[Edu-sig] thought re graphing calculators ...
kirby urner
kirby.urner at gmail.com
Wed Sep 30 05:13:47 CEST 2009
On Tue, Sep 29, 2009 at 5:58 PM, Gregor Lingl <gregor.lingl at aon.at> wrote:
>> The ability to brute force these data points with a self-feedback
>> circuit governed by various expressions, is for computers and
>> computers only. Humans by themselves aren't even in the game. At the
>> very least you'll want an abacus, or lowly calculator if you're a nerd
>> (snicker).
>>
>
> Oh no, when thinking about calculations or even only viewing diverging
> graphs humans not only are in the game but are still its main characters.
>
Yes, in terms of being the architects of computers and the programming
languages that control them, humans are still the stars. Mandelbrot
made his original discoveries at IBM, with what were more like
primitive X-rays in terms of being low resolution, like having only a
float type, not a real decimal type. Of course Fractint found a way
to convert these chaos chains into integers, thereby taking advantage
of algorithms pioneered for this type. Were I to teach fractals in
the classroom, I'd want to use the Decimal type. I'm unclear if this
would mean needing to build a new complex type (independent of the
built-in), with decimals for both real and imaginary parts. This
would be an interesting exercise (lots of __ribs__).
> 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.
> The purpose of my script simply is to show, that what we know as real
> numbers are
> different things (entities) than what we have invented (using nature an
> physical phenomena)
> as machine numbers. These two simply obey different sets of algebraic laws.
> The distributive
> law is not valid for machine numbers with the operations + and *. And this
> statement is true
> independent from the setting for getcontext().prec.
> (Floating point) machine numbers with + and * do not form a field. To
> describe their
> behaviour you have to devise different algebraic structures.
>
I might put it differently. Real numbers are somewhat speculative and
are imagined to run through all these algebraic feedback circuits to
give identical results. The reals have never been implemented
however, are more of an "abstract class" from which we inherit
methods. The subclasses are responsible for doing the actual work. No
one has ever used a real number in a real calculation. They've used
what some call "approximate types" which makes sense if you think of
reals as ideal i.e. we've decided "perfect" inheres in something which
has no existence. The flip side of the coin is: if it doesn't exist,
it's less perfect.
> This is not a gewgaw!
>
> In fact my intentions are much less ambituous. I'd be very glad if even 50 %
> of my
> students accepted seriously that the squareroot of two does not equal to
> 1.41421356237.
> They do not, regardless of the fact that they are able to multiply this
> number with itself
> by hand (!) and to recognize that the result does not equal 2.
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.
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). Recall
I'm Wittgenstein-trained so have a penchant for not abiding by
orthodoxies. Poincare realized the solar system was chaotic long
before the rank and file.
Kirby
>>>
>>> from turtle import *
>>> from decimal import Decimal, getcontext
>>> getcontext().prec = 50
>>>
>>> k = Decimal('3.9')
>>>
>>> N = 250
>>>
>>> That's what I meant with (in principle)
>>>
>>> Gregor
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