# Are there mathematical contexts where “finite” implicitly means “nonzero?”

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!

• As far as I know, there aren't any. – ajotatxe Oct 22 '14 at 16:59
• The dictionary does more than what you say, it says 'not zero' and 'finite' are synonymous which is wrong by any account. – Git Gud Oct 22 '14 at 17:07
• Looking at the dictionary entry you linked to, it seems to list three separate definitions of "finite" as it relates to math: (a) capable of being completely counted (b) not infinite or infinitesimal (c) not zero, which is the "physicists' sense" as Lubin puts it. Definition (c) makes some sense to include in a general dictionary as there are certainly people who use "finite" to mean nonzero, without regard to whether it's mathematically valid. – David Z Oct 22 '14 at 18:29
• @anorton That's correct, which is why the set of all rational numbers is infinite. :-) – templatetypedef Oct 22 '14 at 20:20
• @anorton The definition you're referencing from the dictionary is Dedekind-infiniteness. One way of proving that the set of rational numbers is infinite is to find a bijection from the rationals to a subset of the rationals. In other words, the definition that you were referencing does correctly let you conclude that the rationals are infinite, though it doesn't let you conclude just how big they are. – templatetypedef Oct 22 '14 at 20:48

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.

• The concept of zero is purely mathematical as well. Or at least purely abstract. – Asaf Karagila Oct 22 '14 at 17:30
• Here's one physicist chiming in who definitely considers zero to be a finite number... – David Z Oct 22 '14 at 18:25
• But, you know, in deference to physicists, it’s just as hard to measure a tiny number accurately as it is to measure its reciprocal. I think that in the real world, the logarithmic view makes perfectly good sense. – Lubin Oct 22 '14 at 20:35
• Interesting logarithm theory. This usage of "finite" to mean nonzero has irritated me since the age of 16 (I'm now 51, so that's 35 years of tooth grinding all pent up!) when I first heard my high school chemistry teacher use it in this way. It sounded haughty to me then thus has so ever since: even more so because he was clearly confusing some of the other students when the much more descriptive "nonzero" would have gotten the ideas across much better. I thought for a while (since it seemed to be more of a foible the generation before me until I got older) that it might have been some ... – WetSavannaAnimal May 27 '15 at 12:41
• ... botched rendering of the term "standard" from nonstandard analysis in some ham fisted attempt to show off one's familiarity with that field and the generation before me would have heard of its emergence in the "what's new in science" press when they were undergrads and grads. But then 0 would have equally been a standard hyperreal just as it is finite, so it was all a bit mystifying to me. – WetSavannaAnimal May 27 '15 at 12:46

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.

• I agree about dictionary definitions not being valid in math. The main reason I was asking was to check whether there actually was some part of math where 0 was not considered finite, since it wouldn't surprise me in the slightest if there was some validity to the students' concerns that just happen to be in a branch of math I'm not familiar with. – templatetypedef Oct 22 '14 at 17:20
• +1 for "There is really no point in insisting that a definition in a dictionary has any implication on the mathematical meaning of the word." If students are attempting to learn math from lexicographers instead of actual mathematicians, then something is very, very wrong. – imallett Oct 22 '14 at 17:49
• @Graphics If I were one of those students, I would point out that unless the word 'finite' has been actually defined in class (and for an informal class such as this appears to be, I wouldn't be surprised if this wasn't the case), I would have no objective basis to form a concept of the precise definition of the word, and lacking other materials, consulted a dictionary to see if there was a 'common' English meaning being invoked. After all, mathematicians use many words that they have not explicitly defined, and to a beginner it may not be clear which adjectives are subject to definition. – Mario Carneiro Oct 23 '14 at 0:23
• @MarioCarneiro This is true, and I generally agree. However, I sortof feel like if you're in discrete structures then you should know by now that the real number 0 isn't in-finite. – imallett Oct 23 '14 at 2:40
• @Graphics Then again, maybe these are the same kind of students who aren't sure if zero is even... For some reason the concept of zero doesn't mix well with people's intuitive mental models without a bit of training, so in some sense this complaint doesn't surprise me. – Mario Carneiro Oct 23 '14 at 3:42

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").

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".

$0$ 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.

• When I asked this question, I specifically did so to point out how some definitions end up not describing anything. I totally agree about the root of the paradox. – templatetypedef Oct 22 '14 at 17:17

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.

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.

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.

;-)