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Comment by kragen | Hacker News Reader

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 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_T...

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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/10.1016/j.hm.2017.08.001

> 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.

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.

> √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)).

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.

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.

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

Ultrafinitism does not rule out higher algebraic structures

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.

> 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), there are 965 rational fractions in [0,1] with denominator at most 10 (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.

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

imaginary numbers are not the same thing as irrational numbers

How do you rationalize pi or e?

By fiat, of course. :) (e.g. https://en.wikipedia.org/wiki/Indiana_pi_bill)

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.

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.

To an ultrafinitist, there is no such thing as a number that is inexpressible.

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?

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 that help us describe and predict things in the real world.

But there is plenty of math which doesn't do either. It's just as real as the math that does.

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...

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.

What I described isn't really an algorithm, it's just taking the digits of a number, let's say:

foo=3.14159265...

Where after 5 is some continuing sequence of decimals.

The series of functions is literally just:

foo(0) = 3 foo(1) = 3.1 foo(2) = 3.14...

And to be clear, it's not just like, an algorithm that estimates pi, it's literally just a list of return values that is infinitely long that return more and more digits of whatever the number is. That is actually how he defines pi.

https://youtu.be/lcIbCZR0HbU?si=3YxcHfPlCFrlr5h3&t=2080

pi _happens_ to be computable, and there are more efficient functions that will produce those numbers, but you could do the same thing with an incomputable number, you just need a definition for the number which is infinitely long.

To be clear, I don't think any of this is a good idea, just pointing out that if he's going to allow that kind of definition of pi (ie, admit a definition that is just an infinite list of decimal representations), you can just do the same thing with any real number you like. He of course will say that he's _not_ allowing any _infinite list_, only an arbitrary long one.

That's the key point though, this list isn't infinitely long, and all the numbers in it are rational. And it is an algorithm (specifically, a lookup table).

All the numbers you get this way are going to be rational, and if you require them to be finite, you can't even identify them with any irrational numbers. At least with the computable numbers you get an infinite set of irrational numbers along with the rationals, while still never touching the vast majority of all numbers (the remaining, incomputable irrationals).

He doesn't work with imaginary numbers, either. He treats complex numbers as matrices of rationals.

Which is the same thing for all intents and purposes.

An ultrafinitist is still allowed to call that 'i'.

Still kind of freaked out that a Möbius transform can be expressed as a matrix multiplication.

I 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.

It 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.

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.

Imaginary numbers, quaternions, octonions, Clifford Algebras, etc. can still have finite expressions.

After all, the Cayley-Dickson construction is not an infinite affair.

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