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  1. Any 'physical' quantity is expressed as (generally) a Real Number. Real Numbers are abstract mathematical constructs.

  2. Laws of Physics are written as mathematical equations; where these real numbers are equated to each other (By now we have already left the world of physics and are purely in mathematics)

  3. But when we 'measure' any physical quantity we can only measure up to a level of precision. (And this is due to resolution of the measurement instrument - I'm not talking of quantum effects etc.)

Say for e.g. distance ($d$), speed ($s$) and time ($t$) $d = s*t$ (I know this is a definition and not a law but explains the point try to make here)

But when I measure $s$, $d$ and $t$ and plug those values in - I will never get an equality only an approximation.

I will never know (i.e. measure) what is the value of real number $d$ which represents distance for me (and similarly for any other physical quantities). If I can never know $d$ - What's the point of putting them in equations?

Does this put some fundamental limitation?

Apologies if I'm unable to frame my question well - even I'm not very clear on what am I finding difficult to grasp. Any pointers would be helpful please

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  • $\begingroup$ Actually, very few quantities are expressed as real numbers. For example, $2$ is a real number, but what about $4\:\rm kg$? It’s not a real number; they are incompatible for addition. $\endgroup$
    – gen-ℤ ready to perish
    Oct 30, 2020 at 3:32

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Almost all equations in physics are intended to produce approximations, this is just an unavoidable consequence of the fact that "measurement" in the real world can never be made to arbitrarily high accuracy.

It is also usually uneconomical to attempt to make measurements to extremely high degrees of precision (of course in some cases the intention is to push the limit, for example in particle colliders). In the example you gave, the mathematical relationship $s=d/t$ is exact, but if the numbers you are using in the equation ($d$ and $t$) have some degree of experimental error then you will not get an exact number. What you will get is a range within which the value of $v$ is guaranteed to lie - not a single, exact value.

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  • $\begingroup$ Thanks - If the criteria for law to be "exact" is that it should fulfil the experimental scrutiny then this inherent limitation means you can never say if the law is exact. Also why should we call it "error" maybe our abstraction of physical quantities to real numbers is an error. Do I make sense? $\endgroup$
    – aman_cc
    Oct 30, 2020 at 3:01
  • $\begingroup$ Correct, a "law" in physics does not mean "this is exactly how X system behaves to arbitrarily high degree of accuracy". I would say it is not that our construction of the real numbers is the cause of the "error", it is simply the fact that if we want to measure, say, a distance to an extremely high degree of precision we need to be able to construct instruments that are sensitive enough to do so. Which is not always possible. $\endgroup$
    – Charlie
    Oct 30, 2020 at 12:03
  • $\begingroup$ yes agree - i just claim the it is NEVER possible. hence i wonder if we are introducing some sort of ambiguity (not because of the axiomatic construction of real numbers but by assuming physical quantities are real numbers) $\endgroup$
    – aman_cc
    Oct 30, 2020 at 14:50
  • $\begingroup$ Well, I used the phrase "not always possible" since it is technically possible to measure discrete "numbers" of things completely accurately. But for a general continuous spectrum of measurement arbitrarily accurate measurement is not possible, sure. $\endgroup$
    – Charlie
    Oct 30, 2020 at 16:27

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