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Given an arbitrarily chosen constant of nature (say, the speed of light c), we can confidently say that the fact that it is equal to 299 792 458 meters per second is a contingent fact about our universe (in other words, it is logically possible for c to equal some other value). In fact, it is logically possible for any constant to be equal to any positive real number other than the one it has in the actual world.

Let's also take into consideration the fact from mathematics that if you chose a random positive real number, the probability of choosing an algebraic number is exactly zero.

Given these two premises, does it follow that all of the nature's constants are not algebraic, i.e. transcendental (and hence irrational)?

EDIT: To make things more robust, let's make the following assumption: A Theory of Everything exists, and we are talking about it, it only and its constants.

NOTE: Constants varying with time don't change the essence of the question - if the constant is changing with time, then my question is about its value at a particular point t in time.

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    Constants of nature are measured: thus they are rational. The only way to assert that a number is irrational is through a mathematical proof (like for the square root of 2 and e). – Mauro ALLEGRANZA Jan 19 '18 at 10:05
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    @MauroALLEGRANZA I am adhering to a form of realism here, meaning that constants of nature have some specific value regardless of our measuring them, meaning they can in fact be irrational. – user30487 Jan 19 '18 at 10:09
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    See speed of light: "The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its exact value is 299,792,458 metres per second (approximately 3.00×108 m/s, or 300,000 km/s (186,000 mi/s)). It is exact because the unit of length, the metre, is defined from this constant and the international standard for time." – Mauro ALLEGRANZA Jan 19 '18 at 10:12
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    @MauroALLEGRANZA I am talking about the constants of nature, and you are talking about the definition of a meter. – user30487 Jan 19 '18 at 10:13
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    @FrankHubeny you are indeed making valid points, I would love to hear a full answer from you. If the constants are variable, this doesn't change the question at all - the question will then be about the value of the constant at any given point t in time. I have edited my question. – user30487 Jan 19 '18 at 13:26
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First, the fine-tuning argument suggests that the tunable constants in the Standard Model have only got a narrow range of variation if a universe capable of bearing life is to emerge. So they can't quite be anything.

Second, before we can find out whether a number is irrational we must measure it to infinite accuracy. In mathematics, this is a given. In physics, this is not so. A measurement with infinite precision needs to be carried out.

Thus, given the limits on accuracy given by physical measuring instruments and the actual indeterminacy at the Planck scale it seems to me that to ask whether physical constants are rational or irrational is not a physically meaningful question.

Still, it's an interesting question.

  • "So they can't quite be anything." - it is logically possible for them to be anything. "A measurement with infinite precision needs to be carried out." - I am not concerned with measuring these constants in the real world, I am talking about their exact values in a platonic sense. About your second to last paragraph - according to the mainstream physics, space is continuous, not discrete. Also, what the Planck scale really represents is debatable – user30487 Jan 19 '18 at 9:01
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    @Alex Adamov: What makes you think that they can be so concieved? It's more likely given what physics has found out after the that advent of QM that even platonically speaking, that physical numbers do not have infinite precision. – Mozibur Ullah Jan 19 '18 at 9:06
  • Because there is no logical contradiction in conceiving them so. Elaborate on the QM bit, because I do not agree with you on that either. – user30487 Jan 19 '18 at 9:11
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    @Alex Adamov: yeah, sure - I'm not sure why you feel philosophy ought to be reduced to logic, that merely waving the logical wand makes things philosophical. Kant would say, judgement is far more important. Well, that's no surprise since these are difficult questions without much guidance from experiment since we're talking about the level of structure round about the Planck scale. – Mozibur Ullah Jan 19 '18 at 9:39
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    @Alex Adamov: I can't say a vox populi is really relevant. I'd rather go by what the ongoing current research programmes say: on that basis, most say that the usual continuum description is suspect. Philosophically it's suspect too, what physical presence can a mere point actually have? At most it represents position and not what is. In some sense, mereology which argues for parthood relationship has merit, rather than the set theory with its elements that the real number line is based upon. – Mozibur Ullah Jan 19 '18 at 10:27
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There are two approaches to our physical theories and reality. Either they are ways for us to discover reality or they only model reality. The answer to whether constants in these theories are irrational or rational numbers depends on which view one takes regarding the relationship between physical theories and reality.

Belief in a Theory of Everything that can be discovered about reality would assume that the constants are discovered and that they are real. Although we may not know their exact values we may as well assume that they are irrational numbers because there are uncountably many of these and only countably many rationals. Even if we discover that the constants are changing, they are more likely irrational numbers for the same reason.

One of the goals of science would be to find exactly what the values of those constants are. Attempting to find more precise values for these constants would challenge the technology used to make measurements. It would be like finding the next largest prime number.

In contrast assume there does not exist a Theory of Everything. What can one say then? Then our physical theories are only models that approximate reality. The constants are only useful in our models which are trained on data that we collect. We don’t want to fit that data too tightly or we will not be able to make good predictions using the model on other data. If that is true, then these constants are all rational numbers based on how precisely we make our measurements and how tightly we fit the training data. We can only make those measurements to a finite number of decimal places. Hence they would be only rational numbers.

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The question is based on deficient knowledge of contemporary metrology. As can be read in the site of the Bureau International des Poids and Mesures [1], the units of time (second) and length (metre) are currently defined such that:

  • the ground state hyperfine splitting frequency of the caesium 133 atom is exactly 9 192 631 770 hertz ([hertz] = [1 / second]),

  • the speed of light in vacuum c is exactly 299 792 458 metre per second.

Moreover, in the revised International System of Units (SI) expected to come into force on 20 May 2019 [2], the units of mass (kilogram) and electric charge (coulomb) will be redefined such that [3]:

  • the Planck constant h is exactly 6.626 070 15 x 10^(–34) joule second,

  • the elementary charge e is exactly 1.602 176 634 x 10^(–19) coulomb.

A consequence of this redefinition will be that the vacuum permeability or magnetic constant mu_0 will no longer be exactly 4 x pi x 10^(-7) H/m. So, the actual case is that a physical constant will no longer be trascendental.

References

[1] https://www.bipm.org/en/measurement-units/

[2] https://www.bipm.org/en/measurement-units/rev-si/

[3] http://iopscience.iop.org/article/10.1088/1681-7575/aa950a/pdf

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