From Prime Numbers to Physical Laws - Arbitrariness or Inevitability?

posted: 31-Jan-2025 & updated: 10-Feb-2025

In the movie “Contact,” (starring Jodie Foster, one of my favorite female Hollywood starts), extraterrestrial beings communicate with Earth using pulses that represent prime numbers. This choice is profoundly significant: even in a universe where different physical laws might reign, the concept of prime numbers would remain unchanged.¹ Prime numbers represent a truth that transcends (arbitrary) intelligent beings’ civilization – whatever we mean by intelligent beings.

This observation leads us to a deeper philosophical inquiry about the nature of truth itself: what is inevitable or universal, versus what is merely coincidental or arbitrary? While this distinction extends beyond physics, our physical laws provide an excellent example sufficient for clearly explaining my arguments (if you know what I mean).

Number 2 - mathematical truth vs physical law

(Here I really mean by mathematical truth is some universal or inevitable truth not necessarily pertaining to the mathematics.)

Prime numbers can be universally defined. Two civilizations, having never interacted, would inevitably arrive at the same understanding of prime numbers.² Any intelligent beings capable of understanding quantity would ultimately arrive at the same mathematical truths.

Contrast this with Newton’s law of universal gravitation. This law states that any two objects with mass exert a pulling force on each other, proportional to each of both masses and inversely proportional to the square of the distance between them, that is,

\[F = Gm_1m_2/r^2\]

where \(m_1\) and \(m_2\) are the masses of the two objects, \(r\) is the distance between them, and \(G\) is the (Newton’s) gravitational constant. Unlike prime numbers, this law isn’t inherently inevitable. A certain universe could exist where the force was inversely proportional to \(r^3\) or \(r\), rather than \(r^2\) (if such a force exists).

So in a sense, this number 2 is not inevitability. Of course, it could be argued that, for example, if this number is greater than 2 by any amount, the universe would collapse immediately after the Big Bang, or/and if the number if less than 2 by any amount, the universe would explode, hence we would not exist now. However, let us postpone the discussions of such matters for now (to make the main point I’m trying to make here).

However, there might be something deeper about this exponent of 2!

Inverse square laws

Now let us move our attention to the electric force which enjoys (mathematically) identical properties to those of the gravitational force except that it has two types of forces; pulling and pushing.

Now consider the Gauss’s law, one of Maxwell’s four fundamental equations of electromagnetism, in both integral form and differential form:

\[\int_{\partial V} E \cdot dS = \frac{1}{\epsilon} \int_V \rho dV \quad \nabla \cdot E = \frac{\rho}{\epsilon}\]

where \(E\) is the electric field, \(\rho\) is the charge density, \(\epsilon\) is the permittivity, the integral in the LHS of the integral form is the surface integral, and \(\partial V\) is the boundary of the solid volume, 3-dimensional area, \(V\).

Here the Gauss’s law, in its integral form, tells us something remarkable. Consider the electric field \(E\) created by \(q_1\) at a distance \(r\). To calculate the magnitude of this field, let \(V\) be the volume of the sphere centered around \(q_1\) with radius \(r\). Then the Gauss’s law implies

\[4\pi r^2 E = q_1/\epsilon\]

Now suppose two electric charges \(q_1\) and \(q_2\). Because (by definition) the electric force exerted on \(q_2\) is \(q_2\) multiplied by the electric field induced by \(q_1\), we have

\[F = q_2E = q_1q_2/4\pi \epsilon r^2\]

The very interesting aspect here is the appearance of \(r^2\). While there’s no a priori reason why electric forces should follow the same distance dependence as gravity, both end up with this same exponent. This isn’t merely coincidental (at least for the electric field) – it emerges from the geometric fact that surface area scales as the square of length. This geometric truth would hold in any possible universe, regardless of its physical laws.

(By the way, just for fun, I could make up Sunghee’s law in for the gravitational force as follows:

\[\int_{\partial V} H \cdot dS = 4\pi G \int_V \sigma dV \quad \nabla \cdot H = 4\pi G \sigma\]

where \(H\) is the gravitational field, \(\sigma\) is the mass density, and the gravitational force is defined by \(H\) times a mass. I can make such laws for any such physical or imaginary quantities following the inverse square law. )

Therefore I argue that, on one hand, this number 2 seems to be quite arbitrary (if we do not involve any types of religious arguments, for example, the universe is defined and created by the God, hence the choice is 2 is not by coincidence or arbitrariness, but inevitability dictated by the God, for the sake of the discussion that I want to do here).

On the other hand, however, the number 2 may be the inevitability because it has something to do with the fact that the surface area scales as the square of length which is absolutely independent of the (physical) specificities of the specific universe we live in or for that matter, any intelligent live or non-intelligent beings exist.

The bridge between universal and physical truth

This reveals something profound: while we can never prove (e.g., through measurement) that gravitational or electric forces follow exactly an inverse square law (we can only establish it to some finite precision), the appearance of the square term might have a deeper necessity rooted in the geometry of space itself.

Note that physical laws, in general, can never be “proved” in the same way mathematical theorems can. We might show through extremely precise measurements that the gravitational force lies between, for example, \(Gm_1m_2/r^{1.9999999999999}\) and \(Gm_1m_2/r^{2.00000000000001}\), but we can never empirically prove it’s exactly \(r^2\). Physics, in this sense, is always based on belief supported by observation and experiment. That is, we (and physicists) somehow believe that the force would exactly follow the inverse square law.

However, the recurrence of the square term across different physical laws suggests a bridge between physical and mathematical necessity. The fact that area is length multiplied by length isn’t a coincidence or a physical law – it’s a mathematical truth as universal as prime numbers.

Discussion

So when we ask whether we might live in a universe where electric forces follow an inverse cube law, or where Gauss’s law doesn’t hold, we’re really asking about the relationship between mathematical and physical truth. While many aspects of our physical laws might be contingent, others might be more deeply rooted in mathematical necessity.

This connection between physical law and mathematical truth suggests that even in a universe with different fundamental forces, certain patterns might persist simply because they’re connected to universal mathematical truths. The exponent 2 in our inverse square laws might be more than just a coincidence – it might be a reflection of the deep connection between physical reality and mathematical necessity.

Just as prime numbers would be recognizable to any civilization capable of mathematics, perhaps certain aspects of our physical laws are similarly universal, not because of physics itself, but because of their connection to deeper mathematical truths.

Or is it?
Does it even make sense?

One can easily say “well, if the electric force didn’t (seem to) follow the inverse square law, we would simply NOT come up with such a thing as the Gauss’s law, hence the connection between geometric fact and physical world simply does not exist, hence it is simply NOT inevitable.”

The pi - $\pi$

The pi, \(\pi\), definitely has everything to do with geometry, or at least, you think so, and I’ve thought and think so.

But then, why does it have to appear, e.g., in the Gaussian integral, i.e.,

\[\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}\]

or in various formula for sums of infinite sequences, e.g.,

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}, \quad \sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} = \frac{\pi^2}{8}, \quad \sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}\]

and

\[1 - 1/3 + 1/5 - 1/7 + \cdots = \pi/4?\]

Again, while (of course) I do not have definite or (in any way) compelling explanations for this, I will derive some quantities in various ways, provide multiple perspectives, etc. just to explore this (once again) fascinating time-killing topics!

The Gaussian integral

Let’s just prove the Gaussian integral, that is,

\[\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}.\]

Let \(p = \int_{-\infty}^\infty e^{-x^2} dx\). Then

\[\begin{align} p^2 &= \int_{-\infty}^\infty e^{-x^2} dx \int_{-\infty}^\infty e^{-y^2} dy = \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+y^2)} dxdy \\ &= \int_{0}^\infty e^{-r^2} (2\pi r) dr = \pi\left.\left(-e^{-r^2}\right)\right|_{r=0}^\infty = \pi, \end{align}\]

hence the proof. Here \(\pi\) appears because the circumference of a unit circle is \(2\pi\), or equivalently, \(\pi\) appears in the formula of the area of a circle.

So obviously, we can see the fact that \(\pi\) appears in the Gaussian integral is due to the fact that \(\pi\) appears in the formula of the area of a circle and the fact that the derivative of \(e^x\) is itself, i.e., \(d e^x / dx = e^x\) (which is one of a handful of nice properties that the Euler’s number enjoys).

The Surface areas and volumes of hyper-spheres

Let us digress a bit (or quite a lot) for fun now. ★^^★ We use the fact that the area of a unit circle is \(\pi\) or equivalently, the definition of \(\pi\), to calculate the quantity \(\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}\). Here we do the inverse. Let us find out the proportionality constants for volumes of the hyper-spheres using the result of the Gaussian integral!

Using equivalence of multiple ways of calculating the Gaussian integral

Now we calculate the proportionality constant for the (hyper-)surface-area of \(n\)-dimensional hyper-sphere, which is also called \(n\)-ball, that is, \(a_n\) in the following equation

\[A_n(r) = a_n r^{n-1}\]

where \(A_n(r)\) refers to the surface area of \(n\)-dimensional hyper-sphere with radius \(r\), using the fact that \(p=\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}\).

(This problem is equivalent to finding the proportionality constant for the volume of the hyper-sphere, that is, to find \(v_n\) in

\[V_n(r) = v_n r^{n}\]

where \(V_n(r)\) refers to the volume of \(n\)-dimensional hyper-sphere with radius \(r\) because it always holds that

\[v_n = a_n / n.\]

Refer to Relation with surface area.)

We know \(a_2 = 2\pi\). Indeed, we use this fact to prove the Gaussian integral. (You even can say this is the definition of \(\pi\).)

Now let us (try to) calculate \(a_3\) (which we learned is \(4\pi\) for a fact in high school). Because (we know) \(p = \sqrt{\pi}\), we have

\[\begin{align} \pi^{3/2} &=p^3 = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+y^2+z^2)} dx dy dz \\ &= \int_{0}^\infty e^{-r^2} (a_3 r^2) dr = \frac{a_3}{2} \left( \left.\left(-r e^{-r^2}\right)\right|_{r=0}^\infty + \int_{r=0}^\infty e^{-r^2}dr \right) = \frac{\sqrt{\pi}}{4} a_3 \end{align}\]

hence \(a_3 = 4\pi\) (and \(v_3 = 4\pi/3\)), which confirms what we learned in high school!

Let us try to do the same thing for \(n=4\), that is,

\[\begin{align} \pi^{2} &=p^4 = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x_1^2 + x_2^2 + x_3^2 + x_4^2)} dx_1 dx_2 dx_3 dx_4 \\ &= \int_{0}^\infty e^{-r^2} (a_4 r^3) dr = {a_4} \left( \left.\left(-\frac{r^2}{2} e^{-r^2}\right)\right|_{r=0}^\infty + \int_{r=0}^\infty re^{-r^2}dr \right) = \frac{a_4}{2} \end{align}\]

hence \(a_4=2\pi^2\) and \(v_4 = \pi^2/2\). We did not learn this in high school, but this can be confirmed by Volume of an \(n\)-ball.

Now do the same thing again for \(n=5\):

\[\begin{align} \pi^{5/2} &=p^5 = \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2)} dx_1 dx_2 dx_3 dx_4 dx_5 \\ &= \int_{0}^\infty e^{-r^2} (a_5 r^4) dr = {a_5} \left( \left.\left(-\frac{r^3}{2} e^{-r^2}\right)\right|_{r=0}^\infty + \int_{r=0}^\infty \frac{3}{2}r^2e^{-r^2}dr \right) \\ &= \frac{3\sqrt{\pi}}{8} a_5 \end{align}\]

hence \(a_5 = 8\pi^2/3\) and \(v_5 = 8\pi^2/15\).

Now the above derivations kind of explain why the \(\pi\) term in \(a_n\) (hence, equivalently in \(v_n\)) grows like

\[1, \pi, \pi, \pi^2, \pi^2, \pi^3, \pi^3, \ldots\]

You can explain it like this: the left-hand-side term grows like \(\pi^{1/2}, \pi^{2/2}, \pi^{3/2}, \pi^{4/2}, \ldots\), but the right-hand-side term, which is the result of the integration, contains \(\sqrt{\pi}\) only for odd \(n\)’s, hence the patterns of \(\pi\) in \(a_n\) and \(v_n\).

Using what I learned in high school

Here we concern calculating \(v_n\), i.e., the proportionality constant for the volume of hyper-spheres.

We know that \(v_1 = 2\) and \(v_2 = \pi\), the latter of which is the definition of \(\pi\) in a traditional sense.

But let us derive \(v_2\) from \(v_1\) for the sake of illustration as to how to derive the recursion formula. For note that \(V_1(r) = 2r\).

Then

\[\begin{align} V_2(r) &= \int_{-r}^{r} V_1(\sqrt{r^2-x^2}) dx = v_1\int_{-r}^{r} \sqrt{r^2-x^2} dx \\ &= v_1r^2 \int_{-\pi/2}^{\pi/2} \cos^2 \theta d\theta = \frac{v_1}{2}r^2 \int_{-\pi/2}^{\pi/2} (1+\cos 2 \theta )d\theta \\ &= \frac{\pi}{2} v_1 r^2 = \pi r^2 = v_2 r^2 \end{align}\]

where we substitute \(r\sin \theta\) for \(x\) and the double-angle formula (one of (lots of) trigonometric identities) is used, hence \(v_2=\pi = (\pi/2) v_1\).

Similarly,

\[\begin{align} V_3(r) &= \int_{-r}^{r} V_2(\sqrt{r^2-x^2}) dx = v_2\int_{-r}^{r} (r^2-x^2) dx \\ &= v_2(2r^3 - 2r^3/3) = \frac{4}{3}v_2 r^3 = \frac{4}{3}\pi r^3 = v_3 r^3 \end{align}\]

hence \(v_3 = (4/3) v_2 = 4\pi/3\).

Just do this just one more time to recognize some patterns here.

\[\begin{align} V_4(r) &= \int_{-r}^{r} V_3(\sqrt{r^2-x^2}) dx = v_3\int_{-r}^{r} (r^2-x^2)^{3/2} dx \\ &= v_3r^4 \int_{-\pi/2}^{\pi/2} \cos^4 \theta d\theta = \frac{v_3}{4}r^4 \int_{-\pi/2}^{\pi/2} (1+\cos 2 \theta)^2 d\theta \\ &= \frac{v_3}{4}r^4 \int_{-\pi/2}^{\pi/2} (1+2\cos 2 \theta + \cos^2 2\theta) d\theta \\ &= \frac{v_3}{4}r^4 \int_{-\pi/2}^{\pi/2} (1+2\cos 2 \theta + (1+\cos 4\theta)/2) d\theta \\ &= \frac{3}{8}\pi v_3 r^4 = \frac{\pi^2}{2} r^4 = v_4 r^4 \end{align}\]

hence \(v_4 = (3\pi/8) v_3 = \pi^2/2\).

Now we can see very clearly that new \(\pi\) is introduced to \(v_n\) every other time, and that’s why we see these patterns in \(v_n\), hence (or equivalently) in \(a_n\).

\[1, \pi, \pi, \pi^2, \pi^2, \pi^3, \pi^3, \ldots\]

This does not appear natural!

Having derived the proportionality constants and figure out why we have such patterns, I am still bothered (as I have for too long throughout my life even since I unfortunately *^^* tumbled upon this fact around 2005) by the fact that there is a certain jumps in the pattern.

These numbers should be and in fact are completely independent of a specific universe we live in, or for that matter, specific universes for any entities exist, hence, I think we should see patterns such as

\[\pi^{0.5}, \pi^{1}, \pi^{1.5}, \pi^2, \pi^{2.5}, \pi^3, \pi^{3.5}, \ldots\]

Or is it only me?
Does what I said above even make sense to you?

Sums of sequences

(WIP)

\[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}\] \[\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2} = \frac{\pi^2}{8}\] \[\sum_{n=1}^{\infty} \frac{1}{n^4} = \frac{\pi^4}{90}\] \[\sum_{n=1}^{\infty} \frac{1}{n^6} = \frac{\pi^6}{945}\] \[\sum_{n=1}^{\infty} \frac{1}{n^8} = \frac{\pi^8}{9450}\] \[1 - 1/3 + 1/5 - 1/7 + \cdots = \pi/4\]

Mass-energy equivalence principle

The famous Einstein’s mass-energy equivalence principle implies that when mass is lost in chemical reactions or nuclear reactions, a corresponding amount of energy will be released. What I focus here is though the equation for the equivalence, which is given by

\[E = m c^2\]

where \(m\) is the mass and \(c\) refers to the speed of light.

This says the amount of energy corresponding to \(m\) is (exactly) proportional to \(m\) and the proportionality constant is exactly the square of the speed of light!

This seems to be (in a sense) even more coincidental than the ones I previously mentioned, e.g., the Newton’s law of universal gravitation, because there we have to introduce some (arbitrary) constant, i.e., the gravitational constant \(G\) to relate the two quantities \(F\) and \(m_1m_2/r^2\). The value is approximately \(6.6743 \times 10^{-11}\) in MKS units.

However, the proportionality constant in the mass-energy equivalence equation is exactly the speed of light squared!

But, what does the speed of light have anything to do with energy except that this specific multiplication \(E=mc^2\) makes the units of both sides compatible?

Why does this have to be true?

31-Jan-2025 PST

Sunghee Yun

Philosopher Thinker

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1. By the way, look at recently developed "Hubble tension" brought by new findings made using James Webb Space Telescope (JWST). How do we even know that the same physical laws should hold throughout the whole universe? Also, the truth or falsehood of this argument is irrelevant for of our arguments, e.g., we can say those aliens were from a different universe.  ↩
2. While it's true that prime numbers are defined in terms of integers, and as Bertrand Russell showed, integers themselves required formal definition, the concept of counting itself appears to be universal.  ↩