5 ms·

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 computational

• 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.
• btilly 5y ago
  I am a math guy. There is no way to actually do these calculations in practice that does not at some point boil down to something done on a computer to finite precision. For example if you want to plot planetary orbits you should plug in your most precise available measurements, use Runge-Kutta, and produce rational approximations. There is no way to test our theories in practice that doesn't at some point involve comparing this finite precision prediction with a finite precision observation. Pontificating about the failures of a discrete approximation to be able to be computationally accurate in a continuous world may be fun armchair philosophy, but CANNOT be useful scientifically. Because we can only measure and work with finite precision approximations to that hypothesized continuous reality.