Are there mathematical contexts where "finite" implicitly means "nonzero?"

Question

I recently gave my students in a discrete math class the following problem, a restatement of the heap paradox:

Let's say that zero rocks is not a lot of rocks (surely, 0 is not a lot of rocks) and that if you have a lot of rocks, removing one rock leaves behind a lot of rocks. Prove that no finite number of rocks is a lot of rocks.

A small number of students submitted proofs by induction with the base case starting at one rock rather than zero rocks. We deducted a point for this, saying that this left the case of zero rocks unaccounted for.

Some students replied back to us saying that zero is arguably not a finite number. Some students pointed out this dictionary definition of finite which explicitly excludes 0 as not finite.

My background is in discrete math, and I've never seen zero referred to as not finite. The empty set of zero elements is a finite set, for example. There are no finite groups of size zero, but that's a consequence of the group axioms rather than because 0 isn't finite.

Are there mathematical contexts in which zero is definitively considered to be not finite?

Thanks!

Answer 1

The problem is that physicists are more influential than mathematicians. They routinely consider zero to be a nonfinite quantity, probably because they are thinking logarithmically. If you hang around physicists, you will hear expressions like “very small but finite”.

But the concept of infinity is a mathematical one, not physical, and certainly mathematicians rule in this matter: zero is most certainly finite.

Answer 2

$0$ is doubtless finite.

I'd say that the paradox' root is the imprecise word 'lot'. I think that a 'lot' is a quantity that can't be perceived at glance. I'm sure that most people don't think in $3$ or $4$ when hear the word 'lot', because if there are $4$ stones, we don't need count them to know.

But I insist. $0$ is never an infinite number.

Answer 3

There is really no point in insisting that a definition in a dictionary has any implication on the mathematical meaning of the word. Germs have nothing to do with real world germs, and cardinals have absolutely nothing to do with the catholic church. Normal spaces are not those which are not irrational, and real numbers might not really exist (e.g. if the universe is finite).

In mathematics $0$ is a finite number, and $\varnothing$ is a finite set.

One might argue whether or not $0$ is a natural number, and that might be open to debate between mathematicians. If you define "finite" as having the cardinality of a natural number, and $0$ not to be a natural number, then indeed the empty set is "not finite", and so $0$ is not finite. But it seems like a very artificial thing to say. If you won't include $0$ in the natural numbers, then you'd define finite as empty or having the cardinality as a natural number.

The lesson here is to stick to the definitions, as best as the conventions you follow allow you. Don't get swayed by the natural language meaning of the word.

Answer 4

It seems to be not unheard-of to speak of small but finite quantities in applied mathematical fields. At least in this context, "finite" is obviously meant to mean "nonzero", or perhaps "not infinitesimal".

Additionally, in abstract algebra, it is not unusual to speak about rings/fields of "finite characteristic" to mean one whose characteristic is not zero. (Generally in ring theory, zero often behaves like the limit of ever larger elements, at least intuitively speaking, so it takes on some properties of "infinity").

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Note that the dictionary definition you link to doesn't claim that 0 cannot be finite, period. It lists three different mathematical usages of the word, not three conditions that all have to be satisfied before you can (according to the dictionary) call something finite. Only one of these possible meanings is "not zero".

Answer 5

Zero can be considered to be an infinitely small number, in some cases this is the natural thing to do. It is quite typical for many natural phenomena to be discontinuous when certain effects become exactly zero. E.g., if the viscosity of a fluid is exactly zero, then that's qualitatively different from being small but larger than zero, as in the latter case you can impose more boundary equations. In the limit of small viscosity you get a boundary layer where in the case of exactly zero viscosity you cannot impose that boundary condition. The smaller the viscosity becomes, the more the boundary layer shrinks.

Answer 6

Maybe this is too applied for your question, but there is a context in which finite means "large" or possibly large. That is in elasticity theory, to distinguish the linear theory of small strain from the nonlinear theory of "finite" strain.

Answer 7

Nobody seems to have picked up on definition 2(a) from the cited dictionary entry: “capable of being completely counted.”  Students are taught from childhood (see Wolfram MathWorld, Math/is Fun, Math Goodies, Purplemath, and For Dummies) that counting numbers are positive integers.  If you ask a person, “How many elephants do you have in your pockets?”, most will reply, “Don’t be silly!  I don’t have any.”, rather than “I have zero elephants in my pockets”, because it’s just common sense that you can’t count something that doesn’t exist — and because they’ve been taught that zero isn’t a “counting number”.  Well, then, if finite means “countable”, then it follows that zero must not be finite.

;-)