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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)

#51

Post by kragen »

dhosek wrote: post #39
kragen wrote: post #17

Earlier quoted context omitted.

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 t…

> √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 expressi…

What do those tables say for 59°59'? I'm skeptical that what you're looking at is, strictly speaking, a trigonometric table.

If you want to know how many courses of bricks your ziggurat is going to need, given that the base is 400 cubits across and there are 10 courses of bricks per cubit, you're going to have to round 2000√3/2 to an integer. You can do that with a table of squares, or you can use a decimal (or sexagesimal) fraction approximation, and I guess you're right that it isn't clear that one is necessarily better than the other.

Incidentally, the fact that we write things like 59°59'30" comes about because the Babylonians at least weren't using Wildberger's "spreads" all the time.

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

#52

Post by vlovich123 »

drdec wrote: post #43
vlovich123 wrote: post #32

Earlier quoted context omitted.

How 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?

Math in general needs to have a big blinking "don't confuse the map for the territory" label on it. E.g. when you calculate the area of a plot of land do you take into account the curvature of the Earth? You have to make a bunch of compromises in the first place to even talk about what the area of a plot land means. Math is a bunch of useful systems that we humans have devised. We tend to gravitate towards the ones t…

I agree. I’m just trying to speak to the “realist” argument that Wilderberg presents claiming that some numbers aren’t real and there’s no point talking about them when they come from very practical mathematics let alone the ones that aren’t. In no way was I trying to claim that some part of maths aren’t real - I was trying to understand the consistency of what to me seems like a confusing argument to make.

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

#53

Post by LPisGood »

vlovich123 wrote: post #32
kragen wrote: post #30

Earlier quoted context omitted.

Right, 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.

How 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?

Here is a paper on just how a serious ultrafinitist copes with that https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/...

The short answer is that they deal with such things symbolically.

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

#54

Post by fuzzfactor »

griffzhowl wrote: post #44
numpy-thagoras wrote: post #21

Alright, 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 abo…

An eventual output of a calculation has to be a finite result, but the concepts that we use to get there are often not. The standard way of setting up calculus involves continous magnitudes, hence irrational quantities, and obviously that's used all over physics and there doesn't seem to be a problem with it. I think to make a compelling case for a finitist foundation for maths you would at the least have to construc…

>An eventual output of a calculation has to be a finite result, but the concepts that we use to get there are often not.

This is so true but it can be good if you're flexible enough to try it either way.

With massive tables of physical properties officially produced by pages of 32-bit Fortran it really did look like floating-point was ideal at first. Because it worked great.

The algorithm had been stored as a direct mathematical equation, plain as day, exactly as deduced with constants and operations in 32-bit double-precision floating point.

But when the only user-owned computers were still just 8-bit machines, there was no way to reproduce the exact results across the entire table to the same number of significant figures, using floating point.

Since it's a table it is of course not infinite, and a matrix to boot. A matrix of real numbers across an entire working spectrum.

The algorithm takes a set of input values, calculates results as defined, and rounds it off repeatably in the subsequent logic before output, so everyone can get agreement. The software OTOH takes a range of input values and outputs a matrix. And/or retains a matrix in "imaginary" spreadsheet form for later use :)

Every single value in the matrix is a floating-point representation of a real number, but they are rounded off as precisely as possible to the "exact" degree of usefulness, making them functionally all finite values in the end. This took a lot of work from top mathematicians, computer scientists, and engineers. And as designed, the matrix then carries the algorithm on its own without reference to the fundamental equation.

The solution turned out to involve working backward from the matrix reiteratively until an alternate algorithm was found using only integers for values and operations, up until the final rounding and fixed.point representation at the end. Dramatically unrecognizable algorithm but it worked and only took 0.5 kilobytes of 8-bit Basic code which was a fraction of the original Fortran.

This time the feature that showed up without having to make extra effort was the property of being more precise based directly on increased bitness of the computer, without need for floating-point at all. Of course the Fortran code accomplished this too by the wise use of floating-point but it took a lot bigger iron to do so. And wasn't going to be battery powered any time soon way back then.

>somehwere this finitist foundation disagrees with the results obtained by the standard foundation,

>there's no reason to think the standard foundation is in error.

This is "exactly" how it was. There were disagreements all over the place but they were in further decimal places not representable by the table. The standard was an international standard having carefully agreed-upon accuracy & precision, as defined by the Fortran which really worked and was then written in stone, with any nonmatched output being a notable failure.

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

#55

Post by adrian_b »

numpy-thagoras wrote: post #21

Alright, 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 abo…

I wonder what ultrafinitists do about topology.

Topology, i.e. the analysis of connectivity, is built upon the notion of continuity and infinite divisibility, which seems to be difficult to handle in an ultrafinitist way.

Topology is an exceedingly important branch of mathematics, not only theoretically (I consider some of the results of topology as very beautiful) but also practically, as a great part of the engineering design work is for solving problems where only the topology matters, not the geometry, e.g. in electronic schematics design work.

So I would consider any framework for mathematics that does not handle well topology as incomplete and unusable.

Ultrafinitist theories may be interesting to study as an alternative, but the reality is that infinitesimal calculus in its modern rigorous form does not need any alternatives, because it works well enough and until now I have not seen alternatives that are simpler, but only alternatives that are more complicated, without benefits sufficient to justify that.

I also wonder what ultrafinitists do about projective geometry and inversive geometry.

I consider projective geometry as one of the most beautiful parts of mathematics. When I encountered it for the first time when very young, it was quite a revelation, due to the unification that it allows for various concepts that are distinct in classic geometry. The projective geometry is based on completing the affine spaces with various kinds of subspaces located at an "infinite" distance.

Without handling infinities, and without visualizing how various curves located at infinity look like (as parts of surfaces that can be seen at finite distances), projective geometry would become very hard to understand, even if one would duplicate its algorithms while avoiding the names related to "infinity".

Similarly for inversive geometry, where the affine spaces are completed with points located at "inifinity".

Such geometries are beautiful and very useful, so I would not consider as usable a variant of mathematics where they are not included.

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

#56

Post by adrian_b »

drdec wrote: post #45
dr_dshiv wrote: post #29

Earlier quoted context omitted.

So 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?

If you do not accept that space is infinitely divisible, then the diagonal of a unit square does not actually exist in the space.

A long time has passed since the paradoxes of Zeno of Elea, so now there really is no reason for not accepting that space is infinitely divisible.

The error of Zeno of Elea was that he did not understand the symmetry between zero and infinity (or he pretended to not understand it).

Because of this error, Zeno considered that infinity is stronger than zero, so he believed or pretended to believe that zero times infinity is infinity, instead of recognizing that zero times infinity can be any number and also zero or infinity.

For now, there exists no evidence whatsoever that the physical space and time are not infinitely divisible.

Even if in the future it would be discovered that space and time have a discrete structure, the mathematical model of an infinitely divisible space and time would remain useful as an approximation, because it certainly is simpler than whatever mathematical model would be needed for a discrete space and time.

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

#57

Post by griffzhowl »

alexey-salmin wrote: post #50
griffzhowl wrote: post #44

Earlier quoted context omitted.

An eventual output of a calculation has to be a finite result, but the concepts that we use to get there are often not. The standard way of setting up calculus involves continous magnitudes, hence irrational quantities, and obviously that's used all over physics and there doesn't seem to be a problem with it. I think to make a compelling case for a finitist foundation for maths you would at the least have to construc…

> Even if you did that, you should show somehwere this finitist foundation disagrees with the results obtained by the standard foundation, otherwise there's no reason to think the standard foundation is in error. Well these are probably easy to find even now? E.g the Banach-Tarsky paradox is unlikely to be provable in finitist math which is somewhat of an improvement.

I was thinking more about applications in physics where calculus and irrational quantities are used all the time.

At more advanced levels the theories are based on differential geometry and operators on Hilbert space. I'm not sure if fully worked out finitist versions of these even exist. Where finitist versions do exist, they're often technically more difficult to use than the standard versions, which is the opposite of an improvement in my view.

Whether it's undesirable for your mathematical foundation to prove the Banach-Tarski paradox is debatable. It's counter-intuitive, but doesn't lead to contradictions, as far as is known. It doesn't apply to physics because the construction uses non-measurable sets.

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

#58

Post by kragen »

vlovich123 wrote: post #52
drdec wrote: post #43

Earlier quoted context omitted.

Math in general needs to have a big blinking "don't confuse the map for the territory" label on it. E.g. when you calculate the area of a plot of land do you take into account the curvature of the Earth? You have to make a bunch of compromises in the first place to even talk about what the area of a plot land means. Math is a bunch of useful systems that we humans have devised. We tend to gravitate towards the ones t…

I agree. I’m just trying to speak to the “realist” argument that Wilderberg presents claiming that some numbers aren’t real and there’s no point talking about them when they come from very practical mathematics let alone the ones that aren’t. In no way was I trying to claim that some part of maths aren’t real - I was trying to understand the consistency of what to me seems like a confusing argument to make.

I don't think uncomputable numbers "come from very practical mathematics"! Rather, they come from Gödel, Church, and Turing demolishing Hilbert's program of solving the Entscheidungsproblem once and for all. Possibly, if Hilbert had succeeded, that would have made it "very practical mathematics", or possibly not, but that counterfactual is reasoning from a logical contradiction.

https://plato.stanford.edu/entries/church-turing/decision-pr...

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

#59

Post by dhosek »

kragen wrote: post #51
dhosek wrote: post #39

Earlier quoted context omitted.

> √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 expressi…

What do those tables say for 59°59'? I'm skeptical that what you're looking at is, strictly speaking, a trigonometric table. If you want to know how many courses of bricks your ziggurat is going to need, given that the base is 400 cubits across and there are 10 courses of bricks per cubit, you're going to have to round 2000√3/2 to an integer. You can do that with a table of squares, or you can use a decimal (or sexag…

Those tables don’t give a value for that. They only give values for angles whose trigonometric values can be expressed in terms of rational expressions with radicals (and angles as rational expressions in terms of π).

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

#60

Post by kragen »

dhosek wrote: post #59
kragen wrote: post #51

Earlier quoted context omitted.

What do those tables say for 59°59'? I'm skeptical that what you're looking at is, strictly speaking, a trigonometric table. If you want to know how many courses of bricks your ziggurat is going to need, given that the base is 400 cubits across and there are 10 courses of bricks per cubit, you're going to have to round 2000√3/2 to an integer. You can do that with a table of squares, or you can use a decimal (or sexag…

Those tables don’t give a value for that. They only give values for angles whose trigonometric values can be expressed in terms of rational expressions with radicals (and angles as rational expressions in terms of π).

That sounds like a sort of "trigonometric table" you couldn't use in practice for calculation. You need to be able to look up whatever value you measure with your sextant or theodolite to within its measurement precision. You seem to be using the term "trigonometric table" in a way conflicting with the standard use explained in https://en.wikipedia.org/wiki/Trigonometric_tables perhaps as a sort of pun or joke, or perhaps as a sort of act of activism against hegemonic notions of trigonometry.
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