I did a little digging in curiosity about the Plimpton 322 story, and discovered that essentially Wildberger was already a proponent of rational trigonometry and then figured out that this tablet was based on it, which is the opposite order the news stories led me to believe.
But anyway the aspect of it that has me curious is whether it benefits 3D graphics or anything else that depends on lots of triangle computations. I found a paper that concludes it's more easily computed, but it seems to make some pretty simplistic assumptions (cost of sqrt = cost of sin, all algebraic costs = 0).
Using base 60 makes no sense for computer graphics.
However, doing everything possible in terms of vectors and never using angle measure unless interfacing with humans or legacy software is a generally good idea.
If you want to represent a rotation use a complex number of unit magnitude. If you want to represent a 3-dimensional rotation use a unit quaternion. If you want to store one of these using only 1 or 3 parameters, respectively, take the stereographic projection and optionally reduce the precision of your floating point numbers afterward.
This doesn’t imply turning everything into Wildberger’s rational trigonometry per se, but if you need metrical comparisons then you end up developing/using some of the same machinery.
How curious! I knew about rational trigonometry, and I was thinking the modern proponent of that idea would be so mad others are capitalizing on it without citing him...it so turns out he was a co-author, fancy that.
This is a great article touching on the several ways in which this new Plimpton 322 hype is overblown. There's usually a paper like that comes out every few years :)
If you're interested in learning more about Babylonian math-history I highly recommend Jens Høyrup's Lengths, Widths, Surfaces.
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=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.)
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/S0315086017...
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.
Wildberger is a recurring subject on /r/badmathematics including this recent story. He's almost a crank and has some very very strange views on mathematics. He strongly objects to the use of infinity and proofs and mathematics that involve it, which includes real numbers.
If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.
The kind of crank with a PhD from Yale that has taught at Toronto uni, Stanford etc?
Or is it impossible to have contrarian views to basic tenets of a human-made axiomatic system such as math without being called a blasphemer?
>If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.
Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms, and does that correctly, that's not much different than e.g. non euclidean geometries.
In mathematics education, I always disliked the way inifinity is shoehorned into the different sets of numbers and treated as "just another number", so that you can write shorthands like: 1/∞=0, instead of the more accurate ∀ε>0, ∃c>0: x>c ⇒ 1/x<ε
Infinity is not a number. It's a limit.
Treating it as a number only confuses real understanding of it.
This article seems to boil down to "stop liking what I don't like." Is Wildberger's distaste for real numbers eccentric? Yes. Is there anything fundamentally wrong with what he is proposing? Absolutely not.
As long as rational trigonometry is logically consistent then you really can't argue against it except on aesthetic grounds. I don't see why popsci outlets like Scientific American should feel the need to rally the public against rational trigonometry on emotional grounds.
Mansfeld and Wildberger made a number of trivially provable false claims. I read the SciAm article as 1) focusing on their false claims and 2) not trying to "rally the public" against rational trigonometry at all, much less on emotional grounds.
In particular, Mansfeld and Wildberger (especially as reported in other outlets) are making the claim that "rational trigonometry" is more accurate, and that's a concrete claim with is 1) easily verifiable and 2) wrong.
And that's not even touching on the bizarre claim that "we count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5." There's so much wrong with that, it's hard to know where to start.
I didn't read it that way. They are rallying against the claim that rational trigonometry is "superior" or "more accurate", and not on emotional grounds.
Glad to see Wildberger getting some recognition, even if some of it is mixed. His rational trigonometry is an entirely rigorous and interesting piece of mathematics even if you don't think it's particularly pragmatic.
This is my first time being exposed to this but I wonder. Is the rational trigonometry useful? As in, are there applications in which it would make a tangible impact, for example accelerating algorithms or making learning easier?
Has there ever been a "ancient tablet unravels secrets undiscovered" that contributed to science and not history? I think everyone can safely eliminate this sort of thing in a Bayesian sense.
This isn't quite the same thing, but FWIW, during Renaissance times, scholars would rediscover ancient Greek works and legitimately learn "new" mathematics from them. See for instance Diophantus's Arithmetica, which was translated from Greek to Latin and thus rediscovered by the west.
I had to go through a Babylonian secant table and find the error as an exercise. The actual point of the exercise was in determining how the error in calcuation was made.
Figuring out how science was done is still science, even if all you're doing is figuring out how to do the exact same thing but in a different way.
The story says George Plimpton bought the tablet in 1922. George Plimpton (I assume it's the same one) was born in 1927. I admit he was awesome enough that Jonathan Coulton wrote a song about him, but could he buy a tablet five years before he was born?
They clearly treated it as a base sixty floating point system, even though they wrote each “sexagesimal digit” using collections of two symbols one of which represented 10 times the other. The symbol for 10 was only ever used to represent 10*60^n. Tens were never themselves the unit.
(Though the abstract sexagesimal system evolved out of earlier concrete systems with less standardized mixtures of physical units.)
Perhaps so, but in this case the solution is one that's already well known (that you can express trig functions exactly when you're dealing with right triangles with integer sides).
If you want to do some computation involving a given angle, you're not likely to have the luxury of picking the angle that makes the computation easy using rational arithmetic. The sine of 45° is an irrational number (1/sqrt(2)).
[+] [-] svachalek|8 years ago|reply
But anyway the aspect of it that has me curious is whether it benefits 3D graphics or anything else that depends on lots of triangle computations. I found a paper that concludes it's more easily computed, but it seems to make some pretty simplistic assumptions (cost of sqrt = cost of sin, all algebraic costs = 0).
[+] [-] jacobolus|8 years ago|reply
However, doing everything possible in terms of vectors and never using angle measure unless interfacing with humans or legacy software is a generally good idea.
If you want to represent a rotation use a complex number of unit magnitude. If you want to represent a 3-dimensional rotation use a unit quaternion. If you want to store one of these using only 1 or 3 parameters, respectively, take the stereographic projection and optionally reduce the precision of your floating point numbers afterward.
This doesn’t imply turning everything into Wildberger’s rational trigonometry per se, but if you need metrical comparisons then you end up developing/using some of the same machinery.
[+] [-] noobermin|8 years ago|reply
[+] [-] anateus|8 years ago|reply
If you're interested in learning more about Babylonian math-history I highly recommend Jens Høyrup's Lengths, Widths, Surfaces.
[+] [-] svat|8 years ago|reply
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=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/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/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/S0315086017...
[+] [-] thejynxed|8 years ago|reply
[+] [-] Houshalter|8 years ago|reply
If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.
[+] [-] szemet|8 years ago|reply
He is not alone, this is a known branch of mathematics called finitism: https://en.wikipedia.org/wiki/Finitism
[+] [-] coldtea|8 years ago|reply
Or is it impossible to have contrarian views to basic tenets of a human-made axiomatic system such as math without being called a blasphemer?
>If I understand correctly, he goes even farther and claims even really big but finite numbers don't "exist". And through this he claims to have "resolved" the Goldbach conjecture and other strange things.
Does he just "claim" or does he give proof? Because if he has the proper system, and he proves his theorems based on his axioms, and does that correctly, that's not much different than e.g. non euclidean geometries.
[+] [-] bergoid|8 years ago|reply
Infinity is not a number. It's a limit.
Treating it as a number only confuses real understanding of it.
[+] [-] chroem-|8 years ago|reply
As long as rational trigonometry is logically consistent then you really can't argue against it except on aesthetic grounds. I don't see why popsci outlets like Scientific American should feel the need to rally the public against rational trigonometry on emotional grounds.
[+] [-] Lazare|8 years ago|reply
Mansfeld and Wildberger made a number of trivially provable false claims. I read the SciAm article as 1) focusing on their false claims and 2) not trying to "rally the public" against rational trigonometry at all, much less on emotional grounds.
In particular, Mansfeld and Wildberger (especially as reported in other outlets) are making the claim that "rational trigonometry" is more accurate, and that's a concrete claim with is 1) easily verifiable and 2) wrong.
And that's not even touching on the bizarre claim that "we count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5." There's so much wrong with that, it's hard to know where to start.
[+] [-] AnimalMuppet|8 years ago|reply
[+] [-] agjacobson|8 years ago|reply
[+] [-] akyu|8 years ago|reply
[+] [-] yoz-y|8 years ago|reply
[+] [-] rocqua|8 years ago|reply
[+] [-] arjie|8 years ago|reply
[+] [-] leoc|8 years ago|reply
[+] [-] roywiggins|8 years ago|reply
https://theconversation.com/medieval-medical-books-could-hol...
Not quite so old, but 19th Century ship's logs to learn about weather and climate:
https://www.oldweather.org/
Old supernova observations (okay, maybe not vital scientific information, but):
https://en.wikipedia.org/wiki/History_of_supernova_observati...
Quite old: Edmund Halley used Greek star charts to show that stars move in the sky (relative to Earth). Just very, very slowly.
https://en.wikipedia.org/wiki/Proper_motion#History
[+] [-] mikebenfield|8 years ago|reply
[+] [-] orbitingpluto|8 years ago|reply
Figuring out how science was done is still science, even if all you're doing is figuring out how to do the exact same thing but in a different way.
[+] [-] kqr2|8 years ago|reply
https://www.livescience.com/55230-renaissance.html
[+] [-] smelterdemon|8 years ago|reply
[+] [-] dfboyd|8 years ago|reply
[+] [-] david927|8 years ago|reply
https://en.wikipedia.org/wiki/George_Arthur_Plimpton
[+] [-] unknown|8 years ago|reply
[deleted]
[+] [-] IshKebab|8 years ago|reply
Calling it base 60 makes it sound like they have 60 different digits which would be insane.
[+] [-] jacobolus|8 years ago|reply
(Though the abstract sexagesimal system evolved out of earlier concrete systems with less standardized mixtures of physical units.)
[+] [-] unknown|8 years ago|reply
[deleted]
[+] [-] k_sze|8 years ago|reply
What if?
[+] [-] jacobolus|8 years ago|reply
[+] [-] HillaryBriss|8 years ago|reply
and do you think i listened? i could kick myself. i really could!
[+] [-] tree_of_item|8 years ago|reply
I'm pretty sure the vast majority of mathematics could have been described this way when it was first developed. Hilarious.
[+] [-] _kst_|8 years ago|reply
If you want to do some computation involving a given angle, you're not likely to have the luxury of picking the angle that makes the computation easy using rational arithmetic. The sine of 45° is an irrational number (1/sqrt(2)).
[+] [-] HillaryBriss|8 years ago|reply