Any successful Aristotelian foundations of mathematics needs to account for mathematical objects that are uninstantiated and even uninstantiatable. Examples include (1) positive whole (or “natural”) numbers larger than the number of objects in reality, (2) negative numbers, and (3) infinities.

Uninstantiated natural numbers

As the Aristotelian sees things, we abstract quantity and structure from reality, isolate certain aspects of these (which we call axioms), and extend these abstracted notions beyond our experience. Call these three stages abstraction, isolation, and extension respectively. Even though we can technically distinguish between isolation and extension, in practice these two steps occur together in the same cognitive action. We’ll use the term synthesis to refer to the activity involving isolation and extension. These activities of abstraction and synthesis are not unique to mathematics: we use them all the time. Once we have an concept of a horse and the concept of blackness, for instance, we can consider the combination of these two concepts without having ever seen a black horse. True, the Aristotelian says that “whatever is in the intellect was first in the senses”, but this mustn’t be taken to mean that a concept can exist in the intellect only if it was sensed. Rather it should be taken to mean that sensation provides the raw data from which concepts are abstracted. This is consist with some concepts being synthesized from others.

Once we understand this, then, the problem of uninstantiated whole numbers seems to disappear. Initially we come to see the concept of quantity by considering the relation from an aggregate to a unit. For instance, we consider the relation between a specific aggregate of apples and the unit apple. If we have six apples and six oranges, then the aggregate of apples is related to the unit apple in precisely the same way that the aggregate of oranges is related to the unit orange.[1] It is on account of this that we say that the two aggregates are of the same size. We can label all the various aggregate sizes: 1, 2, 3, 4, 5, … We can also see that all aggregates of size 3 contain aggregates of size 2, all aggregates of size 2 contain aggregates of size 1, and so on. Thus we come to see that there is an ordering amoung these numbers. We can also see that an aggregate of size 3 together with an aggregate of size 2 makes an aggregate of size 5, an 2 aggregates of size 3 together make an aggregate of size 6. Thus we come to understand addition and multiplication, and similarly with subtraction and division (restricting ourselves to just the natural numbers for the time being).

Depending on which mathematician you talk to, 0 will sometimes be considered a natural number and other times not. Typically we will use whatever is convenient at the time. We could get 0 by considering an empty aggregate’s relation to any unit, or by considering a non-empty aggregate’s relation to a unit not contained in that unit (the relation of 6 apples to the unit orange).

At this point we will have experienced a number of aggregates, but there will inevitably be aggregates of sizes that are impossible for us to experience (either because of cognitive limitations or limitations on the number of things in reality). As we saw earlier, however, this does not stop us from having concepts of such aggregates. Through a (usually complicated and messy) combination of abstraction and synthesis we can come to consider any and all natural numbers.

Negative numbers

What about negative numbers? At this point we move from talking about natural numbers to talking about integers, which are whole numbers that are either positive or negative or zero. We might be tempted to try and extend our work above to negative numbers in straightforward way. After all, surely all natural numbers are also integers? Well, kind of.

We said above that natural numbers are relations between aggregates and units. Integers, on the other hand, are relations of difference between two aggregates. Let’s return to our apples and oranges. Say we have 10 apples and 6 oranges. One of the relations between these two aggregates is that if I take away 4 apples from the former, then I will have two aggregates of the same size. More precisely, there will be in a one-to-one correspondence between apples and oranges such that every fruit is matched to some other fruit. This same relation holds between an aggregate of 11 apples and 7 oranges, 9 apples and 5 oranges, and so on. This relation (or any relation co-extensive with it) is the negative integer 4. Now imagine I had it the other way around: 6 apples and 10 oranges, 8 apples and 12 oranges, and so on. These are related in a way inverse to negative 4, since now in order to make the former equal size to the latter we’d need to add 4 apples. This relation is the positive integer 4.

This parallels what we do when constructing the integers out of the natural numbers in first year mathematics courses. Starting with the Peano axioms we get the natural numbers. Then we build the integers up from pairs of natural numbers, where the pair (a,b) intuitively represents the difference between a and b.[2]

Just as we came to understand ordering, addition, subtraction, multiplication, and division with the natural numbers, so we can with the integers. Assume you have two integers x and y. As we have seen, each integer is a relation between two natural numbers, so let x be the relation from a to b andy be the relation from c to d (where a, b, c, and d are all natural numbers), written as x = (a, b) and y = (c, d) respectively. Again, as we have seen, an integer can be a relation between more than one pair of natural numbers, as when the integer -4 holds between 5 and 1, 6 and 2, 7 and 3, and so on. Using this fact we can align a and c, by which I mean the following: because of how the natural numbers are ordered either a > c, a < c, or a = c. If a > c then a – c is a natural number and y = (c + a – c, d + a – c) = (a, d + a -c). If a < c then we do this the other way and get x = (a + c – a, b + c – a) = (c, b + c – a). And finally, if a = c then we needn’t change anything. At the end of this alignment we will have three variables e, f, and g such that x = (e, f) and y = (e, g). Given this alignment, we say that the ordering between x and yis the same as the ordering between f and g. The intuition behind this is as follows: if both x and yrepresent adding or removing a certain amount from an aggregate of size e, then the ordering of the two integers is the same as the ordering between these two results.

Next consider addition. Once again assume we have two integers x = (a, b) and y = (c, d). This time, however, align b and c to give us x = (f, e) and y = (e, g). Then x + y = (f, g). The intuition here is that the addition of two integers is the same as applying the one to the result of the other.

I will leave subtraction, multiplication, and division as an exercise to the reader. Each time you will extend the respective operation from the natural numbers. There is, however, a new operation that arises with integers which we might call “additive inversion”: a is the additive inverse of b if and only ifa = -b. This is fairly simple to get using the notion of relations: for any integer x, x = (a, b) if and only if -x = (b, a).

In summary then, integers are understood as relation of differences between aggregates, and so negative numbers do not pose much of a problem for the Aristotelian. As before, through a combination of abstraction and synthesis we can come to consider any and all integers, even those we haven’t (or couldn’t have) experienced.

Infinities

As you might expect, when we start talking about infinites we need to get more abstract and precise in our approach. One of the hallmarks of modern mathematics is that we seek a universal foundation for the things we study. Often this is some form of set theory, but in the past century we’ve also seen that categories, topoi, types, and others can serve as a foundation equally as well. For any of these foundations, the Aristotelian can do something similar to what we were doing above for numbers. For the sake of simplicity here we’ll just use sets as our foundation, and not worry too much about their details. I will also treat 0 as a natural number here, which is an inconsequential philosophically but helps with presentation. (If you’d prefer to not think of 0 as a natural number, then you can assume we’re talking about non-negative integers.)

We’ve previously explained that when the mathematician speaks of “defining” things in terms of sets, what he really does is establish what we called a “correspondence of aspect using analogy”. This involves “encoding” those things in terms of sets such that the relevant aspects of the things are captured from the perspective of the set. So, for instance, say we wanted to study ordering amoung the natural numbers. We can do this from the perspective of sets by considering the following “definition”:

1. Let 0 be defined as ∅, the empty set.
2. Let any natural number n be defined as {0, 1, 2, 3, … n-1}, the set of all previously defined natural numbers.

When writing this definition out verbosely, we’ll get the following:

• 0 = ∅
• 1 = {0} = {∅}
• 2 = {0, 1} = {∅, {∅}}
• 3 = {0, 1, 2} = {∅, {∅}, {∅, {∅}}}
• …

From this perspective, one number is less than another number when the former is contained in the latter. That is, “1 < 20” is the same as saying that “1 ∈ 20”. This way, we can study the ordering amoung the natural numbers quite easily from the perspective of sets.

Notice that this definition only captures those aspects we want to study about numbers. If we wanted to study a different aspect, we might need a different set-theoretic definition of numbers. On the other hand, two different definitions might be equivalent for the purposes of studying a given aspect. Consider another putative set-theoretic definition of natural numbers:

1. Let 0 be defined as ∅, the empty set.
2. Let any natural number n be defined as {*n-1*}.

So, verbosely, this looks as follows:

• 0 = ∅
• 1 = {∅}
• 2 = {{∅}}
• 3 = {{{∅}}}
• …

Using this definition it would be a lot more difficult to define what “1 < 20” means in terms of sets, but it would be equally as easy to define what “19 is immediately before 20” means as the first definition (namely, “19 ∈ 20”).

The point of all of this, for our purposes, is that not all definitions are equal, and it is this very fact that we exploit when studying infinities. We will focus on two “types” of infinity: cardinal infinities and ordinal infinites. In case you didn’t know there are an infinite number of each of these infinites. (Just let that sink in.)

Now natural numbers have a bunch of aspects, and we study different infinites by focusing on one of these to the exclusion of the others. This restriction effectively enables us to go beyond the finiteness of numbers. Depending on which restriction we make, we get a different type of infinity.

The aspects of numbers include quantity, matching, and ordering. Now both matching and ordering are more fundamental than quantity. This video gives a good explanation of why matching is more fundamental, but basically the idea is that I can know facts about matching or ordering without knowing the quantities involved. For instance, I can know that there are as many people as there are chairs in the room without knowing how many there are of either, and I can know that you finished the race before me without knowing our respective places.

Cardinal infinities

When we choose to focus on the matching aspect of numbers we study cardinal infinities. These are the infinites marked by the Hebrew letter ℵ (aleph). If we have two sets X and Y, there are three possibilities for matching:

1. We can pair elements of X and Y such that every element in X is paired with exactly one element in Y, and there are no elements in Y left over. For finite sets this occurs when the two sets are the same size.
2. No matter how we pair the elements one-to-one, there will always be some elements in Y left over. For finite sets this occurs when Y is bigger than X.
3. No matter how we pair the elements one-to-one, we will never be able to pair every element in X. For finite sets this occurs when X is bigger than Y.

If we just focus on matching we can talk of the “size” of infinite sets, in terms similar to those just listed, but we must avoid thinking that we’ll get exactly the same kinds of results as in the finite cases. In finite cases sizes link to quantities, and it is exactly this link that we remove in order to study infinities. For instance, we can match each natural number to an even number such than none are left over, and so there are “as many” natural numbers as there are even numbers. The cardinal infinites represent the various “infinite sizes” that we could have. ℵ0 is the “size” of the natural numbers and any set for which we can give pair with the natural numbers with no left overs on either side. Thus, ℵ0 is also the size of the even numbers. When speaking precisely, we might say that infinite sets don’t have “size”, but rather cardinality. Cardinality is a notion that captures “matchability” or “pairability”. In finite cases, size and cardinality are the same. Of course, we rarely speak so precisely, and happily use the words interchangeably for infinite cases too.

An early result in set theory from Georg Cantor is that for any set (finite or infinite), the set of all subsets of that set will always be a bigger cardinality than that original set. This means that there are bigger infinities than ℵ0. One case he proved in particular was that no matter how you match up the natural numbers with the real numbers (points on the continuum, or numbers that can be represented with decimal expansions), there will always be some real numbers left over. So if we have a set of cardinality ℵ0, we say that the cardinality of the set of all subsets of that set is ℵ1, and the cardinality of the set of all subsets of that set is ℵ2, and so on.

Notice how the Aristotelian has no problems with any of this, for all we’ve done is the same thing we’ve been doing all along: abstraction and synthesis. In this case we’ve abstracted the notion of matching and synthesised the general notion of cardinality.

Ordinal infinities

We do something similar with ordinal infinites, which focus on the aspect of order. Imagine we went with the first set-theoretic definition of natural numbers given above. What number would set of all natural numbers represent? Presumably none of them, since no natural number is such that all natural numbers is less than it. But from the perspective of order, it would represent what we’d informally take to be the infinite-th position in a list. Just as before we have a general notion of ordinalwhich, when finite, agrees with the usual meaning of position or index, but which can also be used of infinite positions. And just as before we have a specific letter for ordinal infinities: the Greek symbol ω (omega). The first ordinal infinity is ω0, and using our first set-theoretic definition we have that ω0 = {0, 1, 2, 3, …}.

At this point we can see an interesting difference between the two different set-theoretic definitions we gave above: only the former is capable of capturing ω0. We can understand this from two perspectives. Formally, from a set-theoretic perspective the axiom of foundation prohibits infinitely nested sets, and this is exactly what we’d need if we were to give the definition of ω0 on the second account. Informally, from an intuitive perspective because ω0 is the infinite-th position there cannot be a natural number that is immediately before it. But this second definition effectively encodes the natural numbers in terms of the natural number immediately before them (n is defined solely in terms of n-1).

For the Aristotelian, this serves to show that what we can synthesise depends on how we abstract.

Now, just like the cardinals, there is more than one ordinal infinity. Unlike the cardinals, the next ordinal after ω0 is ω0+1 = {0, 1, 2, 3, …, ω0}.[3] Then it’s ω0+2, ω0+3, …, ω1 (=ω0+ω0), ω1+1, and so on.

Again the Aristotelian has no problems with any of this. In this case we’ve abstracted the notion of order and synthesised the general notion of ordinality.

Notes

1. Readers will note that this establishes an analogy of proper proportionality of the form “apple aggregate : apple :: orange aggregate : orange”.
2. We later take equivalence classes of these pairs, which corresponds to the idea that the same difference relation that holds between 6 and 10 also holds between 7 and 11, 8 and 12, and so on.
3. With the cardinals, ℵ0+1=ℵ0. For instance, if we have some set {a, 0, 1, 2, 3, 4, …} which is cardinality ℵ0+1, then we can create a paring from {0, 1, 2, 3, 4, …} to it as follows: 0 → a, 1 → 0, 2 → 1, 3 → 2, … Thus, given how cardinals are defined, {a, 0, 1, 2, 3, 4, …} also has cardinality ℵ0.