Can the laws of physics and the constants of nature exist in a fundamental sense without mathematical realism?

Question

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?

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Appendix

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Examples of fundamental physical and mathematical constants appearing together in Physics

Einstein's field equation

R_μν - (1/2) R g_μν + Λ g_μν = (8πG/c⁴) T_μν

Where:

• R_μν is the Ricci curvature tensor.
• R is the scalar curvature.
• g_μν is the metric tensor.
• Λ is the cosmological constant.
• G is the gravitational constant.
• c is the speed of light.
• T_μν 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.

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Additional food for thought

• e is everywhere: https://www.nature.com/articles/s41567-019-0655-9
• On the Cosmological Significance of Euler’s Number: https://www.ptep-online.com/2019/PP-56-03.PDF
• Significance of π in physics: https://physics.stackexchange.com/questions/161183/significance-of-pi-in-physics
• Was mathematics invented or discovered?
• Is the (surprising) applicability of mathematics to the physical world a brute fact or something crying out for a (theistic) explanation?

Answer 1

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?

Answer 2

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?