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Love it. I remember reading (might have been here) that math and art both get difficult at the same moment for the same reason. When you're a kid, you live in

by geebee 7y ago

Love it. I remember reading (might have been here) that math and art both get difficult at the same moment for the same reason. When you're a kid, you live in the rational world. By this, I mean numbers that can be expressed as the ratio of two integers. Human creations are rational. The volume of a square is a neat, tidy equation. So is the area of a square. You can draw them easily, too, using a ruler and clean nifty lines, and they look great. Squares are all over human creation. You can draw a car with straight lines and squares. Wheels and other things bring in this inconvenient number, pi, that is "irrational", but let's just go with three point blah blah and it'll be fine. At least the curvature is constant. So, where are the squares in nature? Hell, where are the circles. Where is the constant curvature. How do you draw a leaf, a tree, a face? How do you calculate the surface volume of a leaf, or the volume of a tree? All of a sudden, you can't measure it with the numbers you know. There is no neat ratio of integers that will calculate the volume of that tree trunk. Or even the volume under an easily expressed mathematical equation on a graph. In fact once you start measuring nature, rather than the things people make, rational numbers aren't anywhere. All of a sudden, you have to deal with limits, sequences, strange numbers that can be made arbitrarily close to zero as other numbers approach infinity. It turns out every number is "irrational", pretty much nothing is rational. So, instead of irrational, let's call it Real. Where do math and art get hard? When you start to describe things as they are, rather than as we imagine the to be. You know, Real.

com2kid 7y ago
> 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!
- geebee 7y ago
  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.
- pg_is_a_butt 7y ago
  start with a square where the diagonal is 1 unit.
- jasomill 7y ago
  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.
- tripzilch 7y ago
  > 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.
derefr 7y ago
> 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.
- geebee 7y ago
  very small, or arbitrarily small? ;)
  - hashmap 7y ago
    Very small. I think the resolution of this simulation was set to planck length. The screen door effect is hardly noticeable.
UnFleshedOne 7y ago
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. :)
perl4ever 7y ago
"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.
- grafs50 7y ago
  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)
  - lonelappde 7y ago
    Infinite information requires infinite energy.
- lonelappde 7y ago
  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.
  - perl4ever 7y ago
    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.
foobarian 7y ago
Calculus is just Minecraft just on a infinitesimal scale. Plenty of squares to go around!
tripzilch 7y ago
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.
- geebee 7y ago
  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.
slantaclaus 7y ago
Love it. Hate to brown nose the top poster but this comment is better than the article itself

by geebee 7y ago

com2kid 7y ago

> 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!

geebee 7y ago
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.
pg_is_a_butt 7y ago
start with a square where the diagonal is 1 unit.
jasomill 7y ago
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.
tripzilch 7y ago
> 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.

derefr 7y ago

> 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.

geebee 7y ago
very small, or arbitrarily small? ;)
- hashmap 7y ago
  Very small. I think the resolution of this simulation was set to planck length. The screen door effect is hardly noticeable.

UnFleshedOne 7y ago

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. :)

perl4ever 7y ago

"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.

grafs50 7y ago
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)
- lonelappde 7y ago
  Infinite information requires infinite energy.
lonelappde 7y ago
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.
- perl4ever 7y ago
  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.

foobarian 7y ago

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

tripzilch 7y ago

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.

geebee 7y ago
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.

slantaclaus 7y ago

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