All this assumes that numbers do exist in the physical world as some quantities. They don't. Two apples do not bring the number 2 into existence. Our perception of multiplicity at a macro-scale, is at the root of counting numbers, and it depends on defining the inside and outside of a thing. If you cut those 2 apples into 4 pieces, do the 2 apples still exist? If we grind the pieces into paste, do you still identify them as 2 apples?
Also, cuts are just another way of talking about infinity, which falls outside of the domain for logical foundations of mathematics or the physics of universe. So, cuts are just as alien as infinity.
The logical foundations require existence of multiplicity, comparison (smaller, greater), time-driven causalitiy, true/false and existence to be distinct from non-existence etc. All these would fail when you bring in things like cuts or infinity. So no point in putting them to the logic of our world and claiming that we understood something.
You're criticizing a strawman of Platonism. The people who believe that numbers such as 2 do exist as some quantities are Platonists, and they do not believe that numbers exist in the physical world. Rather, Platonists believe that the way the physical world exists is just a pale shadow of the way numbers like 2 exist.
Separately, you are advocating finitism, though apparently without much understanding of the logical issues involved, and you seem to be conflating formalism with finitism.
Nobody has come up with a convincing proof of either mathematical Platonism or finitism or their inverses. Consequently, plenty of mathematicians and logicians working at the frontiers of their fields subscribe to any of the four possible combinations of these doctrines.
Nobody came up with a proof of finitism because the semantics of a mathematical proof can forbid an expression of reality of the physical world. Physical world abhors infinite and mathematics was supposed to be a tool to describe the physical world. Instead, math went into hallucinations about infinity etc. Physics works perfectly well even when you replace infinite with a relatively large number, giving you as much precision as you need. Physics never needed PI to be calculated to millions of digits, or multiple infinities to be defined. Computers are working fine without this poetry of infinite. I'm not sure why infinity can't be redefined as a relatively large finite number.
> Physics works perfectly well even when you replace infinite with a relatively large number
Has this actually been studied? I reckon you'd run into problems pretty quickly if you tried to impose a "maximum" allowed number (call it N). For example, you might set N equal to the number of quarks in the universe. But now you have no way to represent the number of different configurations of quarks, taking into account their possible quantum states (i.e. taking the factorial). But you also have a time dimension (which I assume you're splitting into discrete chunks to avoid infinitesimals). To decsribe the quantum state over time requires your previous estimate multiplied by the number of time points. And so on... ad infinitum!
I can't make any sense of your comment; it seems like it has missing words, circular logic, and fundamental misconceptions, but it's not really coherent enough to tell.
Also, cuts are just another way of talking about infinity, which falls outside of the domain for logical foundations of mathematics or the physics of universe. So, cuts are just as alien as infinity.
The logical foundations require existence of multiplicity, comparison (smaller, greater), time-driven causalitiy, true/false and existence to be distinct from non-existence etc. All these would fail when you bring in things like cuts or infinity. So no point in putting them to the logic of our world and claiming that we understood something.