Is there a deeper meaning when a number is squared?

Question

In my opinion, math is about more than just memorizing equations, it's about numbers that are built in a way that represents our understanding of something.

So I ask this, what does it mean intuitively when you square a number?

Consider the equations in Physics like:

The force between two charged particles $F_m=\frac{G\cdot M\cdot m}{R^2}$

Yes, I'm sure there is a proof out there which describes how this expression was derived, but what I want to know is can you look at the equation and understand what it means when a number is squared?

Or, is this not a way in which we can look at math? I mean to say that, math isn't meant to be interpreted like that.

Another equation to consider is the equation of kinetic energy:

$E_k = 1/2\cdot m\cdot v^2$

I would really like to develop a deeper understanding of math so I can more easily interpret Physics equations.

Answer 1

I doubt there is a general statement you can make about the physical meaning of a quantity squared; such quantities appear in too many different contexts.

But the physical meaning of the $R^2$ in the denominator of universal gravitation equation that you quoted is very deep indeed. It's there because:

1. We live in a three-dimensional universe
2. Energy is conserved

Consider a mass, say the Earth. The gravitational effect of the Earth on some other mass is a constant. Suppose the gravitational effect of the Earth at some distance, say $d$, is a certain amount. At twice the distance from the Earth, distance $2d$, the gravitational effect is spread over a sphere which has twice the radius of the one at distance $d$, and so has four times the area. So the gravitational effect at distance $2d$ is only one-fourth the effect at distance $d$. Similarly the effect at distance $3d$ is spread over a sphere 9 times as large, and so is only one-ninth as great at any point on that sphere. This is the physical meaning of the $R^2$ in the denominator.

The $R^2$ appears in the denominator of many similar force-related formulas. For example, the electrostatic force exerted by a point charge at distance $R$ is also proportional to $\frac1{R^2}$, for the same reason.

Similarly, the intensity of a sound at distance $R$ is proportional to $\frac1{R^2}$, and the intensity of light at distance $R$ is proportional to $\frac1{R^2}$, and the intensity of an explosion at distance $R$ is proportional to $\frac1{R^2}$.

If the universe were four-dimensional, the $R^2$ in the denominator would be an $R^3$ instead; if the universe were two-dimensional, it would be an $R^1$ instead. In similar situations where the universe under consideration is effectively two-dimensional, as when considering the intensity of ripples on the surface of a pond, the intensity is proportional not to $\frac1{R^2}$ but to $\frac 1{R^1}$. Similarly, the electrostatic force exerted by a very long, thin current-carrying wire is proportional to $\frac 1{R^1}$ rather than $\frac 1{R^2}$. An explosion at one end of a long tunnel does not decrease in intensity proportional to $\frac 1{R^2}$; the force at the other end is about the same as the force near the explosion, because the long narrow tunnel is effectively one-dimensional, so the force is proportional to $\frac1{R^0}$, which is constant.

Answer 2

MJD's answer for the square appearance is the classical one and very good, but I wanted to just add some points around your wider question.

First, physicists for hundreds of years have explained the square as the area of the sphere of influence of the force at that distance. Kant even, in his Prolegomena, says that natural laws MUST have this form by logical necessity even though it is a synthetic proposition, and is one of the big examples of synthetic a priori. When electrical forces were found to have the same form, many took it as obvious and well understood.

Now days, though, we know these laws are actually wrong. General relativity gives corrections to gravity that change the expression, as does (the relativistic) Maxwellian unification of electromagnetism (due finite propagation of the force). Kant is pretty much a laughing stock of hubris, and physicists no longer believe you can give those kinds of simplistic explanations based just on what you do to numbers. This is why you won't find many mathematicians willing to jump on answering your question.

Instead, science works the other way. It builds models that describe how to think about phenomena, with an ontology of things and their transformations. Then they derive equations for that model. By selecting different models and checking predictions, they get better equations over time. But they try not to ascribe some magical reason or meaning to why the equations take a given form outside the model.

There is however, some sense in looking at multiplying and dividing quantities. Unit analysis tells scientists what they are measuring. You have to be able to measure things consistently, or measurement is meaningless. If you are measuring mass and want to test it, you must test it against a theory that predicts a quantity convertible to mass. So when you see the $v^2$ in the kinetic energy, understand that you will expect units of kg $m^2 s^{-2}$ for energy, so it's at least understandable. But to get these equations, you need a model, and classical physics uses Hamiltonian or Langrangean models on Poisson manifolds.

Of course, like the gravity equation, the kinetic energy equation was also found wrong years ago, first with special relativity. So again, just be careful looking for meaning outside a model, and expect to be wrong in the long term.