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Comment by dhosek | Hacker News Reader

I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-physical (although it might be a larger subset of numbers than the constructible numbers depending on the structure of space-time’s quantization).

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If reality is quantized and there is a smallest number that is physically relevant, you don't need the reals to break it. Take that smallest number and divide it by two, and now you have a physically meaningless number using only the rationals.

This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.

You're taking an anomalously narrow view of the parent comment. Say the minimum distance is one inch.

You want to say that the concept of half an inch lacks physical representation, but that isn't true. You can easily demonstrate it as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.

dhosek is saying that in a quantized space, there are reals that cannot be demonstrated this way, and he is right, but the same thing is untrue of rationals.

("In a quantized space", by the way, just means that all measured quantities are necessarily integers. That causes all kinds of problems, but "lacking examples of arbitrary rational numbers" isn't one of them.)

> You can easily demonstrate it [half an inch] as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot.

You seem to have your units confused. Half an inch is a distance, while ratios, or comparisons of ratios, are all dimensionless scalars.

Have you ever seen a map with a scale indicator?

"suppose that the minimum distance is one inch. well, that's one of something. so now imagine half of that! there you go: one half. a physically unrealizable number."

Except that you’re assuming that 1 must necessarily correspond to that minimum distance. Keeping that situation, we can just say that 1 corresponds to two inches and then realize 1/2 as that number.

Classical geometry (a la Euclid) allows for constructing a lot of numbers. We get natural numbers pretty cheaply, negative integers through adding in a concept of directionality (zero is a bit of an imaginative leap which is why it was absent from Western mathematics for so long). Constructing arbitrary ratios is possible through similar triangles and square roots through right triangles, but some basic algebraic numbers like cube roots cannot be constructed with a ruler and straight edge (which raises the question of whether, in a quantized universe, whether irrational cube roots actually exist). Of course there’s no guarantee that the quantization is going to be uniform and we also have the ɣ factor of special relativity (1/sqrt(1-v²/c²)) which gives us a non-Euclidean space to complicate things, but it’s not clear that if you can find a value for 1 that allows you to get a measurement for every irrational number.

One could make the argument that the only numbers that actually “exist” are the natural numbers, but the question ultimately is can you model any real number in the physical universe. Modeling ½ is simply a question of picking a unit to be 1 and finding its midpoint (or for that matter, declaring two apples to be “1” and thus a single apple would be “½”, although it’s a bit of a challenge to use apples to model (2-√3)/5

Make a line of 10 apples and declare it to be 2 units long, then make a square that's 15 applies diagonal, finally measure how much longer the 10 apples are than the side of the square.

It's a pretty linear increase in complexity between the math and the apples.

in what way does drawing a diagram have anything to do with physically realizing a number?

I didn't say a diagram? I'm talking primarily about physical objects.

And I'm just explaining how to use the same methods as the comment I replied to.

"Make a line of 10 apples and declare it to be 2 units long" <- what number does this physically realize?

Both ten and two. And it lets you model fifths.

But don't ask me about that. That part wasn't my idea at all. Ask dhosek about that way to do fractions. My contribution was the square root and the subtraction.

Take the smallest number corresponding to a physically meaningful distance in meters, divide it by two, and that number is still a physical meaningful distance if you switch the unit to decameters or kilometers.

Any rational number has some meaning if you just add the right unit, even if the units become increasingly ridiculous. But for reals that trick does not work

Sqrt(2) “any unit that is an integral multiple of the shortest interval”.

But it would get complicated, for any given allowable velocity and allowable length, you’d get more lengths from Lorentzian contraction.

There are really a couple of different ideas being combined: are there an infinite number of quantum states for the universe, are space and time continuous, is the forward direction of time resolved by computable processes.

And even bigger ones lurk: are space and time emergent properties from quantum waveforms that lack an inherent idea of space and time (but things that are highly correlated give rise to a notion of being near each other in “space time”)?

Reality probably isn't quantized in the "on a grid" sense, but rather the "ability to resolve" sense. The more computation you put in the higher accuracy you can get.

What numbers in R would not be possible to express in an infinite universe?

Chaitin's constant. Or rather, you can never see that it has been laid out in your infinite universe.

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