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Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)
- nyc111 1y agoMore detailed video of Plimpton 322 from the authors of the paper https://youtu.be/L24GzTaOll0?si=sNdwKiM7uYXbzVfL https://youtu.be/L24GzTaOll0?si=sNdwKiM7uYXbzVfL
- kragen 1y agoIt's probably worth adding the context that Wildberger's agenda is to ground mathematics in integers and rational numbers, eliminating those pesky irrationals Euclid introduced, because reasoning about them invariably involves infinities or universal quantifiers, which everyone agrees are tricky and error-prone, even if they don't agree with Wildberger's radical variety of finitism. So he was delighted to find a kindred spirit millennia ago in the Plimpton 322 scribe and, presumably, the entire Babylonian mathematical tradition. cf. https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_Trigonometry_to_Universal_Geometry https://en.wikipedia.org/wiki/Divine_Proportions:_Rational_T...
- 7thaccount 1y agoHow do we do things like electrical engineering without imaginary numbers? Is this method an actual improvement?
- michaelsbradley 1y agoimaginary numbers are not the same thing as irrational numbers
- vlovich123 1y agoHow do you rationalize pi or e?
- aarestad 1y agoBy fiat, of course. :) (e.g. https://en.wikipedia.org/wiki/Indiana_pi_bill https://en.wikipedia.org/wiki/Indiana_pi_bill)
- empath75 1y agoYou don't. He basically defines numbers like pi and e not as numbers, but as iterative functions, which you can run to whatever level of accuracy that you want. It's sort of a silly argument, because _all_ numbers can be treated like the output of a function, including the real numbers, so he has basically smuggled in all reals through the back door, because any real number can just be thought of as a function with increasingly precise return values with an infinitely long description, just like pi is.
- MarkusQ 1y agoYou can't get all the reals that way. The reals that can be produced by an algorithm make up a vanishingly small (e.g. countable) subset. Almost all of the reals are inexpressible.
- LPisGood 1y agoTo an ultrafinitist, there is no such thing as a number that is inexpressible.
- kragen 1y agoRight, but to be clear, it's not that ultrafinitists like Wildberger believe that they can express all the real numbers; rather, they believe that those inexpressible real numbers don't actually exist.
- vlovich123 1y agoHow does that work for calculus which regularly looks at the limits of functions as x approaches infinity and has very real real world applications that stem from such algorithms?
- empath75 1y agoHe doesn't work with imaginary numbers, either. He treats complex numbers as matrices of rationals.
- numpy-thagoras 1y agoWhich is the same thing for all intents and purposes. An ultrafinitist is still allowed to call that 'i'.
- dhosek 1y agoStill kind of freaked out that a Möbius transform can be expressed as a matrix multiplication.
- 7thaccount 1y agoI never said they were, but could've sworn that the Wikipedia page or parent comment did (I can't find it now and am questioning my sanity). I couldn't understand how he could try to get rid of them, although this isn't surprising as mathematics is basically magic to me once you get past calculus. I guess this is only about removing irrationals though.
- griffzhowl 1y agoIt depends on the particular construction. You could construct the "complex rational field" by adding i to the rationals with the rule i^2 = -1. That seems to be what Wildberger is ok with. The standard complex numbers involve adding i to the real numbers, which Wildberger doesn't like. I don't know how you'd do electrical engineering with the rational complex field, because electrical engineering and physics in general involves a lot of irrational quantities and calculus, and the standard foundations of these concepts use real numbers. It's really up to finitists to show that there are problems with these methods and that they have a better way of doing things, because so far the standard way seems to work very well.
- fuzzfactor 1y agoElectricity has always been standing by to do the same things regardless of how far your imagination wanders away from where it started.
- clickety_clack 1y agoElectricity is not standing by, it is malevolently trying to burn out your equipment. If you allow your imagination run too far it’ll heat up your equipment and burn it out. You need to increase your capacity to keep your imagination in check.
- numpy-thagoras 1y agoImaginary numbers, quaternions, octonions, Clifford Algebras, etc. can still have finite expressions. After all, the Cayley-Dickson construction is not an infinite affair.
- LeifCarrotson 1y agoThanks for the context - I was baffled at first how the Guardian would run with the tagline "a trignometric table more accurate than any". But it's because the sine of 60 degrees is said by modern tables to be equal to sqrt(3) / 2, which Wildberger doesn't "believe in", he prefers to state that the square of the sine is actually 3 / 4 and that this is "more accurate". The actual paper is at [1]: [1] https://doi.org/10.1016/j.hm.2017.08.001 https://doi.org/10.1016/j.hm.2017.08.001
- margalabargala 1y ago> But it's because the sine of 60 degrees is said by modern tables to be equal to sqrt(3) / 2, which Wildberger doesn't "believe in", he prefers to state that the square of the sine is actually 3 / 4 and that this is "more accurate". Personally I don't believe in either value. I prefer to state that the sine of 60 degrees is 2.7773. I believe that is more accurate.
- kragen 1y agoWell, no, if you look at a trigonometric table, it doesn't say sin 60° = √3/2, because that isn't a useful value for calculation. It'll say something like 0.866025. But that has an error of a little more than 0.0000004. Instead Wildberger prefers saying that the spread (sin²) is ¾, which has no error. It is more accurate. There's no debate about this, except from margalabargala. The news from this paper (thanks for the link!) is that evidently the Babylonians preferred that, too. Surely Pythagoras would have. But how do you actually do anything useful with this ratio ¾? Like, calculating the height of a ziggurat of a given size whose sides are 60° above the horizontal? Well, that one in particular is pretty obvious: it's just the Pythagorean theorem, which lets you do the math precisely, without any error, and then at the end you can approximate a linear result by looking up the square root of the "quadrance" in a table of square roots, which the Babylonians are already known for tabulating. For more elaborate problems, well, Wildberger wrote the book on that. Presumably the Babylonians had books on it too.
- dhosek 1y ago> √3/2 … that isn't a useful value for calculation Some tables do indeed have that value and it is a very useful value for calculation, one that can be symbolically manipulated to get you an exact number (albeit one likely expressed in radicals) for your work. When I used to teach algebra, it was a struggle to get students to let go of the decimal approximations that came out of their calculators and embrace expressions that weren’t simple decimals but were exact representations of the numbers at hand. (Then there’s really fun things like the fact that, e.g., √2 + √3 can also be written as √(5+2√6) (assuming I didn’t make an arithmetic error there)).
- vessenes 1y agoThank you for this expansion. I was about to rabbit hole on how it could be that ratio-based trig (and what is that?) is more accurate than modern calculations. Re: rationals, I mean there's an infinite number of rationals available arbitrarily near any other rational, that has to mean they are good enough for all practical purposes, right?
- kragen 1y agoThat "density" is how Euclid defined the irrational real numbers in terms of the rationals; his definition, cast into modern language by Dedekind, is what we normally use today.
- Someone 1y ago> that has to mean they are good enough for all practical purposes, right? For practical purposes, they’re bad. Denominators tend to explode when you do a few operations (for example 11/123 + 3/17 = 556/2091), and it’s not easy to spot whether you can simplify results. 12/123 + 3/17 = 191/697, for example. You can counteract things by ‘rounding’ to fractions with denominators below a given limit (say 1000) but then, you likely are better of with reckoning with a fixed denominator that you then do not have to store with each number, allowing you to increase the maximal denominator. For example (https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence), there are 965 rational fractions in [0,1] with denominator at most 10 (https://oeis.org/A005728/list https://oeis.org/A005728/list), so storing one requires just under 10 bits. If you use the fractions n/964 for 0 ≤ n ≤ 964 as your representable numbers, arithmetic becomes easier.
- dansmyers 1y agoIf you're interested in ancient math, take a look at Eleanor Robson's accessible paper on the Plimpton 322 tablet: https://scispace.com/pdf/words-and-pictures-new-light-on-plimpton-322-3wekczq7oy.pdf https://scispace.com/pdf/words-and-pictures-new-light-on-pli... Robson's argument is that it isn't a trig table in the modern sense and was probably constructed as a teacher's aide for completing-the-square problems that show up in Babylonian mathematics. Other examples of teaching-related tablets are known to exist. On a quick scan, it looks like the Wildberger paper cites Robson's and accepts the relation to the completing-the-square problem, but argues that the tablet's numbers are too complex to have been practical for teaching.
- curtisszmania 1y ago[dead]
- numpy-thagoras 1y agoAlright, I'll bite: To defend Wildberger a bit (because I am an ultrafinitist) I'd like to state first that Wildberger has poor personal PR ability. Now, as programmers here, you are all natural ultrafinitists as you work with finite quantities (computer systems) and use numerical methods to accurately approximate real numbers. An ultrafinitist says that that's really all there is to it. The extra axiomatic fluff about infinities existing are logically unnecessary to do all the heavy lifting of the math that we are familiar with. Wildberger's point (and the point of all ultrafinitist claims) is that it's an intellectual and pedagogical disservice to teach and speak of, e.g. Real Numbers, as if they're actually involving infinite quantities that you can never fully specify. We are always going to have to confront the numerical methods part, so it's better to make teaching about numbers methodologically aligned with how we actually measure and use them. I have personally been working on building various finite equivalents to familiar math. I recommend anyone to read Radically Elementary Probability Theory by Nelson to get a better sense of how to do finite math, at least at the theoretical level. Once again, on a practical level to do with directly computing quantities, we've only ever done finite math.
- dr_dshiv 1y agoSo what is the length of the diagonal of a unit square, if not square root of 2? It can’t be rational—how is that rationalized by Wildberger?
- numpy-thagoras 1y agoRead Wildberger if you want to know what he thinks. I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation. The whole point is that as a theory for the foundations of mathematics, you do not need to assume numbers with infinitely long decimal expansions in order to do math.
- birn559 1y ago> I can tell you that it is the output of a function, not a distinct entity that exists on its own independently of the computation. Could you elaborate? What is the output of that function if not an entity in it's own? Having studied math with philosophiy minor long time ago I am curious.
- rubycollect4812 1y ago>”He bought it from Edgar Banks, a diplomat, antiquities dealer and flamboyant amateur archaeologist said to have inspired the character of Indiana Jones – his feats included climbing Mount Ararat in an unsuccessful attempt to find Noah’s Ark – who had excavated it in southern Iraq in the early 20th century.” A little off-topic, but as a non native English speaker this sentence in the article made me look up whether there’s scientific consensus that Noah’s Ark has been found and I’d just never heard about it. Turns out there isn’t, and the end of the sentence actually refers to the tablet. Was still a fun rabbit hole to go down.
- 1970-01-01 1y agoTurns out the PIN on the ancient tablet was just 1-2-3-4-5 https://www.cnbc.com/2019/04/10/toddler-locks-ipad-for-48-years-heres-how-to-unlock-it.html https://www.cnbc.com/2019/04/10/toddler-locks-ipad-for-48-ye...
- kbelder 1y agoYou mean I-II-III-IV-V?