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> So, where are the squares in nature? Hell, where are the circles. Where is the constant curvature.

They exist! The crystal structures of molecules are rather Platonic, for example. Nature is elegant when you get very small, requiring fewer and fewer core concepts as you work your way down to more fundamental levels of understanding. (At lower levels those fundamentals might be irrational ones, but still, those few primitives [like spirals in complex space] become the only tools you need.)

The inelegance, then, comes from modelling the interactions of mind-bogglingly huge collections of these fundamental things, at high levels of abstractions, and then expecting your abstraction (which is just that: a formula that allows you to make some useful prediction of these super-high-level interactions) to be as elegant as the fundamental forces operating at the lowest levels.

very small, or arbitrarily small? ;)

Very small. I think the resolution of this simulation was set to planck length. The screen door effect is hardly noticeable.

I could make the opposite argument. In nature/reality there are no irrational numbers.

Say you want to measure the circumference of the visible universe. This is all the space we have, the rest is outside our light cone and we can't interact with it to the point that we can only infer it might exist.

You need about 55 decimals of pi to get this circumference to the accuracy of a Planck length. Or about that many, give or take.

So that's measuring the very largest real thing that could possibly matter, down to the accuracy of the very smallest thing we can conceive of.

So that's it for pi. Any further decimals are strictly theoretical. We can prove they must be these decimals and not others, using math, but it's only theoretical knowledge, these additional decimals serve absolutely zero purpose in nature or reality. You cannot get to them by measuring reality, you can only theorize about what these numbers would be if you could measure to infinite precision, which we can't, because there are limits.

The "Real" numbers is really a misnomer. Even if you don't buy the above accuracy argument, and want to describe nature as something infinite (even though we're strictly limited to interacting with a finite subset of it), then at least agree that it's countable. The real numbers are way too stretchy and insane (see the Banach-Tarski paradox). In nature you can't stretch things infinitely far, nor can you cut up things to arbitrary precision.

And yes occasionally we discover new smaller particles, or sub-particles, but what we don't discover is a continuum. And it would be really weird if we did, because you can do crazy tricks to the Real numbers.

Very interesting response. The thing that keeps me from entirely agreeing (with an admission that I don't have a background in the science you mentioned[1], so I can't really understand it) is that the number is pi - the 55 decimals is just an approximation that can be expressed as a ratio of two integers. The number isn't the rational that can be made arbitrarily close to a limit, the number is the limit. That number is pi. pi is every bit as much a number as 1, 2, 3. So is sqrt(2). So is e.

My understanding is that almost all numbers are real, but not rational. They can be expressed as the limit of a sequence that can be made arbitrarily close to the limit. But the number is not the sequence, the number is the limit of the sequence.

[1] = I had to look up what a Planck is. Should have taken more physics along with the math.

> So is the area of a square.

Try to cut that square in half diagonally and things get irrational really fast!

FWIW I thought calculus made a lot of sense and helped make the world make more sense. Algebra is a fancy set of rules to manipulate rather abstract symbols, calculus actually explains how real things work!

Good example. It's amazing how quickly you leave the rationals, even with a square, you almost immediately need numbers that can't be expressed as the ratio of two integers... historically, was that the first encounter with an "irrational" number (Pythagorean theorem applied to a right isosceles triangle)? I do remember the proof from number theory about 20 years ago, though I could never recreate it now from memory.

And algebra itself is one of those real things that calculus helps to explain!

Take, for instance, the Fundamental Theorem of Algebra, where the proof in terms of elementary complex analysis and/or topology closely related to complex analysis make the truth of the theorem geometrically obvious, the (mostly) algebraic proofs I've seen leave me with little more than the desire to re-check the proof, because I'm not at all certain that something equivalent to the F.T.A. hasn't been implicitly assumed at some point in the proof.

Now it may be "just me" — I've always had an easier time following analytical proofs than abstract algebraic ones — but just thinking of the necessary prerequisites — homotopy between maps defined by complex polynomials vs. what? Galois theory? — I don't think it's just me.

Incidentally, complex numbers are another case where, as with the irrationals, a poorly-chosen name has made simple and quite generally useful ideas seem esoteric to those not already familiar with the subject.

> Try to cut that square in half diagonally and things get irrational really fast!

In theory. In reality you have an integer number of atoms in one half and another integer number in the other.

There are no irrational numbers to measure, and squares made from "continuum material" do not exist in reality.

"once you start measuring nature, rather than the things people make, rational numbers aren't anywhere"

I thought that because nature is made of quantized things, that the opposite is true - all numbers are really rational, and it's irrational numbers that don't exist except in human imagination.

One way to look at why an irrational number cannot describe a physical object-

Suppose you had an object with a variable position in one dimension that could be described with an irrational number. Then that single object can, in principle, store an infinite amount of information in the decimal expansion of that number simply by positioning it and measuring its position.

The information is not encoded in the object but rather in the arbitrary set of coordinates imposed on it. You can always impose a set of coordinates where the object is at the origin, or where its position in all axes is a small integer.

Systems of coordinates don't have to linear, they don't even have to be monotonic. The underlying object is unchanged by a human making a choice of a set of coordinates in which to describe it, nor by switching to a different system of coordinates.

Your keyboard's "Q" key does not know when you think of it in terms of you-centric Cartesian-ish or spherical-ish coordinates. It doesn't know when someone nearby thinks of your "Q" key in a different set of coordinates, or with a different coordinate-origin, and it doesn't matter to the "Q" key or its behaviour even if those coordinates aren't relatable by e.g. a Lorentz transformation. A physicist might care about that. The engineers who blueprinted the keyboard might care about that too, for electronics-timing reasons, for example. The person nearby you might not.

It is that you are using a real valued coordinate system -- chosen by you, rather than by some law of nature, and certainly not by the "Q" key or any of its molecules or subatomic components -- that lets you encode the coordinates of its depressed position as containing some decimal representation of your login password. You are free to choose coordinates in which that information vanishes, just like you can choose a system of coordinates with the origin on the depressed Q key in which the entire works of Shakespeare can be found in the base-36 expansion of real-valued coordinates at some small spot in your left fovea.

> an irrational number cannot describe a physical object

Use a system of coordinates in which the centre of mass of your pineal gland is at the coordinate origin and the tip of your nose is at (x, \pi, z) at all times.

That's not true. Algebraic numbers are irrational but countable.

Simply bisecting a square creates an irrational number. If you refuse that then you must refuse that the sides of a square are integers.

By your logic, a fraction 0.1 1/10 is infinite because it can be expressed as an infinite series of nontrivial powers of pi or even in base 3.

1. Why does bisecting a square (I assume you mean diagonally) create an irrational number? If matter, and space-time itself are not infinitely divisible, then wouldn't that mean any diagonal line is kind of a zigzag when you look at it closely enough? It's not that the sides aren't integers, it's that the diagonal isn't strictly possible.

2. This is not my logic nor what I wrote about someone else's argument. I didn't say that all infinite decimals store infinite information. Obviously you can talk about how much information is stored in a given infinite string but I didn't go there and it's not relevant. It doesn't matter whether you think all real numbers store the same amount of information or not.

I described the claim that if you could physically realize an arbitrary real number you could store unlimited information in the tiniest piece of matter. Which goes against the intuitive idea based on experience that more matter is required to store more information.

I don't think this is true at all.

Position isn't quantized; energy is.

An object having a position described by an irrational number does not store infinite information. The object isn't storing anything at all; it's just sitting at that location.

The person doing the decimal expansion has to store an infinite amount of information. What if the person just uses an infinite-series expression?

Another question - what if I just shift my reference frame to be one irrational unit offset from yours? Does the information encoded by object-positions in my reference frame now change?

I don't think it makes much sense to go smaller than the planck length (10^-35m) because probing smaller distances would create black holes. So in a sense position is quantized according to current understanding of gravity and QFT.

I guess you can say that, but I think what I'm saying is a pretty standard position in QM:

https://physics.stackexchange.com/questions/32665/why-positi...

Is space-time known not to be quantized? Or is anything?

Does a slide rule or an abacus store information, or does it just "sit there"?

From what I remember, the book 'Quantum Mechanics in Simple Matrix Form' explicitly discusses which variables are quantized and which are not, according to standard quantum mechanics. It takes a few chapters to develop the argument.

Yes, an abacus stores information, but not an infinite quantity; the information is only a few bits, and it's interpreted that way because we take a very low-precision view of the state of the abacus by dividing the rungs in half, mapping each side to 0 or 1. This is the key; the units of position used for measurement.

The object itself can be at any 'true' position (including an irrational position) but no measurement device has enough precision to say for sure; so I think the answer is that objects can take any position, but the fact that measurement-devices are limited makes it a moot point.

So what if it does? Unless we can accurately measure with infinite precision somehow, we can't store/retrieve an infinite amount of information.

And even if we could, is there a law of physics that would contradict? (I'm actually asking, I have no idea)

Infinite information requires infinite energy.

Real number is often just another approximation on the way to the true value. You are still counting spherical cows in vacuum.

Area of a platonic circle is a real number, area of an actual circle is discrete, but varies depending on how you draw the border.

You can count atoms and get your volume of the tree in integers that way. Bonus points for being temperature and pressure independent (more so than a volume of the spherical tree in vacuum anyway).

I guess they call those numbers irrational because they are never represented physically and thus are a pure figment of imagination. :)

Love it. Hate to brown nose the top poster but this comment is better than the article itself

Calculus is just Minecraft just on a infinitesimal scale. Plenty of squares to go around!