[Edu-sig] thought re graphing calculators ...
kirby.urner at gmail.com
Wed Sep 30 19:04:50 CEST 2009
On Wed, Sep 30, 2009 at 1:39 AM, Edward Cherlin <echerlin at gmail.com> wrote:
<< trim >>
>> Yes, these have been interesting discoveries and remain highly
>> relevant in the workaday world. The idea of closure makes perfect
>> sense in this world of types (Python is a typed language). Is a * b
>> always going to yield a type the same as a,b? (assuming a,b were of
>> the same type to begin with). If polynomials are the type, then a*b
>> is a polynomial, as is a+b. Not a/b though. Polynomials form a ring,
>> not a field.
> You can take quotients of polynomials if you adjoin one point at
> infinity to either the real or complex field. But you're right, it
> isn't quite a field.
The definition of "polynomial" is quite narrow and the rational
expressions one gets with p/q, p,q both polynomials, need not simplify
to something worthy of that brand name. Many bastard children if you
wanna sound neo-Victorian about it (not saying I do). It's really not
a big trauma if you need to shift types. Early Pythons had a closure
model for integer division such that int / int always had an int
answer -- a kind of pure algebra preserved with //. Nowadays we're
used to the 3.x standard wherein int/int is more like a ring, where
you "fall out" of the integer type simply for taking a multiplicative
inverse of any |integer| except 1 (0 raises an exception).
<< trim >>
>> Could we develop a geometry which does not depend on the metaphysics
>> of real numbers, continuity, infinity? Or still have infinity, but
>> make it more like Poincare's, a direction (like a time axis).
> There are vast realms of such geometries, going back to projective
> geometries over finite fields and the like, and to general topology.
> Lie groups (including the one-point compactification of the complex
> plane) and Lie algebras. Banach spaces. Measure theory. Spaces of
> fractal dimension. Minkowski space. Differential geometry of
> Einsteinian spacetime. Non-commutative geometries in quantum mechanics
> (von Neumann algebras over Hilbert space). Many more.
A feature we're looking for is "accessible to grade schoolers" i.e. we
don't want you already out the other end of some lengthy pipeline
wherein brainwashing has already occurred. I understand that some
elite schools get into jiggering with the fifth postulate (Euclid's)
even pre-college, however I'm more interested in following the lead of
Karl Menger (dimension theorist) and messing with the "dimension"
concept. This discussion fuels debate in the high schools, with
strong players on both sides. Not waiting for any fool PhD to get in
the ring and sound intelligent about a non-Euclidean geometry (is more
the attitude we encourage, among people as young as 15).
>> I'm Wittgenstein-trained so have a penchant for not abiding by
> Likewise, but I trace it back earlier, to the Pythagorean who
> discovered that 2^0.5 is irrational, to Socrates, and to Shakayamuni
Yeah, these dudes were born earlier in time, I agree.
Resolved: "Irrational numbers are of course morally superior to the
rationals as all the best constants (e, phi, pi) are irrational, even
transcendental if we're lucky."
Pro, con or stand aside? Come prepared next Tuesday.
>> Poincare realized the solar system was chaotic long
>> before the rank and file.
> Peano, Hausdorff, Julia, Koch, and many others were also playing with
> fractals starting more than a century ago.
Or you could say Phi (golden mean) is the Phirst Phractal (certainly
the recursivity is there in the algebra).
> Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
> Silent Thunder is my name, and Children are my nation.
> The Cosmos is my dwelling place, the Truth my destination.
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