What is a number, really? Or to put it another way: think of the lights on a traffic light, the segments of the circle in the Mercedes-Benz logo, the beats in one measure of a waltz rhythm, the way Julius Caesar divided up Gaul, the colors on the U.S. flag, a three-liter bottle of Coca-Cola – what do these things have in common? Philosophers have claimed that there is a Platonic essence of “three-ness” that exists in all of these instances, and math educators look for children to develop a “number sense” that allows them to recognize it in all its different manifestations. But what if this is looking at it backward? What if “three,” as a concept, begins in our direct experience of the world? I’m going to talk through this line of reasoning, drawing on previous research on the development of number sense in children, and thinking it out using the semiotic theory of C. S. Peirce.
Think about how kids learn their numbers. My son, one month shy of his third birthday, can count up to twelve or fifteen or so, but he hasn’t learned how to enumerate objects in a set. They’re just a sequence of words that he’s memorized. Once he gets past twenty, the formation of number words will become recursive, so that he can keep generating new words, making the sequence indefinitely long. And that’s essentially all that numbers are: a way of generating an indefinitely long sequence of words. From a semiotic perspective, this is a “first,” an entity that exists on its own, considered apart from any connection to the outside world.
As kids learn the sequence of number words, they also learn to use them to count. That is, they learn to look at a set of objects and assign a number word to each object in the set. The last number word that they get to is the one that describes the set as a whole, and grammatically it becomes a quantifier – instead of “three” being just the word that comes between “two” and “four,” you can talk about “three cookies” or whatever, as a shorthand for “if I assign a number word to each of these cookies, I stop at three.” Semiotically, this is the “secondness” of number, the relation between number and some other phenomenon. And developmentally, this is when kids learn that a set of three items is still three, no matter how they are arranged in space.
The next step is the reification of number, which happens at around the time that kids learn arithmetic. You begin with “three apples plus two apples makes five apples,” but eventually you generalize it, stripping away the apples, so that “three plus two makes five” no matter what you’re counting. The number words now look more like nouns than quantifiers, as if “three” were a thing that existed on its own. Now the “thirdness” of the sign emerges: the relation between relations that allows us to see each occurrence of the sign as a token of a type. In other words, we’ve seen enough sets with three objects that we can see “sets of three objects” as a class, including the waltz, the soda, the traffic light, etc. The noun “three” is now a shorthand for the common property of sets describable with quantifier “three,” which was, in turn, a shorthand for counting up to three.
The final stage is “alienation,” which construes the numbers as entities that exist in their own right, apart from (prior to?) any human engagement with them. This is what allows the number three to have properties of its own: it’s odd, it’s prime, it’s an integer, and so on. (Try phrasing those claims in terms of “if you assign number words to the objects on the table, the last number word you use is three.”) From a semiotic perspective, I would describe this as a process of “rhematization,” which takes a bit more explaining.
To begin with, Peirce considers the semiotic process as more than just a sign indicating its object. Crucially, there’s a third piece, called the interpretant, that indicates the effect of the sign’s being used. To use an example from N. J. Enfield, dark clouds can be taken as a sign whose object is the impending rain, and whose interpretant is my throwing an umbrella in my bag. Peirce also writes about three different mechanisms through which a sign can be linked to its object. An icon, such as a portrait, indicates its object through resemblance; an index, such as wet grass signifying rain, indicates its object through common history; and a symbol, such as a yin-yang signifying Taoism, indicates its object through social convention. This is an oversimplification, of course, and signs can blur the boundaries between these categories; a photograph is an index (created through the action of light on a photosensitive medium) as well as an icon (it looks like its object), and a word, the prototypical symbol, is learned as an index that is repetitively used in the presence of its object.
Since a sign can fall into different categories of this schema, it can be interesting to talk about how it’s perceived in this respect. So, when you look at a photograph of me, it may have originated as an index through chemical processes, but you interact with it as a physical resemblance of its object – that is, as an icon. And when you see it, it brings to mind the category made up of every representation of me that you’ve seen; it’s a token of a type, and that type is “pictures of me.” When a sign is taken as an icon in its interpretant in this way, it’s called a rheme. What I’m claiming is that the alienation step of number sense acquisition can be viewed semiotically as rhematization. The word “three” or the numeral 3 doesn’t actually resemble a set with three objects in it, so it doesn’t begin as an icon; but when you hear or see it, it’s taken as a direct representation of that abstract quality of “three-ness.” It refers to a type, like a blank slate that can be applied to any token collection with three things in it.
So what’s the point of this theorizing? What I’m hoping is that it’ll help me to analyze the sort of tasks that students are given in mathematics classrooms. At various levels, students are led to build up these equivalence classes, to expand the set of tokens that correspond to a given type. So “three” doesn’t only include three cookies on a table; it’s also equivalent to 5 – 2, or to 4x + 7 if x = -1, or to f'(x) if f(x) = 3x – 12. Once the sign is taken as an icon, it opens up the whole world of mathematics.
This has been an attempt to synthesize these two sources:
Peirce, C. S. (1955). Logic as semiotic: the theory of signs. In J. Buchler (Ed.), Philosophical writings of Peirce (pp. 98–115). New York: Dover.
Sfard, A. (2008). Thinking as communicating: human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.