Help understanding quote on theory and knowledge in Gravitation (Misner, Wheeler, & Thorne, 2017)

Question

I came across an interesting discussion at the start of chapter 3 of MWT Gravitation. It reads:

Here and elsewhere in science, as stressed not least by Henri Poincare, that view is out of date which used to say, "Define your terms before you proceed." All the laws and theories of physics including the Lorentz force law, have this deep and subtle character, that they both define the concepts that they use (here $\vec{B}$ and $\vec{E}$) and make statements about these concepts. Contrariwise, the absence of some body of theory, law, and principle deprives one of the means properly to define or even to use concepts. Any forward step in human knowledge is truly creative in this sense: that theory, concept, law, and method of measurement --- forever inseparable --- are born into the world in unison.

I find this quote interesting but am having a hard time fully understanding the views of the authors. My initial guess is that it is saying you can define terms, say momentum: "momentum is the product of mass times velocity." However, this is useless in isolation because without fitting this into a greater theory/concept/law one cannot yet identify the momentum of a moving object?

I suppose my overall reading of the quote may be summarized by:

Until we embed the definition into a theory the definition is meaningless

Am I close to the point that the authors were trying to emphasize? What is your reading of this quote?

Answer 1

In physics everything is a mathematical model. If you ask me what an electromagnetic field is I would answer it is that which is described by Maxwell's equations. If you ask me no, really, what is an electromagnetic field? I would have to say (a) that depends on what you mean by really and (b) for most meanings of really I have no idea. As long as I have a mathematical model that tells me what the EM does I am unconcerned by what it really is.

This is the point Misner et al are making. They are saying that the electric and magnetic fields are defined by the way they interact with charges. That is, we have a mathematical model (the Lorentz force law) that defines the electric and magnet fields to be those objects that interact with charges as described by the law. The mathematical model both defines the electric and magnetic fields and tells us what we will observe when they interact with charges.

Answer 2

That discussion in chapter 3 was so important to Misner Thorne and Wheeler that they revisited it in section 12.3

Point of principle: how can one write down the laws of gravity and properties of spacetime in Galilean coordinates first (§12.1), and only afterward (here) come to grip with the nature of the coordinate system and its nonuniqueness?

And then they refer to and paraphrase the discussion in section 3.1 that you quoted.

I will discuss two cases:
-Ohm's law and definition of electric resistance
-the definition of inertial coordinate system.

In MTW the 'definition' they are talking about extends to quantitative definition.

First example:
What does it take to give a quantitative definition of the electric resistance of a particular material?

The point is: the only way to provide a quantitative definition is to state it in terms of Ohm's law of resistance. $R=\frac{V}{I}$

For electric resistance Ohm's law is doing two things: it describes a property of nature, and it provides the means to give an operational definition of electric resistance.

Now, if there would not be a wider context then that double duty could be a case of circular reasoning. It is not circular reasoning because there is context. Different metals have different conductivity, corresponding to how easily the electrons can move about.

There isn't a way of first formulating a quantitative definition of electric resistance, and only then discover Ohm's law. It's only when you have a theory that you have the means to unambiguously define the concepts that make up the elements of the theory.

For the second example I set up a thought demonstration.

Let's say a solar system began at the very perifery of a galaxy, and some astrophysical process pulled that solar system loose from its parent galaxy, making it careen in to intergalactic space.

Several billions of years later: on an Earth lookalike: an intelligent species evolves out of the life on that planet.

For simplicity let's make that planet completely like our Earth. This Earth is so far away from other galaxies that none of them are visible. So the only celestial bodies that are visible are the other celestial bodies of that solar system, nothing else.

Under those circumstances: can a Kepler still find Kepler's three laws of planetary motion?

On our Earth the astromers had the benefit if an unmoving background reference: the fixed stars. But in this thought experiment no background reference is available.

I argue that in this throught demonstration it is still possible to arrive at Kepler's laws of motion, and subsequently the inverse square law of gravity.

If we grant that the Earth is spinning then we expect it to act as a spinning top. The plane of the ecliptic and the Earth's equatorial plane are at an angle to each other. The intersection of those two planes is a line; we can choose to make that line a reference of (angular) position.

I argue that with that provisional reference in place it is possible to suss out Kepler's three laws, and subsequently the inverse square law of gravity. It will be more difficult than it was for our Kepler and out Newton, but it will be possible.

But then:
The astronomers will recognize the precession of the equinoxes. With an estimate of the moment of inertia of the Earth they arrive at an estimate of the rate of precession, but not an exact value.

Without fixed stars as reference the rate of the precession of the equinoxes cannot be measured; it must be inferred. The way to home in on the non-rotating coordinate system is to apply the inverse square law of gravity. All of the planets of the solar system are perturbing each other (Jupiter doing most of the perturbing of course.) So for all planets some level of precession of their perihelion is occurring. An exhaustive calculation will give for each planet the expected amount of precession. There is only one coordinate system such that all the planets move in accordance with the expected perihelion precessions: that is the inertial coordinate system. (Eventually the astronomers will be able to narrow down the uncertainty margins so far that they can identify an anomalous perihelion precession for the planet Mercury.

Summerizing:
The laws of motion hold good if and only if you use a non-rotating coordinate system. (It is possible to use a rotating coordinate system, but only in the following sense: in order to make that work you must in the calculation provide the rotation rate relative to the inertial coordinate system.)

In the absence of other reference is it still possible to home in on the inertial coordinate system. You infer your own rotation relative to the inertial coordianate system by applying the laws of motion.

I argue that astronomy is actually exceptional, in that there is a background reference: the fixed stars.

What Misner Thorne and Wheeler are arguing, and I think they are totally right: in the vast majority of physics there is no such background reference. The only way to provide an operational definition of the concepts that the laws of physics are making statements about is to apply those very laws.

As to suggestions that the above is circular reasoning:
Physicists design and build high performance machines such as particle accelerators. When you have built a machine, and you start operating it, that is when the rubber meets the road. If the machine performs as designed you know you are doing something right.

I hope I can persuade you to give this a lot of thought. I regard it as one of the deepest insights into what is involved in doing science. Once I knew about it I started recognizing it everywhere.