you said a lot and i probably don't understand but doesn't pi contradict this? pi definitely exists in physical reality, wherever there is a circle, and seems to be have a never ending supply of decimal points.
Is there a circle in physical reality? Or only approximate circles, or things we model as circles?
In any case, a believer in computation as reality would say that any digit of π has the potential to exist, as the result of a definite computation, but that the entirety does not actually exist apart from the process used to compute it.
What does it mean to "exist in physical reality"?
If you mean there are objects that have physical characteristics that involve pi to infinite precision I think the truth is we have not a darn clue. Take a circle, that would have to be a perfect circle. Even our most accurate and precise physical theories only measure and predict things to 10s of decimal places. We do not possess the technology to verify that it's a real true circle to infinite precision, and many reason to think that such a measurement would be impossible.
The accuracy of their volume and radius did not reach the level of a 64-bit float, but it was several orders of magnitude better that of 32-bit FP numbers.
While you cannot build a thing made of molecules with an accuracy better than that of a FP64 number, you can have a standing wave in a resonator, which stays in a cryostat, where the accuracy of its wavelength is 4 orders of magnitude better than the accuracy of a FP64 number, and where the resonator is actively tuned, typically with piezoelectric actuators, so that its length stays at a precise multiple of the wavelength, i.e. with the same accuracy. Only the average length of the resonator has that accuracy, the thermal movements of the atoms cause variations of length superposed over the average length, which are big in comparison with the desired precision, which is why the resonator must be cooled for the best results.
However, it does not really matter whether we can build a perfect sphere or circle. What it matters that modelling everything while using a geometry that supposes the existence of perfect circles we have never seen errors that could be explained by the falseness of this supposition.
The alternative of supposing that there are no perfect circles is not simpler, but much more complicated, so why bother with it?
When talking about whether arbitrarily precise numbers are real in the universe, it extremely matters.
Sadly, atoms exist. In some ways that makes things more complicated, but it's the truth. Anything made of discrete chunks in a grid can't have arbitrarily precise dimensions.
Any rotating/spinning black hole will no longer be a perfect sphere.
But even then, the biggest black hole we think is possible measured down to the planck length gives you a number with 50 digits. And the entire observable universe measured in planck lengths is about 60 digits.
So how are you going to get a physical pi of even a hundred digits on the path toward arbitrary precision?
> > The idea of arbitrary precision is intrinsically broken in physical reality.
There is no contradiction here.