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Although the hype is overblown, there is nothing particularly wrong with the article (except for the hype), which is why it was accepted into _Historia Mathemat

by svat 9y ago

Although the hype is overblown, there is nothing particularly wrong with the article (except for the hype), which is why it was accepted into _Historia Mathematica._ A good introduction to Old Babylonian mathematics (the broader context for this paper) is in a paper by (surprise) Knuth, called _Ancient Babylonian Algorithms._ (You can find PDF links here: https://scholar.google.com/scholar?cluster=10887370978433539328 https://scholar.google.com/scholar?cluster=10887370978433539...) It's a very nice paper, and an example of Knuth's scholarly approach: despite not being a historian in this paper he produced the first translation of many old Babylonian tablets into English, by comparing the published German and French translations and then looking up Akkadian and Sumerian dictionaries to resolve differences! (Unfortunately he misread one of the secondary sources and later published a retraction; ignore the first column on page 672, about sorting (everything about Inakibit-Anu.) Now to Plimpton 322: it has, for various triples (a, b, c) satisfying a^2 + b^2 = c^2, columns containing a, c, and either a^2/b^2 or c^2/b^2. This has been known for decades; what's been unclear is what this table was used for. Some have proposed this was number-theoretic (just a listing of Pythagorean triples), but this doesn't answer why these specific triples were chosen. Some have proposed this was trigonometric, but the concept of an angle is not otherwise known in Babylonian mathematics. Enter Wildberger: for many years now he's championed "rational trigonometry" (trigonometry using only rational numbers). So instead of the sine and cosine he proposes using their squares; instead of angles there's something called "spread", etc. It all works reasonably fine, is not significantly harder to do than regular trigonometry (he claims it's easier, but it's about the same), and you get to work with only rational numbers. (This is also related to a broader project within mathematics which rejects infinite numbers -- see finitism, ultrafinitism.) So the Plimpton 322's fractions of the form a^2/b^2 are exactly like the functions of "rational trigonometry", and the authors of this paper (http://www.sciencedirect.com/science/article/pii/S0315086017300691 http://www.sciencedirect.com/science/article/pii/S0315086017...) make the argument that these tables were used for such a purpose. In one of the recent discussions, a commenter here pointed out this demonstration they (the commenter) made: https://teacher.desmos.com/activitybuilder/custom/59a05b5f50338d34a21e61c0#preview/8b58c98f-a047-46d1-8f53-d102993b11ad https://teacher.desmos.com/activitybuilder/custom/59a05b5f50... If you ignore the "more exact fractions" (which is true under a particular interpretation, i.e. numbers that can be written as finite sequences of integers in the "floating-point" representation), and "more useful than current mathematics" hype from the newspapers, the paper itself is sound and an argument about the nature of this table that is as plausible as any other speculation. They address previous interpretations and discussions, and have a coherent theory of how this table could have been generated or what it could have been used for. It even addresses the objection that Babylonians didn't have the concept of an angle: using exactly the objections raised against the trigonometric interpretation to say they used rational-trigonmetry-like functions instead. :-) The paper is a good example of one aspect of the history of mathematics, the way arguments are adduced for speculation on topics where there's little evidence (have you read the rhetoric in Robson's paper?), and the fun of constructing explanations out of little data. Here it is again: http://www.sciencedirect.com/science/article/pii/S0315086017300691 http://www.sciencedirect.com/science/article/pii/S0315086017...

thejynxed 9y ago
To think any civilization could build structures like the Babylonians did and not have a concept of angles is patently absurd. From what architecture they have that still stands, there is plenty of visual evidence that they indeed calculated for angles in some fashion.
- jaclaz 9y ago
  Yes and no. There is the concept of gradient (or slope). Even today you will find that earthworks (of course it depends on coutry regulations) use normally fractions expressing the gradient. You dig (in normal terrain) sections with a 1/1 slope, depending on consistency, harder soil can be excavated with 1/2, 1/3, 1/4 or 1/5 (w/h). You normally create landfillings /for roads, railways, etc.) with 3/2 slopes, which is a very good example how something that is extremely complex expressed in angles, like arctan 2/3, can easily be represented in practice. As well, acqueducts or sewages, railways and roads are usually calculated longitudinally basing on percent (or per thousand) gradient, like 0.1%, 0.5% (h/l). Vertical walls, like the walls built to contain the terrain on the "mountain side" of roads create - if perfectly vertical - the optical effect of appearing "leaning" towards the road, and usually are built with a 5% to 10% "opening" gradient from bottom to top to avoid that effect (and to save concrete), without anyone actually ever calculating the angle at the base.
- jacobolus 9y ago
  They clearly had a concept of similar triangles, and a concept similar to our notion of “slope” or “tangent” (apparently they actually used the reciprocal of slope, i.e. length per unit depth). However, they may not have had the concept as of ~4000 years ago of “angle measure” per se. You’d have to ask an expert in the history of astronomy when that concept was first developed semi-formally; my understanding is that it was 1000–1500 years later by subsequent Mesopotamian astronomers. Angle measure (another word for circular arclength corresponding to a particular rotation) is actually quite a tricky thing to get right. It’s the logarithm of the rotation (a pure bivector quantity) with the orientation stripped away – if we call our rotation R oriented in the plane of the unit bivector i, it’s θ = log(R)/i. To go from an angle measure back to a “complex number” (scalar + bivector) with which you can do anything useful, you need to put the orientation back in and then exponentiate it: R = exp(iθ). In dimensions higher than 2, you need to use rotation operators via a kind of sandwich product, because composition of rotations is not commutative. This is not an obvious concept, didn’t really get figured out properly until the late 19th century, and still isn’t taught very clearly to students today.

by svat 9y ago

thejynxed 9y ago

To think any civilization could build structures like the Babylonians did and not have a concept of angles is patently absurd. From what architecture they have that still stands, there is plenty of visual evidence that they indeed calculated for angles in some fashion.

jaclaz 9y ago
Yes and no. There is the concept of gradient (or slope). Even today you will find that earthworks (of course it depends on coutry regulations) use normally fractions expressing the gradient. You dig (in normal terrain) sections with a 1/1 slope, depending on consistency, harder soil can be excavated with 1/2, 1/3, 1/4 or 1/5 (w/h). You normally create landfillings /for roads, railways, etc.) with 3/2 slopes, which is a very good example how something that is extremely complex expressed in angles, like arctan 2/3, can easily be represented in practice. As well, acqueducts or sewages, railways and roads are usually calculated longitudinally basing on percent (or per thousand) gradient, like 0.1%, 0.5% (h/l). Vertical walls, like the walls built to contain the terrain on the "mountain side" of roads create - if perfectly vertical - the optical effect of appearing "leaning" towards the road, and usually are built with a 5% to 10% "opening" gradient from bottom to top to avoid that effect (and to save concrete), without anyone actually ever calculating the angle at the base.
jacobolus 9y ago
They clearly had a concept of similar triangles, and a concept similar to our notion of “slope” or “tangent” (apparently they actually used the reciprocal of slope, i.e. length per unit depth). However, they may not have had the concept as of ~4000 years ago of “angle measure” per se. You’d have to ask an expert in the history of astronomy when that concept was first developed semi-formally; my understanding is that it was 1000–1500 years later by subsequent Mesopotamian astronomers. Angle measure (another word for circular arclength corresponding to a particular rotation) is actually quite a tricky thing to get right. It’s the logarithm of the rotation (a pure bivector quantity) with the orientation stripped away – if we call our rotation R oriented in the plane of the unit bivector i, it’s θ = log(R)/i. To go from an angle measure back to a “complex number” (scalar + bivector) with which you can do anything useful, you need to put the orientation back in and then exponentiate it: R = exp(iθ). In dimensions higher than 2, you need to use rotation operators via a kind of sandwich product, because composition of rotations is not commutative. This is not an obvious concept, didn’t really get figured out properly until the late 19th century, and still isn’t taught very clearly to students today.