Uninstantiatables in Aristotelian Mathematics

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.

Analogy at the foundations of mathematics

Consider the Benacerraf identification problem in philosophy of maths: there are multiple different ways of “defining” natural numbers in terms of sets, so there is no way of determining which definition is the “correct” one. This is not just a problem about natural numbers but they’re a useful notion to introduce the problem with. In fact, it’s not even just a problem with sets, since there are even multiple foundations from which to choose to define the natural numbers (I’m thinking about type theory and category theory here, considered as foundations).

Now, I’m broadly Aristotelian about mathematical objects: we abstract quantity and structure from reality, isolate certain aspects of these (which we call axioms), and extend these abstracted notions beyond our experience. One of these abstraction-and-extensions is the natural numbers. Presumably something similar could be said for things like sets, groups, rings, topological spaces, metric spaces, the real numbers, the rational numbers, and so on. So, as far as I’m concerned (as are some other non-Aristotelians, I’m sure), the natural numbers are intelligible wholly apart from any set-theoretic “definition” of them.

So what do we do when we “define” natural numbers (or anything) in terms of sets? For starters, I don’t think we define them at all. Rather we characterize numbers in terms of sets (although we nonetheless talk about this in terms of defining or constructing). But what does that mean?

Consider a concrete example: characterize 0 as ∅ (the empty set), 1 as {∅} = {0}, 2 as {∅, {∅}} = {0, 1}, 3 as {0, 1, 2}, and more generally n = {0, 1, 2, …, n-1} for any natural number n. For the sake of brevity, call these sets which we characterize the natural numbers with the natural sets. Now the natural sets have certain features or aspects, partly due to being sets and partly due to their relations with one another (relations they might not share with non-natural sets). Similarly, the natural numbers have their own features or aspects. There are aspects of the natural numbers that aren’t aspects of the natural sets, and there are aspects of the natural sets which aren’t aspects of the natural numbers. The important thing is that there is a correspondence between some of the natural number aspects and some of the natural set aspects. Call this the correspondence of aspect. The extent and nature of the correspondence will depend entirely on which sets we choose and how we choose to relate them to the numbers. Our considered characterization, for example, quite naturally captures the notion of order between natural numbers. That is, n < k (considered as numbers) if and only if n ∈ k (considered as sets).

So far so good, but we can push this further. What does correspondence of aspect consist in? There’s a mathematical way to make this precise: isomorphism. But we’re talking about the very interpretation of mathematics itself, so that won’t work.

Perhaps we could look to metaphysics, particularly Scholastic metaphysics. Here we find the notion of analogy, and it turns out to be quite useful. Much could be said about this, but roughly analogical predicates fall between strictly univocal and strictly equivocal predicates. They are not used in exactly the same sense when predicated, but they are nonetheless related. An example is the notion “seeing” when we see a table (say) and when we see how a conclusion follows from the premises of an argument. Clearly we don’t “see” in exactly the same way in both cases (like the “ball” in “rugby ball” and “soccer ball“), but yet the two seeings aren’t completely unrelated either (like the “ball” in “matric ball” and “rugby ball“). We say that my seeing the table is analogous to my seeing the conclusion. Much can be said about analogy, but for our purposes it will suffice to note that this particular kind of analogy is called analogy of proper proportionality. In these cases we have a relation between relations: our eyes are to the table (relation 1) as our intellect is to the conclusion (relation 2). We write this as “eyes : table :: intellect : conclusion”.

Ok, now come back to our correspondence of aspect. It seems to me that this consists in analogy of proper proportionality (and perhaps sometimes in other kinds of analogy, but I don’t see where they’d apply at this point so I’m sticking with proper proportionality for now). For instance, the characterization we’re considering captures that each natural set is to ∈ as each natural number is to <. If you feel like conveying this in a particularly obfuscated way for just 0, then you could write “∅ : ∈ :: 0 : <“.

So, in summary, we don’t define numbers in terms of sets (strictly speaking). We establish a correspondence of aspect using analogy between sets and numbers, and study those aspects from the perspective of sets. More generally, when we try use sets as a foundation what we’re doing is finding correspondence of aspect using analogy between various structures or quantities and sets and studying them from that perspective. And depending on the correspondence and sets in view, we might end up studying slightly different collections of aspects of these quantities and structures. Although I suspect for many purposes the differences will be irrelevant.

True mathematical propositions

Platonists believe that abstract objects such as numbers, colours, sets, ideas and so on are actually existing things (often referred to as the “Platonic realm”). So, if you’re a Platonist and a mathematician, you can take the axioms of maths as descriptions of the kind of objects, found in the Platonic realm, that you want to be working with. So, by “set” we mean those objects described by the Zermelo-Frankel axioms (for example). Then say we have some mathematical proposition that says “There exists some set S such that P(S)” where P is some property S could satisfy. If you’re a Platonist, you can take this statement at face value: there really is a set S, such that P(S). I wanted to think about how some other schools of thought could make sense of this claim (or any truth claim).

Another way of interpreting mathematical claims is formalism. For formalists, mathematical claims don’t have any meaning: they are merely strings of symbols (of course we don’t always write them as symbols, but there’s an unspoken assumption that anything we say without symbols can be translated into symbols). The axioms represent the base strings, and there are certain “translation rules” that tell us how to translate valid strings into other valid strings (this is a slightly simplified understanding of formalism but I think it’s sufficient). Clearly, on this interpretation, a true mathematical claim is one for which there exists a sequence of valid transformations from the axioms[1]. For the formalist, then, to say that S exists, is to say that “(∃S)P(S)” is a true claim.

I was thinking about another way a person might interpret mathematical claims (I don’t buy it, but it’s interesting anyway). Of course there seems to some sort of niceness about Platonism, since it allows one to take things at face value. On the other hand, you might (like me) be disinclined to think that there are really these abstract objects in existence. So why not take a sort-of “counterfactual Platonism”? So, we interpret the axioms as statements that would describe specific objects were abstract objects real[2]. Then we can just take the claim that S exists like the Platonists, just with the extra counterfactual bits: S is that object that would exist such that P(S), were abstract objects real.

Notes

  1. I suspect I should be slightly more specific with my definition of “true” here. After all, according to Godels incompleteness theorems, there are true statements that can’t be proved or disproved. But, on formalism, a proof is just a valid sequence of translations of symbols starting from the axioms, so if we took what I said here, then truth and provability would be the same thing. What we would actually need to say is that a string is “true” if it can be validly arrived through string translations (as we said above) but then add (because of the principle of excluded middle) that every syntactically correct string is either true of false. There’s probably a better way to say this.
  2. I suspect we would need to interpret this counterfactual using strict logical possibilities. This is because, plausibly the Brouwer axiom for alethic possibility is true and abstract objects are taken as necessarily existing entities, if they exist (and we would want them to exist in at least one of the possible worlds we’re quantifying over with our counterfactuals, lest the claims be vacuously true; of course this might not be a requirement, but let’s assume for generality that it is). But these two things would imply that they actually exist (let A be the proposition that some abstract object B exists. Then, by assumption, ◊□A. By Brouwer, ~A→□◊~A, and so by contraposition, ◊□A→A), which is what we wanted to avoid in the first place.