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In this section, I shall argue that numerical idealism not only overcomes the traditional problems for units views. It also overcomes the kinds of problems facing Fine’s units view. Then I’ll show how numerical idealism overcomes common objections against the view that numbers are mind-dependent objects.

First, numerical idealism resolves the equinumerosity problem and explains why equinumerous collections have the same number of individuals. We can define an abstraction function which takes the collective idea of some individuals and provides its abstraction, in my sense. Suppose we begin with Tom, Dick, and Harry, on the one hand, and Tina, Daria, and Hilda, on the other. There is the collective ideaTom and Dick and Harryand another collective ideaTina and Daria and

Hilda. The abstraction function essentially replaces each component in a collective idea with a unit, the idea of something which is distinct from whatever things the other units represent. Thus, the abstraction function takesTom and Dick and Harryand provides the abstractionsomething, another something, and something else. And the function takesTina and Daria and Hildaand also provides an abstractionsomething, another something, and something else. The function takes a collective idea and provides an equinumerous collective idea of units.

What ensures that the collective idea of units is the same in both cases? What generally ensures the identity of a collective ideaxand a collective ideayis thatxandyhave the same component ideas. And, generally, if the components are not collective ideas themselves, a component ofxis identical with a component ofywhen they have the same parts. So we might expect a collective idea of unitsxto be identical with a collective ideaywhen each unit of one is a unit of the other. The identity of collective ideas of units is more complicated than this, however.

Let’s go through the cases. 0 and 1 are simple. 0 is the idea nothing, the result of abstraction from any empty collection. 1 is the idea of something, the result of abstraction from any collection with a single thing. There is only one idea of nothing and one idea of something. So the result of abstracting from any empty collection is the same and the result of abstracting from any idea of a single thing is the same.

The complications arise for numbers greater than 1. The result of abstracting from any collective idea of individuals is a well-formed collective idea of units. So we need to show that each well- formed collective idea of units is unique. Each unit in a well-formed collective idea includes an iteration of the idea of being something. And each unit in a well-formed collective idea includes the ideas of being distinct from whatever the other units represent. But there is nothing in particular which each unit represents, and so there is nothing in particular each unit represents something as being distinct from. If some well-formed collective idea were not unique, then there should be some difference in what the indiscernable collective ideas of units represent. But there is not. Each unit from a collective idea of units has universal range; and on any particular occasion, the object of each unit will be distinct from the objects of the other units.

The abstraction problem says that abstraction from a set of individuals either provides an equinumerous set of impure units or a singleton set of a pure unit. Units views require numbers greater than 1 to consist of more than a single pure unit. In order for there to be multiple units, then,

each unit must have some feature that distinguishes it from the others. If, in abstraction, we subtract from the ideas of things we count the ideas of what distinguishes them, then abstraction from two or more things will result in a single unit.

Numerical idealism diagnoses this problem as a simple confusion. Abstraction, the subtraction of mental conjuncts, allows one to represent something’s being distinct from something else without representing any particular thing or any particular thing’s particular features. When we abstract to a collection’s number, we abstract to a collective idea of units. Each unit in numbers greater than 1 is a complex idea. It includes not only an idea of being something, but ideas of being distinct from whatever the other units represent. Each unit is pure: no unit includes an idea of having a certain shape, color, size, or whatever. All such ideas are lost in abstraction. But the units themselves are distinct because they point to each other as not representing what the others represent. On any particular occasion, each unit of such a number applies to something which isn’t what any other unit applies to. The distinctness of units arises from the mere distinctness of what they represent. And abstraction allows us to subtract all the parts of each component in a collective idea until we arrive at a collective idea of units which represents mere distinctness.

Does the current view also resolve the plurality problem—i.e., can it explain how the units which comprise each number are both indistinguishable and distinct? There are no units in 0, and only one unit in 1. So the plurality problem arises for numbers 2 and greater. Each unit in such a number contains the idea of being something and the idea of being non-identical to whatever it is the other units are ideas of. Each unit has an identical idea as a part, the idea of being something. And each unit has a part identical in form, the idea of being non-identical to whatever thing the other units are ideas of. In this sense, each unit in a number is indistinguishable from the other units.

Units from a single number are distinct. Each unit represents something distinct from whatever the other units represent. In order for us to apply a unit from a collective idea to something properly, we must apply it to something different from whatever things we apply the other units to. And for ideasF andG, if on any particular occasion we can only properly applyF to something which is distinct from whatever we applyGto, thenF andGmust also be distinct. Each unit is also distinct from other units. It is easy to show that each unit from some number is distinct from the units of every other number. The unit from a numbernincludes the ideas of being distinct from each of the othern

- 1 things represented by the other units. Units from greater numbers include ideas concerning the distinctness of more thannthings and units from lesser numbers include less of these ideas.

There are three problems with Fine’s view, and numerical idealism solves each of them. The first of the problems is that isn’t clear what variable objects are because it isn’t clear what their essential function of having a range of values is. Second, it isn’t clear why zero is a set even though numbers greater than zero do not seem to be tied essentially to sets. And, finally, since Fine only says that variable objects are abstract objects, it is doubtful whether we have any connection to them. I’ll respond to this last problem in the next section. I’ll conclude the present section with responses to the first two problems.

In Fine’s account, the ability of units to take a value on a particular occasion allows them to number the collection of their values on that occasion. If the unitsu2₁andu2₂take Tom and Jill as their values, respectively, then{u2₁andu2₂}numbers{Tom, Jill}. And any unit’s universal range is what supposedly accounts for each numbern’s ability to number anyn-membered collection whatsoever. Sinceu2₁andu2₂each have a universal range,{u2₁andu2₂}numbers any collection{x, y}for which

xandyare distinct. It is essential to Fine’s units that they have these functions. But it is not clear what it means for a unit to take a value on any particular occasion. And so it isn’t clear what it means for Fine’s units to have a universal range of values, an ability to take anything as a value.

Numerical idealism isn’t subject to these kinds of worries. Instead of “taking a value” no a particular occasion, a unit in my account applies to an individual. And a unit applies to an individual when, first, the individual is identical to itself, and, secondly, when we intend to apply the unit to that individual. The second condition involves an intention on someone’s part to think of a particular individual as “a something.” We apply a unit apiece to Tom and Jill with an intention of thinking of Tom and Jill each as something distinct from the other. Doing so on any particular occasion is an instance of counting Tom and Jill as two things. Of course, 2 numbers Tom and Jill whether we count Tom and Jill or not. Tom and Jill are mind-independently something and something else—and so they satisfy the ideasomething and something elsewhether we intend to apply the collective idea to them or not.

Anyxandywhich are non-identical satisfy the ideasomething and something else. The universal satisfaction ofsomething and something elseby anyxandywhich are non-identical is what accounts for the number 2’s numbering any pair of things. And the universal satisfaction of ofsomething and

something elseby anyxandywhich are non-identical is what allows us to apply the idea to any particular pair of things and truly say of any such pair that it consists of two individuals. Whereas Fine’s account of units leaves their central functions mysterious, numerical idealism explains the central functions of units in terms of the satisfaction of an idea (in the case of numbering) and the intention to apply an idea to the thing or things which satisfy it (in the case of counting).

Why is zero a set? Suppose there is nothing in the barn. The set of what is in the barn, according to Chapter 5, is the collective idea of individuals in the barn. In this case, that idea is the idea of nothing (or none), the empty set. The number of what is in the barn is the collective idea of units which jointly represent how many somethings there are in the barn. In this case, that idea is also the idea of nothing (or none). Zero is the ideanothing, which, according to Chapter 5, is also the empty set.