I'm very far to be competent enough to challenge you (and also to fully understand you), but just to have a clearer picture, could you give me an example in which a natural system needs, say, exactly 197/599 otherwise "something massively important fails"?

In other words, are you trying to disregard irrational numbers by saying that rational numbers are in some way of higher value with regard to physics or are you de-valuing numbers in general?

I am simply asking where in physics something being irrational and not a very accurate to a degree rational approximation makes a difference in the theory.

If we have 197 positively charged ions at location A and 599 negative ions in position B (each plus minus 1 e) separated by 10000 units of length, if we release an electron at position 3644-3645 randomly in that range it will go left most of the time. If we change the 197 or 599 by one either or both, this is no longer true, it will almost always go left or almost always right from a release between 3644 and 3645 depending on what we did to the charges. (or fix the numbers properly if i missed something)

If the electron goes left after 1000 electron releases a majority of the time a trigger starts an explosion, if it doesnt, it wont trigger it!

You change the ratio of charges to something else and it fails to happen.

Do the same with torque in some balance and small balls of similar mass that is within 10^-5 of the standard unit of mass correct.


The problem is really that i have discrete entities anywhere all the time. But where is something like precisely 2^(1/2) and not say a 300 digits rational approximation of it fails the physics?



I honestly also wish to understand in math where remarkable results in number theory or other more tangible conjectures proven finally, higher levels of infinity made the difference beyond the infinity of natural numbers.

BruceZ in emails recently that we discussed some of these things have linked me to an interesting paper about this that i want to understand better.

https://case.edu/artsci/phil/Proving_FLT.pdf

I really do want to appreciate how higher levels of infinity have made the difference in math in areas that seemingly do not have anything to do with higher infinity. I am also curious about the physics theories because i do not yet have any example (to my knowledge or memory at least) that accepting existence of infinity made a difference.

I think the justification for doing math the way it's commonly done today is that it makes the math easier to do. So if we did math differently we simply might not have many of the mathematical results we have now that prove so useful in other fields like physics. While the theoretical results provided by the math may only live in the abstract rather than reality, they point to the numerical methods that are required in any case to get usable numbers.


PairTheBoard

I think some rather exotic math involving infinity is probably involved before you invest billions and build LIGO and win a Nobel.

I'm not sure I've seen anyone answer, but yes, it can be Aleph_(any natural number).

In fact, it can be aleph_(any ordinal whose cofinality is not omega)

Where omega is the size of the natural numbers. This result is a pretty straightforward consequence of forcing, if you look into it.