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What "a measure" means in definitions?

I was trying to find a proper definition of temperature and then I got into some troubles. I always thought of temperature as:

Temperature = a measure of the average kinetic energy of the particles in an object

My problem is how we interpret these two definitions. For example in the first definition, what is meant by "a measure". I can't understand it. Maybe it is silly for others but I just can't get it. It is like temperature is another measure of average kinetic energy. It says "a measure" not "the measure". Is temperature "a member" of the "set of measures of average kinetic energy"?

Compare the first definition of the temperature with the definition of volume and mass:

Volume = the amount of space an object occupies

Mass = a measure of body's inertia

Why it says "the amount" and not "a measure of space an object occupies"?

What is a physical quantity?

Physical QUANTITY = A PROPERTY of a material or system that can be quantified by measurement Wikipedia.

From this definition it is clear that when we refer a physical quantity we refer to a property. So when we refer to mass and we say that is a physical quantity then it must be a property which can be measured. Therefore what is the measure of mass?

It seems definitions in physics are not as formal as mathematical definitions are. I don't look for how we should think of mass (relativity) or the proper definition of temperature (partial derivative of internal energy with respect to entropy). I am asking about the "form" of the definitions. In mathematics and generally in axiomatic systems a definitions serves as an abbreviation. Therefore in physical sciences what that abbreviation refers to?

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  • Physical definitions are more informal, but vagueness resolved by context and abuse of notation and terminology are common shorthands in math too. It doesn't say "the amount" because temperature is only proportional to the average kinetic energy and not equal to it. "A measure" means that some transformation is involved, usually monotone and often linear. Specifics are given by formulas. Or that it is a theoretical quantity attached to experimental measurements indirectly. A mass is measured through weight (gravitational) or acquired acceleration (inertial) in experiments.
    – Conifold
    Dec 3 '20 at 19:47
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    "X is a measure of Y" might also mean that if you know the value of X for a given physical system, that is sufficient to calculate the value of Y for that system without any further information. So for example, momentum is not a measure of mass despite containing mass in its formula (momentum = mass*velocity), because if you know the momentum but don't know the velocity, you won't be able to deduce the mass from the momentum alone.
    – Hypnosifl
    Dec 3 '20 at 22:30
  • ...although in some cases of the use of this phrase, Y is not really a separate named quantity but more like a conceptual description of what X is measuring, like the idea that volume is measuring "the amount of space an object occupies." The statement that mass is a measure of inertia might be similar, it's telling you conceptually that to go about measuring "mass" you should measure an object's resistance to acceleration under a given force (unless we are talking about gravitational mass, which is conceptually measured in terms of the degree of gravitational acceleration of nearby objects).
    – Hypnosifl
    Dec 3 '20 at 22:55
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For what I've learned, the main metaphysical problem within this domain is our capacity to compare. When do you know that a face that is compared to another, "matches"? How does one differentiate between green and red? How does one know that 0.9999... equals 1.0? How do we know that 1.0 is equal to 1, and that two is different than three? etc. Most of such capability is provided by our particular and subjective being.

A measurement is a comparison (mostly quantitative, but not necessarily). So, measurement depends on the subjective capability of comparison.

A quantity is a quantified property. This does depend on quantities.

Temperature is essentially a feeling, it is not "average kinetic energy", although it represents such energy. The first law of thermodynamics was created to give a physical-mathematical sense to such feeling, and it is based on comparisons.

And one more thing: experience is a subjective feature. If you and I have a discussion about a topic, that we call objective, you will have a subjective understanding of it, and so I will. The topic is therefore not strictly objective: it is an agreement of two subjectivities (yours and mine).

Mathematics objectifies knowledge (because it has not a strong dependence of experience, see Immanuel Kant about a priori knowledge), but objectivity in physics is precisely such agreement of subjectivities.

For example, you can measure that the temperature of a liquid is 39.76235 deg.celsius while I measure 39.76224 deg.celsius at the same instant. How can we be objective about that? Even if we use two identical of the most precise thermometers on earth, some digits at the right end will always differ (moreover because the liquid is not static, energy flows in waves due to internal interactions, and it is continuously interacting with the environment). How can we be objective in such case?

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Measurement is a metaphysical problem. We can measure things to the extent we consider them separate. Math is all about the relationships between entities to the extent they are separate. Any measurement depends on what we're trying to use it for. To some extent measurements must be shared because their use is commonly shared. You need someone else's measurement to be the same as yours, or vice versa. IF we consider the end of one thing to be point x and the beginning of the next to be x+1, THEN we can measure them.

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I don't look for how we should think of mass (relativity) or the proper definition of temperature (partial derivative of internal energy with respect to entropy).

That's good, because they are nothing to do with how they are defined.

The first temperature scale, was defined by the inventor of the first practical thermometer, Daniel Fahrenheit, and that's no accident. The scale set two reference points, human body temperature as 100, and the freezing point of brine, as zero. The choice was made so these would be easily & cheaply available, for calibrating thermometers.

Caloric theory, was the dominant account of temperature until the end of the 19thC, so obviously just equating temperature with the modern explanation, doesn't work. Only with the proof atoms exist (Einstein's explanation of Brownian motion provide the first accurate estimate of the size of atoms, so this is recent history), and the work of Boltzmann, was temperature behaviour accounted for in terms of particle kinetics.

Celsius initially used melting & freezing points of water for calibration, at reference pressure. The Kelvin scale, as part of the System Internationale used in almost all scientific work, uses a definition based on more fundamental quantities - mass, length, and time.

In the SI definitions, mass is defined in relation to a reference piece of metal (there are moves to find a reference using something that can generated or checked locally). Length is defined by time for a photon to travel a certain distance. And time is defined by vibrations of a particular atom.

Consider mathematically, what happens when you measure a length. How accurately are you measuring to reference? How accurate is the reference calibrated? How precise is the state calibration is done to? Error bars almost never go away in science (right down to the uncertainty principle which is fundamental), & reducing sources of uncertainty is what a lot of experimental physics is; like discovering the Cosmic Microwave Background, or detecting gravitational waves. Statistics, distributions, averages, that whole branch of mathematics exists for a reason.

There are currently thought to be 19 independent fundamental constants in physics, which include units of scales like those already listed, and dimensionless numbers, which are often ratios, like the ratio of proton to electron mass. These are thought to be parameters, linked to the initial conditions of the universe. In many applications of maths, it is crucial not only to know not only the behaviour of a system, but where it started, to narrow what is under consideration.

In mathematics there is pi, and e, and many other transcendental numbers, which have a very similar nature to the fundamental constants. It is likely they both derive from the same thing, fundamental geometry of our universe.

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