10 ms·

Don’t Fall for Babylonian Trigonometry Hype

216 points by robinhouston 9y ago

anateus 9y ago
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.
chroem- 9y ago
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.
- AnimalMuppet 9y ago
  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.
  - deleted 9y ago
    [deleted]
- Lazare 9y ago
  I disagree. 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.
  - chroem- 9y ago
    >that's a concrete claim with is 1) easily verifiable and 2) wrong I fail to see how it's wrong. If you call Math.sin(x) it will return an iterative numerical approximation of sine. Would you care to elaborate rather than casually dismissing it? > There's so much wrong with that, it's hard to know where to start. Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can't argue against that.
    - AnimalMuppet 9y ago
      OK, as a purely useless concession, I will grant you that base 60 gives you more rational numbers that you can express as terminating decimals. (Uh, not decimal decimals, but... whatever the right term is.) But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Second, this doesn't help you at all if you're taking trig functions of almost all angles, where the result is not rational.
      - chroem- 9y ago
        >But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Fractions are great until you actually need to evaluate them to use their value. Saying that any number can be perfectly represented with fractions is much like saying that any integral calculation can be performed by simply writing a tall S in front of an algebraic expression. You still have to calculate the final result. >Second, this doesn't help you at all if you're taking trig functions of almost all angles, where the result is not rational. That's just it though: we have no closed form way of calculating the values of trigonometric functions. We only have algorithms that describe how to approximate their values with acceptable amounts of error.
        jdmichal 9y ago
        > Fractions are great until you actually need to evaluate them to use their value. What does this even mean? Fractions are already a value, or rather they are a representation of a value. The same as the decimal number 25 is a representation of 2×10¹ + 5×10⁰.
- agjacobson 9y ago
  It's not rational trigonometry. The angles are irrational. In fact, it's useless, and was probably a decorative demonstration of Pythagorean triples.
arjie 9y ago
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.
- smelterdemon 9y ago
  Maybe not science, but the Rosetta Stone?
- orbitingpluto 9y ago
  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.
- mikebenfield 9y ago
  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.
- kqr2 9y ago
  Does the rediscovery of ancient texts that help kick-start the Renaissance count? https://www.livescience.com/55230-renaissance.html https://www.livescience.com/55230-renaissance.html
- roywiggins 9y ago
  Possibly antibiotics: https://theconversation.com/medieval-medical-books-could-hold-the-recipe-for-new-antibiotics-74490 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/ https://www.oldweather.org/ Old supernova observations (okay, maybe not vital scientific information, but): https://en.wikipedia.org/wiki/History_of_supernova_observation 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 https://en.wikipedia.org/wiki/Proper_motion#History
svachalek 9y ago
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).
- noobermin 9y ago
  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.
- jacobolus 9y ago
  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.
  - vog 9y ago
    > doing everything possible in terms of vectors and never using angle measure unless interfacing with humans or legacy software is a generally good idea More generally, use scalar products instead of angles. Calculations based on scalar products are faster, numerically more accurate, simpler, and easier chained. Convert from/to angles only for input/output. Even more generally, always use sane representations internally, and press cumbersome stuff to the borders of your system. Examples: - You should use unicode strings internally, and decode/encode only on I/O. - You should use simple, native data structures internally, and deal with XML representations only at the boundaries. - And so on.
dfboyd 9y ago
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?
- david927 9y ago
  It's his grandfather: https://en.wikipedia.org/wiki/George_Arthur_Plimpton https://en.wikipedia.org/wiki/George_Arthur_Plimpton
- deleted 9y ago
  [deleted]
k_sze 9y ago
What if it's just a page of math homework from Babylonian times? And not a special trig table at all? What if?
- jacobolus 9y ago
  That’s one of the previous interpretations. If you read this paper you’ll find out why these authors think that interpretation is implausible.
tree_of_item 9y ago
> but seems like a solution to a problem that doesn’t exist. I'm pretty sure the vast majority of mathematics could have been described this way when it was first developed. Hilarious.
- HillaryBriss 9y ago
  also ethereum
- _kst_ 9y ago
  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)).
HillaryBriss 9y ago
the title of this post is exactly what my father said to me as a child, every day before i went to school. and do you think i listened? i could kick myself. i really could!
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...
deleted 9y ago
[deleted]
akyu 9y ago
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.
- rocqua 9y ago
  How does he deal with the diagonal of the square? because that needs sqrt(2).
  - jacobolus 9y ago
    He bases his metrical geometry on “quadrance” (squared distance) and “spread” (squared sine) rather than distance and angle measure.
    - gus_massa 9y ago
      Then, how does he deal with the the pentagon? because that needs sqrt(5+sqrt(5)) or something similar with two nested square roots.
      - jacobolus 9y ago
        To deal with pentagons you need to extend the field of rationals by an additional element. Wildberger is kind of ambivalent about finite field extensions.
- yoz-y 9y ago
  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?
Houshalter 9y ago
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.
- coldtea 9y ago
  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.
  - Houshalter 9y ago
    Smart people can be cranks. In fact they are far more likely to be so, perhaps because they overestimate their own ability. See e.g. http://lesswrong.com/lw/j8/the_crackpot_offer/ http://lesswrong.com/lw/j8/the_crackpot_offer/ In any case I didn't call him a crank, but many actual mathematicians do. >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... I might be misremembering this, but I think his argument was just that "computers have tested all possible counterexamples up to some big number, therefore it's true. All this talk of different axiom systems is giving ultrafinitism vastly more credit than it's due. It's not a different but coherent mathematical system. It's a "philosophy" based entirely on refusing to accept a result because it feels counterintuitive. And then coming to vastly more counterintuitive and absurd results as a consequence of trying to resolve it. I mean they really believe that there exists a biggest number and there are no numbers bigger than that. Even regular finitists are kind of absurd, denying the existence of pi or the square root of 2.
IshKebab 9y ago
And stop calling it base 60. It's clearly a mixed base, call it base 6-10. Calling it base 60 makes it sound like they have 60 different digits which would be insane.
- jacobolus 9y ago
  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.)

216 points by robinhouston 9y ago

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.

chroem- 9y ago

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.

AnimalMuppet 9y ago
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.
- deleted 9y ago
  [deleted]
Lazare 9y ago
I disagree. 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.
- chroem- 9y ago
  >that's a concrete claim with is 1) easily verifiable and 2) wrong I fail to see how it's wrong. If you call Math.sin(x) it will return an iterative numerical approximation of sine. Would you care to elaborate rather than casually dismissing it? > There's so much wrong with that, it's hard to know where to start. Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can't argue against that.
  - AnimalMuppet 9y ago
    OK, as a purely useless concession, I will grant you that base 60 gives you more rational numbers that you can express as terminating decimals. (Uh, not decimal decimals, but... whatever the right term is.) But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Second, this doesn't help you at all if you're taking trig functions of almost all angles, where the result is not rational.
    - chroem- 9y ago
      >But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Fractions are great until you actually need to evaluate them to use their value. Saying that any number can be perfectly represented with fractions is much like saying that any integral calculation can be performed by simply writing a tall S in front of an algebraic expression. You still have to calculate the final result. >Second, this doesn't help you at all if you're taking trig functions of almost all angles, where the result is not rational. That's just it though: we have no closed form way of calculating the values of trigonometric functions. We only have algorithms that describe how to approximate their values with acceptable amounts of error.
      - jdmichal 9y ago
        > Fractions are great until you actually need to evaluate them to use their value. What does this even mean? Fractions are already a value, or rather they are a representation of a value. The same as the decimal number 25 is a representation of 2×10¹ + 5×10⁰.
agjacobson 9y ago
It's not rational trigonometry. The angles are irrational. In fact, it's useless, and was probably a decorative demonstration of Pythagorean triples.

arjie 9y ago

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.

smelterdemon 9y ago
Maybe not science, but the Rosetta Stone?
orbitingpluto 9y ago
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.
mikebenfield 9y ago
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.
kqr2 9y ago
Does the rediscovery of ancient texts that help kick-start the Renaissance count? https://www.livescience.com/55230-renaissance.html https://www.livescience.com/55230-renaissance.html
roywiggins 9y ago
Possibly antibiotics: https://theconversation.com/medieval-medical-books-could-hold-the-recipe-for-new-antibiotics-74490 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/ https://www.oldweather.org/ Old supernova observations (okay, maybe not vital scientific information, but): https://en.wikipedia.org/wiki/History_of_supernova_observation 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 https://en.wikipedia.org/wiki/Proper_motion#History

svachalek 9y ago

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).

noobermin 9y ago
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.
jacobolus 9y ago
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.
- vog 9y ago
  > doing everything possible in terms of vectors and never using angle measure unless interfacing with humans or legacy software is a generally good idea More generally, use scalar products instead of angles. Calculations based on scalar products are faster, numerically more accurate, simpler, and easier chained. Convert from/to angles only for input/output. Even more generally, always use sane representations internally, and press cumbersome stuff to the borders of your system. Examples: - You should use unicode strings internally, and decode/encode only on I/O. - You should use simple, native data structures internally, and deal with XML representations only at the boundaries. - And so on.

dfboyd 9y ago

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?

david927 9y ago
It's his grandfather: https://en.wikipedia.org/wiki/George_Arthur_Plimpton https://en.wikipedia.org/wiki/George_Arthur_Plimpton
deleted 9y ago
[deleted]

k_sze 9y ago

What if it's just a page of math homework from Babylonian times? And not a special trig table at all? What if?

jacobolus 9y ago
That’s one of the previous interpretations. If you read this paper you’ll find out why these authors think that interpretation is implausible.

tree_of_item 9y ago

> but seems like a solution to a problem that doesn’t exist. I'm pretty sure the vast majority of mathematics could have been described this way when it was first developed. Hilarious.

HillaryBriss 9y ago
also ethereum
_kst_ 9y ago
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)).

HillaryBriss 9y ago

the title of this post is exactly what my father said to me as a child, every day before i went to school. and do you think i listened? i could kick myself. i really could!

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...

deleted 9y ago

[deleted]

akyu 9y ago

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.

rocqua 9y ago
How does he deal with the diagonal of the square? because that needs sqrt(2).
- jacobolus 9y ago
  He bases his metrical geometry on “quadrance” (squared distance) and “spread” (squared sine) rather than distance and angle measure.
  - gus_massa 9y ago
    Then, how does he deal with the the pentagon? because that needs sqrt(5+sqrt(5)) or something similar with two nested square roots.
    - jacobolus 9y ago
      To deal with pentagons you need to extend the field of rationals by an additional element. Wildberger is kind of ambivalent about finite field extensions.
yoz-y 9y ago
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?

Houshalter 9y ago

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.

coldtea 9y ago
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.
- Houshalter 9y ago
  Smart people can be cranks. In fact they are far more likely to be so, perhaps because they overestimate their own ability. See e.g. http://lesswrong.com/lw/j8/the_crackpot_offer/ http://lesswrong.com/lw/j8/the_crackpot_offer/ In any case I didn't call him a crank, but many actual mathematicians do. >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... I might be misremembering this, but I think his argument was just that "computers have tested all possible counterexamples up to some big number, therefore it's true. All this talk of different axiom systems is giving ultrafinitism vastly more credit than it's due. It's not a different but coherent mathematical system. It's a "philosophy" based entirely on refusing to accept a result because it feels counterintuitive. And then coming to vastly more counterintuitive and absurd results as a consequence of trying to resolve it. I mean they really believe that there exists a biggest number and there are no numbers bigger than that. Even regular finitists are kind of absurd, denying the existence of pi or the square root of 2.

IshKebab 9y ago

And stop calling it base 60. It's clearly a mixed base, call it base 6-10. Calling it base 60 makes it sound like they have 60 different digits which would be insane.

jacobolus 9y ago
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.)