# Question about Physicists' Convention

Many times I hear physicists say "$x$ is finite" when what they really mean is $x>0$ or $x\neq0$.

Why is that?

Zero is a valid finite number. What's wrong with simply saying $x$ is strictly positive?

Since $\infty$ is a mathematical construct which has nothing to do with our physical measurements (it can be used as an abbreviation sometimes in order to say something more subtle, but no measurement will ever yield $\infty$), what sense is there in the distinction between finite and infinite quantities in physics anyway?

• This is not just a physicists' convention. From Wikipedia on the definition of a finite number: In mathematical parlance, a value other than infinite or infinitesimal values and distinct from the value 0. – Qmechanic Oct 18 '16 at 12:29
• 1) I agree that is bad practice to adopt the terminology that you refer. 2) No measurement will ever yield an infinite outcome, but there are measurements that do not yield an upper bound as well (e.g. the total energy of a some systems can be arbitrarily increased). It is not wrong in my opinion to consider the spectrum of the associated observables unbounded, i.e. "going up to infinity". – yuggib Oct 18 '16 at 12:43

## 2 Answers

I think it's because we sometimes talk about quantities that are not the result of measurements, but rather conceptual entities. For example, when you pump a laser, at some point you go through a situation where the upper level population equals the lower level population, prior to inversion. At that moment, you could say the temperature is effectively infinite. You will never stick a thermometer in there that reads infinity, but the concept of infinite temperature is encountered. Or, one might ask if the size of the universe is infinite, even though no size measurement will ever come out infinity.

Similarly with near-zero results. You can talk about "absolute zero" as a concept for temperature, though Nernst's law tells you that you'll never actually encounter it. So if you say "the temperature must be finite", you might mean you can't get a zero result, though I agree that's an imprecise use of the mathematical meaning of "finite." "Nonzero" would suffice, and be more accurate, it's some sort of conventional misnomer, like "blackbody radiation" or "degeneracy pressure."

A speculative guess:

Many interesting quantities in physics are actually ratios, proportionality factors, slopes ... and it is often more or less arbitrary whether the quantity you find in textbooks is one number or its reciprocal. So ratios of $a:0$ or $0:a$ will both be special (and will often require equally special treatment) and you often need a word to denote the usual case that is neither of these.

For this meaning, "finite" seems to have been chosen by the usual messy process of natural-language development, and it has won such acceptance that it is also used for quantities where the quantity itself is actually more meaningful than its reciprocal.