Euler's equation
and so on, it is one thing to give the "easy" explanation
which appeals to intuition, but the mathematical formalism
is something else again. How do you communicate the beauty
of these symbolic practices, other than taking people
through years of routinized practice exercises?


Ed Wall wrote:
Martin and Andy

This is interesting (and an experience I had also with Shrodinger's
wave equation for a hydrogen atom although I was lured away by
education) as I am trying with some of my students (who, so they
believe, have neither strong interests or abilities in mathematics -
elementary school teachers presently or on the way) to develop a sense of beauty within mathematics. Part of this is because their students do have sort of a sense and part of this is because I have been wondering if I can crank it up a notch (not all the way to tensor algebra - smile)
and also, partially negate, their, these teachers, abominable
mathematical experiences.
    I have begun to have a little success as without coaching (and an
hour or so thinking, talking, and working) they seem to be able to
distinguish on their own between two or so acceptable  proofs - ones
they, for the most part, understand and generate - as to the one that
somehow is elegant (whatever that means although I happen to agree with their choice). Assuming that taste is cultural (although there are ways
in which mathematics, one might say, isn't. I don't mean by this
Platonic), I've been sort of bemused by the response. Anyway, it seems
that Vygotsky would have been interested in 'intellectual' taste.


On Apr 24, 2009, at 5:20 PM, Martin Packer wrote:

As an undergraduate I was in a class in which we solved Shrodinger's wave equation for a hydrogen atom (the simplest case, and I think the only salvable one) using tensor notation. I can confirm that it is beautiful mathematics, and it almost prevented me from becoming a psychologist.


On 4/23/09 10:32 PM, "Andy Blunden" <ablunden@mira.net> wrote:

:) Yes, Ed, I found tensor calculus a genuine thing of
beauty. After learning about e^ip=-1 a couple of years
earlier only ijGlm=0 could top it (excuse lack of sub- and
superscripts and Greek letters). But it is not so much the
mathematics that is at issue I think, when someone says
"relativity is simple" but just how the mathematics is
related to experience. Einstein himself wrote an
introduction to the Special Theory which does the whole
thing up to the variation of length with relative speed,
without using mathematics. But tensors are a mathematics
whose object is not physical relations, but differential
equations. That's tricky!

Any way, it's a long time ago for me too!


e^ip means the base of natural logatrithms raised to the
power of the square root of minus one times the ratio of the
diameter to the circumference of a circle, and it = -1
In ijGlm , G is a tensor of space-time, ij are subscripts
and lm are superscripts. But I may have that wrong!

Ed Wall wrote:

     It has been quite awhile since I have taught a course in
special/general relativity (about 20 years); however, the tensor
calculus is, I thought then, a nice way to go about it and brings some things to light that are important on the way to general relativity. Tensor algebra is actually somewhat straightforward by the way, but
is a matter of opinion. However, all of this has now become perhaps a bit off topic (smile) and you are correct that special relativity does
not, at a certain level of understanding, require manipulation of


On Apr 22, 2009, at 10:40 PM, Andy Blunden wrote:

Yes and No. I was using the word "metaphysics" in the way Pragmatists
use it. Strictly speaking, of course, *all* thinking contains
metaphysical assumptions. So in that you and Kuhn are right and I was

Perhaps I could stop using the word Metaphysics to mean the
reification of thought forms into independently existing substances, and others stop using the word Ontology to refer to personal identity
formation? :)

But I disagree with you about your Kantian conclusion that "science is a purely logical". It was this Kantian belief (along with Euclid) that
was overthrown by Einstein. The Logical positivists were wrong of
course, because they interpreted the subject in Kantian terms, as an individual person and their private psyche having direct access to
eternal reason.

Interstingly Einstein disagreed with Bridgman. Einstein said that
within the context of a consistent theory, not every entity in the theory has to be subject to an operational definition. Einstein right, Bridgman wrong. But I think Bridgman got the right idea nonetheless.

Where Hegel and you are wrong, I believe, is the presumption that we are at the end of history (neither of you claim that of course, but it is a valid implication in both cases.) If the nature of time and space can be deduced completely from a critique of the cultural practices at any given time, e.g. in 1807 before the Michaelson-Morley experiment was possible, then obviously the practices whose critique will allow
the Special Theory of Relativity to be deduced "by logic" i.e.,
critique of practice, are impossible. If "science is a purely logical" then that presumes that no further significant developments in social
practices (such as the Michelson-Morlet experiment) can be made.

BTW Ed, I think we have to treat the Special Theory and the General
Theory differently. There is absolutely nothing simple about the
general theory and its tensor calculus!


Martin Packer wrote:
Oh Andy, I'm going to have to disagree with you once again!
At least, I'm going to disagree if by your statement here you mean to
that Einstein was avoiding metaphysics. That was the
interpretation the
logical positivists made, arguing that Einstein had exposed the fact Newtonian physics had hidden metaphysical assumptions, but that, with
operational definitions (Bridgman's term, but his ilustrations
were from
Einstein), Einstein had finally showed that science was a purely
logical (or
if you prefer practical) activity, free from metaphysics. What a mess
has led us into!
I'm on Kuhn's side on this issue: every scientific paradigm has
assumptions embedded in its practices. So we don't have metaphysics
on the
one hand and practice on the other. We have alternative kinds of
practice, each with their metaphysical assumptions. (The
metaphysics of
Einsteinian physics include the assumption that space is something
that can
be curved by a mass, for example.) The merits of each of the
alternatives is
what scientists spend their careers hotly debating. Even what
*counts* as
metaphysics is different from one paradigm to another.
But that's probably what you meant!  :)
On 4/22/09 8:17 PM, "Andy Blunden" <ablunden@mira.net> wrote:
All Einstein did was, instead of regarding time and space as
metaphysical entities existing independently of human
practice, he closely examined the practice of measuring time
and distance. That's all.
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Andy Blunden http://home.mira.net/~andy/
Hegel's Logic with a Foreword by Andy Blunden:
From Erythrós Press and Media <http://www.erythrospress.com/>.

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Andy Blunden http://home.mira.net/~andy/
Hegel's Logic with a Foreword by Andy Blunden:
From Erythrós Press and Media <http://www.erythrospress.com/>.

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