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.
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.