If you think you can describe the physical world without any irrational numbers, when one of the most basic (the ratio of the area of a circle to its diameter) is irrational, I think you're probably mistaken.
Now we do have tools that overcome those limitations somewhat - like limits and stuff but doesn’t remove the need for better tools.
So I somewhat disagree with both of you - we need new, better numbers that are further from math abstraction and closer to actual reality.
Otherwise it is like this story with Poynting vector from Veritasium video - only confuses instead of explaining.
See https://www.jpl.nasa.gov/edu/news/2016/3/16/how-many-decimal... for more.
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.
In 2^64 years, Earth will be inside of a larger body. How many digits of Pi you need to predict that?
How do you know that in the first place? Maybe with rationals science it “will not”, so there is “no problem” at all.
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.
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.
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.)
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.
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.
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.
I guess my argument is, since you can always just pick a rational approximation to Pi, you cannot prove empirically that we live in a universe where more than a finite number of digits of Pi matter. That is, the mathematical irrationality doesn't really matter, physically speaking, since no experiment could ever prove that every digit in Pi actually contributes to the result.
If the universe does have ways to do this, to mix an entire irrational number into a physical outcome, that means hypercomputation is probably possible, since Turing machines definitely can't.
Your simulation might need to ask for an increasingly tighter bound on the real value of Pi. You can totally do this with no more than the usual rational numbers, but it's not equivalent to "just picking some rational approximation" and running with it, because what accuracy/precision you pick is outcome-dependent and it's always possible to request more.
There must be some digit after which no computation will ever access, because it will require more negentropy than the entire universe has to even calculate. The digits after that don't matter to the universe.
Either way though, then doesn't your model of the universe just need an extra parameter, the number of digits to care about? Seems like everything else being equal, the fewer unmotivated parameters in your model, the better. Especially because this would rely on internal details of what happens in the universe, seems unlikely to be true unless this is a simulation.
In other words, cutting your beams to +/- 1/2” may work for each individual beam in a building but that does not imply that your building as a whole can tolerate an average beam length being +.499” above nominal.
The stronger version of the argument is that the length of a steel beam cannot be more precise(-ish) than the radius of an iron atom, so only 10-12 decimal places (in meters) are required to fully describe a steel beam's length. Likewise an actual circle's area isn't a function of Pi, but is rather a 'really large number' regular polyhedron. Which could then be approximated by a fairly pedestrian number of decimal points of pi to atomic precision.
That said e.g. orbits are rather smooth, and could probably be considered to be fairly exact w.r.t. an arbitrary reference.
You start with doing something the most correct way possible on paper and then convert that into the fastest possible method within your allowable bounds on precision and/or convergence. Operational reordering to keep additions in floats with similar exponents is great but you save that concern until it’s time to crunch numbers. When you’re trying to build an entire theory on how something complex works you’ll have a much better time using the available abstractions to manage complexity without getting bogged down in implementation details.
Edit: addressing your point more directly, numerical computation itself must necessarily be done over fixed precision numbers but the tools we use to decide what and how to do that computation come out of theory done over the reals because of those specific properties of the reals. You can make things work over the rationals but the theory is tedious and the results of generally lower utility.
Any issues introduced by using a finite approximation to Pi will eventually be swamped by the uncertainty in the initial conditions. If there's no uncertainty in the initial conditions, there will still be some finite approximation to Pi that will give you results as accurate as you can measure...
-- Apologies to Kronecker