On Fri, Dec 28, 2012 at 7:30 AM, Paul Tanner <email@example.com> wrote:
<< SNIP >>
> If p is actually a proved statement, a theorem, then person B is > simply showing himself or herself to be a fact-denier - and if he or > she persists in such fact-denial, to be a crackpot. >
That's quite an illuminating statement in light of your earlier claims to have "proved" this and that about the sloppy world of voting and health care. In your model, the statements you put forth are "as theorems" (truths of the same caliber) and those who oppose you in argument are in fact fact-denying crackpots and/or slobs.
This attitude might well work to your advantage in debate, if you're able to communicate your utter disregard for the opponent's view in a way that makes you seem professorial. Keep to the tone of the "expert witness" but don't let the jury think you arrogant, as if the trial's outcome were more up to you than them (or maybe it's no jury and all judges -- in NFL debate we lean towards tribunals in the higher "courts" with even more judges at the championship level).
> The problem is, if a person does not first do the careful work needed > to prove that a mathematical result is actually incorrect, then that > person's claim that that result is incorrect - along with his or her > sloppy argument - shows himself or herself to be mathematically inept > and ignorant and a sloppy thinker. >
You're aware from literature classes and/or from life experience that some kind of tension builds suspense and serves as a plot driver. We want to see how things turn out. A "gripping page turner" (as the New York Times might describe some book) is one that keeps us motivated to move through its plot twists, its state changes and transformations.
In setting up this debate between two views, one that the limit is zero, and the other that the limit is an iotum (so long as there's curvature), I provide a motive (like a motor) to break through the ice of:
(a) Descartes' Deficit (rarely if ever taught in US elementary schools, probably taught in all the better Swedish ones **) (b) an algorithm for generating high frequency icosa-spheres (like at EPCOT -- though that's triacontahedron based) (c) vocabulary words like "icosa-sphere" and "frequency" (important in architecture, geometry) (d) lots of standard notation used for talking about limits (a boost if you're delving into calculus, one of the maths many islands / locales / namespaces).
Student listeners to this debate are likely to pick up quite a bit of shop talk, all the better if they see peers using such, and not just those already steeped in this material.
In the better classrooms (e.g. in Sweden), a teacher will have students stand in front and conduct parts of the lesson.
Naturally this requires a classroom in which there's respect and decorum. It's still controversial to place classroom surveillance cameras. &&
"To contradict" is to speak in opposition to. In making this an humanities event that uses math content, we're having this not be your turf. You're not here to serve the state on the punch clock. You wandered in as a member of the audience. The mathematical meaning of "contradiction" is not front row. p & ~p is close enough to what A and B are doing. The judges need the audience to remain politely silent, or you may rap on the chair with your knuckle to indicate you think a point was made well.
> Again: Students need to be taught that in mathematics, they ought not > throw the term "contradiction" around casually - it's just plain > sloppy. To prove a contradiction one does not merely negate a given > statement with a sloppy argument.
They need to become masters of their mother tongue, including the sloppy uses of terms, such as "voting" and "health care", which have no existence in strongly logical mathematics (except in science fiction and fantasy).
To have debaters mine mathematics for a resolution is commendable and both the Aff and the Neg did a good job.
The Aff argued persuasively that there's no limit on how close epsilon might get to 0, where epsilon is size of the "tax" a vertex must pay for the privilege of being a local apex, at the tip of a radial, on a hill with a view.
Give me a large enough frequency, and we continue our asymptotic approach to where we might say "each vertex on a perfect "at infinity" sphere is instantaneously flat i.e. is surrounded by 360 degrees".
The Neg pointed out that there's a discrete iotum of "tax" that comes from just being in a sphere-like polyhedron, inheriting Descartes Deficit (a tetrahedron of degrees) as a builtin characteristic of the ecosystem.
No matter what the frequency, there's a constant 720 "ownership fee" (might be a better term for "tax" in capitalist thinking) that must be contributed by all who have purchase on this globe. Taxpayers may indulge themselves in the illusion that Zero is a reachable ideal, but of course it never is. Inside that epsilon is a grain of sand, a positive amount, ad infinitum, and there's no getting rid of it, or rather, the tetrahedron has been subtracted and is not coming back.
My approach is not all that different from taking 1 == 0.99999... and saying there's dramatic tension here, and using that to motivate exposure to additional concepts. Will that magical "..." be strong enough to give us 9s forever, or will it peter out and leave us asserting a falsehood. Everything hinges on the ability of "..." to really deliver. One needs to cultivate utter confidence to that effect, and to say things like "we're not talking about a *physical* possibility" (in contrast to what?).
That's a known pedagogical technique (andragogical too), to use tension and suspense ("who will win?") as a motivation to pick up more of the relevant terminology. I don't think it's a stupid technique myself, though I understand if you choose to avoid it in your own dramatic role as teacher in a mathematics classroom setting (a very special case condition which 99.9% of us do not experience with much frequency if at all -- and yet we might teach math and logic and rhetoric and PR, including for money).
** those knowing a wee bit of history will remember that Descartes became a controversial member of the Swedish court, as beloved personal tutor to the queen, not unlike the role of John Dee, mentor of Sir Francis Bacon. Descartes was nominally Catholic though he feared the Inquisition enough to encrypt major findings, while Sweden was a Protestant country such that his influence aroused deep suspicion, making his presence there something of a trial http://www.cathnews.com/article.aspx?aeid=19371 (Catholics were suspicious of him too, given ties to Rosicrucianism, which the Wikipedia article doesn't talk about).
&& sometimes prisons have been transformed into centers of study as well, in which case the surveillance cameras just fit right in. One of the famous warden-scholars, of San Quentin, led prisoners through a multi-year course in General Semantics and other topics. http://www.youtube.com/watch?v=ww8sYTfYJlA This may have been when poet Gene Fowler ('Waking the Poet') encountered a bigger world of ideas (reminiscent of the Malcolm X story -- prison is sometimes educational, if enough agreement builds)