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The question I want to ask is: What measures are needed to describe the physical world and what are the fundamental ones of those, in the proper sense of the word fundamental? But that might be too general or unanswerable.

So …

Do scientists (or mathematicians), either applied or theoretical, ever talk about something called fundamental measures—as distinct from what we conventionally describe as fundamental units (i.e. SI base units)?

In trying to conceptualise what might be fundamentally different about different units like distance and angle (coming at it as a high school mathematics teacher), I then asked what are the universe’s fundamental units as far as we know them today. I searched for “fundamental units” and found SI base unit on Wikipedia. I wasn’t satisfied because the ampere is defined in units of charge, mass, time and distance. So technically by definition not fundamental. I understand that by convention (and the practicalities of experimental measurement) charge is not an SI base unit. But to me and many others it seems more “fundamental”. Only this reply to Fundamental units hinted at an elegant but unsatisfying answer that seems to be approaching “unified theory” territory. Other similarly unsatisfying answers move towards philosophy.

As a mathematician, I accept that I can say $1\neq 2$ through the definitions that make up the abstract logical world we call mathematics. So perhaps I am looking for something more in the field of what we define as conceptually different/fundamental.

For example, I have in my mind angle and distance as things that require measure and for which we define units which we conceptually consider to be fundamentally different types. Sure you can define any angle by the length of three sides of a triangle but, “fundamentally” speaking, change of direction is not the same as displacement in a direction. Similarly, while you can describe the effects of electric charge in terms of mass, time and distance and define a unit for a given amount of that effect (ampere), charge itself is still a phenomena that is of a fundamentally different type of measure. Thus I wonder whether semantically fundamental measure is a different concept to fundamental units.

So, are ‘fundamental measures’ a thing? Has this been thought out/through before? Can the question be discussed satisfactorily without invoking too much logic, philosophy or frontier theoretical physics?

The only specific mention of it (without digging deeply into academic publications) is here.

In 1964, Duncan Luce and John Tukey perceived that fundamental measurement is not a physical operation, but a theoretical property:

"The essential character of what is classically considered ... the fundamental measurement of extensive quantities is described by an axiomatization for the comparison of effects of (or responses to) arbitrary combinations of 'quantities' of a single specified kind... Measurement on a ratio scale follows from such axioms... The essential character of simultaneous conjoint measurement is described by an axiomatization for the comparison of effects of (or responses to) pairs formed from two specified kinds of 'quantities'... Measurement on interval scales which have a common unit follows from these axioms; usually these scales can be converted in a natural way into ratio scales.

The last time something similar was asked (fundamental dimensions), the topic was closed for being too broad. So, to keep this open, I am asking specifically about whether a semantically distinct concept referred to possibly as “fundamental measures” is a thing. I also worry someone might banish this topic to the Mathematics site. So, to keep it here in Physics, I want to keep my question on measures needed to describe the physical world.

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  • $\begingroup$ Well, the meter is defined in terms of the second just as much as the ampere is defined in terms of both. Does that mean that the meter is not "fundamental" by your definition? $\endgroup$ – Emilio Pisanty Jan 21 '18 at 3:48
  • $\begingroup$ I don’t think so. Units are a “size = 1” amount of something we measure in the physical world. 1 metre can be defined as the length something travels in a given time. The unit of “metre” may be definable in terms of the unit of “second” but length and time are different kinds of things to measure. $\endgroup$ – lukejanicke Jan 21 '18 at 3:56
  • $\begingroup$ @JohnForkosh Yes that is helpful. The systems of units on that page (e.g. Planck units) give everything in terms of length (m), time (s), mass (kg), temperature (K) and charge (C). So the “fundamental measures” then are length, time, mass, temperature and charge. $\endgroup$ – lukejanicke Jan 21 '18 at 5:24
  • $\begingroup$ Although, is not energy more fundamental than temperature? For me, that opens a can of worms because of mass–energy equivalence in Einstein’s famous equation. But this equivalence, and distance—time equivalence (via relativity), don’t necessarily mean they are actually just different expressions of some more fundamental thing about the universe that can be measured. Or maybe it is, as @Ted Bunn suggests in the topic I linked—to my dissatisfaction. I think not. So maybe I can say that in the same way angle, although it has some equivalence to length, is also a fundamentally different measure. $\endgroup$ – lukejanicke Jan 21 '18 at 5:28
  • $\begingroup$ I agree with you on the Ampere, the coulomb should be a base unit and then the ampere be derived as 1 C/S. There has actually been talk of doing this. It is absurd to take as a definition something that can only be measured when one has two wires of infinite length $\endgroup$ – David Reed Jan 21 '18 at 6:51
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It sounds a bit like you're talking about dimensional analysis.

Essentially, a dimension is an equivalence class of different units which all measure the same quantity. For example, meters, inches, and light-years are different units which all measure a fundamental property called "length".

The three fundamental dimensions (in high school physics, at least) are mass, length, and time ($M, L,$ and $T$). We can build almost all other quantities out of these three.

Using the notation $[x]$ to mean "the dimensions of $x$",

$$[\text{velocity}] = \frac{L}{T}$$

$$[\text{acceleration}] = \frac{L}{T^2}$$

$$[\text{force}] = \frac{ML}{T^2}$$

$$[\text{energy}] = \frac{ML^2}{T^2}$$

so on and so forth. This helps with checking ones work for mistakes, but it's more useful than that. For example, if I asked, "what force is required to keep an object of mass $m$ in a circular orbit with speed $v$ and radius $r$," then there is only one way to combine those parameters to get a quantity with dimensions of force ($ML/T^2$) - the only such combination is

$$ F \sim \frac{mv^2}{r}$$

which is indeed the equation for centripetal force. The same technique can be applied to argue the form of the periods of oscillation for a spring and a pendulum, and why the latter cannot explicitly depend on the mass of the pendulum bob.

In plasmas and fluid dynamics, dimensional analysis is raised even higher through the Buckingham $\pi$ Theorem, which formalizes the use of these techniques in the study of scale-invariant physical systems.

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  • $\begingroup$ Yes this the kind of answer I’m looking for. I will further investigate “dimensional analysis”. Is angle one of those dimensions? $\endgroup$ – lukejanicke Jan 21 '18 at 4:26
  • $\begingroup$ Actually no - angles are dimensionless based on their definition. This can be demonstrated in a number of ways. For example, the power series expansion of $\sin(\theta)$ is $\theta - \theta^3/3! + \ldots$ which only makes sense if $\theta$ is dimensionless. $\endgroup$ – J. Murray Jan 21 '18 at 4:41
  • $\begingroup$ Right. That makes sense. But angle in the physical world is something fundamentally different to measure than length. Even if you can make a construction with lengths to define any angle (in which the length units could cancel out, like in gradient) it is still a different thing to measure in the physical world. $\endgroup$ – lukejanicke Jan 21 '18 at 5:15
  • $\begingroup$ Speed needs a measurement of time and a measurement of distance, both time and distance being more fundamental measures than speed and speed being directly and exclusively constructed from them. Angle similarly needs two measures, but both of distance which cancel each other out, but only if you define angle as a ratio of orthogonally measured lengths. A sloppy way to say it but I mean like in a right triangle. But that reintroduces 90° tucked away in the assumed 2/3-dimensionality of space. So angle, while dimensionless in its units, must still be a “fundamental measure”. $\endgroup$ – lukejanicke Jan 21 '18 at 5:18
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    $\begingroup$ "Fundamental measure" is your terminology, so you are free to apply it to angles if you wish. In the well established context of dimensional analysis, however, angles are pure numbers with no meaningful dimensionality. Also - you say that speed is less fundamental than length or time, but this is not necessarily so. The choice of mass, length, and time as fundamental dimensions is not unique - it is merely conventional. The $\pi$ theorem makes this more explicit (though it is somewhat technical). $\endgroup$ – J. Murray Jan 21 '18 at 5:33

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