Kirby,
My apologies for the delay. I've gotten distracted by some much less worthwhile discussions as well as some worthwhile ones that simply required less of my time and energy than this. I hadn't forgotten you.
Of course, I think I still have a couple of other posts out there waiting for you, so perhaps apologies are unnecessary.
> There is a Nintendo character named Kirby, whom I reference
> in my
> slides at
http://www.myspace.com/4dstudios
That's the one!
I'm happy to be associated with something
> other
> than a vacuum cleaner for a change
The Dysons are much better.
You ever heard of Jack Kirby? He co-created Captain America, The Fantastic Four, The Incredible Hulk, The X-Men, and many others.
http://en.wikipedia.org/wiki/Jack_Kirby
> "Triggering perceptual shifts that restore clarity" might
> be one way
> of characterizing it. That's part of the inherent
> difficulty too, as
> once in the realm of perceptual shifts, gestalts, how does
> one know
> if this "triggering" has occurred?
The discussions of "aspect blindness" in the later work (PI pt. II, RPP I and II, and LWPP I & II passim.)
Often, in cases like we're discussing, it is evidenced in a capacity to put a picture to a new use.
Be careful though. Sometimes we can confuse resistance to ideas with an inability to understand them. Recall the "you just don't get it" of identity politics.
You'll get these pedagogical concerns,
> that exposure to
> an unfamiliar model of area and volume, not based on right
> angles,
> will only interfere with student comprehension of
> traditional content.
That's an empirical question of course. I don't know whether exposure to variant schema would enrich comprehension or cause confusion. Likely, it depends a lot on age.
> Do we teach this in art schools? In some maybe.
I think it definitely has a place in design schools!
> You could see this as countering some unexpressed view that
> "math is
> fascist" in the sense of "my way or the highway".
Again, I wouldn't call this a philosophical issue, per se, but if people misuse the word "fascist" and if their notions of political liberty blind them to the role that authority, unquestioning agreement, and strict training by repetition play in the functioning of our understanding, our language, and our very form of life, then perhaps they need a wake up call and perhaps their notions of liberty are simply bosh.
Seeing freedom as well as constraint in all of this is certainly a good thing. You're right about that.
But people who make such judgments as "math is fascist" are in my experience among the worst dogmatists, obscurantists, and opponents of human freedom and dignity going. As well as being total asshats.
That's not diplomatic though. Obviously.
>
> From the point of view of struggling students, sometimes
> finding math
> oppressive, a little novelty might go a long way towards
> forestalling
> a disconnect.
hmmm. Perhaps.
I wonder though if students like that whose interest one might hope to pique with the new thing ("Ooooh! Shiny!") would have any follow through anyway. That would be nice and I do not rule it out.
> That's the thing though: mathematics as we teach it
> in schools is
> conservative, change-averse. We might think of it as
> the one
> discipline least likely to change. This has to do
> with math's
> reputation for dealing with eternal verities, with nothing
> new under
> the sun, at least where primitive geometry is concerned.
"Has to do with"? Is the conservatism of mathematics explained by such notions? Are such notions rooted in the conservatism? Or are such notions an _expression_ of the attitude with which we approach mathematics and of how we treat mathematics? And isn't how we treat mathematics an important part of the various roles it plays for us?
(Conservatism in mathematics and conservatism in the teaching of mathematics. Conservatism in the teaching of fundamental mathematical concepts and conservatism in the teaching of further extensions. Compare.)
That's
> why I say math teachers
> need help from philosophers, especially those in the
> Wittgenstein
> camp, given their pragmatic operationalism (meaning through
> use).
hmmm
Did you catch my remarks elsewhere regarding operationalism? (It was in a post to JRStern "re: QM without consciousness" or something like that.) I'm wondering if you mean "pragmatic" as a reference to Pragmatism or simply colloquially. And if you mean "operationalism" in a more or less doctrinaire sense?
I certainly would not be inclined to call myself a pragmatist or an operationalist.
> Going back to my volumes table, I'm suggesting what's novel
> is our
> ability to interject more whole number volumes and simple
> fractions
> than previously, even though we're introducing such
> non-rectilinear
> concepts as the rhombic dodecahedron, a space-filler.
> We assign it
> a volume of six, slice and dice it to come up with a
> corresponding
> cube, octahedron and tetrahedron of 1/2, 2/3 and 1/6 the
> volume
> respectively (i.e. 3, 4 and 1) -- the beginning of a
> language game,
> with more pieces to come later.
I find this aspect very interesting.
I've shared some of this with my father, an engineer who, when I was a child, made a hobby of welding models of Platonic solids and the like. He was interested as well.
>
> Having this more sophisticated visual vocabulary coupled
> with less
> intimidating, more memorable whole number relationships, is
> something
> new (since the 1960s). It's new because our Roman and
> Greek
> forebearers considered right angles "normal" even though
> squares
> have no inherent structural stability.
I wonder here about the "even though"...
When you do
> post and lintel
> architecture, rest cross-beams atop columns, you get used
> to thinking
> of structure as rectilinear.
Is this meant to be an explanation? What sort of evidence would support it or falsify it?
The tetrahedron (tetra
> for four),
> although the minimum wire-frame enclosure, more primal than
> the cube,
"More primal" in the sense of "minimal" or in some other sense?
> didn't get as much focus in the early days of western
> civ. Those
> mental habits are difficult to counter to this day.
> Yet what better
> example of challenging a dominant paradigm?
I mentioned in another post, I wonder about putting things in these terms.
Do you mean "paradigm" in anything like a Kuhnian sense? It doesn't seem to me that this approach is incommensurable without the prevalent one in anything but the familiar and epistemologically innocent mathematical sense.
Math
> teachers need help.
>
> However since the invention of microscopy, other more
> powerful
> instrumentation, it has become apparent that nature is
> more
> triangulated in her designs. Our more sophisticated
> visual vocabulary
> is going to help us down the road, as future biologists,
> chemists,
> engineers. The world of sphere packing, of lattices,
> will be more
> front and center, thanks to our more 60-degree based
> approach.
> Stabilize what we have, trail blaze new material.
> It's an exciting
> time to be a math teacher.
>
> That's the PR anyway.
>
Ah.
I've never trusted PR.
> In other words, we're hoping to excite teachers about
> these
> developments, not trigger a "let's not rock the boat"
> backlash right
> from the get go.
Okay, yes. Though I wonder if math teachers as a group are as persuaded by such forms of persuasion as some others might be or if their liable to think, "Sounds like snake oil. Too good to be true. Panacea. We've been burned before."
I don't know enough about the attitudes of math teachers in various parts of the country to do more than entertain possibilities that might warrant caution. Take what you will.
> Also, as soon as one touts something as "better" or "new
> and
> improved", there's some resulting anxiety about introducing
> the
> change, undermining the existing "music of authority" in
> some way.
I think experiences with New Math might offer legitimate reasons caution without needing to posit... um, I'm not quite sure what :music of authority" is meant to suggest.
> This may account for why this material is still unfamiliar
> and not
> widely discussed, even after a half century. There's
> an underlying
> defensiveness perhaps?
I'd be reluctant to speculate about motives here.
The remarks about various transformations, comparisons with tessalations, and so on, were quite interesting in their own right but seem to not connect with my point. (I still found the material quite fascinating though.) I think I made the point better in a subsequent post though, so I'll leave it.
> If the philosophers think what we're doing is OK, i.e.
> passes enough
> tests for being coherent, not breaking the rules, then math
> teachers
> might feel comfortable enough to develop a lot more
> bridging
> exercises.
Why in heaven's name would math teachers give a gods damn about what philosophers have to say and why would any philosopher presume to suppose she could approve or disapprove here? (I know the answer to the latter: some philosophers are a presumptuous lot. But I hope no Wittgensteinian would so presume.)
A philosopher (perhaps a better one than myself) can help in exploring how we relate these new concepts with the old, how we think about the transitions between the different cases.
Whether the new approach is legitimate is a mathematical question and not a philosophical one. It is a question of whether one can translate between the systems, whether conversions between them preserve consistency. From what I can tell, that's not a problem, but a mathematician is far more qualified to answer that.
If the answer to the mathematical question is affirmative, then it may be best not to emphasize talk of "paradigms" (unless the buzz word is just good for PR rather than being offered as a philosophical insight) and such because the real question is one of the allocation of educational resources (a procedural, political question, not a philosophical one - save perhaps to a Rortian).
On that question, I could only entertain possibilities with you. If thinking about mathematics philosophically gives me any particular insight into reservations people might have, that would be a side effect. It wouldn't make it a philosophical matter.
A philosopher might be able to assist in thinking and speaking more clearly about the relationships between the different approaches and in helping others to but - as the reference to PR may remind us - clarity may not be the winning strategy in political matters anyway.
JPDeMouy
============
=========
=========
=========
==
Need Something? Check here:
http://ludwig.squarespace.com/wittrslinks/