6 ms·

That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed

by midjji 5y ago

That often works, but not always, some systems generate sequences of operations which can be symbolically simplified, or equivalently could be exactly computed using reals, but which if computed using any finite precision will fail. A simple example would be solving for the position of a planet in orbit under simple Newtonian gravity, a sufficient number of revolutions latter. For any finite precision the number of orbits can always be increased until failure occurs. Its a cool how few bits of pi are required to compute the circumference of the earth to within an atoms width accuracy. But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later.

btilly 5y ago
That logic is disingenuous at best. Planetary orbits are chaotic. Long before your imprecision in pi is going to significantly mislead you, shifts in mass due to, for example, earthquakes and weather patterns are going to cause orbits to be impossible to predict. There are theoretical systems where the exact value of pi matters. But no physical system is going to match that, and measurement error is going to quickly exceed calculation errors from pi.
- TheOtherHobbes 5y ago
  Pi is only incidentally a number. Pi is an abstraction that represents a certain relationship. (Actually more of a set of relationships.) If you define pi as a specific constant with limited precision - because "that's all physical systems need" - you lose insights into the web of relationships around it. This is a bad thing and makes many kinds of math harder. It's the conceptual equivalent of lossy data compression. You don't want to do it unless you really, really need to. And if you do it, you need to be aware that you're now using approximations instead of abstractions, and those are not the same thing.
- wruza 5y ago
  But when you remove or take these fluctuations into account, you’re still left with an error. This rational model has no chance to ever be correct computationally, unless you cheat and add more detailed ratio every time you see a loop. Also, how exactly will you define rational pi? Let’s start with 3/1, why go any further. If it doesn’t represent reality (draw a circle and measure it with a string), well, strings have vague length anyway. See where this is going? The bigger problem is analyzis will not work, probably. Can you do analyzis with rationals? E.g. f(x)=x^2 isn’t continuous at f(x)=2/1. I’d theorize that there is a way to do finite calc without giving up on “reals”, by using enough FT coefficients (and packing this complexity into sine waves), but it’s the same sort of cheating probably, unless pi is the “origin” number of all “physical” reals. (I’m not a physicist nor math guy, so one can reframe my ideas as questions instead.)
  - midjji 5y ago
    What you are proposing is essentially using a different number basis, and the simplest applicable one would be a basis which expresses numbers using whole numbers{...,-1,0,1,2...} times pi. This allows you to exactly express pi as just 1, and all such using only whole numbers which is nice. It also unfortunately means that simple things like y= 1_{decimal} x would have to be expressed using the relative to basis transcendental number pi i.e with precision problems. And every basis you could chose behave like this. Different basis have different advantages, same as different function basis. The classic example would be that in a standard basis, its easy to add and subtract, but more costly to multiply, divide or factorize, while in the prime basis the former is expensive as hell, but the latter is trivial. As a result, rationals and pi in a sense disjunct domains. You cannot express either using less less than an infinite number of the other, and the same holds for combinations of a rational and pi. Numbers which behave this way relative to each other are more common than the rationals, and pi is just the most common example. It does lead to a rather neat requirement for the fundamental physical constants though. The reasoning goes like this, imagine that a model K2 of physics could be described using two constants, a, b. gravity and the speed of light say. Now lets say we managed to prove that a = 2b and therefore that everything predicted by model K2 can also be predicted by model K1, which just uses the coefficient b. K2 is equivalent to K1, sure, but only one constant would then be fundamentally required, and if K2 is sufficient to describe all of physics, physics would only have one fundamental constant. The same reasoning would hold if a=b^2, and so on. But, if the function required to express a as a function of b requires infinite information, this does not meaningfully apply, as this will always apply to every pair of numbers. Meaning that we know that if the fundamental constants of a model of physics does not lie in disjunct domains in the sense above, there is a simpler version which has fewer constants. For example, since pi has infinite information, if a=pi b, then the simplification cannot be meaningfully made without introducing pi as a fundamental. More generally this also fundamentally means that true physics cannot be expressed using finite precision if ideal grand unified theory as more than one fundamental constant.
    - wruza 5y ago
      I though about a slightly different thing: Fourier series with rational coefficients, which I believe is not the same as simply Q times pi. Sin(x) has nothing to do with pi, apart from the fact it becomes 0 at pi times n, which seems irrelevant. It was based on an assumption that the nature is all-waves, so these components would pack our “synthetic, infinite” R (which we try to get rid of itt) naturally into these waves and then the computation would be left with just coefficients and their relations. The problem of iterations disappears, because instead of collecting a decimal error, we would collect more and more sines, as if they were fundamental, like sort of ‘i’, but many of them. But if you speak and think only sines, then the problem of unfolding them doesn’t exist. sin 3/4x + 1/7 sin 1/3x would be a simplest answer (an “anyternion”), probably written as (1,3/4)+(1/7,1/3) because these are the irreducible resulting wave parameters and sin is just an implied concept. Anyway, it’s a stupid layman’s theory. I’m pretty sure that a parallel/on-demand digit computation in R-based models is much easier than trying to get rid of philosophically infinite things which in practice do not matter that much.
drran 5y ago
> But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later. In 2^64 years, Earth will be inside of a larger body. How many digits of Pi you need to predict that?
- btilly 5y ago
  Sorry, in 2^64 years the Sun will have become a red giant, spent a billion years as that, then turned into a white dwarf. The fate of the Earth by that point is...probably not good.
- wruza 5y ago
  In 2^64 years, Earth will be inside of a larger body How do you know that in the first place? Maybe with rationals science it “will not”, so there is “no problem” at all.
  - drran 5y ago
    We are falling into Great Attractor, which is also falling into Shapley Attractor: https://www.youtube.com/watch?v=PEAY7LNhdxM https://www.youtube.com/watch?v=PEAY7LNhdxM
aulin 5y ago
I believe that has to do with discretization of a math operation (derivative, differential equations) that's inherently continuous. There are algorithms (symplectic integrators? Verlet?) that are perfectly capable of computing orbits without adding any drift.

by midjji 5y ago

btilly 5y ago

That logic is disingenuous at best. Planetary orbits are chaotic. Long before your imprecision in pi is going to significantly mislead you, shifts in mass due to, for example, earthquakes and weather patterns are going to cause orbits to be impossible to predict. There are theoretical systems where the exact value of pi matters. But no physical system is going to match that, and measurement error is going to quickly exceed calculation errors from pi.

TheOtherHobbes 5y ago
Pi is only incidentally a number. Pi is an abstraction that represents a certain relationship. (Actually more of a set of relationships.) If you define pi as a specific constant with limited precision - because "that's all physical systems need" - you lose insights into the web of relationships around it. This is a bad thing and makes many kinds of math harder. It's the conceptual equivalent of lossy data compression. You don't want to do it unless you really, really need to. And if you do it, you need to be aware that you're now using approximations instead of abstractions, and those are not the same thing.
wruza 5y ago
But when you remove or take these fluctuations into account, you’re still left with an error. This rational model has no chance to ever be correct computationally, unless you cheat and add more detailed ratio every time you see a loop. Also, how exactly will you define rational pi? Let’s start with 3/1, why go any further. If it doesn’t represent reality (draw a circle and measure it with a string), well, strings have vague length anyway. See where this is going? The bigger problem is analyzis will not work, probably. Can you do analyzis with rationals? E.g. f(x)=x^2 isn’t continuous at f(x)=2/1. I’d theorize that there is a way to do finite calc without giving up on “reals”, by using enough FT coefficients (and packing this complexity into sine waves), but it’s the same sort of cheating probably, unless pi is the “origin” number of all “physical” reals. (I’m not a physicist nor math guy, so one can reframe my ideas as questions instead.)
- midjji 5y ago
  What you are proposing is essentially using a different number basis, and the simplest applicable one would be a basis which expresses numbers using whole numbers{...,-1,0,1,2...} times pi. This allows you to exactly express pi as just 1, and all such using only whole numbers which is nice. It also unfortunately means that simple things like y= 1_{decimal} x would have to be expressed using the relative to basis transcendental number pi i.e with precision problems. And every basis you could chose behave like this. Different basis have different advantages, same as different function basis. The classic example would be that in a standard basis, its easy to add and subtract, but more costly to multiply, divide or factorize, while in the prime basis the former is expensive as hell, but the latter is trivial. As a result, rationals and pi in a sense disjunct domains. You cannot express either using less less than an infinite number of the other, and the same holds for combinations of a rational and pi. Numbers which behave this way relative to each other are more common than the rationals, and pi is just the most common example. It does lead to a rather neat requirement for the fundamental physical constants though. The reasoning goes like this, imagine that a model K2 of physics could be described using two constants, a, b. gravity and the speed of light say. Now lets say we managed to prove that a = 2b and therefore that everything predicted by model K2 can also be predicted by model K1, which just uses the coefficient b. K2 is equivalent to K1, sure, but only one constant would then be fundamentally required, and if K2 is sufficient to describe all of physics, physics would only have one fundamental constant. The same reasoning would hold if a=b^2, and so on. But, if the function required to express a as a function of b requires infinite information, this does not meaningfully apply, as this will always apply to every pair of numbers. Meaning that we know that if the fundamental constants of a model of physics does not lie in disjunct domains in the sense above, there is a simpler version which has fewer constants. For example, since pi has infinite information, if a=pi b, then the simplification cannot be meaningfully made without introducing pi as a fundamental. More generally this also fundamentally means that true physics cannot be expressed using finite precision if ideal grand unified theory as more than one fundamental constant.
  - wruza 5y ago
    I though about a slightly different thing: Fourier series with rational coefficients, which I believe is not the same as simply Q times pi. Sin(x) has nothing to do with pi, apart from the fact it becomes 0 at pi times n, which seems irrelevant. It was based on an assumption that the nature is all-waves, so these components would pack our “synthetic, infinite” R (which we try to get rid of itt) naturally into these waves and then the computation would be left with just coefficients and their relations. The problem of iterations disappears, because instead of collecting a decimal error, we would collect more and more sines, as if they were fundamental, like sort of ‘i’, but many of them. But if you speak and think only sines, then the problem of unfolding them doesn’t exist. sin 3/4x + 1/7 sin 1/3x would be a simplest answer (an “anyternion”), probably written as (1,3/4)+(1/7,1/3) because these are the irreducible resulting wave parameters and sin is just an implied concept. Anyway, it’s a stupid layman’s theory. I’m pretty sure that a parallel/on-demand digit computation in R-based models is much easier than trying to get rid of philosophically infinite things which in practice do not matter that much.

drran 5y ago

> But equally cool how bad it gets if you try the same to compute the position of earth 2^64 years later. In 2^64 years, Earth will be inside of a larger body. How many digits of Pi you need to predict that?

btilly 5y ago
Sorry, in 2^64 years the Sun will have become a red giant, spent a billion years as that, then turned into a white dwarf. The fate of the Earth by that point is...probably not good.
wruza 5y ago
In 2^64 years, Earth will be inside of a larger body How do you know that in the first place? Maybe with rationals science it “will not”, so there is “no problem” at all.
- drran 5y ago
  We are falling into Great Attractor, which is also falling into Shapley Attractor: https://www.youtube.com/watch?v=PEAY7LNhdxM https://www.youtube.com/watch?v=PEAY7LNhdxM

aulin 5y ago

I believe that has to do with discretization of a math operation (derivative, differential equations) that's inherently continuous. There are algorithms (symplectic integrators? Verlet?) that are perfectly capable of computing orbits without adding any drift.