Mesopotamia/babylon and Summaria
-I really should have started talking about this point yesterday. The point is that for tens of thousands of years, mankind had been doing a certain amount of arithmetic. Then, when they started these constructions, now geometry becomes developed; now, arithmetic evolves into algebra. Or, was it the other way around? Morris Kline in his "Mathematics in Western Culture" points out that if you tried to create mathematics for the very practical reason of navigation while your out in sea, then you're already either about to be dead or you're in a certain amount of trouble(better not get to far from sight of land!). Maybe it was bit of both. Certainly the Jericho tower wasn't based on much mathematics. Maybe it and many other constructions now lost to time inspired some geometry. But, the Egyptian pyramids appear to me more an inspiration from mathematical investigations than the other way around.
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I want to go back a little bit and point out some of Egyptian mathematics and maybe Babylonian if I get around to it.
Back to the point about how humans had been doing a certain amount of number mathematics for tens of thousands of years as proven from the talley sticks found in the like the 1930s. Regardless of how sophisticated they got, let's look at how sophisticated the Egyptians got.
-Numbers establish boundaries in a world where boundaries are hard to define. They define discreteness. Of course, the initial efforts to establish these boundaries were based on vague ideas of them. People from tens of thousands of years ago to the Egyptians didn't have different words for one, two, three. They simply said, one; one, one; one, one, one. So, eight seven wasn't 87, it was one, one, one . . . all the way up to 87; but, by the time you get up to 87, how do you know that this is 87 and ones? This is subtle and may take awhile for it to hit you. Another way to point this out is to say there's a theatre of people. The theatre owner doens't know how many seats is in there(assuming he's some guy tens of thousands of years ago); but, if he sees that therer's a full house, then he knows he has a theatre of people. This is how they understood higher number groupings. They would have some base, either five, ten, twenty, sixty, and to get more like 61, or less like 59, they'd put a one before or after the sixty. They couldnt' systematicaly construct from one to infinity all the numbers, so that they could understand say that five equals 4+1, but also, 3+2. As Jacob Bronowski stresses in his "origin of knowledge and imagination", we solve problems from our current perspective. We either take what we see at face value and get a flat earth; or, we look at underlying structural relations like the planets are doing this retrograde motions and there's these shadows on the moon; and, so maybe we go around the sun and not the other way around? There's a vague surface level understanding of things, and there's a mathematical underlying structural relations understanding of things. And at first, with numbers, people just had the theatre manager understanding of numbers.
-We can see this in Egyptian arithmetic. The symbols for numbers up to ten which has its own symbol are all the same one describing the unit one. They count the number up to ten, as one, one, one, and so on for all numbers up to ten. The universe as Jacob Bronowski argues is an infinity connected whole. But, we as multicellular species cut ourselves off from this infinit reality; we are finite. We can't describe all the numbers off to infinity. We have to make a finite base from which to describe this infinity. The Egyptians certainly used the base ten. We know that cultures for tens of thousands of years used all kinds of other bases. Base two, five, ten, twelve, twenty. even eight. Linguisticaly, nine in 'new number' beyond the base eight. Our finite perspective is also evidenced throughout the rest of Egyptian mathematics(and all mathematical science all the way up to now and the future).
-They multiplied by a doubling process. For instance, twelve times twelve is doubling of twelves and counting how many doubles we've made till say we have the four doublings number 48 and the eight doublings number 96, and then we add those two 48 and 96 to get 144. They could even do a shortcut where they do a halving after doubling. They multiply one number by ten; but, then they half that, and add the result. For division, they reverse this process. Well, actually, they just did a bunch of multiplications systematically, wrote them down as 1120:80=14. They they just wrote tables of the results. The Babylonians also did this with algebra. They wrote tables of squares, cubes; and, to get their answers to algebra problems, they just looked up the tables like we learn how to read off the solution to problems by graphs today. You have an x and y axis; you number them, and put up the date, and to find the solution you follow the x line to a particular number, then follow up the y line till you hit a data point. Both the Egyptians and Babylonians kind of did their math this way!We can further see how the Egyptians had to overcome their finite perspective(and our fundamental human nature all the way to us now) in fractions.- Just like the people tens of thousands of years before them had names for higher numbers but no underlying structural understanding to systematically generate those numbers(like the theatre owner), the Egyptians had names for certain fractions, 1/2, 1/3, 2/3, 1/4, and 3/4. These fractions you can see how they relate in creating the number one. 1/2 gets completed by adding one to the numerator. Likewise, 1/3 and 2/3 complete each other and so do 1/4 and 3/4. To calculate with fractions, they established certain formulas and then reduced. They had /6+/6=/3. They put a bar over a number to indicate say 1/6; but, the finite standardized system of typing I'm using can't represent a bar over a number; so, I have a dash next to the numbers. So for /3+/6=/2; they substituted /6+/6=/3 for the /3 in the "/3+/6=/2" problem I'm showing here to get 1/2! And they built up to higher and higher problem just like that! Understanding Jacob Bronowski's points as I've described in my "origin and nature of mathematical knowledge" article(which is linked at in the 'about me' to the right of the webpage for easier access), and you should just find all this absolutely amazing! They go further. I think this article is long enough.---------------------------------------------------------------------------------------------------------------------------------------Back to Babylonian mathematics. They had different ways of represeting number using wedges which I'm not even going to attempt to describe with the finite perspective of the symbols I'm typing with here! They went much further in algebra than the Egyptians did. I've indicated some of the nature of how they did their algebra. By means of this almost coordinate tablet system(imagine going through the Babylonian tablet library to look up the data and solutions to a given problem!), they solved quadratics, progressions, cubics, systems of cubics. All that amazes enough. They also at least observed that the ratio of the circumference to diameter of different sized circles is the same; and, well, people argue they had pythagorases theorem. Once again, I"m going to argue that they didn't have the geometry;The webpage reloaded and erased everything I said up to this point; so, I'm tired, and I'm going to try to get to the point. I argue that the Babylonians didn't have the Pythagorean theorem and that they got their pythagorean triples(solutions to the right triangle) by means of the number theory of the Babylonians.
- A couple of thins to note maybe out of order in this write up. One major point for why I want to point out Babylonian mathematics is that mathematics seemed to flourish now with cities and the building of major architecture certainly in both Egypt and Babylon. My point here is why then and not for the tens of thousands of years before? Because they hadn't hit on agricultural civilization yet. They hadn't dammed up nature to the point that these problems would come to their attention yet.
There was agricultural civilization for thousands of years(the Jericho tower is the oldest known structure; it predates everything by thousands of years including the Malta ruins). Those who study Babylonian mathematics(I've seen this in more than one place. I read in my fathers "Astronomy: from Stonehenge to Quasars" that the
as - What makes It's interesting that the Egyptians didn't say learn from the Bablonians to keep up with them. This could have been due to the Babylonians keeping it a secret; but, this doesn't make sense in the light of how all the mythologies flowed amongst all the different cultures back then; why combine each others mythologies and not take up each others mathematics? People learn what they learn and stick to what works for them.
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ReplyDeleteFractions presented mankind with some mathematical problems of irregularity. The Egyptians delt with irregular fractions by the algebraic substitution method as described above. The Babylonians had another remarkable way of dealing with 'irrational numbers.'
ReplyDeleteOnce again, the Babylonians made tables of everything, including inverses of numbers. These inverses would solve division problems by multiplying numbers by their inverses. But, many of these inverses had infinit regressions. The Babylonians found which inverses didn't; they called them regular numbers.
Regular numbers are of the form of a/b=1/60^n. The denomitor of 60 has a prime decomposition of 2^2 times 3 times 5. So, when a number has this prime decomposition in its denominator, it can't be reduced any further!
I've found an interesting connection between some mathematics of Socrates and Egyptian/Babylonian(they problably influenced one another)
ReplyDeleteStart out with a large square, call it HEFG. At the midpoints of this squares sides, call them, ABCD, draw perpendiculars, so that they meet at the center of the square, called O. Assume we actually started with one of the little squares, say AEBO. For one, we can say the overall large square is four times the any of the little squares; but, if we triangulate all the little squares, we get another inner square, ABCD that is double of the little square(or any of the little squares).
This was a problem noted by Socrates(through Plato). What's interesting is that if we abstract out the inner square that doubles the size of one of the little squares, ABCD, and include one of the little squares other triangles, so the figure looks like it's got a hat on, that shows that we've got a square root of the doubled square, ABCD. Where one side of the doubled square is C, hence by the area of a square formula, base'x'height, we get c^2, which = 2a^2(where a is one of the sides of the smaller squares. Now, a little algebra, reveales c= radical2a(square root of 2.
For reasons I need to review actually, but it seems the Egyptians for one noted that (7/5) squared, and (5/7) squares gave either the root or halfed it depending on which way you put 5 and 7 in the fraction. (7/5)^2 is 49/25, which is a good approximation to the square root of 2!
- Probably the biggest accomplishment of pre-Greek mathematics is Pythagorean triples.
Start with a^2+b^2=1, factor the left side into (a-b)((a+b)=1 . . . set (a+c)= p/q, and (a-c)=q/p - add and subtract them. Adding first,
a+c=p/q
+a-c= q/p - or, 2a =(p/q)+(q/p), dividing out the 2 gives a=(p/q)+(q/p)/2, and multiplying by pq gives, a= p^2+q^2/2
subtracting,
a+c = p/q
-(a-c)=p/q , or 2c= (p/q) - (q/p), and then dividing by two, and multiplying by pq again gives, c= p^2 - q^2/ 2 , then b= 2pq. B is perhaps a little mysterious and I need to review; but, it is the answer. I thinking plug in a and c, and they probably cancel out leaving 2pq for b.
The Babylonians primarily did their algebra geometrically, just as the Greeks did. Scholars have often thought that the Greeks did geometric algebra because of their encounter with the irrationality of the square root of two. Also, Van Der Waerden often felt that the Babylonians must have gotten their algebra geometrically, but never proved it. Recently, researchers are finding that the case is as Van Der Waerden suspected.
In book two of Euclid's Elements, there's the famous geometric representations of basic algebraic expressions like A^2+2ab+B^2, in terms of a square. Lines are drawn in the square that divide up the square into various rectangles, large and small squares. Fitting algebraic letters to varioius sides, we can read off algebraic expressions like the above. This is indeed how the Babylonians did their algebra, modern researchers are finding! Euclid's Elements indeed preserves this ancient knowledge in a deductive/axiomatic format.
Some of the algebraic expressions/equations they come up with are like (a+b)^2 = a^2 + b^2 + 2ab. Similar expressions can be had as (y-x)^2 + 4xy = (y+x)^2 . . . and by switching out variables like this - xy into square root(xy), y=p^2, and x=q^2, then square root of xy = pq, 4xy = 2pq^2, and the other expression get turned into (p^2-q^2), and (p^2+q^2), so, the Pythagorean triples are related to this geometric algebra!
- Also, the Pythagorean theorem can be derived by these geometric algebra . . . (a+b)^2=c^2 + 2ab , (a+b)^2 = a^2 +b^2 + 2ab, substituting for (a+b)^2 in both, we get c^2 +2ab = a^2 +b^2 +2ab . . . and so the 2abs cancel out, leaving the Pythagorean theorem!
Here, we have an amazing interplay of geometric algebra, the kind of geometric proof almost certainly Pythagoras used to mesmerize(and inspire) his fellow Greeks, and Pythagorean triples and irrational numbers.