this post was submitted on 20 Jun 2024
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Irrational (mander.xyz)
submitted 2 months ago by [email protected] to c/[email protected]
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[–] [email protected] 105 points 2 months ago (1 children)

My only guess as to what this could mean is that since quantum mechanics is quantum, i.e. discrete, the universe therefore cannot be continuous as the reals are. But this is a category error. Just because you could never find an object that is, say, exactly pi meters long, does not mean that the definition of pi is threatened. There's nothing infinite that we can observe, but infinity is still a useful concept. And it works both ways; just because quantum mechanics is our best model of the universe doesn't mean the universe is therefore quantum. 150 years ago everyone believed the universe was like a big clockwork mechanism, perfectly deterministic, because Newtonian physics are deterministic. And who knows, maybe they were right, and we just don't have the framework to understand it so we have a nondeterministic approximation!

[–] [email protected] 24 points 2 months ago* (last edited 2 months ago) (3 children)

We could make an object that is exactly pi meters long. Make a circle of 1 meter in diameter, and then straighten it out. We would not be able to measure the length more accurately than we can calculate it (that might be the largest understatement ever) but to the tolerance with which we could make a 1 meter diameter circle, you should have the same tolerance to the circumference being pi.

[–] [email protected] 27 points 2 months ago (2 children)

I mean, you only need 39 digits of pi to calculate the circumference of a circle with a diameter the size of the universe to the width of a hydrogen atom. So no matter how detailed you get it's impossible to determine if a circles circumference is anywhere close to exactly pi.

To ops point, you could set up your thing theoretically and we can math out that it should be pi. But we could not make that object.

[–] [email protected] 4 points 2 months ago* (last edited 2 months ago) (2 children)

Right, by my point is that your accuracy and precision are the same whether you are making a 1 meter length object or a π meter length object. Your meter stick is not accurate to the width of a hydrogen atom, either.

But if we accept the precision of our manufacturing capabilities as "close enough," then it is equally as close to exactly π as it is to exactly 1.

In other words, to say we cannot make an object that is π meters is to say we cannot make an object that is any specific length.

[–] [email protected] 3 points 2 months ago* (last edited 2 months ago)

Not to reiterate what other people have said here. But you can make an object 1 meter long by defining that object as 1 meter (hell, you don't have to, but you can define 1 meter as the length that light travels in a specific amount of time or something silly). Then, to create something two meters long, you can have two of those one-meter lengths. To make something π meters long, you would need infinite precision, that is not true for 1 meter or even 1/3 as you mention later in this thread.

There is no way to divide anything into exactly π length. There is an easy way to divide something into a number that can be expressed as a fraction, such as 1/3, or any fraction you care to come up with, even if it can be represented as .3 repeating.

[–] [email protected] 2 points 2 months ago (1 children)

The precision of our manufacturing capabilities might be limited as QM has this discreete nature. It might be limited in this universe. So pi may only exist theoretically

[–] [email protected] 2 points 2 months ago (3 children)

But you could make that same argument for a lot of fractions. 1/3 doesn't exist because you cannot divide a quantum in three. 0.333 repeating means that eventually you have to divide an indivisible foundational particle in thirds.

[–] [email protected] 3 points 2 months ago (1 children)

If you have three particles, 1/3 of that is one particle. No need to divide an indivisible particle.

[–] [email protected] 1 points 2 months ago (1 children)

But if I don't have three particles, 1/3 requires division.

[–] [email protected] 3 points 2 months ago

Right, but you can have exactly a third of some group of particles. You can't have exactly pi of some group of particles I think is what they were saying

[–] [email protected] 2 points 2 months ago* (last edited 2 months ago)

The other guy said good about one out of three known particles. That's what make it rational!

The problem is that something that doesn't exist in our universe or reality doesn't disprove anything in mathematics. Mathematics is abstract. It is rules built up on rules. It does not care about reality or anything

[–] [email protected] 1 points 2 months ago

You can divide a thing made up of any multiple of 3 number of things into three. Say, divide twelve eggs by three that's four eggs, rational division is justified by "I could have multiplied some numbers beforehand so now I can divide", it's the inverse of multiplication, after all.

But that only applies to rationals: The issue is that there's no integer you could multiply pi with that would result in an integer... otherwise pi would be a rational number which it isn't.

[–] [email protected] 2 points 2 months ago

Sure. But how about a plank length instead of width of H atom. That would be great.

[–] [email protected] 10 points 2 months ago (3 children)

No, by our current understanding there is no length smaller than a Planck length, and any distance must therefore be divisible by an integer. That is, the length is made up of discrete quanta. Pi, or any other irrational number, is by definition not divisible by an integer, or it would be a ratio, making it rational. This has nothing to do with the accuracy or precision of our measures.

[–] [email protected] 12 points 2 months ago

Planck length isn’t the smallest possible distance. It’s simply the smallest distance at which our current understanding of physics still holds up. Beyond that, our current models break down, but our current models are very incomplete

[–] [email protected] 10 points 2 months ago

Mmmmmm don’t know about that.

The Planck length is the minimum resolvable accuracy of the universe. That doesn’t mean it’s a building block like the electron is.

[–] [email protected] 5 points 2 months ago

I believe you're mistaken. A Planck length is the minimum length we can extrapolate down before physics gets weird, but that doesn't mean it is the smallest possible length anything can be.

And an irrational number does exist as a discrete unit, it simply cannot be described as a fraction. Case in point, if you could create a spherical particle that was exactly 1 Planck length across, it would have a circumference of exactly π Planck lengths.

By your logic, such a theoretical particle could not exist because the circumference includes an irrational number in the size of the body.

[–] [email protected] 3 points 2 months ago

Two issues:

  1. Planck length The resolution of the circle would be too low to be precisely pi. (I'm not sure whether everything moves only in Planck lengths or whether we live in a voxel world but either way it's not as precise as however many digits of pi we know.
  2. Matter waves Assuming that the circle has energy (which it must due to Heisenberg's uncertainty principle(I haven't actually read this but I assume that like a vacuum according to the unruh effect matter can't be devoid of energy)) the particles making up that circle would have a wavelength in which they can interfere with other particles and also have an area in which they might be.
[–] [email protected] 49 points 2 months ago (2 children)

There is no fact about reality that can ever threaten facts about mathematics. Mathematic definitions exist independent of reality.

[–] [email protected] 22 points 2 months ago (1 children)

In fact, you can build a system of definitions that very clearly doesn't exist in real world, like hyperbolic geometry.

[–] [email protected] 16 points 2 months ago (1 children)

Sometimes we find that obscure pure mathematics does describe reality when no one expected it to. Riemannian geometry is one such example.

[–] [email protected] 4 points 2 months ago

Complex numbers, and a bunch more things too

[–] [email protected] 16 points 2 months ago

Kurt Gödel:

[–] [email protected] 25 points 2 months ago (20 children)

But irrational numbers aren't the same as imaginary numbers. Also, there are irrational imaginary numbers. And quantum physics loves using imaginary numbers. So that sentence in the image is nonsense, right?

[–] [email protected] 3 points 2 months ago

“Imaginary” was merely poor word choice from long ago.

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[–] [email protected] 23 points 2 months ago* (last edited 2 months ago) (1 children)

What? You use these words, but I do not think they mean what you think they mean.

Quantization is probably the result of vibrational modes, that doesn't mean irrational numbers don't exist, just that we can't measure an infinitely precise value. Tau and root-two exist, they arise naturally in the most basic geometric shapes.

[–] [email protected] 1 points 2 months ago (1 children)

This sounds really interesting but I'm afraid it's a bit high level for me. Can you explain how vibrations would cause quantisation? I'd also be happy with a link to the correct Wikipedia article or a paper which explains it. :)

[–] [email protected] 13 points 2 months ago

The premise here is completely wrong.

[–] [email protected] 6 points 2 months ago (2 children)

I know those words, Someone please explain .

[–] [email protected] 3 points 2 months ago (1 children)
[–] [email protected] 8 points 2 months ago (2 children)

Imaginary numbers are a rotational operator.

You don't need quantum mechanics to observe rotation in the real world.

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[–] [email protected] 1 points 2 months ago

Yes please!

[–] [email protected] 2 points 2 months ago (2 children)

https://www.sciencenews.org/article/quantum-physics-imaginary-numbers-math-reality

Nontechnical article explaining the concept.

[–] [email protected] 9 points 2 months ago

But that's imaginary numbers, not irrational numbers.

The issue between quantum physics and irrational numbers is different than the use of imaginary numbers: Irrational numbers have infinite decimals, while quantum physics is quantized.

[–] [email protected] 6 points 2 months ago (1 children)

So because quantum mechanics is well modeled by imaginary numbers, the existence of quantum particles threatens the definition of irrational numbers? That doesn't make any sense.

[–] [email protected] 4 points 2 months ago

Yes, it does not make any sense. If the link above is what it appears from the summary, some students unknowingly attempted to square the circle.

https://en.m.wikipedia.org/wiki/Squaring_the_circle

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