thought for the day/the classical Greek world/part 2; the mathematics mostly

I'm rather surprised when somebody said this book was basicaly about early
number systems; what? Van Der Waerden must be rolling in his grave at that!
Maybe in the first chapter on Egypt and the second chapter and even the third
chapter on Babylonian mathematics. But, what this book really amounts to is
showing how the Greeks mathematics was for the most part just a deductive
reasoning and axiomatizing of mankinds first efforts to create 'functional'
knowledge as opposed to just vague brush the problem under the rug theories that
is religion(God is basicaly a vague concept; it is the algebraic X standing for
"I don't know" and "I don't want to know;) Now, the beginnings of this
'functional knowledge' was numbers as shown in Van Der Waerden's account of what
the Egyptians accomplished; but, they did accomplish some geometry. Then, the
Babylonians acomplished some more geometry and algebra. But as I've stated
before, the biggest thing this book shows is how the Greeks, at least at first,
established those results logicaly.

Van Der Waerden tries as I've
indicated to show the extent of classical mathematics. The only mathematics he
doesn't address are the tally bones found around his time. I'm surprised he
didn't get around to maybe making a second edition to amend his effort here. Oh
well; it's a small problem probably more due to the amount of knowledge at his
disposal back then. I would recommend "Pi in the Sky" for the second chapter;
this second chapter in "Pi in the Sky" is worth the book alone as far as I'm
concerned; all the other chapters in my opinion are better talked about is say
the works of Morris Kline. Let me get to the Egyptian mathematics though.


I can't believe somebody can say they like mathematics and not find Van
Der Waerden's account of Egyptian mathematics fascinating and to maybe say, wow,
maybe they were ingenious in the way they handled their problems. To me the
whole science of planetary astronomy and how we figured it out is the classical
example of how mathematical science is done. We have to figure things out from
our current perspective. If you just take what you see at face value, you think
that the earth is flat and all objects have a natural state of rest. Genius, or
mathematical insight, is about seeing behind appearances. Nature is an
infinitely detailed whole; we make fine distinctions that are not totaly true;
science and mathematics establishes boundaries and says 'know that the results
are only valid within the errors indicated.' If you don't find the mathematical
process fascinating alone, then you don't appreciate matheamtics and science,
and you cannot find the ingenious solutions that the Egyptians came up with to
making a workable number system.

Here's just an instance. Ancients who
first started trying to come up with numbers(not coverd by Van Der Waerden in
this book as indicated above) would have words to describe certain numbers like
one, two, maybe up to five. Sometimes, they'd have words just for one, two, then
five, or ten. And, they didn't think of two, five, or ten or higher as one, two,
three, four, five; they just had a bag of something and they'd say if I can
match up the objects in this bag with the number of sheep you've presented me,
then I have this bag of sheep(whatever number of objects in the bag). This can
be more vividly pointed out by saying, a guy working the theatre doesn't know
how many seats is in his theatre(if he doesn't know numbers); but, he knows he
has a theatre of seats; now, if he has a full house of people in there, then he
knows he has a theatre amount of people in there whatever their number. This is
how they understood five, and ten, and higher numbers; they didn't know how to
relate the amount of objects in higher numbers with the smaller ones . . .
because to them, each higher number was a similar unit. When they matched bag
with sheep, they diddn't count one, two, three, but one, one, one, one, one; and
that was five. Look at Van Der Waerden's account of Egyptian number symbols; he
has a symbol of one, ten, and other higher numbers; when they wanted eight
units, they put out eight ones!

To further point out how you can see
that we create our knowledge from our current perspective just note that one of
the major difficulties faced by all ancient peoples was how to represent each
new higher number(once they got past just unit, unit, unit representation; they
found another problem as described in this paragraph); do you keep creating new
symbols for each new higher number? Well, it gets hard to come up with some new
creative and distinct symbol for each new higher number; this is why the place
value number system is so special and important(amongst other reasons). Because
of our finite cutting of the universe, we have to find a finite set of symbols
to represent a potentialy infinit number of 'numbers'!

That's more or
less enough about the remarkable things Van Der Waerden finds in Egyptian
mathematics and my defence of his efforts to explain the great creative efforts
of even the Egyptians. Of course the Babylonians went way beyond them in
algebra. Remarkably it seems that the Egyptians had a better accounting of the
number pie!

It's to the Greeks that mathematics becomes more than just
rules of thumb. Van Der Waerden shows the great scholarly work done in his time
of who wrote which books of Euclid's elements and the amazing content in
Euclidn's "Elements." Euclid's "Elements" is special not just because it was one
of the greatest early axiomatic deductive logic books, but because it
axiomatized and deduced all the great efforts of countless nameless
mathematicians from the Egyptians to the Bablonians(and various peoples tens of
thousands of years before agricultural civilizations efforst to deal with the
universe). Without it, all that mathematics would have withered away; or, the
perhaps subconscious effort to come out of the dark ages may never have happened
at all.

Van Der Waerden then goes on to describe Archemedes and
Apollonius primarily. Most people today wouldn't know David Hilbert, Riemann,
Gauss, Leonardo Euler, Lagrange, Abel, Galois, Hamilton, Dedekind, Cantor(of
transfinite number fame and not the four volume history of matheamtics books)
Klien, Lie, Frobenius, Poincare, Noether, Emil Artin, Weyl(the last three and
David Hilbert are the intellectual group Van Der Waerden is associated with),
Wedderburn, Weber, Serre, Atijah, Grothendieke, Connes, Thurston(the last five
are some contemporary still living matheamtical giants). 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!

Next up is Appollonius who actually
calculated pie better than Archemdes. Other accomplishments was the
parallelogram law for vectors(I always attributed this to Simon Stevens in the
1700s; sorry Simon!); i have Appollonius's conics and hope to get through it;
so, I skimmed this part of the book; there's things to be said here that are
underrated and maybe Van Der Waerden didn't stress. My point about how we figure
things out from our current perspective. I don't think that people before the
Greeks didn't know about ovals and such, but the Greeks came to the conics in a
round about way! The conics turned out to be a solution to their playing around
with loci! One of Appollonius's accomplishments has to do with some loci
problems; he also comes up with the envelope of the ellipse. He more or less
hints at algebraic geometry; if he had the algebra of the Arabs even, he may
vary well have gone the whole way. I would note that if you read E.T. Bell's
"Men of Mathematics", each chapter is about a matheamtical great; it goes first
chapter, the Greeks; second chapter Descartes! This is kind of the point of Van
Der Waerten's book; he essentially shows the wonders of the mathematics pre,
maybe not Descartes, but pre-renaissance reawakening mathematics.

I
think I'll note a couple of things that I never heard of before that absolutely
blew my mind was the perspective drawing found in the ruins of Pompie and the
mathematics of the Astrolabe. To say the least, I can't believe I've never heard
about this perspective drawing back at least first century A.D! And, the
matheamtics of the astrolabe; they map the stars onto the disk by means of
stereographic projection! Here's something that I think Van Der Waerden misses;
Ptolemy's does some of his initial trigonometry by means of the Platonic Solids!
Just like in trigonometry classes when your taught that the only trig values
findable(precalculus; well, unless you want to do all this geometric algebra of
Ptolemy) is by means of the isosceles triangle; Ptolemy actualy uses the
Platonic solids! Also, plane trigonomery was figured out by stereographic
projection of spherical trigonometry!

It's surprising to me that Van Der
Waerden doesn't mention the prime number theorem in Euclid's "Elements". I don't
know if maybe he just felt like he had nothing new to say about it, like who
proved it; or, maybe he felt those things are known and he didn't need to rehash
those things.

Lastly, I'll get back to perhaps a really biggie(one can't
possibly expect anyone to recall and account for everything in a book such as
this; so, I don't mean to say this is a knock on Van Der Waerden!) is the
relation of classical mathematics to modern mathematics; for instance, Descartes
figured out coordinate geometry by means of loci; but, also, the whole use of
higher curves to solve the three delian problems and the construction problems;
it's known that the three problems need to be solved by means of higher curves;
and in terms of abstract algebra - by means of algebraic extentions. As Herman
Weyl argues, knowing the problems of classical matheamtics is a good place to
know the the problems of todays matheamtics(at least some!).

I thought
I'd further add that I still didn't mention every big, medium, and small ideas
and points Van Der Waerden makes in this book; this shows how much great
material is in the book!