Hacker News Reader

Comment by moloch-hai | Hacker News Reader

I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms.

We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In practice, in fact, we use floating-point numbers, not reals, when doing actual calculations, and use numerical analysis to stay well clear of nonsensical results. If you tried to rely on BT in a real calculation, you would find a lot of NaNs and Infs.

View on HN

In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete.

We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing?

> we use floating-point numbers, not reals, when doing actual calculations

I don't believe only calculations made by computers to be "actual calculations". Humans were calculating thousands of years before computers were invented, using integer or real numbers (e.g pi).

No human or computer has ever used an actual real, transcendental number in any computation. It would take infinitely long.

I know what you mean, but Euler's identity is a counterexample to what you literally said, isn't it?

I don’t think so. Programs that use numbers approximate e and π. It’s only programs that manipulate symbols that can use their exact values.

All programs manipulate symbols. All bits are is symbols.

It's just sometimes we use those symbols to represent a subset of the integers (e..g. with the popular binary notation)... or a subset of the rationals (e.g. with the popular floating point notation)... or a subset of the reals that happens to include a transcendental number because we decided that some symbol (or combination of symbols) represents some particular transedental number.

That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.

I absolutely agree with you from the outset . . .

. . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula.

Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π

But surely that's over simplification?

Is this magic by wizards?

You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them.

It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.

Just to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end").

It's another of those intuition challenging moments in math.

It's the wackiest result I know of, and seems overwhelmingly more profound than BT.

> Wherever it comes down to actual numbers

Wait, are you saying e and pi aren't "actual numbers"?

I am not a Platonist, or any sort of theist.

e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world.

I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no better than 113/355 meters long.

> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature.

Thank you! I was beginning to think I was the only one who believes this. So many people seem to mistake the map for the terrain.

I probably confused matters with my undefined expression "actual numbers". e and pi are as actual as i or 0.

I think Turing introduced "computable numbers" which are a lot like the reals, but countable. You can write a program that produces each. So it includes integers, rationals, polynomic irrationals, and lots of transcendentals. But the set any particular person (or computer) operates on over the course of their existence is not just countable, but finite and not very large. You have to have expressed a representation of each in it at least once. We can only express a small number of numbers over the course of a life. Computers can do more of them. You might claim all those that programs you wrote have expressed.

And, of course, the number has to have finite expression. Representing pi as a summation is finite. But presenting a numerical computation involving it has to be an approximation if it ever to finish.

How is "The circumferance of an idealized circle divided by its diameter" not a finite expression of π? Saying something cannot be expressed finitely in an integer-based numeral system, and saying that it admits no finite representation are two radically different statements.

Despite it being a non-starter from a pragmatic standpoint, we could for instance easily imagine a novel numeral type that encodes the set S = {a + b·π where a and b are integers} (we can encode integers quite easily and all we need to reposesent such a number in silico is to encode a and b). Using such a numeral type, we are able to do exact arithmetic if our operations are restricted to addition and subtraction (and if we are content with fractional representation of numbers as being considered "exact", we can also do division and multiplication although we would have to work within the larger set S' = { (a + b·π) / (c + d·π) where, a, b, c, and d are integers and c·d ≠ 0} rather than within S).

> Despite it being a non-starter from a pragmatic standpoint

That was my point, thanks.

> e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature.

It’s like that for all numbers, not just the fancier ones! I have never experienced a 1.

Even spiders and newborn babies count; it is a natural animal behavior. Axiomatic systems, not so much.

I don't think you have to be a Platonist to allow that numbers that do not exactly correspond to humanly measurable quantities can yet be part of a calculation.

Another example would be Binet's formula for Fibonacci n[1], which though it involves Phi, sqrt(5), etc., always evaluates to an integer, given integer n.

The existence of these irrationals in Platonic heaven doesn't follow from the formula, which afaik depends only on the usual rules of algebra and logic, though I admit I'm not sure what constructivists make of this type of calculation.

1: see https://r-knott.surrey.ac.uk/Fibonacci/fibFormula.html or https://mathworld.wolfram.com/BinetsFibonacciNumberFormula.h... for example

It feels to me like you're redefining what a number is to be very different to what anyone with a maths background would say a number is. Essentially you're saying that neither e nor pi are numbers?

They are not numbers you have ever used in an actual calculation. You always approximate them anyplace they don't cancel out.

There is a countable subset of the reals called "computable numbers". We only ever use a subset of computable numbers in real calculations.

in my view they never were "numbers" (as in constants), they are functions

The only numbers that may exist in nature are the non-negative rationals. Negative numbers are widely useful, but they don't exist in any meaningful way: they're an abstraction over reality. Irrational numbers definitely don't exist, and don't get me started on so-called "imaginary" numbers -- they're no less present in nature than the negative numbers are.

But the concepts are useful, and the symbols have meaning. The article mentions normal words having different meanings in mathematical language, and that impacts even every-day use of maths.

Banach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity.

I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)

My understanding is that Planck Length is not a limit to the smallest possible thing; it's a limit to the most precise possible measurement. So there could in fact be lots of things happening at sub-Planck Length scales, but we could never observe them.

Atoms move according to path integrals which would be at least countably infinite in number even if time and space were discrete because they are allowed to loop back on themselves.

The main reason that we use real numbers is because we need real numbers to do calculus, at least in the normal way. And to do physics, we need to do calculus. You can keep approximating integrals as sums of very large number of terms, but it gets unwieldy to handle symbolically.

Real numbers almost always give the same result as more cumbersome methods, and we are mostly only interested in such results, so it is surprising when something wacky shows up to remind us of the tightrope we walk.

> It relies on uncountably-infinite division of an object

But the theorem claims finite division and not infinite?

In R³, given a solid ball B of radius R it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R

Finitely many pieces, but on infinitely variable boundaries. It was clever to make the proof allow a finite number in that place. Without, it would have attracted no attention.

It's still astounding to me that the 'fractal trick' starts working in R3 and not already in R2.

Agreed, each added dimension adds wacky surprises.

Packing spheres into a cube leaves you a pretty comprehensible space in the middle. Pack hyperspheres into a hypercube, and all hell breaks loose.

Keep in mind that "pieces" might not mean the same thing you think of it as meaning.

Reply on news.ycombinator.com