Some questions about the babylonian tablet in the news...

[Kyriakos]

https://phys.org/news/2017-08-mathem...nian-clay.html



Hm, i read the article and have some questions, as a non-mathematician, of course.



I gather this is not a theorem (proof) of the pythagorean a^2+b^2=c^2, and the tablet had been theorized in the past to just have some such pythagorean triplets, albeit large numbers. And it uses a base60 numbering system.
But: given there was no theorem on this in Babylonian times, what is the new discovery/importance about? The article mentions that the approach seems to be about ratios (of what?) instead of "angles or circles", but again where is the theorem allowing for this to be used? Afaik the first theorem (which was a theorem of ratios moreover...) is attributed to Thales, and is the one about ratios of parts of sides of triangles when created by the same line cutting them all. So where is the babylonian math basis for using such ratios? (and if it isn't using those ratios, what is it using, and with what proof? cause otherwise it was already known for a long time that this tablet had pythagorean triplets ).

[Magister Militum Flavius Aetius]

They did discover the theorem. That Pythagoras "discovered" the theorem is sort of a myth. It could be argued that he perfected the Greek version of Trigonometry from which we base our own.

The difference is that the Babylonians discovered trigonometry via trigonometric ratios. Not in the Greek sense where it was discovered to be related to the "unit circle" (i.e. a triangle is any three points on a circle).

[Kyriakos]

I think you aren't right there (going by mathematicians in other forums i asked; my own uni degree in in philosophy, i just am aware of some of the ancient mathematics due to my studies and work), given a theorem is proof in an axiom-set system, and apparently no one claims the ancient babylonians had developed mathematical proof (the new evaluation on this tablet isn't about this either). Ie their calculations are based on what by practice looked to be (and is) a consistent pattern, and they extrapolated from that. First one recognized as forming a math proof is (argued by various ancients like Aristotle, as well as Euclid later on) Thales, who used the division of the circle into 360 degrees (babylonian afaik) but formed an axiom-bound system and provided proof of his statement on ratios (and other things about triangles and circles).

So, apparently, the new discoveries re this tablet aren't about proof or not (there wasn't any), but about base60 tied stuff. Maybe some people can shed light on this?

[John Doe]

I think the tablet is to be seen the same way as an abaque, like the tables of Log our parents used before electronic calculator came in. The tablet is more representative of the degree of complexity of Babylonian administration. Mathematics is a component of human thinking, the importance of the tablet is more anthropological than pure math. A good parallel would be the discovery of the Rhind Papyrus, it gave a good insight of Egyptian problem solving tools and methodology.

As for the base60, I'm not sure what's the importance.

[Kyriakos]

If you allow going on a tangent ( ) maybe using base60 in this trigonometry happens to get rid of most of the irrational numbers? (square roots in base10 used in greek math as well).

[John Doe]

The OP article talks about ratios, so it's fractional numbers that are involved, not irrationals. Some people says it's because 60 is the lowest number you can divide by 2,3, 4 and 5 without getting a fraction and that's may be why they used base60 for everything.

Article

[pacifism]

One thing worth pointing out is that base 60, or sexagesimal, is still used today in navigation. It's partly because 60 is such a composite number, so there are more whole number ratios when counting is done in that way. Whether or not ancient Mesopotamians new about this four thousand years ago, I don't know.

[Iskar]

Fascinating find.

From what I gather, it is certainly not a "proof" of Pythagoras' Theorem by any stretch, as it just enumerates a finite amount of pythagorean triples (i.e. numbers satisfying the a²+b²=c² equation). They seem to have known the theorem, though, as the way of recording the triples in orderly fashion makes a systematic use in construction, geodesy, etc. plausible.

[The point of the theorem is that given three numbers a, b, c satisfying the a²+b²=c² equation any triangle constructed with a, b, c as side lengths will be a right-angled triangle. So if you want to ensure right angles in buildings or land demarkations, you only need to construct such triangles by measuring the sides in the right way. This principle was long known and remained in use for thousands of years, even up to late-medieval masons in Europe, that used a 13-knot-rope to create right angles, which makes use of the most basic triple (3, 4, 5).]

Besides just trial and error there is an easy way of generating such pythagorean triples using coprime integers and if we believe the analysis on this site, then it is highly likely the author(s) of this tablet knew of this procedure, which is an astonishing mathematical feat, even if they did not record the abstract proof of the principle.

[To be fair, the Greek "proof" of the theorem - as with almost any other proof before ~1900 - does not really conform to our requirements of formal cogency and notional uniformity. It is in fact "just" a cleverly drawn picture by which a geometric proof "without loss of generality" - i.e. calculating a non-abstract but generic, non-special example - is possible.]


The point of hexagesimal notation of numbers here seems to be intermediately related, in so far as the wider range of finite fractions in base 60 (in addition to 1/2 and 1/5, 1/3 is finite as a floating point number in base 60, so 1/6, 1/12, 1/15, 1/30 and their powers are as well), allowed the author(s) of the tablet to conveniently note more triples than would meet the eye in base 10.

[Kyriakos]

^The idea to be rid of irrational numbers appearing due to roots, is also present in Plato's Dialogue between Socrates and Theaetetos (and Theodoros, of the eponymous spiral), where the suggestion is that (again) any integer as side becomes a surface of a rectangle, and thus roots revert to the corresponding integers.

[Roma_Victrix]

Yeah, fascinating find for sure, but it doesn't offer a mathematical proof for it, unlike Pythagoras.

[Iskar]

Well, the proof of the theorem is pretty simple and is basically this picture:


The way the pythagorean triples are listed on the tablet makes it highly unlikely the author(s) did not understand the underlying regularity of the theorem. To be fair they probably did not have a concise notion of "proving" theorems, which in a way emerged with Greek philosophy, although, as I said, very few of any of the arguments published before ~1900 satisfy our modern notion of a cogent proof.

[sumskilz]

The point of hexagesimal notation of numbers here seems to be intermediately related, in so far as the wider range of finite fractions in base 60 (in addition to 1/2 and 1/5, 1/3 is finite as a floating point number in base 60, so 1/6, 1/12, 1/15, 1/30 and their powers are as well), allowed the author(s) of the tablet to conveniently note more triples than would meet the eye in base 10.

In cuneiform, 1 and 60 are written identically. They are the same symbol. You can only tell them apart by context, but you get the idea that they are essentially the same. There are different "whole" measures for different commodities, but all whole measures are subdivided into 60 shekels. If 60 is written with the divine determinitive before it, then it means Anu, the father god/creator god/judge of all in the oldest version of the Babylonian pantheon.

[Iskar]

Ah thanks, I was not entirely clear about that. It does make a lot of sense, though, as it makes any finite fraction indistinguishable from an integer, so there'd be lots of obvious triples to note down in base 60 cuneiform.

If you disregard the later addition of 0, then we write 1 and 1[0] the same as well. Where the Babylonians used context to determine what power of 60 they were at we just over centuries developed the system with zeros and a comma to encode that information umambiguously.

Interesting thing about "the divine one/sixty". Did they consider fractions of Anu?

[sumskilz]

Interesting thing about "the divine one/sixty". Did they consider fractions of Anu?

Seems like it. Enlil is 50, Ea is 40, Sin is 30, Shamash is 20, Ishtar is 15, and Adad is 6. If you use the deity numbers to divide a length of string, with Anu being the full length, then you end up with what appears to have been their musical scale based on their surviving musical instruments. The scale seems to be the predecessor of all the European, Indian, Middle Eastern, and North African heptatonic scales. Of course such scales could have been arrived at independently based on the math, but it seems unlikely considering all those places also have musical instruments which descend from Near Eastern precursors.

[Iskar]

That is truly fascinating. The Babylonians just climbed up a good number of steps on my personal appreciation ladder.

[Kyriakos]

Sounds ( ) like more babylonian usurpation of discoveries attributed to Pythagoras Music notation based on dividing chords to neat fractions, that is.

[Roma_Victrix]

Sounds ( ) like more babylonian usurpation of discoveries attributed to Pythagoras Music notation based on dividing chords to neat fractions, that is.

EXACTLY! Pythagoras is under assault; may Αθηνά protect him! And I'll be DAMNED if some no-good Babylonian tries to marry my daughter.

[Kyriakos]

^Careful now, Roma, you must remember that in courts there existed the "Vote of Athena" herself, which was meant to break any tie between the judges, and ALWAYS went in favour of the accused

So Pythagoras will still win even in case of a tie.

[Iskar]

...unless he has to pass a field of beans to get to court.

[Kyriakos]

^I generally fear to ask what that alludes to