Wednesday, June 1, 2011

thought for the day

What happened between the Spanish arabs and Europe was before what happened
to(mostly) Italy and the fall of the Byzantine empire. The dates are obvious;
the Arabs more or less fell around 1000 A.D.; the Byzantine empire fell around
1500 A.D.

The interaction between the Spanish Arabs and the Europeans
caused a pre-renaissance; it caused the crusades; it caused the Inquisition; the
knowledge that came out of Arab spain could only be copied so far; meaning, only
so many people could learn of all this knowledge; it was more for geeks like
Adelard who went out of their way to go into spain; real knowledge seekers; not
just guys who get hit on their head when some cultural force finds them. But,
this initial renaisance also was more about translation and commentary such as
Francis Bacon(or Roger bacon; i still get those two confused); they learned
about Aristotle and Plato; they certainly saw Euclid's elements; but . . . have
you looked at Euclid's elements? If you were even an Arab back then much less
some biblical copyist who first opens up a copy of Euclid's "Elements", you'd
probably burn it, throw it away, or loose it on the shelves(they didn't have any
form of order in the libraries of the christian monastaries back then; books
just got lost!); i mean, this thing starts out with a list of axioms, then
definitions; at which point, your like 'so what!?' Then, it goes straight into
theorem proving! If you don't have a clue about what deductive logic means and
how it works, you are lost by theorem one! Your only clue is Aristotle and
Plato; and, if you havn't gone through their mass of works and found the clues,
and gone through a period of reflection to note, dah! deduction! That must be
what all this theorem proving is in Euclid's "Elements", then, once again, that
literature was completelly meaningless to the majority of Europe . . .

And then, the black plague destroyed any hope of this initial
renaissance to capitalize on this knowledge from Arab Spain. Then came a second
influx from the Byzantine empire!

What happened when the Byzantine empire
fell was that the printing press happened about the same time, the scholars from
the Byzantine empire knew what deductive logic meant in the first place. And, in
the end, Galileo was influenced to do his studies by all the scholars that came
before him about Aristotles physics. If it hadn't been for all that time and
effort to figure out what all that Greek knowledge meant, Galileo would have had
to spend all that time figuring out what all that meant.

This is why the
killing of Archimedes was such a tragedy; the Romans did not learn from him when
they could have. This is why the destruction of Athens was also a tragedy;
because, you can't just expect some culture that hasn't figured these things out
to figure it all out; it has to go through periods of development to even begin
on knew research, or to even understand something like the mathematics the
Greeks invented(a deductive approach to all the Babylonian mathematics; and
then, the Greeks went a little bit farther; although, from the Babylonian
viewpoint . . . except for the algebra of the Babylonians . . . was far beyond
anything they could ever imagine; trig, number theory . . . the infinitude of
prime numbers . . . conic sections, irrational numbers, Archimedes proof and
calculation as oppossed to mere measurement of Pi, some intial work on
integrations and differentiations, the balancing point of irregular shaped
surfaces, the use of higher curves to solve the three Greek classical problems;
the trisection of angles, the doubling of cubes, the squaring of the circle).
This is the tragedy of the dark ages; this is why the dark ages were the dark

I thought I'd put a bit of my review of Van Der Waerden's "Science
Awakening" in here to update my account of Archemedes!

'Most people have
heard of Archemedes; but, do they have any idea of why he is so highly regarded?
Highly douteful! For one thing(and this goes for all Greek mathematicians back
then), all mathematics was done in terms of an awkward geometrized
algebra(Eudoxes was highly regarded by Greek mathematicians like Plato because
he perfected it and made is fairly workable). So, when Archemedes proves
deductively how to calculate Pie(a double reductio absurdium proof!), he then
calculates it with this geometrized algebra! It's hard to appreciate the
difficulties that are brought in here that leaves those who look at this
absolutely intellectualy drunk; it's like when you show the trigonometry(and
Archemedes practicaly shows the way towards developing a trigonometry in his
calculation of pie) that comes from an isosceles triange; you find that one of
the sides requires a radical expression; now imagine having to do this with
geometrical algebra, and you should be feeling just . . . ; I've yet to explain
what makes this all amazing actually. The Babylonians at least observed that two
different sized circles have the same ratio of diameter to circumference. This
was proved in Euclid's Elements. C=(pie)D. But, what is pie(meaning the
numerical value)? As I've stated, Archemedes relates the area to the legs of a
right angle triangle. Basicaly, he's related the two dimensional property of
area to that of the one dimensinal constant of the circumference. Now, because
the area of a triangle is 1/2hb, we get (pie)r(squared). Archemedes goes on to
calculate the circumference of the circle as already described. But, then
Archemedes goes on to use much of the same strategy he used to show the way
towards calculating pie to calculating the area and volume of a sphere; he
relates the second and thired dimensions together! Euclid's "Elements" shows the
plane and solid geometry of his day before(and much else like number theory);
but, the solid geometry is flawed in areas; Archemdes comes up with this
dimensional analyses solution to the theory of solid geometry! That alone puts
him above most!

But! Archemedes goes on to do primitive calculus(using
geometrical algebra), he calculates the center of balance of odd shaped
figures(like obtuse and scalene triangles), he uses arithmetic and geometric
progressions to handle large numbers(and to deal with much of his irregular
surface results); he solved the area of triangles which is normaly attributed to
Heron(trigonomery students should know what I'm talking about). Some stuff that
was knew to me from Van Der Waerden that I didn't know before are Archemedes
construction of the hexagon which turns out to not be constructable by
straightedge and compass(a Plato restriction which isn't mathematical valid but
does have some mathematicaly interesting things; more on this later); Archemedes
constructs it with conics. Van Der Waerden relates that Pappas notes that
Archemedes constructed and explored much semi-regular solids(you can't be a
mathematician or consider yourself a mathematica enthusiasts if you don't know
what I'm talking about here). By this time, you should be putting Archemedes up
there with Gauss and many others!'

No comments:

Post a Comment