Don’t Fall for Babylonian Trigonometry Hype
blogs.scientificamerican.com
blogs.scientificamerican.com
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).
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.
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.
If you're interested in learning more about Babylonian math-history I highly recommend Jens Høyrup's Lengths, Widths, Surfaces.
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...
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.
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.
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.
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.
>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.
A conceptual circle and a real circle differ markedly because of the lumpiness of matter, quantisation, uncertainty. Every real circle is an approximation (yay Plato!).
Pi doesn't appear to me to be real, as in writ in matter/energy -- it's a ratio of measurements of an imaginary article, circles don't exist in our experiential 4-space.
Irrational numbers like root-of-2 just fall out of basic maths though, there's not going to be a real line with that measure just as there's no line with a predefined unit measure (ie unless you define that line at that point in time as your unit, but a real line isn't even straight).
I'd be really interested in refutation of this -- it's one of those positions I've held since Uni, at least, but never explored as I've never seen it challenged/supported nor named.
Infinity, axiom of choice, .. are all tools. We use them because their presence gives us other helpful results. None of it has anything to do with "lumpiness of matter", "matter/energy", "experiental 4-space", "real line".
Refutation as in "you overlooked this standard point that undermines your position", like maybe "polarised light describes a perfect circle", or "the gravitational field of a black-hole (when shaved) is a sphere".
AFAIK there's nothing truly infinite, nor perfectly geometric, that we can experience in our physical spacetime -- I'd like to hear from those who can show me that's wrong.
The world itself isn't axiomatic, if certain theories/systems that are unfalsified show parts of it are infinite/etc. -- despite that just being a model -- that's an interesting result.
No, but there's no requirement that the useful/only way to do math is with "platonic" / ideal math objects either.
But even if we knew for sure it was, so what? Theorems about real numbers apply perfectly well to finite approximations. You don't gain anything by restricting yourself to finite math, and you lose quite a bit and make a lot of things much harder.
Time and space are effectively discretised, Planck scale and time. Even aside from that: for considerations of physical measure we always use something with an explicit size, a wavelength of light say, as our rod which limits our precision.
For example, a proton has three real quarks. But the mass of the three quarks is only the 1% of the mass of the proton https://en.wikipedia.org/wiki/Quark#Mass the rest is crap, or in more technical terms https://physics.stackexchange.com/a/81284 . When you measure with more energy, you see more and more virtual particles, infinite virtual particles.
At low energy the effect of the virtual particles is negligible (except in a very few weird cases like the https://en.wikipedia.org/wiki/Lamb_shift ). At high energy the probability of the virtual particles is bigger, so their effect is easier to measure.
I like this drawing of that shows the firs three steps of diagrams with virtual particles that are necessary to calculate the magnetic moment of the electron: http://www.strings.ph.qmul.ac.uk/~bigdraw/feynman/slide3.htm...
When we say "a human hand has 5 fingers", we use the abstract concept of "finger" to make that description. It is abstract because each "finger" is actually unique. It's just a handy approximation we use (pun intended) for descriptive goals.
Basic maths is not reality. Therefore no, irrational numbers do not "fall out of it".
Edit: format
Circles appear in the shape of that same s orbital, though, so I don't see how to avoid pi.
The parent argument works here as much as for fingers though, those 2 electrons are identical but their spin differs. All electrons differ in the Standard Model by either spin or location.
Even without classification we can create arbitrary groupings of physical objects, those groups have cardinalities. (Unless your materialism denies the existence of more than one thing/substance; philosophy is fun, eh.)
I'm for pragmatism in such things however.
So why describe reality with an ideal and infinities and not based on fixed/arbitrary precision but limited math?
He is unorthadox in his pushing of rational numbers, sure, but he gets through an extremely large amount of maths correctly. It is well within his job domain reflect on odd academic questions about what happens outside the world of human measurement.
He is not being called a blasphemer, he is being dismissed with reasons. There is a difference.
When everybody else believed that euclidean geometry was the only one possible, someone that insisted otherwise would also be labelled a crank (and he would be right in calling others wrong).
I am sure you understand that being labeled a crank is not evidence of actually having greater insight. Instead of arguing from a dubious analogy of uncertain applicability, you would be much more convincing if you first took the specific objections to Wildberger's position from the article, and refuted them.
He is not alone, this is a known branch of mathematics called finitism: https://en.wikipedia.org/wiki/Finitism
I am certainly happy to be corrected, but I don’t think we have found any event in nature that requires infinities to explain it.
1 = 0.999... 1/2 = 0.49999...
etc.
In some ways it's a property of the representation, not something intrinsic to the number.
Ultimately it comes down to a trade off. Which axiomatic system does one to work in ? Introducing infinity does have its fair share of warts (by warts I mean consequences that flies in the face of intuition).
You can truncate the series at an appropriate place of your choosing. But yes, \Pi as a number does not even exist unless you add the Reals, so that would need infinity. What I am saying is, one can work with the Rationals.
Infinity is not a number. It's a limit.
Treating it as a number only confuses real understanding of it.
The quarrel Wildberger has is not with the proportion 1:0, but with the concept of a limit.
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 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.
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.
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.
That evaluates exactly to 25. Now try to pour me a glass containing exactly 1/7 gallons of milk without making an approximation.
* Hell, wait, what am I going to do with 24 6/7 gallons of milk?
I assume your "cleverness" is in the assertion that I cannot "prove" 1/7 of a gallon using existent tools because such tools would report the result in a base-10 decimal, which would be necessity have to be approximated because 1/7 is non-terminating in base-10.
Of course, I could instead prove it by weight. Or by combining 7 such equal portions and resulting in a single gallon. The latter being an example of exactly why fractional representation of rational numbers is useful.
EDIT: As an aside, by any chance are you one of the set of people who argue that .99999999... repeating infinitely is not the same value as 1?
EDIT2: This just reminded me a lot of the barometer question, as I think of various ways I could demonstrate 1/7 a gallon. Of course, some things are off the table due to being a heterogeneous substance at the molecular level. Some additional options would be on the table if it were homogeneous, like water.
This is an exceptionally silly argument. Now, I happen to live somewhere where the metric system in use, so allow me to translate your challenge. So you're saying that it's harder to pour 540.773 ml of milk (ie, 1/7th of a gallon) than 757.082 ml of milk (ie, 1/5th of a gallon)? (Notice how both of those are numerical approximations? How strange...)
There's nothing harder or easier about pouring 757.082 ml compared to 540.773 ml, nor do we have some magical pouring tools which allow one number to be poured with greater or lesser accuracy than the other. Nor does it suddenly become possible to pour it with more accuracy if you use a measuring cup with ounces on it. :)
You seem to be under the hilarious misapprehension that it's somehow easier to pour something if you happen to be measuring it in units that happen to have a non-repeating decimal representation in whatever base you happen to be working in. That's not really how reality works.
(And I'm not even going to start on what "exact" means when dealing with a non-molecularly homogenous fluid.)
Real world examples are not going to help your argument here, because the real world can only be interacted with via approximations. 1/7th of a gallon of milk is every bit as much of an approximation as 25 gallons.
(Yes, what helps is having a measuring device calibrated with markings for the amount you're trying to pour. But I have several measuring cups that have markings for 1/3 cup, even though there is no exact base-10 representation of that fraction.)
Right. Now what's being argued here is that instead of measuring an angle X, calculating the sine of X, and then (probably) rounding the result to a convenient decimal approximation, we should measure an angle X, round it to an angle Y with a convenient non-repeating decimal representation, and calculate the sine of Y. (That's the whole point of Mansfield's and Wildberger's analysis of the tablet; that it was a trig table that only gave answers for angles with convenient answers, rather than, as modern trig tables do, giving it for all angles.)
So instead of measuring an angle as 29.88 degrees, calculating the sine of 29.88 as 0.49818510533, then rounding to 0.4982 or whatever, we should instead calculate the sine of 30 degrees, and get the wonderful totally accurate "exact fraction" 1/2.
1/2 is totally accurate representation of the sine of 30 in a way that 0.4982 is not a totally accurate representation of the sine of 29.88, yes... But 0.4982 is a much more accurate representation of the angle you've actually measured.
From an engineering point of view (which offhand seems like the only one where accuracy matters), normal trig is obviously, obviously better than rational trig, because normal trig is more accurate in the ways that matter. The more sigfigs I have in my measurement of my angle, the more decimal places I can leave in my numerical approximation of the answer. It may always be inexact in a way that 1/2 is not, but it'll also always be exact in a much more important way. :)
> Such as? 60 has more prime factorizations than 10
So it does, but that's not the claim being made. Rather, the claim was: "We count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5."
To which one might note that:
1. All fractions are exact, by definition. 1/3 is an exact fraction; it's exactly 1/3, in exactly the way 1/2 is exactly 1/2. The fact it's decimal representation is repeating does not impact this.
2. Mansfeld apparently means "fractions with terminating decimal representations" when he says "exact", which is a weird and non-standard usage. But even so, you can construct tons of fractions in base 10 that have non-repeating decimal representations, like 2/5ths, or 5/8th.
3. Possibly he means "...which only allows you to construct two fractions using a numerator of 1 and a denominator consisting of a number smaller than the size of the base which has non-repeating decimal representations..." (which would be a pretty weird thing to focus on) it's still wrong; it's missing, eg, 1/4th.
4. Finally, he reckons that in base 60, 7/60 + 30/3600 aka 1/8 aka 0.125 is an "exact fraction". That sort of factorization is legitimate, but obviously if that's an exact fraction so is 2/10 + 5/100 aka 1/4th. Or come to that, 6/10 aka 0.6.
In short, the claim is gibberish. It's using a phrase with no clear definition in a way which is internally inconsistent. There is no way that 1/8th is an "exact fraction" in base 60, but only 1/2 and 1/5 are "exact fractions" in base 10.
(Also, you're using the word "irrational" wrong.)
Let's start with an apparent nitpick. You say "rather than, as modern trig tables do, giving it for all angles". But no table gives all angles, it's always sample. For modern tables, the samples tend to be regularly spaced, while the Babylonians apparently chose to sample the rational points. Which is better likely depends on the use case.
It's not obvious that "normal trig" is better from an engineering point of view. At least not for all engineers. If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many. That is, it's fairly easy to make rational ratios between the opposite. It's then handy to know what mathematical properties the resulting angle has. Or else to be able to look at your table and pick ratios that have some useful property. You could not do this with a "normal" trig table, or even a modern a hand calculator.
That's true, and my wording was poor. But during the heyday of trig tables, they were very thorough, giving you results for angles (typically) very, very close to whatever you were trying to calculate, and to usefully high number of decimal places. (I'm old enough - just - that I actually used them in school.)
But more to the point, we don't use trig tables any more, we calculate the number on the fly to any arbitrary desired degree of accuracy. That's always going to be the best choice; there's no tradeoff here.
> If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many.
Also quite true, but of course, no temple wall is exactly N cubits high; the measured height is at best an approximation, so the resulting ratio of height to width (or whatever) is at best an approximation. All we need from our trig tables (or calculator) is enough accuracy. And in practice, that's not hard.
The reality is that, as the linked article says, "it seems like a solution to a problem that doesn’t exist". When we actually used trig tables to build bridges, we didn't have issues with trig tables, and now we have a much better solution. Trig tables that, as you put it, sample the rational points just doesn't seem like it's better for any use case.
If you happen to need an angle measure for whatever reason (e.g. interacting with legacy systems) you can then compute it to however many binary digits are appropriate for your needs.
Actually, I can argue with that. Irrationality is a property of the number, not the representation. An irrational number cannot be written as a terminating "decimal" in any base. And, of course, a rational number can be written as an integer fraction in any base.
https://math.stackexchange.com/a/625481
EDIT: Ignoring, of course, doing silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory. Also, I wonder if every number except pi and its multiples are non-terminating in base-pi...
Hmmm...
Anybody know any resources related to that? I'm interested in the properties of base-pi.
I found this: https://math.stackexchange.com/a/2188852
Strange bases are pretty cool. Any integer can be converted into base-sqrt(2) by interleaving zeoes into its binary representation. E.g. 5 in base-sqrt(2) is 010001, 21 (0b10101) is 0b0100010001, etc. Possibly useless, but neat.
Also: https://www.quora.com/Can-we-have-a-number-system-with-the-b...
Knuth considers transcendental bases (ee is a transcendental number) in The Art of Computer Programming – in the section called Positional Number Systems. One of the consequences of using irrational bases is unfortunately that numbers need not have a unique representation.
EDIT: Wow, this is awesome: https://www.quora.com/What-is-10-in-base-pi
Specifically the answer from Dave Williamson. It's ridiculously comprehensive. He shows how to calculate it using pure math, using a calculator, by writing a program, and demonstrates a partial proof on some of the properties.
(Surprisingly, there seems to be less content on this on mathworld)
I do not see how it is easily verifiable. Rational trigonometry, as I understand, does not use angles, so it is not clear for me how to use it in a lot of cases. I belive there are ways to do it, but it needs some research or reading Wildberger... So, its not seems for me as "easily".
> 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.
To my mind, it is just imprecise wording from Wildberger side. Or maybe it precise in his rational way of thinking, but we, accepting irrational numbers, see it imprecise.
I belive, that he meant to say, that 1/2 and 1/5 is some kind of "base" fractions and all fractions that can be written presice is `(1/2)^m*(1/5)^n`, where m and n are non-negative intergers. And to write precise is means to write in decimal representation. I cannot see any other explanation of Wildberger words which would not imply that Wildberger just plain stupid, which is improbable.
Really, I like the idea: we know how to solve a lot problems with irrational numbers, but does it possible to solve them without irrational? How far mathematics can go without irrational numbers?
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.
https://theconversation.com/medieval-medical-books-could-hol...
Not quite so old, but 19th Century ship's logs to learn about weather and climate:
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.
Calling it base 60 makes it sound like they have 60 different digits which would be insane.
(Though the abstract sexagesimal system evolved out of earlier concrete systems with less standardized mixtures of physical units.)
What if?
and do you think i listened? i could kick myself. i really could!
I'm pretty sure the vast majority of mathematics could have been described this way when it was first developed. Hilarious.
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)).