# Q: How much of physics can be deduced from previous equations/axioms?

The original question was: I was wondering quite how much of physics (mainly regarding classical mechanics but all branches!) can be deduced from previous equations/axioms. I don’t mean going as far back as what axioms we have to take in maths, but I’ve always loved proofs deriving equations from others. And obviously some values need to be experimentally derived, like g=9.81, or the value of a coefficient of friction. But you can still derive equations which feature these without knowing their value.  So how was F=ma derived etc?

Physicist: That is really profound!  It depends on where you’d be willing to draw the line.  In terms of what you might learn in a physics class: most of it.  In each physics class (although it doesn’t always seem like it) there are usually only a handful of equations to learn, and many of those can be derived (in later classes).

In that particular example, force is actually defined to be mass times acceleration.
Before Newton the words “force” and “energy” were thrown around, but not defined rigorously.  There’s a long tradition of scientists grabbing words that have a vague meaning in everyday language and giving them (forcing upon them) solid mathematical definitions, like: energy, jerk, information, temperature, significant, curl, almost-everywhere, …

But the basic laws like “for every action there’s an equal but opposite reaction”, and the associated equations, are probably a lot like the g=9.81 thing; they need to be experimentally discovered.

For example (when m is mass and v is velocity),

m, mv, and mv2 (matter, momentum, and kinetic energy) are  conserved, but mv3 is not conserved, and doesn’t even get a name.  There’s no particularly good reason why mv3 isn’t conserved, or why mv2 is.

You can go back and forth, and argue the why’s, but at the end of the day there will always be a list of true but un-derivable laws called “first principles”.  These are just the simple, generally agreed upon, but unprovable statements of physics.  Like “charge creates an electric field” or “mass creates gravity” or “momentum is conserved”.

You can find a few unprovable-but-true statements by saying almost anything, and then asking “why?” over and over until you start to loop.

So, technically everything is either an axiom or is derived from one.  Most of the big advancements in physics come from discovering the new, and absolutely not derivable, axioms of the universe.  Like “the speed of light is constant, no matter what” or “there’s a limit to how well momentum and position can be simultaneously measured”.  Nobody saw those coming.

What’s terrible is that it’s (often) impossible to tell which laws and universal constants are fundamental, and which are derived from other fundamental things.  For example, near the end of the 19th century Maxwell (of Maxwell equations fame) derived the speed of light, c,  from $\epsilon_0$ and $\mu_0$, the electric and magnetic permittivity of space (which describe how strong the electric and magnetic forces are).  So you’d expect that one, or both, of $\epsilon_0$ and $\mu_0$ must be fundamental, and c must be derived.

However!  In the 20th century Einstein rewrote electromagnetism in terms of relativity (in fact, this was the topic of the original relativity paper).  Using relativity and Coulomb’s law (which describes the relationship between distance, charge, and electric force) you can derive anything you’d want to know about magnetism, including $\mu_0$.  So clearly c and $\epsilon_0$ are fundamental, while $\mu_0$ is derived.

Point is: it’s hard to say what’s derived and what’s not, but what we can say is that there are a bunch of laws that just kinda “are”, and can’t be explained by more fundamental laws.  At last count there are a couple dozen fundamental constants (like the ones in the example above) that can’t be explained in terms of each other, or any laws.  Although most physicists would probably agree that there are plenty, I’ve never heard of anyone actually attempting to count the number of fundamental laws that are out there.

Trying to count the fundamental laws seems like a good way to goad physicists into a fist fight.

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### 12 Responses to Q: How much of physics can be deduced from previous equations/axioms?

1. Neal says:

Maybe it’s interesting to ask, “Which axiomatic approaches are equivalent?”

For example, you can start classical mechanics with one of three fundamental approaches: solving Newton’s 2nd law, minimizing the action, or extremizing the energy. All three imply each other, so the approaches are equivalent. They’re the same physics in different clothes.

2. It seems there’s some kind of misconfiguration in your blog, because I’m seeing “latex path not specified” red-on-yellow warning messages where equations should appear.

3. Questioneer says:

You made the statement that “the speed of light is constant” is an “absolutely not derivable axiom of the universe.”

By “constant” did you mean “unchanging over time”? If so, then we probably haven’t been measuring it that long, compared to the age of the universe, to state this as an empirical fact.

If you meant that the speed of light is the same for all observers regardless of their relative motions, then I’d like you to reconsider your statement.

When I taught relativity to my senior high school students, I tried to give them an intuitive grasp of why Einstein’s claim that the speed of light is the same for all observers must be true. At the same time, I wanted the students to be impressed by Einstein’s tremendous ability to think, seeing as he was a theoretical physicist rather than an experimental physicist. I showed them how he might have figured out the assertion from first principles.

If one starts with the axiom that all motion is relative, not absolute–that is, that it is impossible to tell if you are moving relative to space–the speed of light MUST be the same for all observers.

My argument was by analogy to waves in water. A wave passes a boat. The sailor measures the depth of the water, the temperature, the air pressure, the degree of saltiness, etc. and consults a reference to find the velocity of water waves in that medium. He knows the length of the boat and has a stopwatch, so measures the velocity of the wave. If he computes the same value as the reference value, he knows he’s not moving relative to the wave’s medium (the water.) An observer in motion relative to the water would compute a different value.

Thus, it is possible to determine if you are moving relative to a medium by measuring the speed of a wave in the medium.

Light’s medium is space itself. Thus you should be able to tell whether you are moving relative to space itself by measuring the speed of light (waves) through space. But, by the relativity principle, absolute motion relative to space itself is impossible to determine. Thus all observers must get the same value, the value you get when you measure the speed of light at different times in the day (a la Michelson & Morley).

Of course, I have just substituted one axiom (the relativity principle) for another (the constancy of the speed of light.) However, I find the argument that shows that the speed of light must be the same for all observers to be intellectually pleasing. I find it helps convince the students that, despite its initial counterintuitive nature, the assertion should be true. And when experiment confirms it…WOW!

By the way, I just discovered your site today. It’s terrific…hard to get away from. I’ve already sent the link to many friends. Thanks.

4. Philip Lewis says:

Questioner above made me wonder that if the value of c is the same for all observers does this also mean that that that value has never changed over all time? In other words could the speed of light once been faster or slower than it appears now whilst still appearing constant for all observers at the moment of measurement?

5. The Physicist says:

You can’t rule out that the speed of light could change, but there is absolutely no indication that it ever has.

6. Ron says:

As someone who has debated theology extensively, I must say that the only times I have ever encountered anyone claiming a changing value of C regardless of any supporting data, is when I deal with people desperate to rationalize away our viewing of stellar objects further away from the earth than what would be possible with the biblically deduced age of the universe of 6,000 years.

7. The Physicist says:

However, the conversations tend to be more hypothetical and exploratory. It may seem silly, but some of our most cherished, basic assumptions (backed by observation) have proved to be false in the past: flat/stationary Earth, universal time, counterfactual definiteness, …
That said, it’s no good to expect all of the well established laws of physics to be false.

8. Ron says:

This is more a question of empiricism vs. rationalism, if it could ever be said that there was a competition between the two schools of thought..lol. The shining stars of empiricism are the observation led physical sciences like physics, chemistry, and biology to name a few, and the rock star of rationalism is mathematics.

Rationalism uses axioms as the starting point for all logical deductions made subsequently. Definitions derived from axioms are then used to logically derive further definitions and ideas. As long as the internal logic is sound, everything within a rational system is always true internally to that system. Mathematics is a rock star in that not only is it internally true, it’s conclusions seem to match the external reality of the real world. Not that any rational system has to match reality, strictly speaking, they don’t. They just wouldn’t be all that useful. An example of this would be if the axioms of mathematics ultimately led to the conclusion that squares made the best wheels, it would still be true internally, but not useful in the real world.

Conversely, empiricism doesn’t use axioms as it’s starting point at all. The very nature of physics, chemistry, and the like are to explain observations in their respective fields of study. That being said, observations are the critical starting point for empiricism. However, that doesn’t mean that rationalism systems like math can’t be used to relate observations to each other. As long as that rationalism system stands up to reality(ultimately tested by empiricism), much can be gained from doing so.

We are fortunate to live in a world where both lines of thought have been so well developed. The demanding standards of the scientific method have propelled the quality of our observations. The state of mathematics today gives us an outstanding tool for analysis and to even make non-intuitive predictions of future observations.

9. s says:

I’m under the impression that Noether’s Theorem requires that momentum (mv) and mass/energy (m,mv^2) be conserved if the laws of physics are the same over time and in different places, while mv^3 is not. While this does simply trade one axiom for another, it seems like a more “fundamental” “first principle” to a layman.

RE: “the basic laws … need to be experimentally discovered … There’s no particularly good reason why mv^3 isn’t conserved, or why mv^2 is.”

10. It is possible to generate a mathematical model of reality that starts from first principles and after a series of extensions reaches a level that shows features and phenomena that we know from observing reality.
This model is NOT CLAIMED TO BE A PHYSICAL MODEL. It is a completely deduced model. Much of its content cannot be observed.