It’s significant when an old problem gets solved, but it’s even more significant when the intellectual landscape shifts so thoroughly (albeit slowly) that an old problem ceases to seem like a problem at all. A good example of this phenomenon is what happened to the problem of zero. And if you’re thinking “What problem?”, that just shows how thoroughly the winning side of the zero war carried the day.^{1}

**FROM PLACEHOLDER TO NUMBER**

Many civilizations (notably India) had a number that we can recognize as the modern zero, but ancient Greek civilization didn’t; for those incorrigibly philosophical Greeks the very idea of a *something* (a symbol) representing a *nothing* seemed paradoxical and out of place in mathematics.^{2} The Greeks didn’t have any need for a digit 0, because Greek numerals, like Roman numerals, aren’t based on a positional system; unlike the modern digit “5”, which can mean five, fifty, five hundred, or more depending on how many digits succeed it, a Roman “V” can only mean 5.

In the following centuries, many practical-minded people in many countries realized the virtues of positional notation, wherein a digit like “5” can mean different things depending on its context. Arabic numerals (based on Indian numerals) were an especially handy way of encoding numbers, and caught the attention of the Italian mathematician Leonardo of Pisa (better known as Fibonacci), who wrote: “The nine Indian figures are 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0, any number may be written.” Leonardo brought Arabic numerals to Europe, where they caught on among merchants. Over time, as Arabic numerals became familiar, the difference between the figures 1 through 9 and the sign 0 receded in importance, and a lone 0, unchaperoned by other digits, became increasingly acceptable as a kind of honorary number. European mathematicians became comfortable with zero-as-a-number in the 1600s, and opposition mostly died out in the 1800s.

Nowadays the pictorial device of the number line, found in classrooms throughout the world, makes zero and negative numbers vivid in a way that mere formulas never could, as it trades the philosophical question “What is zero?” for the practical question “Where is zero?” With the acceptance of the number line, the tale of zero became the ultimate rags-to-riches story: once a lowly placeholder for digits, zero has risen to grandeur as the queen of the number line, enthroned at its very center.

The saga of how zero gained worldwide acceptance highlights the international nature of the mathematical enterprise. If you want to know what happened, I suggest you read “Zero: The Biography of a Dangerous Idea” by Charles Seife or “The Nothing That Is: A Natural History of Zero” by Robert Kaplan.

But I want to raise the question, who needs zero? After all, the counting numbers are for counting. If you’re counting the elements of some collection (or, as we often say, of some *set*) and the total you get is zero, the collection whose elements you were counting doesn’t have any elements, so you had no business trying to count its elements in the first place.

For that matter, who needs empty sets (that is, sets that contain no elements)? It’s the business of sets to contain elements; a set that contains no elements isn’t doing its job, so why should we pay any attention to it, or honor it by calling it a set at all?

**THE DISSIDENT CLERGYMAN**

Even as recently as the mid-nineteenth century, there were prominent people whose answer to the question “Who needs zero?” was “Nobody!” I’m thinking of the English clergyman-mathematician William Frend, whose “Principles of Algebra” strenuously avoided use of zero and negative numbers. Frend also wrote a burlesque of zero in the form of a Rabelaisian fan-fiction in which some academics, representing Frend’s zero-loving intellectual adversaries, visit King Pantagruel with the following pro-zero proclamation:

‘It has therefore been most clearly proved that as all matter may be divided into parts infinitely smaller than the infinitely smallest part of the infinitesimal of nothing, so nothing has all the properties of something, and may become, by just and lawful right, susceptible of addition, subtraction, multiplication, division, squaring, and cubing: that it is to all intents and purposes as good as anything that has been, is, or can be taught in the nine universities of the land, and to deprive it of its rights is a most cruel innovation and usurpation, tending to destroy all just subordination in the world, making all universities superfluous, leveling vice-chancellors, doctors, and proctors, masters, bachelors, and scholars, to the mean and contemptible state of butchers and tallow-chandlers, bricklayers and chimney-sweepers, who, if it were not for these learned mysteries, might think that they knew as much as their betters. Every one then, who has the good of science at heart, must pray for the interference of his highness to put a stop to all the disputes about nothing, and by his decision to convince all gainsayers that the science of nothing is taught in the best manner in the universities, to the great edification and improvement of all the youth in the land.’

The king gets his revenge on the prating professors by offering his guests a great banquet and then serving them huge platters and tureens of nothing at all.^{3}

Frend was able to reject zero because in the context of the mathematics of his day, most sensible questions had non-zero answers. Conversely, questions for which the answer is zero could be construed as being on some level non-sensible. Consider for instance the question “What is the volume of a square?” Nowadays we’d be inclined to say “The volume of a square is zero”, but it’d be just as reasonable to say “Squares don’t have volume.” Likewise, when his contemporaries said that the solution to *x*+3=3 is zero, Frend said that the equation “*x*+3=3″ had no solution. Meanwhile, everybody else was freely applying zero and negative numbers and it drove Frend to distraction. The fact that their reproachable methods, when applied to sensible questions, gave irreproachable answers only heightened his frustration.

Avoiding zero is harder nowadays because of new fields of mathematical inquiry that didn’t exist in Frend’s day, or whose state of development was fairly crude.^{4} I’m thinking for instance of combinatorics, one of my favorite kinds of mathematics, which is about counting collections of objects. Some of these collections turn out to be empty, in which case the count of elements is zero. If Frend were alive, he’d try to configure combinatorics so as to avoid empty sets.^{5}

My short answer to the question “Who needs sets containing no elements?” is that we may not need sets that we know ahead of time to be empty, but we do need to be able to work with sets without worrying all the time about whether they’re empty or not, and we need tools that will let us do this. When we create systems of thought for handling sets in their full generality, we will deform the framework we construct if we build into it an exclusion of empty sets. The good we might do by excluding such silly sets is more than offset by the extra trouble we go to in order to enforce the exclusion.

**INCLUDING/EXCLUDING ZERO**

My favorite example of the economy of thought afforded by allowing sets to be empty comes from a mathematical principle called the Principle of Inclusion-Exclusion, which in its simplest case looks like this:

#(*A* **∪** *B*) = #(*A*) + #(*B*) – #(*A* **∩** *B*)

Before I explain the notation, I’ll give an illustrative application: How many integers from 1 to 100 are divisible by 2 or 5 or both? Of the 100 integers ranging from 1 to 100, half (that is, 50) are divisible by 2 and one-fifth (that is, 20) are divisible by 5, so we might hastily think that the answer to the question is 50+20, or 70. But that total double-counts the integers that are divisible by both 2 and 5; to fix our total, we need to subtract the number of integers we double-counted. That is, we need to subtract from 70 the number of integers from 1 to 100 that are divisible by 10. The number of such integers is one-tenth of 100, or 10, so the correct answer to the original problem is 70–10, or 60.

The general formula operating here is that if you have two sets *A* and *B*, you must have

#(*A* **∪** *B*) = #(*A*) + #(*B*) – #(*A* **∩** *B*)

where *A* **∪** *B* means the set of elements that belong to *A* or *B* or both (aka the *union* of *A* and *B*), *A* **∩** *B* means the set of elements that belong to both *A* and *B* (aka the *intersection* of *A* and *B*), and #(*S*) means the number of elements in the set *S* (where in the formula *S* can be *A* **∪** *B*, *A*, *B*, or *A* **∩** *B*). And this is where 0 can enter the party as an uninvited guest, because sometimes it may turn out that *A* and *B* have no elements in common, so that *A* **∩** *B* is an empty set with 0 elements, even if this wasn’t clear at the start.

If you reject the use of empty sets and the number 0, there is a way out for you: you can have *two* principles, one of which says that

#(*A* **∪** *B*) = #(*A*) + #(*B*) – #(*A* **∩** *B*)

when *A* and *B* have one or more elements in common, and the other of which says that

#(*A* **∪** *B*) = #(*A*) + #(*B*)

when *A* and *B* have no elements in common.

The problem with this approach becomes clear when we extend the Principle of Inclusion-Exclusion to handle the case where there are more than two sets in the game. For instance, say we have three nonempty sets *A*, *B*, and *C*, and we want to know how many elements there are that belong to one or more of the sets. If we’re comfortable with empty sets, we can use the formula

#(*A* **∪** *B* **∪** *C*) = #(*A*) + #(*B*) + #(*C*) – #(*A* **∩** *B*) – #(*A* **∩** *C*) – #(*B* **∩** *C*) + #(*A* **∩** *B* **∩** *C*)

without having to worry about whether one or more of the sets *A* **∩** *B*, *A* **∩** *C*, *B* **∩** *C*, and *A* **∩** *B* **∩** *C* might actually be empty. But if we were empty-set deniers and disallowed zero as a number, we would need nine different versions of the formula,^{6} each omitting various terms corresponding to sets that are empty in that particular case. Including zero as a number, and the empty set as a set, allows us to unify nine different cases in one single formula.

**AGAINST (AND FOR) PHILOSOPHY**

I haven’t had a chance to dig into Frend’s writing, so I feel I haven’t earned the right to opine about how he would respond to the above if I could resurrect and converse with him. (Maybe some of you know more about Frend. And maybe some of you know a modern-day zero-denier! If so, I’d be interested in hearing more about them.) Still, I can’t help speculating. I expect that the Reverend would grant the convenience of zero in much the same way as he’d grudgingly concede the convenience of theft, but that he would argue that convenience cannot compensate us for conceptual sin. I imagine him invoking the paradox that dividing both sides of the true equation 1 × 0 = 2 × 0 by zero gives the false equation 1 = 2, and mocking the standard response “You can’t divide by zero” as a laughable contrivance whose ad hoc nature is a desperate attempt to disguise the philosophic incoherence of the concept of zero.

I actually think Frend is partly right in his critique of zero, and to the extent that he’s wrong, it’s partly because he’s applying some of the tools of philosophy in a setting where they’re not useful. When we do philosophy, it’s important to clarify our concepts (truth, beauty, justice) before we try to reason about them, and the concept of zero leads us to the concept of nothing which can lead us to paradoxes if we’re not careful. (“Nothing is better than complete happiness, and a ham sandwich is better than nothing; therefore a ham sandwich is better than complete happiness.”) But in mathematics, premature attempts to reach philosophical clarity can get in the way of progress both at the individual level and at the cultural level. Sometimes you have to just start talking before you understand what you’re talking about. For a mathematician, the meaning of zero comes from the role it plays in number systems, and our comfort with the concept derives from the experience of using it. This shouldn’t seem strange if you consider how one’s mind forms concepts about the world (milk, bed, house) in one’s earliest years. The very young child isn’t disconcerted by questions like “Is almond milk really milk? Is a sofa a bed? Can a car be a house?” Everyday concepts have fuzzy boundaries that on some level make them philosophically incoherent, but that kind of incoherence doesn’t get in the way of our using them.

Then again, that kind of fuzziness wasn’t what Frend objected to about zero. He maintained that zero didn’t denote anything. For him, numbers were closely allied to the notion of quantity, and the kinds of quantities he was concerned with were things like length, area, duration, and weight that in commonsense applications are positive. To convert Frend to my way of thinking, accepting not just zero but negative numbers as well, I might try to bring him into a context in which zero and negative numbers have tangible meaning. For instance, I might bring him to a large tub of water and coax him into doing experiments involving blocks made of different materials, some heavier than water and others lighter. When we attach two or more such blocks together, the composite object sometimes sinks and sometimes floats and occasionally exhibits neutral buoyancy, and the natural mathematical system for predicting what will happen is the number line, with sinking on one side of the line and buoyancy on the other and neutral buoyancy as the point where the two half-lines touch. But I don’t think he’d be swayed. He’d say that weight and buoyancy are separate quantities that may offset each other but should not be confused with one another.

Perhaps I would try parody instead, and use Frend-ish philosophical methods to attack the idea of fractions. “You cannot have half a horse, for if you cut a horse in half it ceases to live, thus losing an essential part of its nature; half a horse is no horse at all.” I don’t think this would change his mind either. If his friend and son-in-law Augustus De Morgan couldn’t convert him, what chance do I have?

In the history of mathematics, philosophical qualms about other new mathematical entities paralleled Frend’s discomfort with zero and negative numbers; as we’ll see, when other generalizations of the number concept (such as complex numbers or Cantor’s infinities) were introduced, some people objected. But over time these qualms weakened, even as mathematicians’ fancies became ever more extravagant, to the point where Conway’s surreal number system, with its infinities and infinitesimals, inspired no objections at all. That’s because we now understand number systems not as things that reside in the world but as constructs in our minds that we use in trying to understand the world. A construct need not be meaningful in every context to be useful; it need only be meaningful in *some* context. Consider that –1 is not a meaningful answer to “How many cows are in that field?” but is a meaningful answer to “What’s the temperature?” To object that –1 is incoherent because it fails to count things is to make the mistake of thinking that numbers like –1 are objects in the world rather than constructs in our minds. And to the extent that philosophy teaches us to honor such distinctions, I’m all for it.

**ZERO IS AS ZERO DOES**

If a main innovation of math in recent centuries was coming up with new kinds of numbers to play with, a main innovation in the philosophy of math was the insight that specifying the rules new sorts of numbers follow is a sufficient grounding for those exotic number and number-analogues. Indeed, knowing how to play with them is in a sense knowing what they are.

What led mathematicians to take this position? The story of zero is part of what led us here. Looking at zero in isolation can lead to disorientation, existential anxiety, or perhaps even terror (a box that contains zero cookies also contains zero monsters and hence is only one small step away from containing one monster). But this just shows that looking at zero in isolation is the wrong way to look at it. Instead of focussing on zero on its own, one should look at how zero interacts with other numbers via the familiar operations of addition and multiplication. Once we figure out how to extend these operations on positive integers to allow 0 to participate, we have a well-defined game, even if the precise nature of this new piece, this 0, is initially obscure. As we play the game and get comfortable using zero, questions of its “essential nature” recede in importance or even become meaningless. A pure mathematician getting derailed by the question “What is zero?” would be like a chess player getting derailed by the question “What is a pawn?” The rules of chess tell us what pawns do, and what a pawn *does* in a sense is what a pawn *is*. As long as our rules are consistent, they constitute a game, and one gets good at the game by playing it. Likewise, with math, as long as our assumptions are logically consistent, our system of assumptions becomes *about* something, namely, ghostly creatures of the mind that satisfy those assumptions by definition. The way one gets a feeling for those ghosts is by interacting with them in some suitable arena-in-the-mind. Then these constructs become enlivened, and one begins to get a sense of what they are, not as a prerequisite for reasoning about them, but in the course of living with them, as a young child lives with milk, beds, and houses.

What I’m describing here is the operational view of zero that situates zero within a number system and defines its ontology relationally. In this setting zero is the amalgam of all the properties it enjoys, like 0+1=1 and 0×1=0. And how is 1 defined? In the same relational fashion, as the amalgam of all its properties. Philosophically this might seem like a vicious circle, defining numbers in terms of properties that relate them to other numbers. But maybe the lesson here is that a strictly linear metaphysics can’t do justice to situations in which relations between entities are the most interesting thing about them and the ontology of individual entities is both baffling and irrelevant.

I believe I once read a book, whose author and title I no longer recall, that credited some late medieval (or maybe Renaissance) magician-alchemist-mathematician for the operational view of numbers. Specifically, John Dee or Nicolas of Cusa or someone like that claimed that only God can truly know numbers, so that we humans should focus on how to calculate with numbers and leave the metaphysics to God. If any of you know what magician (and what book) I’m recalling, please let me know! That magician’s advocacy of the operational approach to Number opened the door to all sorts of new number systems, and ironically allowed human mathematicians to usurp some of the Divine power to create things out of nothing.

*Thanks to Dave LeCompte.*

**ENDNOTES**

#1: Maybe “war” is over-dramatic. Still, to this day, some people consider zero to be a natural number and others don’t, so that when one uses the symbol **ℕ** (denoting the set of natural numbers), one is normally expected to declare which camp one belongs to. This is understood by all parties to be a mere matter of notation and terminology; the sets {1, 2, 3, …} and {0, 1, 2, 3, …} are both accepted as valid sets, and the only issue is which one deserves primacy. But this polite detente depends on the fact that both parties accept 0 as a mathematical entity. Such was not always the case.

#2: In fact, the Greeks didn’t even think 1 should be considered a number, but rather as the building block from which proper numbers like 2, 3, etc. were built.

#3: For more about this sort of menu, see my essay about the null salad.

#4: Of course, the reason mathematics has moved on to these new territories is tied up with the acceptance of zero.

#5: Some readers may object to my use of the plural in the phrase “empty sets”. I’m not entirely comfortable with it either. Over a century ago, mathematicians adopted an extensional approach to sets, wherein two sets are equal when they have the same elements. This means that a set of cookies that happens to contain no cookies is the same set as a set of monsters that happens to contain no monsters; they are both instances of *the* empty set. One might for some purposes prefer an intensional kind of set theory, in which one can distinguish between two empty sets based on what kinds of elements they are permitted to, but don’t, contain, but that’s not how mainstream set theory works.

#6: At issue are the different ways in which some or all of the sets *A* **∩** *B* , *A* **∩** *C* , *B* **∩** *C* , and *A* **∩** *B* **∩** *C* might be empty. When *A* **∩** *B* **∩** *C* is nonempty, each of the other three sets must be nonempty; that accounts for one of the nine cases. Otherwise, if *A* **∩** *B* **∩** *C* is empty, each of the other three sets might be empty or nonempty independently of the others; this accounts for the other 2×2×2 = 8 cases. How many distinct cases arise if we have four sets, or more? Is there a general formula for the number of cases in terms of the number of sets? I don’t know, but I expect the number of cases increases doubly-exponentially.

**REFERENCES**

Martin Gardner, “Negative Numbers”, in “Penrose Tiles to Trapdoor Ciphers”.

Robert Kaplan, “The Nothing That Is: A Natural History of Zero”.

Charles Seife, “Zero: The Biography of a Dangerous Idea”.

Matt LehmanPeople of a certain age fondly remember this catchy number.

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jamesproppPost authorI’d forgotten about that song! Thanks for the trip down memory lane. And speaking of songs about zero: https://youtu.be/L8pySVYytRs

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PaulI strongly urge you to re-number your endnotes to start with 0, if only for this post!

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jamesproppPost authorA very natural idea!

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Naren SundarI like your inclusion-exclusion example. It reminds of how I dislike opening an absolute value operation like |x – y| where it leads you to two possible cases.

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Dave LeCompteTypo in the properties of zero, or maybe I don’t know as much about its additive properties as I thought: “zero is the amalgam of all the properties it enjoys, like 0+1=0 and 0×1=0.”

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jamesproppPost authorYikes! Thanks for catching that.

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