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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.

Urb
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Mateusz Grotek
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  • The experimental evidence suggests that the reality is based on rational numbers, not real numbers. – safesphere May 28 '18 at 04:33
  • On what basis are you suggesting a connection between these "zig-zag" shapes and atoms ? – StephenG - Help Ukraine May 28 '18 at 04:50
  • Related: https://physics.stackexchange.com/q/33273/109928 – Stéphane Rollandin May 28 '18 at 05:49
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    @safesphere: Can you give more details, please? – Mateusz Grotek May 28 '18 at 09:55
  • @StephenG: E.g. if we assume atoms are e.g. approximately ball-shaped (I know it is not true for many reasons (QM included) but it doesn't make a difference, unless they are e.g. cuboids etc.), one cannot pack balls to create a flat surface. So if you calculate the surface area of a body it is not the surface area of a plane figure, but slightly bigger. In real life the difference is even bigger because of complicated shapes of molecules, defects and many other reasons. – Mateusz Grotek May 28 '18 at 09:56
  • @StephenG I understand that for many applications of surface areas it does not make any difference (because what we are really concerned with is not a surface area, but e.g. a flux), but maybe there are some for which it does. Unfortunately I cannot give any examples for this at the moment. – Mateusz Grotek May 28 '18 at 09:56
  • @safesphere I'm intrigued and would like more information about this bold assumption as well. It seems to contradict that a right-angled triangle with unit sides has hypotenuse $\sqrt{2}$. If I draw one on a piece of paper it seems fairly real. – Sputnik May 28 '18 at 10:37
  • @Sputnik You can't measure a real triangle with abstract symbol like $\sqrt{2}$. In reality you measure using a ruler (e.g. laser based) that gives a result in a notation with some basis, such as binary or decimal. When you express your hypotenuse in a decimal notation, it is a rational number. When you program anything in a computer, you use rational numbers. There are no real numbers in physical reality, only rational numbers. 1.414213562373095 is a rational number an you'll never be able to measure your triangle so precise. – safesphere May 28 '18 at 13:58
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    "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 $287.355,\mathrm{g}$. – dmckee --- ex-moderator kitten May 28 '18 at 16:14
  • This can be of some interest: https://arxiv.org/abs/1803.06824 – EigenDavid May 29 '18 at 14:30
  • Related MO.SE question: https://mathoverflow.net/q/63320/13917 – Qmechanic Jun 02 '18 at 10:54
  • I feel like this question, both before and after rewording, is based on a false premise. There are plenty of hints that the differentiable-manifold picture of spacetime breaks down at some point, possibly around the Planck scale. If it breaks down at any scale, then physics with real or surreal numbers is just an approximation, and picking one over the other is a matter of convenience/familiarity, not correctness. – benrg Jul 12 '22 at 18:27
  • The question is whether the quantum theories (e.g. the quantum field theory or string theories) use real numbers, smooth surfaces etc. Even if the spacetime isn't a differentiable manifold, are e.g. strings, loops, or whatever the basic objects of the theory are modeled using reals (and not e.g. surreals) and/or smooth thihngs (and not e.g. zig-zags)? – Mateusz Grotek Jul 29 '22 at 20:25

2 Answers2

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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.

Anders Sandberg
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  • 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. – Mateusz Grotek May 28 '18 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.

Slereah
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  • 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. – Mateusz Grotek May 28 '18 at 10:20
  • 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. – Slereah May 28 '18 at 12:31
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    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. – dmckee --- ex-moderator kitten May 28 '18 at 16:19
  • @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. – Mateusz Grotek May 29 '18 at 01:01
  • @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). – Mateusz Grotek May 29 '18 at 01:01
  • @dmckee In other words the input data are rationals with some uncertainty. The theory produces some results based on the inputs. Another part of the theory (uncertainty propagation theory) tells us how accurate are these predictions. Then we measure the result and compare with the predictions and check if it is inside the range of uncertainty. – Mateusz Grotek May 29 '18 at 01:02
  • Also the circumference of the circle can't be known that wy without assuming the geometry of space – Slereah May 29 '18 at 05:03
  • @MateuszGrotek Again, simply because the recorded measurement is written in terms of rationals does not mean (a) that a measurement is a rational (it is a distribution) or (b) that the underlying physical quantity is a rational (you do not know which value in the range that our knowledge allows is correct, and irrationals are among those allowed (indeed there are more of them than rationals). – dmckee --- ex-moderator kitten May 29 '18 at 05:13
  • @dmckee I think that you use a word "measurement" in a slightly different meaning than the one I use. They are both ok, but we need to agree about the terms if we want to discuss them first. For me measurement is the value shown by an instrument together with the accuracy of the instrument. We both agree that this is and can only be rational. – Mateusz Grotek May 29 '18 at 10:01
  • @dmckee It means the input data for calculations in the theory are rational as well. The "underlying physical quantity", from my perspective, seems to be a theoretical term, not an observational term. I totally agree that irrationals are in the range, but surreals are in it as well. – Mateusz Grotek May 29 '18 at 10:02
  • @Slereah Actually the theory I had in mind assumes that the space is close to Euclidean in the small scale, so this measurement can falsify the theory if it turns out that the circumference is not (6.3±0.4) cm but something else. This toybox example is quite nice for this reason as well. – Mateusz Grotek May 29 '18 at 10:05
  • @Slereah I have not answered to your original comment yet, so please, let me do it now. You are right that no theoretical terms are necessary, and they can be removed e.g. by Craig's reaxiomatization. But there is also a different issue. If we modify the theory by changing some parts of it e.g. real to surreal numbers, it might turn out that the new theory gives some different results for observables. – Mateusz Grotek May 29 '18 at 10:12
  • @Slereah Of course you can remove theoretical terms from both theories concerned (with reals and with surreals), but it does not mean the resulting theories will be the same. That's why it makes sense to consider what impact surreals have for the observationals, and that is actually a part of my original question, because I asked if there is an experimental confirmation. I should probably ask a different question. Is there an experiment which can distinguish the theory with reals from the theory with surreals. If not, then your reasoning is correct and it does not matter. But maybe there is. – Mateusz Grotek May 29 '18 at 10:14