Surreal numbers and "zig-zag" shapes

Question

This question has been reworded.

1. Is there an experiment which can distinguish between mathematical models of physical space based on real numbers and models based on other types of numbers e.g. surreal numbers? If it exists, has it been performed and what are the results? The following paper on arXiv provides some physical consequences of using surreal numbers but none of them seem to be able to be tested experimentally: Some Mathematical and Physical Remarks on Surreal Numbers. Similar questions have been asked on StackExchange with different wording: Why model space with real numbers?, Justifying the use of real numbers for measuring length
2. When surfaces (in the mathematical meaning) are considered in physics, they are usually assumed to be smooth. E.g. a surface of identical electrostatic potential around a point particle is considered a smooth sphere. If we calculate the surface area of this sphere we will get the known result $4\pi r^2$. But if the surface is a really a "zig-zag" (examples of "zig-zags" are given here: https://www.youtube.com/watch?v=D2xYjiL8yyE) it can have a much different surface area. Even if for this particular example the identical potential sphere is a real sphere and not a "zig-zag", there are many other examples of mathematical surfaces in physics (e.g. event horizons, surfaces of identical probability in quantum mechanics etc). Is there an experiment which can distinguish between smooth surfaces and surfaces which are "zig-zags"? A slightly related, but different question has been asked here: Is the consideration of spacetime as a smooth manifold only an assumption?

For reference only, the original question is given below:

In physics it is usually stated that a particular piece of mathematics should not be applied unless there is an experimental confirmation. For this reason I have the following two questions:

1. What is an experimental confirmation that the physical space is based on real numbers and not e.g. surreal numbers?

2. What experiment has confirmed that all shapes considered in physics are not "zig-zag" shapes (by a "zig-zag" shape I mean a shape surrounded by a path similar to the ones shown here: https://www.youtube.com/watch?v=D2xYjiL8yyE). Is there an example of a shape that turned out to be a "zig-zag" shape? Isn't matter fundamentally "zig-zag" shaped, because of the atoms? For this reason, does it make sense at all to talk about surface areas in physics? I think some physical calculations are based on the concept of a surface area.

Answer 1

I have not seen anybody claiming you should not use a particular kind of mathematics unless there is experimental reasons to do so - after all, general relativity (using Riemannian spacetime) was introduced using thought experiments and then experimentally found to describe reality. Instead, what people tend to push is that you should not introduce more complex math than is needed to describe what we can observe (or think we can observe with a future experiment). Using surreal numbers in physics is making things overly complicated. This is basically Occam's razor.

Note that "simple" is sometimes contested. Does physics really run on continuous real numbers (or complex ones), or the apparently simpler countable natural or rational numbers? Maybe only computable numbers? Here what really matters is whether these choices of theory actually make a difference that could be noticed empirically, and whether they lead to more useful theories. Quantum mechanics "won" by showing that the chunkiness of quantization gave new properties that continuous spectra did not have, and these properties turned out to be measurable.

Answer 2

A physical theory is roughly composed of two objects : theoretical terms and observational terms[1]. The theoretical terms are composed of all entities that cannot be measured directly, such as the wavefunction, energy, etc, while the observational terms are the ones that can be measured directly, such as length.

As far as I can tell, observational terms are always real numbers, and even then always rational terms. I can't really measure an infinite quantity on some apparatus, nor a quantity with infinite precision.

On the other hand, the theoretical terms have no limitation as to what they are made of. And indeed, I've seen some attempts to use a variety of them, such as quantum field theory built from hyperreal numbers (although not surreal numbers, I'm not sure there is much benefit to it). The important part is that the rules of correspondance (the mapping from theoretical terms to observational terms) exist, so that if you have theoretical terms that aren't real numbers, they are mapped correctly to real observables.