[Edu-sig] thought re graphing calculators ...

[Edward Cherlin]

On Tue, Sep 29, 2009 at 8:13 PM, kirby urner <kirby.urner at gmail.com> wrote:
>> Since say 5000 years humans have devoloped the concepts of numbers,
>> calculations and
>> algebra. They have discovered, that calculations obey certain algebraic laws
>> like
>> a*(b+c) = a*b + a*c and the like. Finally they have devoloped the concepts
>> of
>> algebraic structures like rings, fields etc.
>>
>
> 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.


> My model of most antiquated education regimes is as follows:  brow
> beat the kids when they're still young and undefended, easy to bully,
> weed out those that question authority too much, keeping those who
> obey.
>
> The newer models (since constructivism) get more philosophy in early
> and train kids to vigorously debate and question, on the theory that
> older people are always a source of obsolete ideas that must be
> filtered, as well as positive ideas worth perpetuating.  Deference
> simply on the basis of age is a recipe for disaster in any
> civilization.  Learn to question authority, as a survival skill.

Even in math.

Peano "proved" that all models of the natural numbers are isomorphic,
but it turns out that you can't carry out that proof in any countably
axiomatizable system. This opens the door to non-standard arithmetic
and analysis. Conway found a non-standard arithmetic that extends to
games of perfect information. We aren't done with this idea.

> As a tip to teachers, I advise against defensiveness on behalf of some
> supposed monolith or cathedral i.e. lets think of "maths" in the
> plural, as they do in the UK.
>
> If some school of thought wants to pioneer a contrarian discourse
> that's not completely supportive of the last 100 years or more, so
> what?  We celebrate consistency and coherency, not uniformity.
>
> 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.

> Recall
> I'm Wittgenstein-trained so have a penchant for not abiding by
> orthodoxies.

Likewise, but I trace it back earlier, to the Pythagorean who
discovered that 2^0.5 is irrational, to Socrates, and to Shakayamuni
Buddha.

> 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.

> Kirby

-- 
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.
http://earthtreasury.org/