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Egyptian, Babylonian, and Greek Mathematics
Read Science Awakening Egyptian Babylonian and Greek Mathematics B L van der Waerden Books
"This in an opinionated and confident history of Greek mathematics. On issues of interpretation and historiography, a few sections are original and many are based on the classical German literature, not readily available to the modern reader. The problem is with the mathematics, where van der Waerden is very unhelpful. He follows the standard practice of simply reproducing or lightly paraphrasing the original sources. A typical section on Archimedes, which almost entirely lacks commentary, simply reproduces a number of proof and ends "All this is found in Archimedes, in essentially the same words" (p. 220).
This is especially ironic since van der Waerden himself portrays as a main reason for the demise of Greek geometry "the difficulty of the written tradition" (p. 266): "An oral explanation makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation of the strictly classical style. The proofs are logically sound, but they are not suggestive. One feels caught in a logical mousetrap, but one fails to see the guiding line of thought. As long as there were no interruption, as long as each generation could hand over its method to the next, everything went well and science flourished. But as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became extremely difficult to assimilate the work of the great precursors and next to impossible to pass beyond it." (p. 266). This is exactly how we feel. It seems obvious, then, that a good history of Greek mathematics will aspire to recreate the qualities here ascribed to the oral traditions. Unfortunately, van der Waerden is generally happy to reproduce the "logical mousetraps" and leave it at that.
One further point is important to remember in this context. Today's mathematics tends to be congenial to "logical mousetraps" and shun intuition and applications. In Antiquity it was probably the other way around: it was precisely the mathematicians who were striving for more intuition and applications, while philosophical and other prejudices were exactly opposed to such a development. This inversion of the current roles is indicated in Plutarch: "Plato himself censured those ... who wanted to reduce the duplication of the cube to mechanical constructions, because [it is based on] a non-theoretical method; for in this manner the good in geometry is destroyed and brought to naught, because geometry reverts to observation instead of raising itself above this and adhering to the eternal, immaterial images in which the immanent God is the eternal God." (p. 163). In other words: the surviving mathematical texts are "censured" by non-mathematicans on non-mathematical grounds, whereas the subject matter screams out for a less formalistic treatment."
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Science Awakening Egyptian Babylonian and Greek Mathematics B L van der Waerden Books Reviews :
Science Awakening Egyptian Babylonian and Greek Mathematics B L van der Waerden Books Reviews
- 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! - This in an opinionated and confident history of Greek mathematics. On issues of interpretation and historiography, a few sections are original and many are based on the classical German literature, not readily available to the modern reader. The problem is with the mathematics, where van der Waerden is very unhelpful. He follows the standard practice of simply reproducing or lightly paraphrasing the original sources. A typical section on Archimedes, which almost entirely lacks commentary, simply reproduces a number of proof and ends "All this is found in Archimedes, in essentially the same words" (p. 220).
This is especially ironic since van der Waerden himself portrays as a main reason for the demise of Greek geometry "the difficulty of the written tradition" (p. 266) "An oral explanation makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation of the strictly classical style. The proofs are logically sound, but they are not suggestive. One feels caught in a logical mousetrap, but one fails to see the guiding line of thought. As long as there were no interruption, as long as each generation could hand over its method to the next, everything went well and science flourished. But as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became extremely difficult to assimilate the work of the great precursors and next to impossible to pass beyond it." (p. 266). This is exactly how we feel. It seems obvious, then, that a good history of Greek mathematics will aspire to recreate the qualities here ascribed to the oral traditions. Unfortunately, van der Waerden is generally happy to reproduce the "logical mousetraps" and leave it at that.
One further point is important to remember in this context. Today's mathematics tends to be congenial to "logical mousetraps" and shun intuition and applications. In Antiquity it was probably the other way around it was precisely the mathematicians who were striving for more intuition and applications, while philosophical and other prejudices were exactly opposed to such a development. This inversion of the current roles is indicated in Plutarch "Plato himself censured those ... who wanted to reduce the duplication of the cube to mechanical constructions, because [it is based on] a non-theoretical method; for in this manner the good in geometry is destroyed and brought to naught, because geometry reverts to observation instead of raising itself above this and adhering to the eternal, immaterial images in which the immanent God is the eternal God." (p. 163). In other words the surviving mathematical texts are "censured" by non-mathematicans on non-mathematical grounds, whereas the subject matter screams out for a less formalistic treatment. - A comparable book might be an Introduction to Algebra book. Complete with descriptions and examples of usage (and of course history). If you're interested in knowing more about Babylonian sexagesimal notation, Egyptian fractions, poetic numbers, Pythagoras, Plato or Hellenistic science interest you then this is a good source for the material and thankfully it has been translated into English. While I believe the translation was accurate, this can be a sleeper if you're not fond of numbers or their manipulation.
