## Mathematics, Substance and Surmise: Views on the Meaning and Ontology of Mathematics (2015)

### An Aristotelian approach to mathematical ontology

Donald Gillies^{1 }^{ }

(1)

Department of Science and Technology Studies, University College London, Gower Street, London, WC1 6BT, UK

**Donald Gillies**

**Email: **__donald.gillies@ucl.ac.uk__

**Abstract**

The paper begins with an exposition of Aristotle’s own philosophy of mathematics. It is claimed that this is based on two postulates. The first is the embodiment postulate, which states that mathematical objects exist not in a separate world, but embodied in the material world. The second is that infinity is always potential and never actual. It is argued that Aristotle’s philosophy gave an adequate account of ancient Greek mathematics; but that his second postulate does not apply to modern mathematics, which assumes the existence of the actual infinite. However, it is claimed that the embodiment postulate does still hold in contemporary mathematics, and this is argued in detail by considering the natural numbers and the sets of ZFC.

1 Introduction. The Problem of Mathematical Ontology

The problem of mathematical ontology is the problem of whether mathematical entities exist, and, if so, how they exist. It can be illustrated by considering the example of the natural numbers, i.e. 1, 2, 3, … , n, … . Do such numbers exist? Simple considerations suggest that we should recognise their existence. For example, anyone, if asked whether there are numbers between 4 and 7, would reply: “Yes, there are, namely 5 and 6”. But, if there are numbers between 4 and 7, it follows logically that there are numbers, i.e. that numbers exist. On the other hand, admitting that numbers exist often gives rise to feelings of unease, since numbers appear to be curious shadowy entities very different from familiar everyday objects such as trees and boulders, tables and chairs, cats and dogs, other people, etc.

These feelings of unease have led some philosophers to argue that numbers do not after all exist, since talk of numbers is always reducible to talk of more familiar things. This view is often called *reductionism*,__ ^{1}__ and is perhaps most easily explained by considering statements about the average Englishman. Let us suppose it is true that p, where p = the average Englishman has 1.87 children. It is reasonable to suppose that every true statement, such as p, apparently about the average Englishman, is reducible to a logically equivalent statement in which no mention is made of the average Englishman. For example, we can construct a statement (call it p*) logically equivalent to p as follows. Let n = the number of Englishmen, and m = the number of children of Englishmen, then define p* as m/n = 1.87. The reductionist argues that accepting p as a true statement does not commit us to accepting the existence of that shadowy pseudo-entity, the average Englishman with his curious family, for p is reducible to a logically equivalent statement p* which does not mention the average Englishman. Similarly the reductionist would claim that statements apparently about numbers are reducible to logically equivalent statements, which do not mention numbers. Thus we do not have to accept that numbers really exist, and these curious shadowy entities fortunately disappear.

Of course it is not so easy to establish reductionism regarding the natural numbers, as it is to establish reductionism regarding the average Englishman. Can we really, for every statement about numbers, construct a logically equivalent statement, which does not mention numbers? And, if we can, do these logically equivalent statements mention other entities, which are just as suspect as numbers? Despite these difficulties, there have been several ingenious attempts to carry out the reductionist programme regarding numbers. However, the degree of their success remains controversial.

The ancient Greeks are noted for their studies of mathematics and philosophy. So we would expect them to take an interest in the problem of mathematical ontology, and this is indeed the case. Plato’s best-known theory is his postulation of a world of forms, and he introduced this theory partly in order to deal with the problem of mathematical ontology. His most famous account of the world of forms is in *The Republic* [__27__]. According to Plato this world consists of objective ideas which have an eternal existence outside space and time, and which cannot be perceived by any of the usual five bodily senses. Our minds, however, after a long and rigorous intellectual training in philosophy and other subjects, can gain knowledge of the world of forms, which Plato therefore calls the intelligible world. Plato mentions as part of his intelligible world, the objects of arithmetic and geometry, namely (*Republic*, VI, 510): “odd and even numbers, or the various figures and the three kinds of angle”. Mathematical statements are true on Plato’s account, if they correspond to what is really the case in the world of forms.

Platonism is not confined to the ancient world, but has had its adherents in more recent times. Two very distinguished modern philosophers of mathematics who have adopted a Platonic position are Frege and Gödel. Yet the majority of modern philosophers have felt uneasy about Platonism. The postulation of a mysterious transcendental world of objective ideas outside space and time has seemed to them too strange and metaphysical to be acceptable. The doubts of such philosophers must have arisen in ancient Greek times as well, because Plato’s own most brilliant student, Aristotle, made an attempt to bring the world of forms back to the material world.

Aristotle did not adopt the nominalist position, which has been developed in more recent times. He agreed with Plato that both the forms and mathematical objects had an objective existence. However, Aristotle did not agree with Plato that the forms and mathematical objects existed in a transcendental world outside of space and time. He argued that they did exist, but in the familiar material world.

I will give a more detailed account of Aristotle’s views on mathematical ontology in the next section, but they can be summarised, in my words rather than his, as follows. Mathematical objects and other abstract entities do indeed exist objectively, but not, as Plato claimed, in a transcendental world which is separate from the material world. They exist rather embodied in the material world. For example (an example of mine rather than Aristotle’s), the number five exists embodied in quintuples such as the toes of my left foot. Wittgenstein gives a very elegant illustration of this position, writing [__32__, p. 29]:

“A man may sing a song with expression and without expression. Then why not leave out the song – could you have the expression then?”

Wittgenstein’s point is that the expression cannot exist without the song. It has to be embodied in the way the song is actually sung. This is a particular instance of the general Aristotelian thesis concerning abstract entities.

Aristotelianism in modern philosophy of mathematics has not been such a popular approach to mathematical ontology as either Platonism or reductionism. However, there have been some adherents of this position. Franklin’s 2014 [__16__] book is a striking recent example. My own interest in Aristotelianism in mathematics began in the mid-1980s. I was then studying the views of Maddy [__22__] and Chihara’s criticisms of them [__12__]. At the same time, I was reading a good deal of Aristotle in order to prepare a lecture course, and it occurred to me that Aristotle’s approach might help to develop the positions of Maddy and Chihara. I have written various papers on this theme over the years, and I hope in the present paper to put all this work together in a more complete and finished form.

There is, however, a difficulty concerning the application of Aristotelianism to contemporary mathematics. Aristotle’s analysis of the problem of mathematical ontology was undertaken for ancient Greek mathematics, and this is very different from contemporary mathematics. Many points in Aristotle’s account, most notably his discussion of the nature of infinity, were quite appropriate for ancient Greek mathematics, but are contradicted by the mainstream assumptions of current mathematics. Still, it seems to me that the general Aristotelian approach, as just characterised, can be applied to contemporary mathematics.

To deal with this issue, the plan of the paper is as follows. In the next section (__2__), I will give a detailed account of Aristotle’s own views on mathematical ontology, and will argue that they were quite appropriate for ancient Greek mathematics. Then in section __3__, I will examine the important respects in which contemporary mathematics differs from ancient Greek mathematics, and the changes in the Aristotelian position, which these differences make necessary. The rest of the paper (sections __4__–__7__) will then develop a detailed account of an Aristotelianism suitably modified for current mathematics.

2 Aristotle on Mathematical Ontology

Aristotle’s views on this question were developed from a criticism of Plato. He states Plato’s position on mathematical objects as follows (*Metaphysics*, I.6, 987^{b} 14–18):

“Further, besides sensible things and Forms he (i.e. Plato – D.G.) says there are the objects of mathematics, which occupy an intermediate position, differing from sensible things in being eternal and unchangeable, from Forms in that there are many alike, while the Form itself is in each case unique.”

At first this passage is puzzling because, as we argued earlier, Plato in *The Republic* [__27__] seems to locate mathematical objects in the world of forms, and not as intermediates. However, Aristotle supplies an argument, which explains what is going on here. Let us consider the equation 2 + 2 = 4. Here we find two 2 s which are added together. However, there is only one form of two. This is what Annas calls *the Uniqueness Problem*, and she illustrates it with a geometrical example. Annas [__1__, p. 25]:

“The real argument for geometrical intermediates is … the Uniqueness Problem. A theorem mentions two circles intersecting.”

However there can be only one form of *the* Circle. Indeed in *The Republic*, Plato speaks (VI, 510) [__27__] of “*the* Square and *the* Diagonal”. It thus seems likely that Aristotle gives a correct account of Plato’s later views. The Uniqueness Problem could have come to light in discussions in the Academy, and this may have led Plato to change his mind regarding mathematical objects. Later in the paper, I will discuss the Uniqueness Problem briefly from a modern point of view.

In fact if Plato really changed to the view that mathematical objects are intermediates, this would have been a development of ideas already expressed in *The Republic* [__27__]. In the section on the divided line, Plato locates mathematical objects in the world of forms, but nonetheless stresses that they are in that part of the world of forms, which is closest to the visible (or sensible) world. This is because mathematicians make use of objects in the visible world such as groups of pebbles or diagrams when thinking about mathematical objects. As Plato says (VI, 510):

“The diagrams they draw and the models they make are actual things … but now they serve as images, while the student is seeking to behold those realities which only thought can apprehend.”

Aristotle criticises Plato’s theory of forms and his theory of mathematical objects in the same way. This common criticism is that abstract entities do not exist separately from, but rather in, the material world. I will first give two passages in which this approach is applied to forms. Let us start with *Metaphysics*, M(XIII).2, 1077^{b} 4–8 [__5__]:

“For if there are no attributes over and above real objects (e.g. a moving or a white) then white is prior in definition to white man, but not in reality, since it cannot exist separately but only together with the compound (by compound I mean the white man).”

I take Aristotle here as accepting the premise that “there are no attributes over and above real objects”; so there is no entity ‘movement’ or ‘a moving’ apart from moving bodies. Similarly whiteness cannot exist separately, but only in compounds such as a white man, or a white snowball.

This point of view is further expounded in the following passage (*Metaphysics* M(XIII).3, 1077^{b} 23–27) [__5__]:

“… there are many statements about things merely as moving, apart from the nature of each such thing and their incidental properties (and this does not mean that there has to be either some moving object separate from the perceptible objects, or some such entity marked off in them) …”

The point here is that we can consider a body *qua* moving without having to postulate some abstract entity ‘a moving’ separate from the body.

Let us now turn from forms to mathematical objects. First of all it will be useful to examine what Aristotle meant by ‘mathematical object’. He is very clear about this saying (*Metaphysics*, M(XIII).1, 1076^{a} 18–19) [__5__] that by “mathematical object … I mean numbers, lines, and things of that kind”. Aristotle did not recognise the existence of negative, rational, real or complex numbers. So by ‘number’ he means more or less what we have called ‘natural numbers’. However, there is a slight difference between his numbers and our natural numbers, i.e. 1, 2, 3, …, n, …. The difference is that Aristotle, in common with many ancient Greek thinkers, did not regard ‘1’ as a number. As he says (*Physics*, IV.12, 220^{a} 27) [__6__]: “The smallest number, in the strict sense of the word ‘number’, is two”. The reason for this is explained at *Metaphysics*, X.1, 1053^{a} 30, where he says: “number is a plurality of units”. The unit itself, i.e. 1, is not a number, or to put it another way (*Metaphysics*, N(XIV).1, 1088^{a} 4–6) [__5__]: “one means a measure of some plurality, and number means a measured plurality … Thus there is good reason for one not to be a number”. Of course, the ancient Greeks did not have 0 as a number either, although today, it is just as acceptable to begin the natural numbers with 0 as with 1.

This clarifies what Aristotle meant by numbers. The next point is that he definitely regarded numbers as existing. Indeed he says (*Metaphysics*, M(XIII).3, 1077^{b} 31–33) [__5__]:

“So since it is true to say without qualification not only that separable things exist but also that non-separable things exist (e.g. that moving things exist), it is also true to say without qualification that mathematical objects exist …”

Indeed, speaking of mathematical objects, he says (*Metaphysics* M(XIII).1, 1076^{a} 36) [__5__]:

“So our debate will be not whether they exist, but in what way they exist.”

So in what way do mathematical objects exist? Aristotle sometimes formulates his view by saying that such objects are not separate from things, which exist in the perceptible world. He explains this with an analogy to male and female in the following passage (*Metaphysics*, M(XIII).3, 1078^{a} 5–9) [__5__]:

“Many properties hold true of things in their own right as being, each of them, of a certain type – for instance there are attributes peculiar to animals as being male or as being female (yet there is no female or male separate from animals). So there are properties holding true of things merely as lengths or as planes.”

In some places Aristotle appears to contradict himself on this point, since he speaks of mathematicians separating mathematical objects, which were supposed to be not separate. Thus he writes (*Physics*, II.2, 193^{b} 24–194^{a} 10) [__6__]:

“The next point to consider is how the mathematician differs from the physicist. Obviously physical bodies contain surfaces and volumes, lines and points, and these are the subject-matter of mathematics. … Now the mathematician, though he too treats of these things, nevertheless does not treat of them as the limits of a physical body; nor does he consider the attributes indicated as the attributes of such bodies. That is why he separates them; for in thought they are separable from motion, and it makes no difference, nor does any falsity result, if they are separated. … geometry investigates physical lines but not *qua* physical …”

However, it is obvious that there is no contradiction here, since Aristotle is clear that mathematical objects are separable in thought but not in reality. Still he speaks rather confusingly of separating what is not separate in the following passage which is a good illustration of his general position (*Metaphysics*, M(XIII).3, 1078^{a} 21–30) [__5__]:

“The best way of studying each object would be this: to separate and posit what is not separate, as the arithmetician does, and the geometer. A man is one and indivisible as a man, and the arithmetician posits him as one indivisible, then studies what is incidental to a man as indivisible; the geometer, on the other hand, studies him neither as man nor as indivisible, but as a solid object. … That is why geometers speak correctly: they talk about existing things and they really do exist …”

Suppose an ancient Greek arithmetician is studying men. He might want, for example, to count the number of men taking part in a running race. Each man is regarded as one and indivisible, or, to put it another way, the unit used to measure the plurality is a whole man. With this unit, the number taking part in the running race might be 6. A geometer might also study a particular runner *qua* solid object. He might measure the proportions of the runner’s body to see whether they corresponded to the proportions of ideal beauty, whether the runner *qua* solid object could be fitted into a circle and square, etc.

Aristotle summarises his position by claiming that mathematical objects really exist, but they do not exist in a separate realm, as Plato claimed. They cannot be separated from familiar material objects, or, rather, they can only be separated from such objects in thought but not in reality. Using a slightly different terminology from Aristotle himself, I will refer to this as the *embodiment postulate*, which is the postulate that mathematical objects exist not in a separate world, but embodied in the material world.

Now there is a difficulty with the embodiment postulate, which Aristotle himself recognised. Numbers do appear to be exactly embodied in the material world. For example, the number of toes on my left foot is exactly 5. Provided the ancient Greek arithmetician had counted correctly, the number of runners in the race would have been exactly 6, and so on. If we turn to geometrical entities, however, it seems that they are embodied in the material world approximately but not exactly. Thus, for example, a stretched string is approximately, but not exactly, a geometrical straight line. Aristotle gives a number of examples of this type in *Metaphysics*, III.2, 997^{b} 35–998^{a} 6:

“neither are perceptible lines such lines as the geometer speaks of (for no perceptible thing is straight or round in the way in which he defines ‘straight’ or ‘round’; for a hoop touches a straight edge not at a point, but as Protagoras used to say it did, in his refutation of the geometers), nor are the movements and spiral orbits in the heavens like those of which astronomy treats, nor have geometrical points the same nature as the actual stars.”

All this is very convincing. Hoops or wheels are approximately circles, and a ruler is approximately a straight edge. If this were exact, then the ruler would touch the hoop or the wheel in one and only one point. Obviously this is not the case. As regards the orbits of the planets, we can consider the modern position. The orbit of a planet, such as Mars, is a very good approximation to an ellipse, but it is not exactly an ellipse. There are small perturbations in the orbit, owing to the gravitational attractions of the other planets of the solar system. In many astronomical calculations, stars can be treated as geometrical points, but they are obviously not precisely points. I will call this problem the *approximation problem*. It obviously raises a problem for Aristotelianism. If geometrical entities are only approximately embodied in material things, this suggests that they may, after all, really exist in a separate Platonic realm.^{2}

That concludes my account of Aristotle’s embodiment postulate. There is, however, another important part of his philosophy of mathematics, which is his treatment of infinity. I will now give an account of this. Aristotle’s general position regarding the infinite is that it has a potential existence but not an actual existence. He states this very clearly in the following passage (*Physics*, III.6, 206^{a} 17–21) [__6__]:

“the infinite has a potential existence.

But the phrase ‘potential existence’ is ambiguous. When we speak of the potential existence of a statue we mean that there will be an actual statue. It is not so with the infinite. There will not be an actual infinite.”

Aristotle clarifies this further by giving a definition of the infinite (*Physics*, III.6, 207^{a} 6–8) [__6__].

“Our definition is then as follows:

*A quantity is infinite if it is such that we can always take a part outside what has been already taken*.”

Aristotle illustrates this definition by applying it to the infinity of numbers and to the infinity of magnitude, i.e. the continuum. He distinguishes the infinite by addition from the infinite by division. Numbers are infinite by addition, but not by division (*Physics*, III.7, 207^{b} 2–14) [__6__]:

“in number there is a limit in the direction of the minimum, and … in the other direction every assigned number is surpassed. … The reason is that what is one is indivisible … Hence number must stop at the indivisible … But in the direction of largeness it is always possible to think of a larger number … Hence this infinite is potential, never actual … its infinity is not a permanent actuality but consists in a process of coming to be …”

The numbers are infinite by addition, because, given any number however large, it is always possible to produce a larger number by adding one. They are not, however, infinite by division, since one, the unit, is taken in arithmetic to be indivisible.

We might expect magnitude or the continuum to be infinite both by addition and division. For consider a finite line segment AB. We can always extend AB to C, where AC is longer than AB. This would seem to give the infinite by addition. Similarly if we divide AB by a point C between them so that AC and CB are both shorter than AB, we do have to stop at that point, but can divide AC by a point C’ between them so that AC’ and C’C are both shorter than AC. This makes AB infinite by division. Surprisingly, however, Aristotle claims that magnitude is infinite by division, but not by addition and thus is the opposite of numbers. He writes (*Physics*, III.7, 207^{b} 15–21) [__6__]:

“With magnitudes the contrary holds. What is continuous is divided *ad infinitum*, but there is no infinite in the direction of increase. … it is impossible to exceed every assigned magnitude; for if it were possible there would be something bigger than the heavens.”

This shows where the problem lies. For Aristotle the Universe is finite and bounded by the sphere of the fixed stars. Suppose a geometrical proof required extending a finite line segment AB to a point C such that AB = BC. If AB where so large that 2AB were bigger than the diameter of the Universe, this extension would not be possible, because C would have to lie outside the Universe. At first sight this seems to create a problem for mathematicians, who, in the course of their proofs do want to extend a line as much as they like. However, Aristotle replies that this apparent difficulty can always be overcome by shrinking the original diagram to a size where the required extension of the line becomes compatible with the dimensions of the Universe. As he says (*Physics*, III.7, 207^{b} 27–34) [__6__]:

“Our account does not rob the mathematicians of their science … They postulate only that the finite straight line may be produced as far as they wish. It is possible to have divided in the same ratio as the largest quantity another magnitude of any size you like. Hence, for the purposes of proof, it will make no difference to them to have such an infinite instead, while its existence will be in the sphere of real magnitudes.”

Aristotle’s notion of the potential infinite does indeed seem to be adequate for ancient Greek mathematics, even for the developments of Euclidean geometry and Diophantine equations which occurred after his lifetime. Arithmeticians required numbers as high they pleased, but did not need to consider the sequence of numbers as a completed whole. Geometers in their proofs needed to extend lines as much as they wanted, and to divide them as much as they wanted, but they never needed to postulate infinitely long lines. Aristotle’s potential infinity would have fitted ancient Greek geometry better if he had been prepared to allow that the Universe was potentially infinite. However, his device of shrinking the diagram made his potential infinities adequate for the geometers. It might be thought that the actual infinite could appear in ancient Greek geometry if it was allowed that a line is composed of points. However, Aristotle denied explicitly that this was the case (*Physics*, VI.1, 231^{a} 25–26) [__6__]:

“a line cannot be composed of points, the line being continuous and the point indivisible.”

There is, however, one seeming exception to the claim that the actual infinity does not occur in ancient Greek mathematics.__ ^{3}__ This occurs in Archimedes’ treatise on

*The Method*. In his proof of Proposition 1, Archimedes considers a triangle CFA. He takes an arbitrary point O on AC, and draws a line MO parallel to AF and meeting CF in M. He then says [

__4__, p. 572]: “the triangle CFA is made up of all the parallel lines like MO”. Aristotle had denied that a line can be composed of points, but here Archimedes gives a two-dimensional version of the same principle by claiming that a triangle is composed of parallel lines. All the parallel lines like MO which make up the triangle CFA constitute an actual infinity which here appears in a work of ancient Greek mathematics.

However, Archimedes says at the end of his proof of Proposition 1 (p. 572):

“Now the fact here stated is not actually demonstrated by the argument used; but that argument has given a sort of indication that the conclusion is true. Seeing then that the theorem is not demonstrated, but at the same time suspecting that the conclusion is true, we shall have recourse to the geometrical demonstration which I myself discovered and have already published.”

Archimedes’ treatise on *The Method* was considered by him as a work of heuristics rather than genuine mathematical demonstration. Its results indicated that some theorems might be true, but did not give genuine proofs of them. So, although Archimedes did indeed consider the actual infinite, it was only as a heuristic device, and not as part of legitimate mathematics. Thus our conclusion that ancient Greek mathematics did not use the actual infinite remains true, despite this example from Archimedes. Archimedes use of the actual infinite in this particular context is the exception, which proves the rule.

If we turn to contemporary mathematics, however, the picture changes. As I will show in the next section (__3__), contemporary mathematicians work within a framework of assumptions, which imply that the actual infinite exists.__ ^{4}__So, for contemporary mathematics, one of the postulates of Aristotle’s philosophy of mathematics, namely the postulate that there exists only the potential, but not the actual, infinite, must be dropped. We can, however, still hold on to his other postulate—the embodiment postulate. This, however, has to be applied in a different, and, in some respects, more complicated way. It has to be shown, for example, that some actually infinite sets are embodied in the material world. This is no easy matter for sets of very high cardinality, such as we find in Cantor’s theory of the transfinite. As a preliminary to examining these problems in detail, we need to explain how contemporary mathematics differs from ancient Greek mathematics, and this will be done in the next section.

3 How Contemporary Mathematics differs from Ancient Greek Mathematics

As we have seen, when Plato and Aristotle examined the question of mathematical ontology, they considered, as mathematical objects, numbers and geometrical entities, such as points, lines, figures, etc. If we want to examine the ontology of contemporary mathematics, what should we consider as the relevant mathematical objects? This is a difficult question, since contemporary mathematics is much more complicated than ancient Greek mathematics and makes reference to many curious objects. However, we can cover most, though perhaps not all, objects of contemporary mathematics by limiting ourselves to just two kinds of object. These are: (i) *the natural numbers*, which are more or less the same as the numbers of Plato and Aristotle, except that we have 0 and 1 at the beginning of the sequence; and (ii) *sets* in the sense characterised by one of the accepted systems of axiomatic set theory. There are in fact two leading systems of axiomatic set theory, namely ZSF, named after its developers Zermelo, Skolem and Fraenkel, and NBG, named after its developers von Neumann, Bernays and Gödel. ZSF and NBG are more or less equivalent, but ZSF is perhaps slightly simpler, and, maybe for this reason, seems to be the one, which is more generally used nowadays. If we add to ZSF the axiom of choice, which again is usually done nowadays, we get the axiom system ZSFC, which is standardly abbreviated to ZFC. ZFC is the most commonly adopted version of axiomatic set theory at present, and there is even an almost canonical version of ZFC, which is the one formulated by Cohen in his 1966 [__13__]. I will therefore consider the ontology of sets as they appear in ZFC. It should be added that ZFC takes as its underlying logic classical 1^{st} order predicate calculus.

The importance of the concept of set in modern mathematics is undeniable, but what is curious is that this concept was very little used by mathematicians until the second half of the 19^{th} century. Mathematicians dealt with magnitudes, and it was thought that these could be represented geometrically, usually as lengths along a straight line. However, a movement grew up in the 1860s to arithmetise analysis—that is to purge analysis of any dependence on geometrical concepts, and to base it on purely arithmetical concepts. The result of this movement was what was called the *arithmetisation of analysis*. It consisted essentially of replacing the old theory of ‘quantities’ or ‘magnitudes’, which could be represented geometrically, by a new theory of real numbers, which were defined in terms of rational numbers. Since rational numbers could be defined as ordered pairs of integers, and integers as ordered pairs of natural numbers, this based the theory of real numbers on that of natural numbers, that is on traditional arithmetic. Several mathematicians produced definitions of real numbers in terms of rational numbers around 1870, but the ones, which became best known, were those of Cantor and Dedekind, which were published in 1872.

The problem with the arithmetisation of analysis was that, while it did indeed eliminate geometric considerations, it did not found analysis purely on arithmetic, but rather upon arithmetic and set theory. Every one of the various definitions of real number involved a consideration of sets containing an infinite number of rationals. Now such infinite sets could well be regarded as just as problematic as, if not more problematic than, the geometrical evidences which had been eliminated. It was a question of out of the geometric frying pan into the set theoretic fire. Moreover, an infinite set of rationals is an example of an actual infinity, and so, if we acknowledge the existence of such sets, we have to abandon Aristotle’s restriction of infinity to the potential infinity.^{5}

Despite, or perhaps because of, these problems, the introduction of actually infinite sets proved very stimulating for the development of mathematics. Cantor worked out his new theory of the infinite in the two decades following 1872, and published a complete account of it in his papers of 1895 and 1897 [__10__]. His friend Dedekind, with whom Cantor corresponded, also worked on the theory of infinite sets in this period. The title of Dedekind’s book of 1888 [__15__] (*Was sind und was sollen die Zahlen?*) suggests that it is a book about natural numbers, but in fact Dedekind has a lot to say in the book about sets (or systems, as he called them). Some of the principles regarding systems, which he formulated, later on became axioms for set theory. Moreover Dedekind stated and at least partly proved some important theorems regarding infinite sets.

A mathematical theory of infinite sets was thus developed between 1872 and 1897, but then disaster struck when contradictions were discovered in this new theory. The first contradiction to be published was Burali-Forti’s paradox of the greatest ordinal, which appeared in 1897. Then came Cantor’s paradox of the greatest cardinal. Developing the reasoning behind these contradictions, Russell discovered that it led to a contradiction within logic itself—Russell’s paradox.

Despite the shock generated by the discovery of these paradoxes, it did not take mathematicians long to find a way round them. In 1908 Zermelo published an axiomatic version of set theory, which was designed to avoid the contradictions while still allowing the theory of infinite sets of Dedekind and Cantor to be developed. Actually Zermelo’s original axiomatisation did not allow the full theory of Dedekind and Cantor to be developed, and further work was needed. Skolem and Fraenkel replaced Zermelo’s axiom of separation by the axiom of replacement in 1922, and in 1925 von Neumann showed a way in which Cantor’s transfinite ordinal and cardinal numbers could be introduced. However, by 1930 axiomatic set theory had been created in much the same form as it exists today.

ZFC is no longer compatible with Aristotle’s theory of the infinite as always potential, because it explicitly postulates an axiom of infinity, stating that an infinite set exists. This is Axiom VII in Zermelo’s 1908 paper. Twenty years earlier, Dedekind had tried to prove the existence of an infinite set, but his ‘proof’ was much criticised by, among others, Russell; and, as it seemed to involve reasoning very similar to that which had led to the contradictions, it was abandoned. As infinite sets were needed for the definition of real numbers, there was nothing for it but to assume their existence as one of the axioms. Moreover, this axiom is independent of the other axioms of set theory, since these are all satisfied in the universe of finite sets.

From the axiom of infinity it follows that the set of all natural numbers (N say) = {1, 2, 3, … , n, …} exists. Thus the natural numbers are no longer a potentially infinite sequence 1, 2, 3, …, n, … . There is an actually infinite set of them (N). In this respect therefore contemporary mathematics goes beyond ancient Greek mathematics. Aristotle’s theory of the infinite as potential is no longer adequate for mathematics, and, if contemporary mathematics is accepted, we have to accept the existence of the actual infinity.

I will next argue that nearly all the objects of contemporary mathematics can be defined in terms of natural numbers and the sets of ZFC. This justifies limiting ourselves to these two types of object, when studying the ontology of contemporary mathematics. While this may not be entirely adequate, at least it makes a good start with the problem. I will begin my survey with a classic text written by two leading mathematicians, namely Birkhoff and MacLane, *A Survey of Modern Algebra*. This was first published in 1941 [__8__] shortly after set theory had become established. The book then went through many editions and its approach is still that which is generally used today. The authors begin with the integers i.e. 0, ±1, ±2, … , ±n, … , though later in the book they remark that they could just as easily have started with the positive integers, i.e. the natural numbers. Rational numbers are introduced as ordered pairs of integers, and real numbers by Dedekind cuts (Ch. IV, section 5, pp. 97–99). Complex numbers are next introduced as ordered pairs of reals. At a more abstract level, groups are characterised as sets of elements on which is defined an operation satisfying various laws. An operation is of course a function, and functions can be defined in terms of sets. A similar approach is adopted for other abstract concepts such as rings, fields, Boolean algebras, etc.

What about geometry? The most natural approach to geometry is the analytic approach in which a point is defined as an n-tuple of real numbers; and lines and other figures are defined by equations. For n = 1, 2 or 3, diagrams still remain a useful guide, but such “geometric evidences” now have a subordinate role. Let us consider, for example, a line segment of unit length. The points of such a segment would be real numbers x such that 0 ≤ x ≤ 1. Such a line segment, a continuum, would consist, in the contemporary approach, of the set of its points—a set which is of course well-defined within ZFC. Thus contemporary mathematics contradicts Aristotle’s claim (already quoted) that (Physics VI.1 231^{a} 25) [__6__] “a line cannot be composed of points”. Lines in contemporary geometry are sets of points, and are, in this sense, composed of points.^{6}

Analysis is developed in the same way as algebra, with abstract concepts such as topologies being defined in terms of sets. Another striking example is probability theory. In 1933 [__21__] Kolmogorov published his axioms of probability, which he formulated in a set theoretic framework. Essentially the same approach is still adopted today, over 80 years later, by those who work on mathematical probability.

These are my arguments for accepting the natural numbers and ZFC as fundamental to the ontology of contemporary mathematics. Let us now consider what arguments might be raised against this position.

The first counter-argument makes use of Gödel’s incompleteness theorems. These show, it could be said, that ZFC is not adequate for contemporary mathematics. Let us consider the implications for ZFC of the two theorems, starting with the second incompleteness theorem. This theorem shows that the consistency of ZFC cannot be proved without using assumptions, which are stronger than those of ZFC. Now this is undoubtedly the case, but is it a serious problem? Mathematicians have been working on ZFC for over 80 years now, but no contradiction has yet come to light. This surely suggests that ZFC is indeed consistent. Suppose moreover that a contradiction in ZFC was discovered in the next few years. When the contradictions in the infinite set theory of Dedekind and Cantor came to light in 1897, it took only a few decades for the damage to be repaired, and their theory to be put on a footing, which has held good until today. Surely if a new contradiction came to light, the damage it caused would again be repaired quite quickly. Of course, in the contemporary world, we have to give up the old idea that mathematics is completely certain. There are many reasons for thinking that mathematical knowledge, like all other kinds of knowledge, is doubtful to some degree.

Let us now turn to Gödel’s first incompleteness theorem. This states that, if ZFC is indeed consistent, then there is a theorem of arithmetic which cannot be proved within ZFC, but which can be proved by an informal argument outside ZFC. This result does indeed show that ZFC is not adequate for the whole of contemporary mathematics. Even leaving aside Gödel’s first incompleteness theorem, there are parts of contemporary mathematics, which do seem to go beyond ZFC. Some set theorists have suggested strengthening ZFC by adding new axioms. Usually these are stronger forms of the axiom of infinity, which postulate the existence of very large transfinite cardinals. Some category theorists have made the more radical suggestion that the whole framework of set theory is unsatisfactory and should be replaced by a category theoretic framework. More recently it has been suggested that homotopy type theory might provide a more satisfactory framework for contemporary mathematics than ZFC. Moreover some very complicated proofs of modern mathematics seem to make implicit use of assumptions, which are stronger than those of ZFC. Such a claim has, for example, been made about Andrew Wiles’ proof of Fermat’s Last Theorem. What can be said about all these objections to regarding the sets of ZFC as fundamental for the ontology of contemporary mathematics?

Well, first of all, it should be observed that ZFC is unlikely to remain an adequate framework for mathematics for all time. Euclidean geometry provided an adequate framework for mathematics for many centuries, but eventually developments in mathematics meant that Euclidean geometry was no longer adequate as an overall framework, though it still continues to be used in some contexts. A similar fate is likely to overtake ZFC at some time in the future. However, it should be added that such revolutionary changes in mathematics occur relatively rarely. During most of its history “normal” mathematics has been conducted within a framework of assumptions, which could be called, using Kuhn’s term, a paradigm. Nowadays despite all the dissatisfaction of category theorists and others, that paradigm is largely defined by ZFC. If a mathematical proof can be carried out within ZFC, nearly all mathematicians would accept it as valid. If the proof required assumptions, which went beyond ZFC, it would appear more doubtful. The success and resilience of ZFC over a period of more than eighty years is somewhat surprising if we reflect on the rather inauspicious circumstances in which it was first developed.

So far I have considered objections to ZFC, which consider ZFC to be not strong enough to provide a framework for contemporary mathematics. However, there is another set of objections, which regard ZFC as too strong. ZFC, as I pointed out above, was designed as a system within which Cantor’s theory of transfinite numbers could be developed. However, many mathematicians of the Cantor’s time, and also some today, regard Cantor’s theory as very dubious, not genuine mathematics, and perhaps even as meaningless. An example of a famous mathematician who opposed Cantor’s ideas is Poincaré, who wrote in 1908 [__28__] (p. 195): “*There is no actual infinity*. The Cantorians forgot this, and so fell into contradiction”. Several mathematicians, notably Brouwer, who were sceptical about Cantorian set theory, tried to develop forms of mathematics which would prove to be a viable alternative to the mainstream approach based on ZFC and standard 1^{st}-order logic. Brouwer developed intuitionistic mathematics, Poincaré’s neo-Aristotelian approach led to predicative mathematics, and there are other forms of constructive mathematics besides. These new systems of mathematics are not useless by any means, because they introduced new ideas, which have been made use of by the mainstream. However, Brouwer’s revolutionary objective of replacing standard mainstream mathematics by another kind of mathematics proved to be a failure. This is another illustration of how difficult it is to carry out revolutions. Though revolutions do occur from time to time, many would-be revolutionaries are doomed to disappointment.

Another objection to the approach I am adopting might be raised, this time by strong advocates of ZFC. They might argue that there is little point in considering the natural numbers separately from ZFC, since the natural numbers can be defined within ZFC as the finite ordinals. This is the approach taken by Cohen who writes [__13__, p. 50]:

“a very reasonable position would be to accept the integers as primitive entities and then use sets to form higher entities. However, it can be shown that even the notion of an integer can be derived from the abstract notion of a set, and this is the approach we shall take. Thus in our system, all objects are sets. We do not postulate the existence of any more primitive objects.”

However, Gödel’s first incompleteness theorem does seem to me to provide an argument against the view that all mathematical objects are sets. Stating the implications of this theorem of Gödel rather informally, we can say that the natural number concept has the ability to go beyond any characterisation of it using a formal system. The ultimate reason is that all the parts of any formal system (F say) can be labelled using the natural numbers, and this labelling shows a way in which the natural numbers can be used outside F, and so go beyond any formal characterisation of natural number within F. So the natural number concept contains within itself a principle by which natural extensions of the concept can always be found. The set concept lacks this capacity for natural self-extension. This is a reason for considering among fundamental mathematical objects, natural numbers as well as sets.

Our discussion of contemporary mathematics has shown that one part of Aristotle’s philosophy of mathematics must be given up in the modern context, namely his claim that infinity is always potential. Our project therefore is to take the other part of his philosophy of mathematics, namely the embodiment postulate, which states that mathematical entities do exist, but they exist as embodied in the material world and not in a separate realm. In the rest of the paper I will argue for this claim for natural numbers and sets, as characterised by ZFC. Naturally the hardest part of this claim to establish will be that infinite sets and Cantor’s transfinite numbers are embodied in the material world. Before we come to this, however, it will be as well to consider first the simpler case of finite sets, and this will be dealt with in the next section.

4 Finite Sets

Our project then is to show that numbers and sets are embodied in the material world. To carry this out, it may be as well to begin with a brief explanation of what will be meant by ‘material world’. Some authors use ‘physical world’ as a synonym for ‘material world’, but I intend to use ‘material world’ in a broader sense. The physical world can be characterised as the world studied by physics. However, the material world should include not just the physical world, but the natural world or world of nature. The natural world is the world studied by any of the natural sciences, that is by chemistry, biochemistry, biology, as well as by physics.

The next important point about the material world is that it consists not just of objects, such as electrons, stones, molecules, cells, plants or animals, but of objects standing in various relationships to each other.__ ^{7}__ These relationships are to be regarded as just as real as the objects themselves. However, relations are abstract, and so abstract entities have been introduced into the material world.

An example from biology illustrates this point of view. A plant growing in the ground, and an insect flying through the air, may, at first sight, seem two distinct and separate material things, but yet in reality they are strongly interconnected. Many flowers cannot survive without being pollinated by insects while some insects cannot survive without obtaining nectar from flowers. Note also that these relationships have nothing to do with human consciousness since they existed long before the appearance of men and women.

Let us now pass from relationships to sets. As a result of objective relations between them, some things are bound together to form sets, which really exist in the material world. This can again be usefully illustrated by a biological example, namely a colony of bees. This consists of one queen, 40,000 to 50,000 workers, and a few hundred drones. Well-known relationships exist between these various types of bee. The queen lays the eggs. The workers collect nectar, turn it into honey, and tend the young. The drones have the sole function of fertilising the queen and are, with the approach of winter, expelled (see [__11__], Ch. 1). Once again these relationships have nothing to do with human consciousness, since, as Chauvin writes (1961, Introduction, p. 11):

“Ants and bees were already in existence 40 million years ago at least, and scarcely differ from those we know today. … And *Homo sapiens* has had hardly 150,000 years of existence …”

Other examples of such naturally occurring sets are: the planets of the solar system, the electrons in an oxygen molecule, the oxygen molecules in the Earth’s atmosphere, the leaves of a tree, the trees of a wood, the cells of a living organism, and so on.

So far we have considered the natural world, but some aspects at least of the social world of human beings can be considered as part of the material world. Here, of course, caution is needed, since humans have sensations, feelings, thoughts and consciousness, which, according to many philosophers, are not material. Indeed some higher animals have sensations and feelings, and perhaps thoughts and consciousness as well. So we must also be careful how much of the biological world we consider as material. Still there are many social events, much of which can be considered as definitely material in character. Consider again our earlier example of a running race between 6 men in ancient Greece. This involved 6 male human bodies in rapid motion during a particular time period. It therefore had a distinctly material aspect. In general many aspects of the social world, and most aspects of the biological world, can justly be considered as part of the material world.

In the world of nature, we gave examples of situations in which the relations between various things bind them together into naturally occurring sets. The same phenomenon occurs in the social world, though here it is perhaps better to speak of socially constructed sets rather than naturally occurring sets. A good example of a socially constructed set in the social world is a nation. The inhabitants of a modern nation state are connected together by a complicated series of relationships. These relationships, though not physical in any direct sense, are nonetheless very real, since, for example, they enable one nation state to wage war against another, as Germany did against France in 1870.

Another example of a socially constructed set is the set of Russian billionaires. This set was constructed socially in the last 20 to 25 years by somewhat curious social processes. However, this set is now definitely in existence and exerts an influence on the rest of humanity in many striking ways.

The naturally occurring sets and social constructed sets, which we have so far considered, are all embodied in the material world as our Aristotelianism requires. Benacerraf in his 1973 [__7__] (see particularly p. 415) draws attention to a problem in Platonism. Suppose mathematical entities exist in a transcendental world outside space and time, it is not at all clear how we can interact causally with them. But if causal interaction between them and us is impossible, how can we get to know anything about them? This could be called: ‘Benacerraf’s causal interaction problem’. It is clear that Aristotelianism offers a solution to this problem, for, if mathematical entities are embodied in the material world then we can interact causally with them. Indeed all the naturally occurring and socially constructed sets so far mentioned can interact causally with human beings.

One particular kind of causal interaction is perception. So we can raise the question of whether it is possible to perceive sets. Now clearly some sets are not perceptible. No one can perceive, at a glance so to speak, the whole of the German nation or the set of oxygen molecules in the atmosphere. However, there are some sets, which do seem to perceptible, such as the leaves on a tree or the runners in a race. Maddy in her 1980 argues for the view that it is possible to see a set. She considers [__22__, p. 178] the case of a cook looking at three eggs in an egg carton and argues that the cook sees a set of three eggs. This seems to me correct, but Chihara in his 1982 makes a striking objection to it. Chihara considers the example of his desk, which has been cleared of all objects except an apple. According to Maddy, when he looks at this apple, he is seeing not just the physical object, but also an abstract object namely the unit set of the apple. However, the unit set of the apple appears to generate exactly the same sensations as the physical apple. Why therefore should we ever claim that we are seeing such an abstract object in addition to the physical object? I will now try to answer this objection of Chihara’s by first giving a more detailed analysis of the process of perception.^{8}

Let us take the example of a man looking at a saltcellar on a table. At first sight this seems to be a very simple case of a physical object interacting causally with a human. In fact, however, what interacts causally with the man is a whole complex of physical objects standing in various relations to one another. This becomes clear if we suppose that there is also a vase of flowers on the table. If the saltcellar is to the left of the vase, the man will see both objects. If the vase is in front of the saltcellar, the man will only be able to see the vase. In the first case, the man can as validly say: “I see __that the saltcellar is to the left of the vase__” as “I see the saltcellar”. In other words it is just as correct to say that the man is seeing the relationship between the saltcellar and the vase (an abstract entity) as that he is seeing the saltcellar (a physical object). What is really affecting the man’s senses is an interrelated complex of physical objects. If he focuses on just one of these and says: “I see a saltcellar”, he is, in effect, picking out one element from the complex. To regard the saltcellar as independent of the other objects to which it is related is a distortion. This distortion does not lead us too far wrong in macro-physics, but may be at the root of some of the problems about the micro-world.

This analysis is connected with the familiar point that all observation is theory laden. Thus when our observer attributes his sensations of sight, and perhaps touch and taste as well, to an enduring physical object (a saltcellar), he is interpreting his sensations in terms of a theoretical scheme involving physical objects located in space and time. This theoretical scheme is by no means unsophisticated, and, if it is legitimate to use it to interpret sensory experience, why should it not be legitimate, on some occasions, to use an enriched theoretical scheme, which involves relations between physical objects, or sets of physical objects, as well as just physical objects. The use of such an enriched interpretative theory might be justified if it could be claimed that, for example, a set of physical objects (as opposed to the physical objects themselves) had a causal efficacy in the process under investigation.

With this in mind, let us return to the controversy between Maddy and Chihara. Suppose Maddy is in the middle of baking a cake, which requires two eggs. She opens the egg carton, and observes to her horror that there is only one egg. Now she will have to go to the shop to buy some more. Maddy really sees the egg as a unit set since she is comparing it with the desired couple. The unit set of the egg has, moreover, a causal efficacy since it has led Maddy to form the plan of leaving her house to buy some more eggs.

At this moment the doorbell rings. It is Chihara, who has come to visit his former pupil. As he enters the kitchen he notices an egg lying on the table, but, knowing nothing of the cake-making plans, he sees it simply as a physical object. For Chihara, it is indeed no more than a physical object. This example shows that two different observers viewing the same scene can nonetheless see different objects. What is really affecting their senses is a complex of interrelated physical objects. One observer can quite legitimately pick one element out of this complex, while the other can, with equal legitimacy, pick a different element out of the complex.

In this section, I have argued that some finite sets are indeed embodied in the material world. Such sets can interact causally with human beings, and indeed it may be possible to perceive sets in some cases. In the next section, I will argue that the same holds for natural numbers.

5 Natural Numbers

I take numbers to be properties of sets. If a set is embodied in the material world, then the number of its members, being a property of the set can also be considered as embodied in the material world.

Maddy also holds that numbers are properties of sets (see her [__23__], p. 502), but she argues that there is an important difference between numbers and sets, and this must now be discussed. I have argued, in agreement with Maddy, that some sets at least can be assigned a definite spatio-temporal location. Maddy argues, however [__23__, pp. 501–2], that numbers cannot be located in space and time in the same way as such sets, because numbers are universals. This point can be explained by adapting an example of Maddy’s to British conditions. Suppose on a Saturday afternoon that games of football are taking place (one hopes peacefully) up and down the land. Each team, or set of eleven men, has a definite spatial location at a given time, but the number eleven does not seem to have such a spatial location. It is manifested as much by one team in one ground as by another team in another ground. It would therefore be a mistake to say that a particular number has a definite spatio-temporal location. There does, however, seem to be another possibility; we could say that a number is instantiated at a particular spatio-temporal location if there is a set there to which it applies, and that therefore the number 11 is scattered about space-time (like water, or hydrogen) rather than having no space-time position at all, i.e. being outside space-time.

This approach gives an answer to the uniqueness problem which, as we saw earlier, seems to have pre-occupied Plato and Aristotle. The uniqueness problem is that there should only be one number 2, but that in the equation 2 + 2 = 4, there seem to be two numbers 2. The answer suggested by the analysis of the previous paragraph is that there is indeed only one number 2, but that it can be instantiated by different sets. The equation 2 + 2 = 4 makes an implicit reference to sets. It says that if 2 is instantiated by set S_{1}, and also by set S_{2}, where S_{1} and S_{2} are disjoint, then 4 is instantiated by S_{1}US_{2}. The apparent plurality of the number 2 is really the plurality of the underlying sets, which are implicitly assumed.

The next question is whether we can see numbers. I have already argued that we can see some relations and some sets. Can we also see some numbers? Let us take a concrete example. Suppose I am looking at a bunch of three bananas. I can see various relationships between the bananas. They are attached to the same stalk, for example. These relationships bind the bananas together into a set, and I have argued that this can correctly be considered as visible. But is the number three also visible in this situation? This question is a difficult one, but I am inclined to answer: “Yes”. Is it not legitimate to say that I see the redness of that rose, or that I see water in that pond? But then I ought also to be allowed to say that I see the number three.

Even if this is correct, however, it is neither necessary nor plausible to maintain that all numbers can be directly perceived by the senses. This is clearly not true of, for example, the number of Germans, or the number of oxygen molecules in the Earth’s atmosphere. Moreover we often have excellent reasons for believing in the existence of objects, which cannot be directly apprehended by the senses. Such objects may, for example, be postulated by scientific theories, which have been very rigorously tested and confirmed in a whole variety of circumstances. Indeed this is really the situation as regards many numbers.

6 Infinity: (i) Denumerable Sets and the Continuum

I have so far considered finite sets and natural numbers. A critic might object: “Your Aristotelian account may look plausible in such elementary cases, but what about the more sophisticated entities of advanced mathematics? How, in particular, are you going to deal with infinite sets of very high cardinality?” Indeed Gödel says explicitly [__19__, p. 483]: “ … the objects of transfinite set theory … clearly do not belong to the physical world”. “Surely”, it might be argued by our hypothetical critic, “Gödel is correct here. It might be claimed that finite numbers such as 1, 2, … are embodied in the material world, but we cannot, without absurdity, maintain that ℵ_{1}, ℵ_{2}, … , ℵ_{n}, … ,ℵ_{ω}, … exist in the material world”. “On this rock”, our critic might conclude, “your Aristotelianism is bound to founder. After all, Aristotle himself only dealt with the potential infinity, and never with the actual infinity”.

My reply to this objection is that transfinite set theory does not constitute a difficulty for Aristotelianism. On the contrary, the approach developed here affords a very simple and plausible account of transfinite set theory.^{9}

In order to introduce infinite sets into the material world, I will argue that continuous intervals of points are physically real. The points could be points of space, or of time, or of space-time. Now when should we regard an entity as physically real, or as a constituent of the physical world? My answer to this question is as follows. Suppose some of the best theories of physics, which we have, i.e. theories which are very well confirmed by observation and experiment, contain symbols referring to the entity in question, then we should regard the entity as part of the physical world. To put the point another way, we should regard as physically real those entities, which are postulated by our best confirmed theories of physics.

Now almost all of the best theories of physics of the present day introduce a four vector (x, y, z, t) where x, y, z are said to be spatial co-ordinates and t a temporal co-ordinate. Moreover it is always assumed that x, y, z, t are continuous variables which take real numbers as values. From this it seems reasonable to conclude that spatio-temporal continua are physically real, just as we conclude that electrons or atoms are physically real.

It is important to note here that there is no *a priori* reason why space and time should be continua. After all it was once thought that energy was continuous, but then the quantum theory was introduced and proved more successful at explaining physical phenomena. Thus there is no reason why a quantum theory of space and time should not, one day, be introduced and prove more successful than the present continuum theory at explaining physical phenomena. If, and when, that happened we should have to give up our present assumptions regarding physical reality—which are anyway tentative and conjectural. In fact, however, no quantum theory of space and time has, so far, proved at all successful. On the contrary the assumption that space and time are continuous appears in all the well-confirmed theories of physics, and so is itself a very well-confirmed assumption. Thus we have excellent grounds for supposing that spatio-temporal continua are physically real.

Against this it might be argued that in fluid mechanics it is customary to assume that water and other liquids are continuous, although we know perfectly well that they are composed of molecules. Similarly, it could be claimed, we should regard the assumption that space and time are continuous merely as an approximation, and not as telling us anything about physical reality.

My reply is that the two cases are not analogous. In the case of fluid mechanics, we have a deeper theory, which tells us that the fluid is composed of molecules, and that the assumptions of continuity are only approximations, which, moreover, break down in situations which can be specified. In the same way, we should not, in the light of modern physics, regard Newtonian gravitational forces acting at a distance, as physically real, because we have a deeper theory (Einstein’s General Relativity) which shows that Newtonian theory is only an approximation, and which, moreover, specifies situations in which considerable divergences from Newtonian theory will occur. Now, if there was a deeper quantum theory of space and time, then the continuous theory of space and time would be in the same situation as fluid mechanics or Newtonian theory; but this is not the case. The continuum theory of space and time is everywhere successful, and, as things stand at present, there is no deeper theory, which corrects it. We are thus justified, so I claim, in taking the continuum theory of space and time as a guide to what is actually the case in the real physical world.

These then are my reasons for supposing that spatio-temporal continua are physically real, and are, moreover, of the character assumed by physical theories, that is to say consist of sets of points which are representable by real numbers. This, however, gives us in the material world, infinite sets whose cardinality is that of the continuum, i.e. **c**, where

It is now an easy matter to define denumerable sets, which are embodied in the material world. For example, the real number π occurs in many contexts in physics, and so can be regarded as embodied in the physical world. Therefore the set of all digits of π, a denumerable set, is also embodied in the physical world. So we can say that both ℵ_{0} and **c** occur in the material world.

This line of argument also answers an objection, which might be raised to our earlier treatment of finite sets and natural numbers. It could be said that the examples given only establish the material embodiment of relatively small finite sets and natural numbers. What about natural numbers, which are much greater than the number of particles in the Universe as so far observed? Such numbers and the corresponding finite sets are certainly assumed by mathematicians, but they may lack any material embodiment of the kind we considered in sections __4__ and __5__. This is true enough, but of course there will obviously be a set of points whose members have any specified natural number, however large. Hence this difficulty is overcome.

So far we have given an Aristotelian account of the infinite numbers ℵ_{0} and **c**, but what of the other Cantorian alephs ℵ_{1}, ℵ_{2}, … , ℵ_{n}, … , ℵ_{ω}, …? They need to be dealt with in a rather different manner, which will be explained in the next section.

7 Infinity: (ii) Other Transfinite Cardinals

In order to deal with the other transfinite cardinals, it will be helpful to take into account another aspect of mathematical entities. So far I have emphasised that natural numbers are embodied in the world of nature. However, such numbers can also be considered as human social constructions. They were constructed by devising symbols such as ‘1’, ‘2’, ‘3’, … , and giving these symbols meaning. Regarding the nature of meaning, I will assume the view, which Wittgenstein presents in his later philosophy. Wittgenstein’s theory is that [__31__, p. 20]: “the meaning of a word is its use in the language”, and he further analyses language as consisting of a number of interrelated language games. By a ‘language game’ he means some kind of rule-guided social activity in which the use of language plays an essential part. He himself introduces the concept as follows [__31__, p. 5]:

“I shall also call the whole, consisting of language and the actions into which it is woven, the ‘language game’.”

And again [__31__, p. 11]:

“Here the term ‘language-*game*’ is meant to bring into prominence the fact that the *speaking* of language is part of an activity, or of a form of life.”

Wittgenstein illustrates his concept of language game by his famous example involving a boss and a worker on a building site [__31__, p. 3]. The boss shouts, e.g. ‘slab’, and the worker goes off and fetches a slab. Wittgenstein’s point is that the meaning of the word ‘slab’ is given by its use in the activity carried out by boss and worker. Actually Wittgenstein’s term ‘language game’ seems hardly appropriate, since his first example of a language ‘game’ is not a game at all, but work. It would seem preferable to speak of ‘language activities’—thus leaving it open whether the activity in question is work or play.

If we accept this theory, it is easy to give an account of how one sort of abstract entity—meaning—is constructed by human activity. For some sign S to acquire meaning, it suffices that S comes to have a generally accepted use in a social activity or activities. Thus by setting up social activities in which signs play an essential part, human beings create a world of abstract entities (meanings). These meanings are the product of human social activity. So, for example, by introducing the numerals ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, …, and giving them a use in various social activities, numbers are created. If numbers have been created, and the boss on the building site shouts ‘five slabs’, the worker will (probably) bring five slabs. In this way the construction of numbers assists in the construction of houses. Numbers are a sort of abstract tool.^{10}

However, there are some limitations to the view of mathematical entities as human constructions. This issue can be clarified by introducing another important idea from the philosophy of language. This is Frege’s distinction between sense and reference [__18__].

Frege illustrates this distinction by his well-known example of the morning star and the evening star. The phrases ‘morning star’ and ‘evening star’ have different senses. The first phrase has the same sense as ‘the star which sometimes shines very brightly in the early morning’ while the sense of the second phrase is the same as ‘the star which sometimes shines very brightly in the early evening’. However, our astronomical knowledge tells us that both phrases, despite their difference in sense, refer to the same celestial body, namely the planet Venus.

If we accept Frege’s distinction, then it is clear that Wittgenstein’s later theory of meaning deals with sense, but not at all with reference. Consider the phase ‘The Sun’. By being given a use in a host of human social activities, this phrase acquires a meaning or sense. In virtue of this sense, the phrase refers to a large incandescent body. However, this large incandescent body has nothing to do with human social activities. It existed long before there were human beings, and its existence has been little affected by human life.

Let us now try to relate this to the Aristotelian view of mathematical ontology, which has been developed so far. It is one of the simpler claims of chemistry that in the air there are a large number of oxygen molecules of the form O_{2}. The set of electrons in an oxygen molecule is thus an example of a naturally occurring set. Moreover, the set of electrons of a particular oxygen molecule in the air is an embodiment of the number 16. Yet oxygen molecules existed before there were human beings, and human social activity has not altered the number of electrons in an oxygen molecule. This shows that number is not a purely human construction, but is also an aspect of the non-human world of nature. If therefore we are going to give an adequate account of the existence of numbers, we must combine, in some way, the view of numbers as human constructions with the view of them as aspects of the non-human world of nature.

To do so, let us consider the question: “Did human beings construct the number 16?” One could say in reply that human beings did invent the system of signs of which ‘16’ is part and by using this sign system, they gave a meaning or sense to ‘16’. Thus humans did construct the sense of ‘16’, or the concept of 16. However, in virtue of the way the natural world is, this sense picks out a reference in the natural world. It refers, for example, to the number of electrons in an oxygen molecule. The reference of ‘16’ existed and exists in the natural world quite independently of human beings, and to this extent the number ‘16’ was not a human construction. Note, however, that the sense of ‘16’, or the concept of 16, are also embodied in the material world. They are embodied in the human social activities, which give the term ‘16’ meaning, and so are embodied in a part of the social world. Thus the Aristotelian account applies to the sense of ‘16’ just as much as to its reference, though in a different way.

We can now use these ideas about sense and reference to give our account of the Cantorian alephs ℵ_{1}, ℵ_{2}, … , ℵ_{n}, … , ℵ_{ω}, … . It is useful to compare these alephs to the natural numbers 1, 2, 3, … , n, … . In both cases we have symbols, such as ‘16’ in the first case and ‘ℵ_{5}’ in the second. These symbols acquire meanings or senses by being used in language activities. The symbols of the finite numbers all have references. For example the number 16 is embodied in the naturally occurring set of electrons in an oxygen molecule. By contrast the transfinite alephs have as yet no application in physics, which would give them a reference in the material world. So the transfinite alephs have no reference. Thus for example ‘ℵ_{5}’ has a sense because it has a use within the language activity of Cantorian set theory. This activity may have few participants, but it is nonetheless a perfectly definite social activity carried out in accordance with clear and explicit rules. On the other hand ‘ℵ_{5}’ has no reference.

Let us next examine the consequences of this analysis for the question of mathematical truth. An Aristotelian view of mathematical entities allows the correspondence theory of truth to be extended from physics to mathematics. The electrons of an oxygen molecule exist in the material world, and statements about them are true if they correspond to what is the case in the material world. Similarly, since in the Aristotelian view, numbers and sets are embodied in the material world, a statement about numbers or sets is true if it corresponds to what is the case in the material world. Note that this account of mathematical truth allows us to defend the law of excluded middle without appealing to Platonism.

It is a consequence of this view of mathematical truth that there are no truths about the transfinite cardinals ℵ_{1}, ℵ_{2}, … , ℵ_{n}, … , ℵ_{ω}, … .__ ^{11}__ However, this is not a defect of this approach, but an advantage rather. All we need do, as far as these transfinite cardinals are concerned, is to speak not of a proposition being true but of its being a theorem in a particular version of set theory. Thus the correct claim that

would no longer be regarded as a truth about transfinite cardinals in the way that

is a truth about finite cardinals. It would be regarded as a theorem of the development of the theory of transfinite cardinals within say ZFC. The advantage of this approach is that many theorems of the theory of transfinite cardinals and ordinals do in fact depend on what version of set theory is being used. A theorem may hold within one version of set theory but not another. Thus it is better to make the appeal to some underlying set theory explicit rather than to use an objective notion of truth, which does not apply in this case.

This view of mathematical truth sheds light on the question of whether the continuum hypothesis will ever be shown to be true or false. Cohen showed in his 1966 that the continuum hypothesis is both consistent with and independent of the other axioms of ZFC. The continuum hypothesis states that ℵ_{1} = **c**, and so it contains a term (‘ℵ_{1}’) which lacks a reference. It does not therefore have a truth value at the moment, and will only acquire one if ‘ℵ_{1}’ does acquire a reference by, for example, being given an application in physics. There is no guarantee, however, that the transfinite cardinals ℵ_{1}, ℵ_{2}, … , ℵ_{n}, … , ℵ_{ω}, …. will ever be used in physics or some other successful theory. So the continuum hypothesis could remain undecidable for the indefinite future.^{12}

There are several general consequences of this approach, which need to be made explicit. To begin with, on the present Aristotelian view, the axioms of ZFC cannot be regarded as true, but only as acceptable. If the axioms of ZFC were true, then any consequence of those axioms would have to be true as well. However, we have argued that ℵ_{4} + ℵ_{5} = ℵ_{5} should be regarded as neither true nor false rather than as true, even though it is a consequence of the axioms of ZFC. The axioms of ZFC are acceptable in roughly the following sense. If a statement (S say) follows from the axioms of ZFC, and if S only refers to entities which exist, in the sense of being embodied in the material world, then S can be regarded as true.

The status of the axioms of ZFC is connected in turn with the fact that ZFC is only partially interpreted. The existence of a full Tarskian interpretation of ZFC is only possible if Platonism is correct. In that case there is a Platonic world of sets, and the axioms of ZFC would be true or false depending on whether they held for this Platonic world or not. On an Aristotelian position, however, there is not a full interpretation of all the sets of ZFC. Only those which can be shown to be embodied in the material world can be regarded as having a genuine reference. There are, however, many sets which are definable within ZFC, but which cannot at present be shown to have a material embodiment. The symbols for such sets, e.g. ℵ_{4}, do not have a reference, that is to say they do not have a Tarskian interpretation, since Tarski’s semantics are referential in character.

It is useful to consider this account of ZFC in relation to two well-known philosophies of mathematics, namely fictionalism and if, then-ism. Let us start with fictionalism. According to fictionalism, mathematical entities do not exist, and so mathematical writings are similar to literary fiction, which can take the form of novels, plays, etc. Literary fiction describes characters and their doings, but these characters are purely imaginary and do not exist in the real world. For example, consider Shakespeare’s play: Hamlet. This presents the actions of a number of interesting characters such as Hamlet himself, Polonius, Ophelia, Laertes, etc. However these characters are purely imaginary and never existed in the real world. Similarly a fictionalist would claim, mathematics deals with entities such as numbers, sets, etc., and describes their properties, but these entities are purely imaginary and do not exist in the real world.

Now it is obvious that fictionalism, as a general philosophy of mathematics, must be rejected from the present Aristotelian point of view. According to this Aristotelian approach, many mathematical entities such as numbers, sets and geometrical figures do exist because they are embodied in the material world. Moreover, many mathematical propositions such as 4 + 5 = 9 are true because they correspond to what is the case in the material world. However, from our present point of view, fictionalism, though not correct in general, does hold for some mathematical entities, such as, notably, the transfinite cardinals ℵ_{1}, ℵ_{2}, … , ℵ_{n}, … , ℵ_{ω}, … . These really are fictions like Hamlet, Polonius, Ophelia, etc. We could perhaps refer to them as metaphysical fictions rather than literary fictions.

Fictionalism has close connections with if, then-ism. Let us take an arbitrary mathematical statement S. According to if-then-ism, we can never claim that S on its own is true. The most we can claim is that there is a set of axioms A, such that ‘If A, then S’ is logically true. Once again it is clear that if, then-ism cannot be accepted from the present Aristotelian point of view. As we remarked, the approach leads to the conclusion that many mathematical statements, e.g. 4 + 5 = 9 can be accepted as true without deducing them from any set of axioms. On the other hand, from the present point of view, if, then-ism is correct for some mathematical statements. Statements such as ℵ_{4} + ℵ_{5} = ℵ_{5} cannot be accepted on their own as being true. The most we can say of such statements is that they can be deduced from the axioms of ZFC, or some similar version of axiomatic set theory.

If, then-ism can be applied not just to the metaphysical fictions of mathematics, but to literary fictions as well. To do so, we simply take any work of literary fiction, and regard all the statements in the work as our axioms. We then accept any conclusion which can be deduced from these axioms. So, for example, we can take all the statements in a standard edition of Hamlet, and see what further statements can be deduced from them. In this way we can conclude that Ophelia is the sister of Laertes, and the daughter of Polonius, but that it would be incorrect to claim that Ophelia is the wife of Hamlet. Of course some statements about Ophelia are undecidable by this method. We cannot establish, for example, whether or not Ophelia had measles as a baby. Now some famous literary critics have indeed applied this if, then-ist method. For example, Bradley drew a surprising conclusion regarding Hamlet. It is usually supposed that Hamlet, being a young student, is about 20 years old. However, Bradley deduces from the text of the play that Hamlet is exactly 30 years old (see [__9__], pp. 407–9). The evidence for this is in Act V, Scene i, where Hamlet has a conversation with the gravedigger. The gravedigger says that he started his present occupation on the day when Hamlet was born, and has been doing the job for 30 years. Moreover, he digs ups the skull of Yorick and says it has been lying in the earth for three and twenty years. Hamlet remembers Yorick carrying him on his back when he was a boy, which confirms that Hamlet must at least be in his late twenties. However, other literary critics have felt that Bradley went too far in these if, then-ist deductions. No doubt Shakespeare did try to create a consistent fictional world, but perhaps he really did intend Hamlet to be a young student of about twenty and did not notice the inconsistency of this with the statements made by the gravedigger. After all, a few inconsistencies can creep into a literary work without the average reader or member of the audience noticing and feeling perturbed.

It is interesting to compare the situation here with the metaphysical fictions generated by ZFC. Of course we cannot be sure that ZFC is consistent, though there is some evidence that it is. However, if an inconsistency were discovered in ZFC, we could no longer accept this axiomatic theory. Because it is based on classical 1^{st} order logic, if it is inconsistent, then any statement of the theory could be proved as a theorem, which is obviously unsatisfactory. Consequently, if an inconsistency were discovered in ZFC, the mathematical community would immediately set to work to produce a new system, which avoided inconsistencies of that type. Literature, by contrast, is more tolerant of inconsistencies, and, if an inconsistency is discovered in a famous novel, it does not have to be immediately rewritten to remove the flaw.

Still there are definitely some points in common between literary fiction and abstract set theory. Perhaps the closest comparison is not between ZFC and a pure work of fiction, but rather between ZFC and a historical novel. Let us suppose that an author tries to create a historical novel with both characters which really existed (RCs), and other characters which are purely imaginary (ICs). The author is conscientious, and makes sure that any scenes involving just RCs are historically correct and well supported by documents as regards the words and actions attributed to the RCs. However, naturally the scenes involving RCs and ICs, or just ICs, though they might be historically plausible, are not literally true. (These hypothetical rules of practice are quite close to those actually adopted by Sir Walter Scott.) Now, if we take any statements involving just RCs from this novel we can regard it as true, but if a statement involves ICs as well as RCs, we should regard it as a fiction, i.e. on Frege’s approach, as neither true nor false. Here the analogy with ZFC is very close. If a theorem of ZFC contains only reference to entities which have been shown to be materially embodied, i.e. actually to exist on the Aristotelian criterion, then the theorem can be regarded as true. If the theorem contains reference to some entity, such as ℵ_{4} which has not been shown to exist by the Aristotelian criterion, then the theorem should be regarded as neither true nor false. Of course this analogy is not entirely accurate. It is always possible that, through the development of science, ℵ_{4} is shown to exist in the physical world. However, it is hardly likely that a character imagined by a historical novelist is found by later historical research to have really existed and carried out the actions described in the novel.

I have argued that ZFC generates some metaphysical fictions which are surplus to the requirements of applied mathematics. If this is so, it might be argued, surely we should start a programme for paring down ZFC so that the modified system produced only what was required by applied mathematicians, and no surplus structure of a metaphysical character. Such an argument does not seem to be correct. There are two reasons why we should tolerate the metaphysical structures generated by ZFC. First of all ZFC is a remarkably simple and elegant system. It consists of only 8 axioms and 1 axiom schema. From this, virtually all the mathematics needed by applied mathematicians can be developed with little difficulty. ZFC is a remarkable achievement, carrying out for 20^{th} century mathematics, what Euclidean geometry did for ancient Greek mathematics. Each system provided a simple and elegant framework within which the mathematics of its time could be developed. Now if we attempted to pare down ZFC so that the modified system generated only those sets needed for applied mathematics, and no surplus structure, the result would inevitably be a more complicated and less elegant system. Moreover what would be gained by eliminating the metaphysical structure of the Cantorian alephs? This structure, even if it is not used in practice, does no harm.

One could even argue that these metaphysical fictions might do some good. The view of the Vienna Circle that metaphysics is meaningless and should be eliminated has now largely been abandoned. Popper criticised this view and pointed out that metaphysics could be not only meaningful, but also positively beneficial for science by providing a heuristic guide for the construction of scientific theories. It is still, of course, not to be excluded that the Cantorian alephs will play some role in a future successful theory of physics.

Thus some metaphysical fictions should be tolerated within mathematics. Yet, at the same time, such toleration should perhaps not go too far. Earlier we mentioned suggestions about modifying ZFC by introducing stronger axioms of infinity, such as, for example, an axiom postulating the existence of an inaccessible cardinal. However, as we have seen, ZFC, as well as generating the infinities which are actually useful in physics, namely ℵ_{0} and **c**, generates many more infinite numbers which have as yet proved of little use in physics or other branches of science. Thus we have so to speak already got more infinite numbers than we really need, and so to postulate the existence of further such numbers hardly seems a good idea. It would only be adding to the metaphysical and fictitious side of mathematics, without contributing anything of practical use.

Acknowledgements

I read an earlier draft of this paper at a seminar in La Sapienza, Roma on 16 February 2015, and I am very grateful for the comments I received on that occasion. Particularly useful were some comments from three experts on Aristotle (Silvio Maracchia, Diana Quarantotto, and Monica Ugaglia) who helped me with a number of points. After the seminar, I had a very useful discussion with Carlo Cellucci concerned mainly with the problem of applying the Aristotelian approach to ZFC, and the possible use of ideas of fictionalism and if, then-ism. This led to many improvements in section __7__ of the paper. In addition I was fortunate to receive extensive comments from quite a number of people to whom I sent a copy of an earlier draft. These included: Ernie Davis, James Franklin, Andrew Gregory, Ladislav Kvasz, Penelope Maddy, Anne Newstead, Alex Paseau, and Brian Simboli. Their input was very helpful in preparing the final version of the paper.

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Footnotes

__1__

Actually reductionism is only one form of *nominalism*, which is the general view that mathematical entities do not exist. Nominalism is defended in Azzouni [__3__]. It is often connected to the so-called *fictionalist* view according to which mathematics is similar to literary fictions such as Shakespeare’s *Hamlet*. Azzouni discusses this view in his 2015, and it is also discussed in section __7__ of the present paper.

__2__

Andrew Gregory stressed the importance of the approximation problem to me in conversation. It is one of the factors, which have led him to prefer Platonism to Aristotelianism regarding abstract entities. The approximation problem is also mentioned by Azzouni in section 2 of his 2015. Azzouni regards it as a strong argument in favour of nominalism.

__3__

This was pointed out to me by Silvio Maracchia and Anne Newstead. See Maracchia [__25__].

__4__

Exactly when the actual infinity entered mathematics in a significant way is an interesting but difficult historical question. The years between 1500 and 1800 saw the rise of algebra, analytic geometry and calculus. Did these developments introduce the actual infinite? Ladislav Kvasz has suggested plausibly (personal communication) that the actual infinite is already in Descartes. Jeremy Gray’s contribution to the present volume has some points relevant to this question. Gray says [__20__]: “At no stage did Apollonius, or any other Greek geometer, generalise a construction by speaking of points at infinity”. However, he goes on to show that points at infinity were introduced by Desargues in 1639. Perhaps this could be considered as an example of the actual infinity, though this is not clear. In what follows, however, I will not discuss this question in detail, but limit myself to the developments from the 1860s, which gave rise to contemporary mathematics.

__5__

The change from the potential to the actual infinite is also discussed in Stillwell [__29__], particularly section 3. Stillwell’s chapter also gives more details about the set theoretic matters which are discussed briefly in the remainder of this section.

__6__

For a detailed and informative account of the similarities and differences between Aristotle’s theory of the continuum, and those of modern mathematics, see Newstead [__26__].

__7__

Cf. Wittgenstein [__30__]: “The world is the totality of facts, not of things”.

__8__

What follows is my own attempt to answer Chihara’s objection. Maddy’s rather different answer to the objection is to be found on pp 150–154 of her 1990 book which gives an overall account of her views on the philosophy of mathematics at that time. It is interesting that she describes her position as [__24__, p. 158]: “more Aristotelian than Platonistic”.

__9__

In his [__17__], Franklin discusses how to deal with “large infinite numbers” from an Aristotelian point of view. His approach is somewhat different from the one developed here.

__10__

The point of view of this paragraph is rather similar to that of Avigad in his 2015. Avigad writes [__2__]: “What I am advocating is a view of mathematics as a linguistic artefact, something we have designed, and continue to design, to help us get by in the world”.

__11__

I am here assuming Frege’s view that if a referring expression in a sentence lacks reference, then the proposition expressed by that sentence lacks a truth value.

__12__

I had a discussion with Lakatos on this point in the late 1960s when I was doing my PhD with him. Lakatos thought that the continuum hypothesis would be decided one day, and, when I asked him why, he replied that this was because of the growth of mathematics. Assuming human civilisation continues, mathematics will undoubtedly continue to grow and to develop new concepts, which will be successfully applied to the material world. However, the direction of this future development is uncertain, and it may not consist of further development and successful application of the theory of transfinite cardinals, but rather of the development and successful application of some completely new concepts and theories. Thus the continuum hypothesis may never be decided.