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Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

theguardian.com

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Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#11

Post by margalabargala »

LeifCarrotson wrote: post #6
kragen wrote: post #3

It'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 kind…

Thanks 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/1…

> 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.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#12

Post by empath75 »

vlovich123 wrote: post #8
michaelsbradley wrote: post #5

Earlier quoted context omitted.

imaginary numbers are not the same thing as irrational numbers

How do you rationalize pi or e?

You 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.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#13

Post by fuzzfactor »

7thaccount wrote: post #4
kragen wrote: post #3

It'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 kind…

How do we do things like electrical engineering without imaginary numbers? Is this method an actual improvement?

Electricity has always been standing by to do the same things regardless of how far your imagination wanders away from where it started.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#14

Post by MarkusQ »

empath75 wrote: post #12
vlovich123 wrote: post #8

Earlier quoted context omitted.

How do you rationalize pi or e?

You 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 i…

You 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.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#15

Post by dansmyers »

If 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-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.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#17

Post by kragen »

LeifCarrotson wrote: post #6
kragen wrote: post #3

It'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 kind…

Thanks 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/1…

Well, 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.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#18

Post by kragen »

vessenes wrote: post #7
kragen wrote: post #3

It'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 kind…

Thank 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?

That "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.

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#19

Post by amai »

LeifCarrotson wrote: post #6
kragen wrote: post #3

It'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 kind…

Thanks 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/1…

What does Wildberger then think about i = sqrt(-1)? Is this also "not accurate" enough?

Re: Mathematical secrets of ancient tablet unlocked after nearly a century of study (2017)

#20

Post by clickety_clack »

fuzzfactor wrote: post #13
7thaccount wrote: post #4

Earlier quoted context omitted.

How do we do things like electrical engineering without imaginary numbers? Is this method an actual improvement?

Electricity has always been standing by to do the same things regardless of how far your imagination wanders away from where it started.

Electricity 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.
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