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Numbers are ideas. Which ideas are they? And what are ideas, in the first place? I’ll answer these questions in reverse order.

When I think of Obama, for instance, there is, first, the token mental act. Somehow or other, at some place and time, I point my mind’s eye towards Obama. The token mental act is not what I’m calling an idea. At the other end of my thought is the intentional object, the thing my thought is about. In this case, the object of my thought is Obama, the man. Though ideas are sometimes the objects of an idea, an idea is not generally its own intentional object. So an idea is neither the token mental act nor the idea’s intentional object. The mental act is directed towards Obama through the intentional or representational content of the mental act. This content is an idea of Obama. Letideas be the intentional or representational contents embodied within mental acts. The claim that numbers are ideas should then be understood as the claim that numbers are a particular sort of intentional or representational content.

Ideas have two kinds of structure which are relevant for identifying numbers with ideas. The first is a kind of parthood structure. Complex ideas have other ideas as parts. Take the complete idea of Obama as he is in the actual world, for example. This idea has many parts, including the ideas of being human, being the 44th president of the U.S., being born in Hawaii, having two daughters, and being married to Michelle.42 As I claim in Chapter 4, ideational parthood is a kind of mental conjunction. The complete idea of Obama is a large conjunction of ideas. It is the idea of beingthis andthatand so on. The ‘and’ here signifies the relevant notion of conjunction.

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An idea Ixis anabstractionfrom Iywhen every conjunct of idea Ixis also a conjunct of idea Iy

but at least some conjunct of Iy is not one of Ix’s conjuncts. For example, the idea of being human is

itself a conjunct of the complete idea of Obama, but obviously not every conjunct of the complete idea of Obama is a conjunct of the more general idea of being human. Hence, the idea of being human is an abstraction of the complete idea of Obama. So is the idea of being identical to something. No matter something’s nature, it is identical to itself and hence identical to something. So the idea of any particular thing includes not only that it is identical to itself but also identical to something. I’ll call the idea of being identical to something the idea of being something. The idea of being something is a mental conjunct of the idea of any particular thing. Since the idea of being something is a conjunct of the idea of any particular thing but the idea of any particular thing includes more ideas besides, the idea of being something is an abstraction of an idea of any particular thing.

The second kind of relevant structure is collective. Collective ideas are the mental analogue of plural referring terms.43 Whereas plural referring terms refer to some individuals, say, Tom, Dick, and Harry, a collective idea is an idea of individuals as distinct individuals. A collective idea is an idea of this thing and that thing and that other thing, and so on. The ideaTom and Dick and Harryis a collective idea of Tom, Dick and Harry.

Consider a collective idea of Tom and Harry. The idea of Tom includes that he is not identical to Harry; the idea of Harry includes that he is not identical to Tom. Let’s subject this collective idea to two consecutive steps of abstraction. First, there is an abstraction from this collective idea which subtracts from the idea of Tom everything except for the idea of being something and the idea of being distinct from Harry. This resulting abstraction also subtracts any ideas within the idea of Harry which concern Tom. So in this first step, the idea of Harry no longer includes the idea of being not identical to Tom. Instead, the idea of Harry includes the idea of being not identical to the indeterminate something which is not identical to Harry. At this first step, the abstraction represents something (which isn’t identical to Harry) and Harry.

The second abstraction subtracts from the idea of Harry everything except for the idea of being something and the idea of being distinct from the other something. This second abstraction represents a thing and another distinct thing. This final abstraction is the idea of something and something

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else. The idea of something and something else is a collective idea. Each component includes the idea of being something. And each component includes the idea of being distinct from the other something. Notice that the collective idea represents the distinctness of a thing and another, not the distinctness of the ideas themselves. One component does not include the idea of being distinct from the other idea, but the idea of being distinct from whatever the other component represents. This idea of something and something else is the number 2, the number of individuals the collective idea represents or is about. And the number 2 is an abstraction of any collective idea of a particular thing and another particular thing.

Here are the first few numbers in the number series:

0 = the ideanothing

1 = the ideasomething

2 = the ideasomething and something else

Hence, ‘the number 0’ is another name for the ideanothing; ‘the number 1’ is another name for the ideasomething; and so on.

Each number is a collective idea of units. And each unit is an idea with incomplete content. First, consider once more the unit which constitutes the number 1. This unit consists of the idea of something. Like the incomplete idea of being a man in general, which is an idea of no particular man, the idea of being something is an incomplete idea of a thing in general. Unlike the haecceities from Chapter 5 whose content determines an object, there is no particular thing which is the object of the idea. Like the incomplete idea of being a man in general, which each man individually satisfies, the incomplete idea of being something is an idea which each thing indivdually satisfies. Since anything at all by itself satisfies the idea of being something, we’ll say that the idea hasuniversal range. On any particular occasion, one may apply an incomplete idea to anything which satisfies the idea: one may apply the idea of a man to some particular man, the idea of an animal to a particular animal, or the idea of something to a particular thing. On these occasions, the things to which we apply incomplete ideas are their objects. Because the content of an incomplete idea doesn’t specify any particular thing as its object, which thing is its object depends on some intention to apply the idea to

a particular thing (and not to any others). The idea of being something has universal range because it may take any particular thing as its object. The number 1 (a single unit) has universal range. On any particular occasion, we may apply it to any particular thing and that thing will satisfy the idea which is the number 1.

Numbers greater than 1 consist of at least two units. Each unit in any number greater than 1 is an incomplete idea of a thing in general distinct from whatever things which are, on any particular application, the objects of the other units. Each unit still has universal range. But, on any particular application to some objects, the additional content of each unit restricts which things serve as objects for the other units. For the number 2, for instance, each unit has universal range. But on any particular application, the object of one unit will not satisfy the other unit (since it otherwise would be identical to something the idea represents as being distinct from). Given the content of that second unit, whatever satisfies that unit will be distinct from the object of the first unit. So on a particular application of the number 2, the object of each unit will be distinct from the object of the other. This is why the number 2 numbers pairs of things only.

Like traditional units views, the present view requires infinitely many different units. The number 1 consists of a single unit. 2 consists of two additional units. 3 consists of three more units. And so on. Unless there is some reason in principle to deny that there is a collective idea whose components are units from different numbers, there will be infinitely many collective ideas which consist of two units, infinitely many collective ideas which consist of three units, and so on. Does this proliferation of collective ideas of units imply that there are infinitely many number 2s, 3s, and so on? No. The collective ideas which are numbers arewell-formed. A unit in the number 3 includes the idea of something distinct from something and some other thing, whereas each unit in the number 2 includes the idea of being distinct from some other thing. Suppose we pair a unit from the number 3 with a unit from the number 2 to form a new collective idea. Such an idea will include the idea of something which is distinct from the object of the other unit; and the other unit will include the idea of something which is distinct from the things a pair of other units represent. But there is only one other unit in the collective idea for its own represented thing to be distinct from it. It represents distinctness from the something represented by an additional unit, one which nowhere appears in the collective idea. This collective idea misfires in what it represents—it is ill-formed. A collective idea of units is well-formed when each unit is such that it represents a thing distinct from all and only

the things represented by the other units. If any unit fails to represent something which is distinct from something represented by any other unit, then a collective idea of units is not well-formed. The natural numbers are well-formed collective ideas.

Here is how numerical idealism would explain counting. Suppose I give you a barrel and ask you to open it and tell me how many things are inside. If there is nothing in there, you observe that there is nothing in the barrel. You form the collective idea of every (mid-sized thing) in the barrel—nothing—and say there are zero things in the barrel. If there is a single book in the barrel, you observe the book and form the collective idea of what’s in the barrel, the idea of the book. Then you move to the abstraction of that collective idea, the idea of something. You then say that there is one thing in the barrel. Or if there is a pair of books in the barrel, you instead form the collective idea of the one book and the other. Then you form the abstractionsomething and something else. You say there are two things in the barrel. What explains why you’re in a position to say how many things there are in the barrel after abstracting from the collective idea of what’s in the barrel? The abstraction you grasp in each case is itself the number of things in the barrel.

This view, which is here presented partially and informally, is a variant of the units view. The number 1 is a single unit, the idea of being something. And the components of the collective ideas which are numbers greater than 1 are also units. How does this sort of idealism fare? I develop the view further as I consider various objections.