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Can the laws of physics and fundamental constants of nature exist without fundamental mathematical constants, operators, and equations also existing?

In other words, can there be fundamental physical laws and constants in physics without mathematical realism being true?

Doesn't the ubiquitous presence of mathematical constants such as π (pi), e (Euler's number), and i (the imaginary unit) in physics suggest that some form of mathematics exists in a fundamental sense, and therefore that mathematics is discovered rather than invented?



Appendix


Examples of fundamental physical and mathematical constants appearing together in Physics

Einstein's field equation

Rμν - (1/2) R gμν + Λ gμν = (8πG/c4) Tμν

Where:

  • Rμν is the Ricci curvature tensor.
  • R is the scalar curvature.
  • gμν is the metric tensor.
  • Λ is the cosmological constant.
  • G is the gravitational constant.
  • c is the speed of light.
  • Tμν is the stress-energy tensor.
  • π is the mathematical constant representing the ratio of a circle's circumference to its diameter.

The first law of black hole mechanics

δM = κ/8π δA + Ω δJ + Φ δQ

Where:

  • π is the mathematical constant representing the ratio of a circle's circumference to its diameter.
  • δM is the change in mass of the black hole.
  • δJ is the change in angular momentum of the black hole.
  • δQ is the change in charge of the black hole.
  • δA is the change in area of the black hole's event horizon.
  • κ is the surface gravity of the black hole.
  • Ω is the angular velocity of the black hole's horizon.
  • Φ is the electrostatic potential at the horizon.

Time-dependent Schrödinger Equation

iħ ∂ψ(x, t)/∂t = -ħ^2/2m ∂^2ψ(x, t)/∂x^2 + V(x, t)ψ(x, t)

Where:

  • i is the imaginary unit, representing the square root of -1.
  • ħ is the reduced Planck constant, equal to h/2π, where h is Planck's constant.
  • ∂ψ(x, t)/∂t represents the partial derivative of the wavefunction ψ with respect to time t. It describes how the wavefunction changes over time.
  • ∂^2ψ(x, t)/∂x^2 represents the second partial derivative of the wavefunction ψ with respect to position x. It describes how the curvature of the wavefunction changes with position.
  • m is the mass of the particle. It determines the particle's response to changes in the potential energy.
  • V(x, t) is the potential energy function. It describes the potential energy experienced by the particle as a function of position x and time t.
  • ψ(x, t) is the wavefunction of the particle. It is a mathematical function that describes the quantum state of the particle, encoding information about its position, momentum, and other physical properties.

Euler's formula and the wavefunction equation

Euler's formula:

e^(iθ) = cos(θ) + i sin(θ)

Where:

  • e is Euler's number, the base of the natural logarithm, approximately 2.71828.
  • i is the imaginary unit.
  • θ is the angle in radians.
  • cos(θ) is the cosine function, representing the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
  • sin(θ) the sine function, representing the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Wavefunction equation

ψ(x, t) = A e^(i(kx - ωt))

Where:

  • ψ(x, t) is the wavefunction.
  • A is the amplitude.
  • k is the wave number.
  • ω is the angular frequency.
  • x is the position.
  • t is the time.

Additional food for thought

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  • 1
    Beautiful question. I would love to know what most expert physicists think about this one. I'm sure there's some disagreement, and a lot of uncertainty, but which way does their intuition lean?
    – TKoL
    Apr 2 at 9:18
  • 1
    As someone who's only into novice physics as a hobby, my intuition tells me that mathematical truths are more deeply fundamentally true than even physical truths. It is more deeply true that 2*3=6 than it is true that I have five toes on my left foot.
    – TKoL
    Apr 2 at 9:20
  • 4
    There are many forms of "mathematical realism", from Platonism to Aristotle's hylomorphism and its modern descendants in the spirit of property dualism, structuralism or Husserlian ideation. What "exists in a fundamental sense" means is very obscure, in some of those one could say that laws and relations exist in a derivative sense, through physical manifestations only. As for ubiquitous presence of mathematics, nominalists have their own explanations in the spirit of Kant - what we consistently find in our experience is what our own cognitive apparatus and practices put there to shape it.
    – Conifold
    Apr 2 at 9:42
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    Are you asking "if 'physics realism' is true, must mathematical realism also be true"? As in: if physics laws "actually exist", do maths concepts also "actually exist". Or are you asking whether "if the universe behaves in consistent ways that we can model using physics, must mathematical realism be true"? I don't think either form of realism is true. The universe does whatever it does, and that's independent of our mathematical or physics models that describe what it does (and that doesn't need to "actually exist").
    – NotThatGuy
    Apr 2 at 9:52
  • @NotThatGuy I'm happy with both interpretations (they are both quite insightful).
    – Mark
    Apr 2 at 10:04

2 Answers 2

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As is the case with many questions on this and related topics, the real difficulty in making any progress is that the word 'exist' and phrases such as 'exist in a fundamental sense' are inherently vague. To take General Relativity, clearly yes, the relationship summarised by the field equation between the various physical quantities is a relationship that exists in the Universe. But there are countless mathematical relationships that don't have counterparts in the physical Universe. For example, if you replace the number 8 in the field equations by some arbitrary function, you will produce a different set of field equations that describes nothing real, as far as we know.

What you can say quite definitely is that we live in a Universe which displays a high degree of conformity to certain patterns or relationships that can be described mathematically, and that the language of mathematics can be used to describe other patterns and relationships which are not observed. Whether or not you then want to represent that by saying mathematics is 'real', seems to me to be just a question of semantics.

However, if you do decide to justify treating mathematics as 'real' because of its correspondence with the physical Universe, where does that leave all the other kinds of mathematics that has no bearing whatsoever on the Universe? You can, for example, define mathematically consistent geometries that do not correspond to reality- are they, then, 'unreal' on the same grounds that you have decided other maths is real?

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    regarding the last bit about different geometries, I guess one could make the case that if they can somehow be reduced to/interpreted in some arithmetic theory we regard as 'real', then they should also be regarded as real, at least 'virtually real'
    – ac15
    Apr 2 at 20:55
  • I really appreciate your comment on "existence." I'd say something like "the relationship appears to be instantiated in reality". This just means that we can experience that instantiation. The term "exist" is too loaded. The risk is that either the user has an atypical/obscure concept of existence or they mean a common or naïve sense of existence without thinking about it carefully.
    – jdods
    Apr 4 at 1:19
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    @ac15: Correct. Exactly the same reason why ACA0 can be regarded to have genuine meaning in reality even if one is a finitist who only accepts potential infinity, because the sets ACA0 asserts to exist can be interpreted as nothing more than arithmetical properties, which are mere syntactic objects.
    – user21820
    Apr 23 at 7:52
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Can the laws of physics and fundamental constants of nature exist without fundamental mathematical constants, operators, and equations also existing?

Kinda depends on what in particular you mean by "the laws of physics" and how the nature of reality actually IS (as opposed to what we assume it to be), but theoretically: yes.

For example, picture a weirdly shaped potato showing next to no perfect symmetry or neat regularity that you'd expect from math and you could still think of it and describe it with a perfect sphere. It won't be perfect, but a) it doesn't need to be and b) none of our physical laws actually are, they all come with the asterisk of (* within a margin of error). It's only really the theory that is neatly mathematical, though whether the theory is a close enough approximation of reality or is an actual analog of reality, describing and not just approximating it, is yet to be determined.

So sure if the nature of reality actually follows patterns and is based upon mathematical concepts and fundamental constants then physics would merely discover math. But if the reality is fuzzy, ugly and chaotic, then physics would merely apply the theoretical construct of math in order to... well "order" it. You know make it neat, explainable, regular and so on.

And π and e are really neat. Like π is basically your mathematical description of something that is cyclical. Whether that circle happens in space, in time (periodic repetition) or in space and time (spirals and waves). And if you know that a periodically swinging pendulum or a "harmonic oscillator" is basically THE go to toy example to start the description of just about anything in theoretical physics, it really comes as no surprise that π is EVERYWHERE.

And it makes sense if you picture any kind of attractive force, you could picture it as a valley, so what happens if you push an object out of it's equilibrium state of resting at the bottom of the valley? Right, it will start oscillating like a pendulum and if dampened after some time return there.

So whether you go to classical mechanics, electrodynamics or even quantum mechanics you'll always find your harmonic oscillators as the most basic assumption.

Also guess what the roots of unity amount to, which can be expressed as complex numbers (those with the "i"): Cyclical groups

And if you haven't guess it already, guess what Euler's formula is: https://en.wikipedia.org/wiki/Euler%27s_formula#/media/File:Euler's_formula.svg

Cycles, cycles and even more cycles. If you now link that to the golden ratio that apparently is linked to how aesthetically pleasing we perceive something: https://arxiv.org/pdf/2301.09643.pdf It really comes as no surprise why we see it everywhere.

Now to be fair "e" is just generally awesome as it's not only cyclical with regards to it's combination with "i" and π but also with regards to it's derivative. So the derivative of e^x is e^x. Which is literally rocket science. As it allows for solving differential equations (such as the rocket equation) or as radioactive decay where the rate dN/dt of change of material N depends on the amount of material to begin with so dN/dt = const * N. Or in case of rockets the problem that the speed of the rocket depends on the mass of the rocket and the acceleration, which depends on the rate of fuel being burned and exhausted which depends on the amount of accessible fuel, which contributes to the mass of the rocket that should be propelled upwards.

So if you look at it only qualitatively you reach a paradoxical state where the more fuel you burn the faster the rocket goes, but the more heavy your rocket is, the slower it goes. So that the fuel that makes the rocket go faster (by being burned) also makes it slower (by adding to it's mass).

So yeah it is possible that we see it everywhere because it actually is everywhere and because the underlying structure of reality is like that. Or we could see it everywhere because we WANT to see it everywhere because it's something that we've understood, that is useful or at least that gives an aesthetically pleasing impression of perfection.

The difference is that for physics, it's an assumption, for math it's a certainty. The laws of physics are our best guess at how the world works and if they fail, they simply weren't good enough and we need to find better ones, while for math, they kinda would be derived from the axioms or are themselves axiomatic, so they can't really be incorrect, but if they are, they'd simply describe an "alternate reality" (that overlaps but is not identical to ours).

Also with regards to circles, do perfect circles actually exist in nature? And I don't mean really really really good circles, but PERFECT circles, like mathematically perfect circles?

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  • Nice +1. But what's the last para claiming? I would have thought it's a trivial No. I mean what's a widthless line to start with?! You must mean something more than that...
    – Rushi
    Apr 2 at 12:09
  • Math is the set of numbers that map to the real number line sequence and everything else is a relation that maps to a number on the number line. It is kind of weird 1/2 = 0.5 is just two numbers that map to another number on the number line. Pi is just a ratio to an inexact number on the number line and e is another special number on the number line. Figuring out why those numbers seem to be inherent in reality rather than special ratios in our mind is above my paygrade. Are numbers and relations only ideals in our minds? Resonance of oscillator(s) maps to real numbers via dynamic unit circle. Apr 2 at 18:37
  • @SystemTheory In math, we've got plenty of objects which don't map to real numbers. Consider, for example, the set of all functions from the reals to the reals. Apr 2 at 19:01
  • @Holden Rohrer - Point taken. I don't know about functions from the reals to the reals. I know an operational amplifier can be designed to change the slope of a line from input to output which maps voltage as real numbers to other real numbers in a so-called analog computer. I wonder if relations that do not map to real numbers are derived from similar number relations anyway such as square root of negative one is not a real number. But we find it useful in the context of dynamic relations such as phase relations in electricity or other rotating machines. Some folks say reality is a computer. Apr 2 at 19:39
  • @SystemTheory The 3-adic numbers do not correspond to the reals. Neither do the complex numbers – though you can think of a complex number as a pair of reals, if you like.
    – wizzwizz4
    Apr 4 at 21:40

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