How many instances of some mathematical object (e.g. an integer) are there?

Question

My question is simple: How many instances of a particular mathematical object are there and what's the reason for it? One? Two? Infinite? Undefined? For example, how many instances of the integer "1" are there? Is the first "1" in the equation "1 + 1 = 2" the same, identical, unique "1" as the second "1", or are they two distinct objects of the same concept of "1"? Why is it that or the other way?

Answer 1

I fancy myself a philosopher and mathematical logician, and I would like to expand upon Ethan Bolker's comment. It is true that questions about the nature of mathematical objects as "things" rarely plays a role in ordinary mathematics; but I think that the reason why this is the case deserves to be addressed.

The problem with this question is that the notion of a "mathematical object" is ambiguous.

In the case of physical objects within a shared range of perception, we can think of certain linguistic tokens (symbols, words, sound, etc.) as labels for the objects themselves. In philosophy and linguistics, such objects are called the "referents" of the tokens. However, many "linguistic objects," such as those which appear in mathematics, do not refer to physical things. Because of this, we cannot meaningfully de-reference them in a way that everyone can agree upon.

The question of whether or not tokens like "$1$" have referents at all is a central subject of the ontology of mathematics. Depending on the point your point of view, we might say that there is exactly one object referred to by the symbol "$1$," that each instance of the symbol "$1$" signifies a different object, that "$1$" doesn't refer to anything at all, or that the thing referred to by "$1$" is dependent on some set of parameters which vary across time.

From an extremely conservative "objective" standpoint, each answer to the question is only an opinion.

However, mathematics itself consists solely of the symbols used to represent it, and the rules used for manipulating those symbols. Since these rules do not change regardless of what the symbols refer to, we can still "do" maths without knowing what it's "really" all about.

It is similar to a game of chess: we might ask the question "how many knights are really on the board?" We could answer "the same as the number of pieces," "only two, because each piece of the same colour represents the same person," or "none, because knights are people, and there are no people standing on the board." Regardless of how or even if we answer the question, we can still play a game of chess.

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That said, one thing we can all agree on is the number of tokens. This is because tokens - whether pieces on a chessboard or symbols on a page - are physical objects.

In this case, the symbol "$1$" appears twice in the formula "$1+1=2$."

Answer 2

The idea that there can be two instances of something does.not make sense. If you have two things either you can tell them apart, and they are two different things, or you cannot, and they are the same thing.

Answer 3

There is no $1$ answer to this, pardon the pun. Opinions of mathematicians on the nature of the integer "$1$" vary from: some Platonic ideal of the essence of oneness, uniquely determined for all time; the set $\{\{\}\}$; any generator of a monoid freely generated by a single element; the size of any set, any two functions into which are equal; and probably most frequently, no opinion whatsoever. The statement that $1+1=2$ is not affected by any of this, since it is true no matter what you believe the three numerals metaphysically designate.

Answer 4

What you are asking about is called the problem of universals.

One table, one chair, one house, the digit 1, the Roman numeral I and the word one are clearly not all the same. Yet we perceive that they each possess a common property of "oneness". Do you believe there is an entity that possess this property of "oneness" and no other properties ? Then you are a Platonic realist. Do you believe "oneness" is a mental or psychological concept that is useful when categorising the real world but has no independent existence of its own ? Then you are an Aristotelian realist.

Answer 5

Suppose that you're a woodworker who needs to cut a board to be 2 feet long. You have a drawer full of 1-foot-long rulers. Consider two different ways of measuring the board:

1. You grab two rulers and lay them end to end. This physically represents $1 + 1 = 2$, with the two 1's denoting distinct objects that are equal (to within an imperceptible manufacturing tolerance) in terms of length.
2. You grab one ruler, make a pencil mark where it ends, and continue measuring from the mark. This is also $1 + 1 = 2$, but the two 1's are for the same physical object, which is used twice.

So, without a clarification on what “objects” the two instances of “1” refer to, it could be interpreted either way.

Answer 6

Let's focus on the natural numbers: 1, 2, 3, 4, 5, and so on. In one way of thinking, these objects were born when people realized that a net holding three fish, a family with three children, a bag of three spearheads, and a bundle of three cords had something important in common. If you give a fish to each child, bind a spearhead with each cord, hang a fish from each cord, or anything like that, you'll find that things pair perfectly, with nothing left over. Eventually, people settled on a standard three—the words "one", "two", "three"—to measure all these threes against. In arithmetic, all threes are interchangeable: if adding three shells to your bag of gifts will make a gift for each of your cousins, then adding three flowers or three stones will do the same (arithmetic tells you all about how many gifts there are, but nothing about how well they might be received). All of the collections above are instances—or representatives, as mathematicians like to say—of the number three. In a calculation, any one can stand in for any other.

Representatives built from words are especially convenient because you have them with you all the time: you can use the words "one", "two", "three" to recognize another representative of three wherever you are, without carrying any special equipment. Representatives built from notches on a stick or knots in a cord are durable and easy to transport, and therefore useful for record-keeping. Representatives built from drumbeats or strokes of a bell can travel long distances almost instantly, making them useful for communication and synchronization. What counts as a representative is a bit subjective, because it depends on what you recognize as a collection and how you split it into distinct elements. In practice, though, trained people almost always agree on what's a three and what isn't. The same goes for ones, twos, sixes, sevens, and all other natural numbers.

Representatives of small numbers, like three or eight, are so easy to produce that their supply is best considered unlimited. Representatives of larger numbers, like a hundred or a thousand, are quite hard to produce, but a look at the night sky or a handful of sand might convince you that their supply is still unlimited in principle. To work with large numbers, people these days rarely use representatives directly; they resort instead to tricks like treating a bent notch as a bag holding ten straight notches, or treating the same notches as bags of different sizes depending on where they're written relative to each other. (I'm sorry that was so vague; I'm reaching the limits of my understanding of modern arithmetic here.)

Many other mathematical objects fit the picture of an abstract property or ideal form that many concrete representatives have in common. Shapes, for example, work this way: a straight pole, a taut cord, and the horizon of the sea share the common form of a line; a bubble, the sun, the moon, and the light through a pinhole share the common form of a circle; and a sunflower's seeds and a snail's shell share the common form of a spiral. Here, you need more training to recognize a representative, and the judgment is more subjective. The important thing is that representatives are still interchangeable: when I go to measure a field or draw the foundation of a building (I'm a geometer by trade), I know that a straight pole and a taut cord can stand in for each other, so I'm free to use whichever is more available or convenient.