[C] [Wittrs] Re: Re: help the math teachers?
- From: kirby urner <kirby.urner@xxxxxxxxx>
- To: wittrsamr@xxxxxxxxxxxxx
- Date: Sun, 20 Dec 2009 09:02:54 -0800
Here's another diagram I've been sharing with the math teachers, looks
like something from LW's RFM in some ways:
This is from my debate with the aforementioned midwesterner.
http://www.flickr.com/photos/17157315@N00/4199404344/
What it aims to show is you might use a triangle to visualize A x B.
How is the meaning of "multiplication" affected? The answers are the
same, i.e. 3 x 10 = 30. But instead of a grid of 3 x 10 squares,
we're working against a grid of triangles and using those as our
units.
Likewise with tetrahedra. A x B x C has a tetrahedral interpretation.
I think these are the kinds of grammatical shifts Wittgenstein would
have us investigate. It's not like we offer any proofs that right
angles are the only way to go. We do, however, use right angles to
peg our "dimension talk" of one, two, three dimensions (then more,
confusing people with talk of a "fourth orthogonal" etc., when doing
multi-dimensional polytopes).
What I endeavor to get across is that "4D" (as in four dimensional)
does *not* have one fixed meaning (note "meaning" -- why this is in a
Wittgenstein thread).
We have at least two different "namespaces" or "language games" in our
culture in which 4D has an established meaning:
3D + Time = 4D = world lines, Einstein's namespace (relativity etc.).
3D + 1D = 4D = tesseract, Coxeter's namespace (polytopes etc.)
Coxeter himself is clear these are two completely different lineages
(contexts) for 4D. See pg. 119 of Regular Polytopes (I might quote it
later).
Then late in the 20th century came this additional namespace which I'd
call a philosophical namespace, which no one studies except for me and
a few people, despite all the honors and awards, applications,
international reputation of its principal author etc:
4D = tetrahedron, Fuller's namespace.
The latter is not starting with the premise that ordinary space is 3D,
as represented by three mutual perpendiculars identified as height,
width and depth respectively. Fuller writes:
527.702 Geometers and "schooled" people speak of length, breadth,and
height as constituting a hierarchy of three independent dimensional
states -- "one-dimensional,""two-dimensional," and "three-dimensional"
-- which can be conjoined like building blocks. But length, breadth,
and height simply do not exist independently of one another nor
independently of all the inherent characteristics of all systems and
of all systems' inherent complex of interrelationships with Scenario
Universe.
Fuller is more like Wittgenstein in saying something like "imagine a
tribe that considered the tetrahedron to be its chief measure of
volume, as well as the signature shape of volume in general, the
minimum tent or representation of enclosure. When they say their
space is 4D, not 3D, they're putting distance between their culture
and our cube."
Implied in the above description is that our tribe is different (why
we have to "imagine" that other one): we worship the cube ("our
cube"), almost forget what a tetrahedron is half the time (a "three
sided pyramid" right?).
Tracing the meaning of "4D" is, I think, a fascinating exercise and
needs a lot more commentators as we seek to move forward, pass the
torch in a coherent way.
The art historian who has done some of the most research on this is
Linda Dalrymple Henderson, in this award-winning title:
The Fourth Dimension and Non-Euclidean Geometry in Modern Art
(Princeton University Press, 1983; new ed., MIT Press, 2009), by Dr.
Linda Dalrymple Henderson.
It's an area ripe for Wittgensteinian philosophy though, not just art history.
You have real world applications on the ground, i.e. people like me,
trying to introduce the elements of a new digital math curriculum.
You also have reason to bring up Remarks on the Foundations of
Mathematics and to go, from there, to the rest of the corpus (it all
hangs together pretty well, Tractatus included, for message if not
approach).
Say, I'm wondering to what extent Rich Text Formatting will come
through on Wittrsamr and echoing lists. I should probably stick with
plaintext, right?
Kirby
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