Re: corrupting the youth

[Jack Uretsky <jlu@HEP.ANL.GOV>]

On Sun, 1 Sep 2002, John S. Denker wrote:

> Jack Uretsky wrote:
>
> > The Harke paper
> >   http://www.harke.org/ps/intro.ps.gz
> >
> >  is directed to more profound
> > problems than are normally faced in undergraduate mechanics,
>
> ... which is why I have repeatedly not recommended it
> as a starting point.
__________________________________________________________________
I thought that was the recommendation, that's why I went there.
What is the recommended starting point?

> Perhaps the problem is that people who get good
> at Clifford Algebra quickly get too good, and soon
> race ahead, beyond the reach and almost beyond the
> sight of the rest of the community.
__________________________________________________________
That's one possibility.  Another is that the Clifford algebra approach
offers no new insights.


> I think the pedagogica agenda should be to do
> a few simple things with it and stop there.  Perhaps
> start with qualitative gyroscopic precession, as
> illustrated by
>   http://www.monmouth.com/~jsd/physics/gif48/add-bivectors.gif
> and stop there.
______________________________________________________________________
Well, that's a very pretty picture, but it doesn't tell me how to tell an
undergrad who hates working with symbols that this is an easier way to
calculate areas.


> Maybe a year later calculate areas using ||A/\B||.
________________________________________________________________________
But (using the Clifford algebra postulates), A/\B and AxB are one and the
same animal.  ixj is a pseudovector whether or not the Clifford
fundamentalists want to agree that it is aligned along the direction k.

Calling the magnetic field a bivector (specified by a surface) doesn't
have the impact of using a compass needle to map out a field line (we're
talking about pedagogy, no?).  And the B-field is just as real as the
E-field, it's just that magnetic poles are difficult to isolate.  Let's
have a little democracy around here.

> Maybe a year after that get serious with rotors,
> electromagnetism, and all that.
>
> >         So,IMO, foisting this stuff on helpless undergraduates (and their
> > teachers) qualifies as corruption of youth in the worst sense of ....
>
> A) That rather depends on what one means by "this
> stuff".
>
> B) I think the real problem is not with the youth, but
> with the old fogies who long ago spent a week getting
> comfortable with the cross product, and who don't want
> to spend a week learning something new.
>
> The youth will have to spend a week on one thing or
> the other, and it seems more-than-likely that a week
> spent learning wedge products will be much more
> valuable than a week spent learning cross products.  I
> don't know that for sure, but I haven't seen anything
> resembling evidence to the contrary.
__________________________________________________________
That may be the real crux of our differences.  I think that the apparent
unification that is offered by the Clifford algebra approach, involves
levels of abstraction that will make the typical American undergrad much
more uncomfortable than the conventional approach.

Clifford fits well into the Bourbaki approach to mathematics which, as far
as I am concerned, will  reach its apex when all of mathematical knowledge
is condensed into a single incomprehensible symbol.


> > the positive
> > subalgebra (to be defined) of the Clifford Algebra of V(4,(4)) is just the
> > ordinary vector algebra that you have all used since childhood.
>
> Practically everything you've ever heard of can be
> rediscovered as some subalgebra of some Clifford
> Algebra.  That means I don't need to separately learn
> about vectors, spinors, complex numbers, quaternions,
> pauli algebra, dirac algebra, etc. etc. etc. -- I just
> learn Clifford Algebra and get all that stuff for free.
_________________________________________________________________
For free?  How many high school teachers on this net are likely to want,
or be able to reproduce, the derivation that I gave in my posting?  And,
after reproducing it, how many would find it easy to teach?  I find the
choice of the word "free" to be a real hoot!

> If you already know all that stuff, maybe you don't
> have much to gain from Clifford Algebra ... but it
> would be absurd to assume that the youth already know
> all that.  And in any case it doesn't detract from the
> simplicity and elegance and power of the Clifford Algebra.
________________________________________________________________________
I don't regard my derivation to be a demonstration of "simplicity and
elegance and power".  There was nothing to tell me that V(4,(4)) was the
right set to start with.  There was nothing to tell me in advance that a7
was going to turn out to be the identity.   And I had to do a lot of work
to show that at the end of the day the 7 elements of the Clifford Algebra
reduced to the the basis vectors of Gibbs and Hamilton.

I'll soon be replacing my signature line with my favorite quote about
Bourbaki.
                                                Regards,
                                                        Jack

--
"But as much as I love and respect you, I will beat you and I will kill
you, because that is what I must do.  Tonight it is only you and me, fish.
It is your strength against my intelligence.  It is a veritable potpourri
of metaphor, every nuance of which is fraught with meaning."
                     Greg Nagan from "The Old Man and the Sea" in
                     <The 5-MINUTE ILIAD and Other Classics>