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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.

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  • $\begingroup$ The experimental evidence suggests that the reality is based on rational numbers, not real numbers. $\endgroup$
    – safesphere
    May 28, 2018 at 4:33
  • $\begingroup$ On what basis are you suggesting a connection between these "zig-zag" shapes and atoms ? $\endgroup$
    – StephenG - Help Ukraine
    May 28, 2018 at 4:50
  • $\begingroup$ Related: physics.stackexchange.com/q/33273/109928 $\endgroup$
    – Stéphane Rollandin
    May 28, 2018 at 5:49
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    $\begingroup$ @safesphere: Can you give more details, please? $\endgroup$
    – Mateusz Grotek
    May 28, 2018 at 9:55
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    $\begingroup$ "experimental evidence suggests that the reality is based on rational numbers" Nonsense. That we record finite precision from measurements with limited instruments does not say anything whatsoever about the nature of reality. In particular writing down a measurement of $287.35\,\mathrm{g}$ does not tell me that the sample had exactly that mass, but rather that the instruments determination of the mass of the sample was better represented by that figure than by any other I could have written down. On a digital balance it could have any value between $287.345$ and $287.355\,\mathrm{g}$. $\endgroup$
    – dmckee --- ex-moderator kitten
    May 28, 2018 at 16:14

2 Answers 2

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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.

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  • $\begingroup$ Yes, my argument should be based on Occam's razor - a piece of mathematics should not be applied unless it (1) makes correct empirical predictions, and (2) makes theory more simple. I am not sure if introducing e.g. surreal numbers or assuming "zig-zag" lines can create any testable empirical predictions which differ from the usual definitions, but I have found the following paper mentioned in the update of the question. $\endgroup$
    – Mateusz Grotek
    May 28, 2018 at 10:25
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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.

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  • $\begingroup$ Thank you for your answer. I think that directly measurable quantities are not even real numbers but actually only rational numbers. But it is possible for a theory to make real numbers or e.g. surreal numbers necessary at some deeper level for different reasons. E.g. if we assume only rational numbers then the spatial symmetries might break down, we cannot apply calculus etc. I wonder if there are any deeper reasons for e.g. surreal numbers, but it is a separate question. $\endgroup$
    – Mateusz Grotek
    May 28, 2018 at 10:20
  • $\begingroup$ It is never necessary. There always exists a way for a theory to get rid of theoretical terms (the Craig reaxiomatization). It may be more practical to use surreal numbers, but as far as I know, I can't really see any practical applications for them that couldn't be done with a simpler system. $\endgroup$
    – Slereah
    May 28, 2018 at 12:31
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    $\begingroup$ To assign measurements to the category of rational number is to make the error of ignoring the uncertainty that comes with them. I get it that "we write down a terminating decimal fraction number, so that's a rational". But in fact we should understand every measurement as having an uncertainty: these are not simple numbers no matter how we treat them in the classroom, and when you begin philosophising about their meaning you should remember that. $\endgroup$
    – dmckee --- ex-moderator kitten
    May 28, 2018 at 16:19
  • $\begingroup$ @dmckee I think it is clear that what can be observed are results of measurements by using some measurement devices. All such results are at most rational. In my opinion the theory of measurements and errors is also a part of the model. $\endgroup$
    – Mateusz Grotek
    May 29, 2018 at 1:01
  • $\begingroup$ @dmckee Taking simple geometry as an example, a fully and correctly stated physical theory of physical geometry (not purely mathematical) will tell us for example, that if we draw a line of (1.00±0.05) cm (measured with a standard ruler) and we construct a circle with this line as a radius (by using a sufficiently accurate tool), and then we measure the circumference of the circle, the result will be (6.3±0.4) cm (measured with an opisometer of sufficient accuracy). $\endgroup$
    – Mateusz Grotek
    May 29, 2018 at 1:01

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