So, my friend was talking about how an electronic (musical) keyboard works, saying “the processor scans the switch states, so even if you press F and G at exactly the same time, there is a finite interval between the moment F plays and the moment G starts playing, but it’s so small that your human ear and brain do not perceive the difference.”
I corrected him: “you mean a non-zero finite interval”.
He said that would be redundant, that zero is not a finite number. I disagree. Is to say that zero is not finite the same as saying zero is infinite, i.e. zero equals infinity, or maybe that zero is an infinitesimal? Does the answer change depending on whether you’re talking mathematics or physics? Is zero a valid finite number for apples but not for time?
I’m not a number theorist but zero certainly behaves like a finite number. It is a definite place on the real number line and on the complex number plane. It is also subject to the ordinary mathematical operations of addition and subtraction since the definition of zero is that there is a number, b, such that a + b = a.
I was about to chime in and say that as a mathematician I’d never heard of zero not being finite, but then out of curiousity I went to dictionary.com. Lo and behold:
So it appears your friend may have the dictionary on his side. I guess 2(a) is the key thing here. Personally, I wouldn’t have thought “finite” and “infinitesimal” were mutually exclusive, but it’s not something I’ve ever really thought about.
Under the standard definition of “finite,” zero certainly is a finite number. However, expressions like “a finite amount of time,” meaning “a nonzero amount of time” are semi-common. It’s used to say that, while the amount of something might be small, it’s more than zero. The use of the expression comes from the infinitesimal notion in calculus. It would take a fair amount of explanation, but when calculus was first discovered, the proofs for it used a mathematical notion called “infinitesimals.” The mathematics for working with infinitesimals was pretty vague, so in the nineteenth century a completely different way of doing proofs in calculus was invented using deltas and epsilons, and that’s how calculus is presently taught. Actually, it is possible to make the notion of infinitesimals mathematically precise, and this was done by Abraham Robinson in the mid-twentieth century. In any case, you should just take the phrase “a finite amount of X” as being an idiomatic phrase in which the word “finite” means something different from its usual meaning.
I would argue that nonzero finite interval is redundant not because nonzero equals finite, but because nonzero equals interval. It’s an amount of time between two set points; if the amount of time is zero, there is no time between these points.
On the other hand, the “finite” part was redundant, too, inasmuch as you can’t have an infinite intervale between two notes in a piece of music that has already been played.
If I may clarify, the friend said that he’d not used the word “nonzero” because that would have been redundant. However, he said:
Given that context, were the interval equal to zero, the sentence would be absolutely bizarre; similarly, were the interval infinite, the sentence would be absolutely bizarre. Therefore, the word “finite” is redundant in the sentence.
Normally, this isn’t a big deal (language redundant: help understand!) but if he’s gonna bring up the issue, he oughtta be consistent, is all.
Of course, this in the technical sense, and not common usage. An open interval [a, b] is the set of all points x with a < x < b. If a = b, the interval is a single point.
I hadn’t considered that the word finite might have different meanings in common usage and in technical usage.
I googled the phrase “nonzero finite”, and it returned some IEEE documents discussing floating point computation, among others, so those people consider zero to be a finite number.
In physics, zero is not considered finite (but neither is it infinite), as a matter of convenience. My understanding is that it is because you can’t actually measure a zero interval, even in principle. The best you can do is say something is less than a given value (an upper bound). This is similar to measuring an “infinite” value: you can only state a lower bound.
So a physicist talking about a “finite” value is only indicating that it could be measured in principle. (And to answer the inevitable question: I’m talking about finite intervals here. You could label a point on a measurement scale as “0” and measure something at the “0”. But that’s only a reference point, not an interval.)
You can also think in terms of a logarithmic scale. Both zero and “infinity” are off-scale. Any real measurement is going to fall somewhere on that log scale. Zero and infinite are not possibilities.