Monday, December 13, 2010

Math- Old School

In the last month we have reported on NEW RESULTS by Williams, Katz and Guth, Sanders, and Pinkerton and Setra. For a change of pace lets look at some really OLD math from a really OLD book- the Bible. (NOTE- this has nothing to do with whether the Bible is true, just that its old.)

In Genesis 18 God wants to destroy Sodom and Gomorrah. Abraham wants to argue against this. Here is a paraphrase using modern terms.

GOD: If there are 50 righteous people in Sodom then I will spare the city.

ABRAHAM: What if there are 45? 45 is pretty close to 50.

GOD: Okay. If there are 45 righteous people then I will spare the city.

ABRAHAM: What if there are 40? 40 is pretty close to 45.

GOD: Okay. If there are 40 righteous people then I will spare the city.

ABRAHAM: You know, uh, 30 is pretty close to 40.

GOD: Okay. If there are 30 righteous people then I will spare the city.

ABRAHAM: 20 is only 10 less than 30 so how about....

GOD: Okay. If there are 20 righteous people then I will spare the city.

ABRAHAM: 10 is only 10 less than 20 so how about....

GOD: Okay. If there are 10 righteous people then I will spare the city.

ABRAHAM: Good. (He stops bargaining. Perhaps he shouldn't have--- They couldn't find 10 and the city was destroyed.)

I think Abraham should have used smaller increments as he went down since going from 20 to 10 is cutting it in half which sounds like a lot. But who am I to argue--- he got God down to 10 which is impressive even though it didn't work.

This reminds me of two paradoxes:
  1. The Small Number Paradox. All numbers are small. Proof by induction. Clearly 0 is small. If n is small then adding 1 can't make it much bigger, so n+1 is small. Hence all numbers are small.
  2. The Sorties Paradox also called The Heap Paradox. 1,000,000 grains of sand makes a heap. If n grains of sand make a heap then so do n-1 grains of sand. Hence 0 grains of sand make a heap.
Is the passage in Genesis 18 really math? In a very primitive form I would say yes. Is there any older source for anything that resembles math?

21 comments:

  1. Actually, the Bible isn't really all that old in the grand scheme of things. :) Genesis is thought to be from around 600 BC. But there's a fragment from Babylonia with math on it that's from around 1800 BC.

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  2. Of *course* the answer to "Is there any older source for anything that resembles math?" is OBVIOUSLY YES.

    For one thing, Euclid's Elements was written around 300 BC — that's about 300 years before Christ was born, and seven centuries before the oldest surviving Christian Bible. And Euclid is of obviously much greater mathematical sophistication than anything suggested by the Bible.

    If you want to argue that the Old Testament is much older than Christ, then consider that "The oldest material in the Hebrew Bible – and therefore in the Christian Old Testament – may date from the 13th century BCE" (sourced statement on Wikipedia), so that's about 1300 BC. And you have:

    * Babylonian mathematics which in 2000–1600 BC had arithmetic, algebra including quadratic and some cubic equations, compound interest, geometry including Pythagorean theorem etc., and lots of other well sophisticated mathematics.

    * Egyptian mathematics, which included Egyptian fractions, solving linear equations, areas of triangles, circles, volumes of cylinders, etc.

    * Some Indian and Chinese mathematics, which may be older than that book of Genesis.

    All of this is more than counting numbers.

    I don't know if posts like this are intentionally meant to provoke, born out of ignoring the history of mathematics, or out of a religious belief that the Bible must be the oldest book (because it talks of Creation?)

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  3. Soooo Bill,
    your belief system is from the same book that perpetuates such an embarrassingly low level of thinking?

    I know, I know. Off topic. SCNR.

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  4. BTW, I'm reminded of this "exercise" in Carl E. Linderholm's hilarious book Mathematics Made Difficult:

    “6. 'And Abraham answered and said, Behold now, I have taken upon me to speak unto the Lord, which am but dust and ashes: Peradventure there shall lack five of the fifty righteous: wilt thou destroy all the city for lack of five? And he said, If I find there forty and five, I will not destroy it. And he spake unto him yet again, and said, Peradventure there shall forty be found there. And he said, I will not do it for forty's sake .. .' [16] How did Abraham have the nerve to get into this conversation, and why was the Other Person so patient?”

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  5. If bargaining counts as math, then certainly the construction of ancient objects such as the pyramids in Egypt and Stonehenge would qualify.

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  6. Fairly common negotiating trick. Come to agreement on a number, then add lots of proportionally small fees on top. $20,000 for the car. Oh, there's also a $50 stocking fee. Oh, there's also a $25 undercoating cost. Oh, the warranty is...
    Proportionally, the additions are small, so you're disinclined to renegotiate the main price over them.

    In this case, God would know exactly how many righteous live in the city, so he's just humoring Abraham. "Yup, you're so clever talking me down from 50 to 10, good luck finding them."

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  7. (NOTE- This comment was emailed to me by the moderator
    but never appeared in the APPROVE THIS COMMENT thing
    on the blogger, so I (GASARCH) post it directly. IF you made this
    comment please re-make it so we can get the original rather
    than this version which may have some format changes.)

    (The ID seems to be ''sure''.)

    Of *course* the answer to "Is there any older source for anything that
    resembles math?" is OBVIOUSLY YES.

    For one thing, Euclid's Elements was written around 300 BC that's
    about 300 years before Christ was born, and seven centuries before the
    oldest surviving Christian Bible. And Euclid is of obviously much
    greater mathematical sophistication than anything suggested by the
    Bible.

    If you want to argue that the Old Testament is much older than Christ,
    then consider that "The oldest material in the Hebrew Bible and
    therefore in the Christian Old Testament =E2=80=93 may date from the 13th
    century BCE" (sourced statement on Wikipedia), so that's about 1300 BC.
    And you have:

    * Babylonian mathematics which in 2000 BCE had arithmetic, algebra
    including quadratic and some cubic equations, compound interest,
    geometry including Pythagorean theorem etc., and lots of other well
    sophisticated mathematics.

    * Egyptian mathematics, which included Egyptian fractions, solving
    linear equations, areas of triangles, circles, volumes of cylinders,
    etc.

    * Some Indian and Chinese mathematics, which may be older than that
    book of Genesis.

    All of this is more than counting numbers.

    I don't know if posts like this are intentionally meant to provoke,
    born out of ignoring the history of mathematics, or out of a religious
    belief that the Bible must be the oldest book (because it talks of
    Creation?)

    ReplyDelete
  8. Posts like this are meant to inspire discussion of very old mathematics, as a contrast to the last few posts which have been on very new mathematics.

    How old the Bible is is actually a hard question which can be (and has been) studied by people without a religious interest in the question. Reputable people have said things from 600BC to
    2000BC.

    However, I'm more interested in old math
    (and your post DID have lots of that for
    which I thank you) then going off topic into how old various people think the Bible is.

    GASARCH

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  9. The Bible has some unsung math: fractal recursion. In Genesis it is said that God created man in his own image; often this is interpreted as an indication that God is somehow humanoid with two arms and two legs and so on. But it can equally well go the other way: Man has powers of creation equal to God. Ever notice that in Genesis Chapter 2, Adam is put to sleep, but nowhere does it say that he actually wakes back up? We're figments of Adam's dreamworld, baby!

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  10. Terence Tao had a very nice talk in his blog about the small number and the sorites paradox. He contributes the paradoxical nature of this statements to the use of informal languages, such as English. Notions such as "heap" and "small" are not defined precisely. I like to think of them as "non-uniform" notions: they change according to the size of the objects we are concerned of.

    Also, perhaps we can formalize the Abraham-God interaction as a form of computation? We can call this new class GA and assume that Abraham asks everytime for the cost to be halved, so it would be GA=MA[logn] or something like that .

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  11. For me the interesting mathematical bit is "how much of the population was 10". If it was 10 out of a few hundreds (not unreasonable for those days) then it was not a negligible percentage (one could also presume that only adult males were counted).

    By the way, later Jewish traditions claim that the single righteous man in Sodom was Lot, which was evacuated.

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  12. My wife Constance—who has a degree in Egyptology—informs me that the Egyptians circa 1800 BC left behind records of student problems that show a fairly sophisticated understanding of algebra, in which unknown quantities to be solved-for were called, not "x", but rather "the heap".

    Here is an example from the so-called Rhind papyrus from that era: "What is the size of the heap if the heap and one seventh of the heap amount to 19?"

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  13. god created the natural numbers.

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  14. There is a paradox similar to the heap paradox, but has to do with bald men. Specifically, if a man has no hair on his head he is bald. Similarly if he adds just one hair to his head, he's still bald.

    If you are bald, someone with just one more hair than you is also bald. So every man is bald, no matter how much hair is on his head.

    To relate this to theoretical computer science, we make much the same mistake with the class P. Specifically we were looking for a class of languages closed under calls to itself - namely we want it to be true that if efficient code calls an efficient subroutine the result is efficient code. Of course, just like a bald man with a head full of hair, algorithms with high degree polynomial runtimes (or even giant constant) are not (practically) efficient. This isn't a new criticism, of course. I hope the comparison to bald men is.

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  15. @Ross: You say that the class P captures more than efficient computation, practically speaking. Well, that justifies the definition even more, because the point is to separate P from NP.

    If it turns out that P=NP, or if there is slightest evidence that P=NP, then sure, time to revise the notion of what is efficient.

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  16. There is a wonderful law review article on a related subject, titled n Guilty Men.

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  17. Yet another anonymous commenter12:45 PM, December 14, 2010

    Another instance of Abraham (Abram) and mathematics: He uses a fair division process when he divides his land with Lot. Genesis 13:8-12.

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  18. The Shulba Sutras dealing with geometric constructions are dated to approximately the same sage. They have such things as tryinh to approximatelay sqare the circle(by assuming π = Sqrt(10).

    The architectural calculations done for the Pyramids are peobably much older.

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  19. My only question is, if you weren't ignoring the history, why ask trollish questions like "Is there any older source for anything that resembles math?" even though the answer (even taking the oldest date you gave for (part of) the BIble) is so obviously yes? :-)

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  20. Sure- fair point,
    my question obviously requires
    knowing BOTH old math and
    when the Bible (actually Genesis)
    was written.

    some points
    1) Even assumng the Bible was written
    in 2000BC, is there math before it.
    YES, thanks to the commenters who
    gave examples.

    2) Was the Bible around in Oral form
    before being written down? Yes,
    but very hard to know how old
    this form of the Bible is.

    3) Is the story in Genesis that I describe the first recorded version
    of the Heap Paradox?

    ReplyDelete
  21. There's a verse in Genesis (41:33) that's applicable to academia in general: "Now therefore let Pharaoh look out a man discreet and wise, and set him over the land of Egypt." In other words, the first piece of expert advice is always to hire an expert.

    As for the claim that Genesis only dates back to 600 BCE, one reason to be suspicious of the traditional dating is that some documents refer to events that happened after their supposed date. If we assume that such was a common practice (it is a plausible explanation for the second half of Isaiah), a lengthy document which included numerous prophecies that did not refer to a very important event was probably written before the event.

    The Torah makes no mention of the Divided Monarchy. I think we can date it to the time of Solomon or before even using a skeptical style of reasoning.

    ReplyDelete