1 Introduction

Philosophers have always been tantalised by the notion of INFINITY and the complicated puzzles that it raises in various philosophical contexts. The nature and characteristics of the infinite and how (if at all) it can be instantiated in the world have been the subject of long-standing philosophical discussions. Philosophers of different eras and traditions of thought have engaged with the infinite through various approaches and from different perspectives. But there is no doubt that some of the most exciting episodes of such engagements have occurred in the medieval traditions of Jewish, Christian, and Islamic philosophy. Philosophers from these traditions discussed a wide variety of issues regarding the notion of INFINITY and its instances in the world (if any). Medieval encounters with the notion of INFINITY have various aspects and can be approached from different angles. Medieval arguments for the impossibility of one or another sort of infinity form one such aspect. Some of the most significant ideas about infinity, which have played a crucial role in the evolution of our understanding of this notion, were introduced and/or developed in the context of the medieval arguments for finitism. In the wide spectrum of these arguments, those that are related, in one way or another, to the problem of the possibility of infinities of different sizes seem to have significant historical and philosophical connections to our modern concept of infinity. Nevertheless, many aspects of the historical development of such arguments and their philosophical significance are still unexplored. This Element aims to shed light on previously uninvestigated corners of medieval finitism by discussing two main groups of the most important medieval arguments that engage with the notion of INFINITIES OF DIFFERENT SIZES.Footnote ¹ Given this specific scope, I refrain from engaging with medieval arguments for infinitism in general or for the existence of infinities of different sizes in particular.Footnote ²

My focus in this study is primarily on the mathematical aspects of medieval finitism. However, it is important to note that extensive discussions of finitism can rarely (if at all) be found in medieval mathematical works. Medieval scholars usually investigated the infinite in either the works of theology and metaphysics (in connection to issues like the eternity of creation, arguments for the existence of God, the infinity of a chain of causally related elements, and the infinity of the objects of God’s knowledge or power) or the works of physics (in connection to issues like the infinity of the world, the infinity and continuity of motion, the infinity of power, the atomistic structure of the material world, and the existence of vacuum). That is why the primary concern of many medieval arguments discussed in this Element is not mathematical. Nevertheless, we cannot reach a comprehensive picture of the historical evolution of the notion of MATHEMATICAL INFINITY without careful analyses of these arguments.

This Element is structured as follows. Section 2 illustrates the definition and some of the characteristics that medieval philosophers typically considered for infinity under the influence of the ancient Greek philosophers and, in particular, Aristotle (d. 322 bce). In the same section, I also discuss some (though by no means all) significant distinctions regarding the various types of infinities that medieval philosophers employed to develop their theories of infinities. Without a precise understanding of those distinctions, we cannot easily detect subtle differences among diverse medieval approaches to finitism. Different versions of what I call ‘the Equality Argument’ are discussed in Section 3. This argument relies on the assumption that there cannot be infinities of different sizes. Although this assumption does not sound true from our contemporary perspective, it was accepted by many ancient and medieval philosophers. Section 4 provides a detailed analysis of another influential finitist argument, which is usually called ‘the Mapping Argument’. The mature version of the Mapping Argument was presented by Ibn Sīnā (d. 1037) – who was referred to in the Latin tradition by ‘Avicenna’ – through the refinement of an earlier, less accurate version by al-Kindī (d. 870). The philosophical significance of the main ideas developed in the context of debates concerning the soundness of these arguments and their relevance to our contemporary conception of mathematical infinity will be discussed in Section 5, where this Element concludes.

Before closing this introduction, I must clarify that although this Element addresses all three medieval Jewish, Christian, and Islamic traditions of philosophy, my primary focus is on the Islamic tradition. This is not only because I am more familiar with this tradition but also because of two other things. First, in the secondary literature in Western languages, medieval Arabic-Islamic theories of infinity are studied no more than their Jewish counterparts and far less than the Christian ones. Second, and more importantly, the most significant discussions of the Equality and Mapping Arguments in Jewish and Christian philosophy are historically posterior to and, in many cases, inspired by earlier discussions of these arguments in the Islamic tradition. In each subsequent section, I analyse the views of medieval thinkers in historical order. As we will see, Muslim figures take precedence in many of these sections. Admittedly, many sophisticated discussions of infinity in the other two traditions have had no anticipation in the Islamic tradition. For example, many of the arguments discussed in the fourteenth-century Latin philosophy (usually considered the most important period of the medieval debates about infinity) have no counterparts in the Islamic tradition. However, I do not discuss those arguments in this Element because, as I have already mentioned, I am mainly concerned with the Equality and Mapping Arguments.

2 Definition and Characteristics of Infinity

Infinitude is limitlessness. However, limitlessness can be understood in two different ways. As Fakhr al-Dīn al-Rāzī (d. 1210), a Muslim theologian and philosopher, puts it:

According to this passage, limitlessness can be understood in two different senses. A thing can be limitless because it lacks quantity. Such a thing is not capable of having a limit. Thus, it would be a category mistake to talk about the limit of it. In the same sense that talking about the colour of justice is a category mistake, talking about the limit of God or of a point is a category mistake. The limitlessness of such things must be taken in the way of simple negation. Limit is by no means attributable to such things. By contrast, things that possess quantity can, in principle, have a limit. Now, if such a thing – for example, a line – has no limit, the limitlessness of it must be understood through metathetic affirmation. To better grasp the distinction made in the passage, consider the sentences ‘justice is colourless’ and ‘the glass is colourless’. The former sentence – assuming that it is true – expresses a simple negation because justice cannot have a colour. Colour is by no means attributable to justice. However, the latter sentence can be interpreted as expressing a metathetic affirmation because the glass has no colour while, in principle, it could have a colour.Footnote ⁴ In our discussion of infinity, we are concerned with things that are limitless in the sense of metathetic affirmation. These are the things that are unlimited, though they could have been limited.

T1 alludes to the general point that, in the framework of the Aristotelian categories, infinity must be considered an attribute of quantities. Quantities are either discrete or continuous. Number and language are examples of discrete quantities; and line, surface, body, time, and place are examples of continuous quantities (Categories 6, 4b20–25).Footnote ⁵ Thus, roughly speaking, infinity must be considered an attribute of magnitudes and multitudes.Footnote ⁶ According to Aristotle, ‘something is infinite if, taking it quantity by quantity, we can always take something outside’ (Physics III.6, 207a7–8). This definition seems to be universally accepted by medieval philosophers. Some of them explicitly endorsed this definition – or some paraphrase of it – in their works. To give a couple of examples, Ibn Sīnā (Reference McGinnis2009, The Physics of The Healing, chapter III.7, § 3) contends that infinite things are those which ‘whatever you take from them, you always find something outside of them’.Footnote ⁷ Instead of appealing to a repetitive process of taking from infinity, Ibn Sīnā defines infinity by a single take. He says that infinity is such that no matter how big what you take from it is, something remains. Although it is not explicitly stated, it must be assumed that what is taken is itself finite. The same definition, with a slightly different phrasing, is endorsed by Fakhr al-Dīn al-Rāzī (Reference al-Rāzī and al-Baghdādī1990, Eastern Investigations, vol. 1, p. 297–98): an infinity is such that ‘when you take whatever amount of it that you wish, you find something outside of it without the need for returning [what is taken]’. By adding the phrase ‘without the need for returning’, he probably means that you can take more and more from an infinity, and even if what is taken is never returned, there always remains something other than what has been taken so far.

Other medieval philosophers either did not provide any explicit definition of infinity or offered other definitions that are somehow compatible with the Aristotelian definition of infinity. We will soon visit some of such definitions. Nevertheless, to the best of my knowledge, there was no significant criticism of the Aristotelian definition of infinity in medieval philosophy. At least not when we are only concerned with physical and mathematical infinity without touching on other things, for example, the qualitative infinity of God. The Aristotelian definition seems to be compatible with all the medieval discussions of infinity that are investigated in the following sections.

Aristotle makes two crucial distinctions about infinity. One between infinity by addition and by division, and the other between actual and potential infinity (Physics III.6, 206a14–25). Roughly speaking, a totality is infinite by addition if and only if it is (or at least can be conceived as being) formed by the successive addition of parts each of which has a similar finite quantity (or, less technically, size). For example, a straight line AB that starts from A and extends infinitely in the direction of B is infinite by addition because it can be conceived as being formed by the successive addition of a segment of a finite length, such as d (Fig. 1a).Footnote ⁸ On the other hand, a totality is infinite by division if, with no limit, it can be successively divided into smaller parts. For example, a finite line CD can be halved infinitely many times by being successively divided at D₁, D₂, D₃, … so that, for every n≥1, CD_n=2CD_n+1 (Fig. 1b). CD is infinite by division but not by addition. To explain the idea of infinity by division using the aforementioned Aristotelian definition of infinity, it can be said that a magnitude is infinite by division if and only if, no matter how many times you divide it into smaller parts, it is always possible to make a further division. The distinction between infinity by addition and by division provides a conceptual tool to differentiate the talk of infinitely big things from that of infinitely small things. This explains the natural association of the notion of INFINITY BY DIVISION with that of CONTINUITY. But what we are concerned with in this Element is mainly infinity by addition. More precisely, the primary aim of the two types of finitist arguments discussed in this study is to reject the possibility of certain sorts of infinity by addition.Footnote ⁹

Figure 1 Infinity by addition and infinity by division.

In broad terms, if the process of addition or division by which an infinity is formed is already completed and all the parts or components of that infinity coexist simultaneously, then that infinity is actual; otherwise, it is potential. If a magnitude is being extended infinitely by successively adding segments of a certain length while its current length is finite, then that line is only potentially infinite. Similarly, if a finite magnitude is, in principle, divisible into infinitely many parts but is not yet so divided, its infinity (i.e., the infinitude of the multitude of all its division) must be considered potential, or so Aristotle suggested.Footnote ¹⁰ As we will see in the following sections, the distinction between actual and potential infinity plays a crucial role in the medieval accounts of infinity. However, it is important to note that not all medieval philosophers share similar interpretations of the notions of ACTUALITY and POTENTIALITY. As a result of various modifications that medieval philosophers proposed to these notions, there are examples of infinite totalities that are taken as actual infinities by some philosophers and as potential infinities by others.Footnote ¹¹

Along with these distinctions about various types of infinity, we should take note of some of the most important characteristics that medieval philosophers considered for infinity. In particular, there are two ideas about infinity that many medieval views regarding the size of infinity are developed either based on or in reaction to them. The first idea goes as follows:

In the next section, we will see that the long history of the discussions of infinity in which this idea or something in its vicinity is presupposed goes back at least to Lucretius (d. circa 55 bce). EI was accepted as an incontrovertible axiom by many medieval philosophers. One might think that, for example, the infinite benevolence of God is not comparable to an infinite line considered in geometry. They are not of the same species. Nor can they be compared to each other quantitatively. Thus, it does not make sense to ask which one is greater, or so one might contend. However, different infinite geometrical lines are of the same species and comparable to each other. Therefore, if EI is true, we must conclude that no infinite line can be greater or lesser than the others. They are all equal to each other. To put it more cautiously, they are all of the same size. In general, in the context of the forthcoming discussions, when it is said that two things are equal, it merely means that those things are equal in terms of quantity.

A rationale behind EI could be that if something is infinite, it must be limitless. Moreover, if something is limitless, it must, in a sense, encompass everything. So, nothing can be greater than an infinity. Not even another infinity can surpass it. As it is stated by John Philoponus (d. 570) in his Against Aristotle on the Eternity of the World (Reference Philoponus2014, fr. 132, p. 144), it is ‘impossible that ⟨anything⟩ should be greater than the infinite, or that the infinite should be increased’.Footnote ¹² Also, EI might be motivated by some conceptions of the infinity of God. In theological contexts, it is usually assumed that the only real and absolute infinity is God. God is the unsurpassable being with respect to whom everything else is limited and finite. If so, absolute infinity is a unique thing that cannot be greater or lesser. That is perhaps why some theologians – for example, the Franciscan Roger Bacon (d. 1292) in his Opus Majus (1928, vol. II, p. 798) – have defined infinity as ‘that to which nothing can be added’.

The second crucial idea about infinity, which can be detected in the background of remarkable ancient and medieval discussions of infinity, states that infinity must be treated as a number:

It is noteworthy that, in medieval discussions of infinity, when a multitude of things (e.g., numbers, points, human beings, and revolutions of celestial bodies) is referred to, it is typically considered collectively as a whole whose smallest parts are its members.Footnote ¹³ Accordingly, although every member of a multitude is a part of it, the other way around does not necessarily hold. For example, the multitude of odd numbers is a submultitude and, consequently, a part of the multitude of natural numbers, though not a member of it. This is roughly how the notions of MULTITUDE, SUBMULTITUDE, and MEMBERSHIP mentioned in NIM and in the following analyses must be read.

It must also be emphasised that NIM has no specific implication for the ontological status of the infinite numbers it is referring to. The mere claim that there is an infinite number, describing how many members a certain infinite multitude has, does not say anything about the nature of that infinite number. In particular, it does not commit us to a realist/Platonist account of the ontology of infinite numbers. The claim in question can be read coherently in a nominalist way. As a result, people with different views regarding the ontology of mathematical objects can share the same view regarding the plausibility of NIM.

If we accept NIM, then EI can be understood as implying that there is a unique infinite number, say I, such that the number of the members of every infinite multitude is I. No infinite multitude can have more or less than I members. However, it is worth noting that NIM on its own does not imply the existence of either any infinite multitude or any infinite number. For example, as we will shortly see, some medieval philosophers endorse NIM and argue that there can be no infinite multitude of objects because there can be no infinite number.

One might think that presupposing NIM is necessary when applying EI to infinite multitudes. However, this is not the case. If we believe that multitudes are comparable (in terms of greatness or, simply, size) solely through the comparison of the numbers describing how many members they have, then EI is not applicable to multitudes unless we accept NIM. However, it is coherent to contend that multitudes are comparable to each other even if some of them lack numbers describing how many members they have. This suggests that, in principle, one can deny NIM without denying the applicability of EI to infinite multitudes/magnitudes. That is why I formulated these principles independently. Having said that, we know from modern logic and set theory that if we accept that there is a way to compare the sizes of infinities to say whether or not they are equal to each other, then it is possible to introduce infinite numbers. One way to do this is by employing an abstraction principle. Thus, if we find that principle plausible (or, more precisely, if we accept that applying such an abstraction principle to infinities is legitimate), the existence of comparable infinities implies the existence of infinite numbers. It must be noted, however, that infinite numbers can be defined even without using such principles.Footnote ¹⁴ Nevertheless, we do not need these technical considerations to discuss medieval accounts of infinity that were much more naïve than ours.

EI and NIM have been addressed in many ancient and medieval exchanges on infinity. As counterintuitive as they might seem from the viewpoint of contemporary mathematics, they were endorsed by many ancient and medieval philosophers. In particular, these principles were employed in some of the most influential medieval arguments against the existence of actual infinity. In the next section, we discuss an argument for finitism in which EI plays a crucial role. An extensive investigation of the engagements of several medieval philosophers with this argument can help us have a better understanding of medieval approaches to the problem of infinities of different sizes.

Before commencing our discussion of the first argument for finitism, it must be noted, as a general caveat, that there are so many subtle differences between various medieval views regarding the possibility of actual infinity that cannot be accurately represented by the coarse-grained distinction between finitism and infinitism. For example, not everyone who is finitist about the material realm also denies the possibility of the presence of infinities in the mind and/or in the realm of the mind-independent immaterial entities (e.g., souls). Moreover, finitists do not necessarily have similar conceptions of actual infinity. As a result, an infinity that is actual for a philosopher might not be considered actual by another. That is why one philosopher might reject the existence of an infinity accepted by another while both insist that actual infinity does not exist. These differences are perfectly reflected in the finitist arguments proposed by medieval philosophers. Regrettably, space limits prevent us from covering many such arguments. Our focus would be on different versions of two finitist arguments which play a pivotal role in the history of philosophical investigations about infinities of different sizes.

3 The Equality Argument

As I mentioned earlier, the history of the arguments against the existence of actual infinities by appealing to EI goes back to at least Lucretius. Arguing in favour of atomism, he writes:

The most plausible reconstruction of the argument presented in T2 seems to be something like the following:

1. (1) If there is no indivisible part, then every object is composed of an infinite number of parts.

2. (2) If every object has an infinite number of parts, then the minutest objects and the whole universe are composed of an equal number of parts.

3. (3) The minutest objects and the whole universe are not composed of an equal number of parts.

Therefore:

1. (4) It is not the case that there is no indivisible part.

EI seems to be a hidden assumption of this argument. In particular, (2) does not hold unless EI is true. The mere fact that the whole universe and the minutest objects are composed of an infinite number of parts does not imply that they have the same number of parts unless we assume that all infinite multitudes are of the same size.

Plutarch (d. circa 120) has offered a similar argument in favour of atomism in which EI is more explicitly presented. In a harsh criticism of Stoic anti-atomism, Plutarch writes:

Arguments of the same spirit were later developed in the context of the debates concerning the temporal origination of the world. Criticising Aristotle and Proclus (d. 485), John Philoponus has offered a series of arguments against the eternity of the world. According to some of these arguments that are based on EI, the world is not eternal in the sense of having no temporal beginning because otherwise there would have been infinities of different sizes.Footnote ¹⁵ Since EI rules out the possibility of such infinities, we have to reject the eternity of the world, or so Philoponus contends. An argument developed based on this general pattern goes as follows:

Like many other ancient and medieval thinkers, Philoponus believes that the beginninglessness of the world implies the beginninglessness of the generation of every species, including human beings. This means that if the world is beginningless, the collection of all the members of any species that have lived until a specific time is infinite. T4 aims to show that this consequence is controversial. If the world had no beginning, the number of men living before Socrates would have been infinite. However, the number of men who have lived until the present time is greater than the number of men living before Socrates. As a result, the beginninglessness of the world implies the existence of infinities of different sizes. This contradicts EI. So, the world must have a beginning. Moreover, if the world has no temporal beginning, not only the number of men who have lived until the present time but also the number of horses or dogs or members of any other species is infinite. This implies that the number of men and horses who have lived so far is double the number of men who have lived so far. Similarly, the number of men, horses, and dogs who have lived so far is triple the number of men who have lived so far. This again contradicts EI because it implies that there are infinities of different sizes. Accordingly, the assumption of the eternity of the world must be rejected.

There is no doubt that Philoponus’s arguments in T4 appeal to EI. But it seems to me that he also presupposes NIM. In this passage, Philoponus refers to numbers associated with infinite collections by stating that if the world has no beginning, then, for example, ‘the number (ὁ ἀριθμός/ho arithmos) of men … is infinite’. This leaves the impression that he accepts that the existence of infinite multitudes implies the existence of infinite numbers. Equivalently, it might mean that Philoponus accepts NIM (or something close to it). However, one could object to this line of reasoning by reading ‘arithmos’ differently. More specifically, one might take ‘arithmos’ simply as ‘collection’, ‘plurality’ or, to use our own terminology, ‘multitude’. In this reading, the claim that the number of men is infinite implies nothing other than that the multitude of men is infinite. However, the latter claim does not imply the existence of an infinite number. Accordingly, the mere fact that Philoponus says that the number of men is infinite does not justify the claim that he endorses NIM. Thus, we need other pieces of evidence to establish Philoponus’s commitment to NIM.Footnote ¹⁶

It seems to me that a convincing sign of such a commitment is that Philoponus talks about the infinite and the possibility of multiplying it. It is legitimate to ask what the reference of ‘the infinite’ is in T4. It is obvious that ‘the infinite’ does not refer to a certain infinite collection of things (e.g., men, horses, or dogs). Philoponus states that by considering the multitude of the horses that have lived so far, in addition to the multitude of the men who have lived so far, we are doubling the infinite. He then adds that by considering similar multitudes of other species we can multiply the infinite many times over. Even if we can make sense of multiplying a multitude, what Philoponus talks about in T4 is not multiplying an infinite multitude itself. Rather, he talks about multiplying the size of an infinite multitude. What he refers to by ‘the infinite’ is the size of an infinite collection. On the one hand, accepting EI, he believes that all infinite multitudes must be of the same size. So, the size of any infinite multitude must be the infinite. On the other hand, by considering the multitudes he mentioned, we can think of infinite multitudes whose sizes are multiplications of the infinite. To get rid of this contradiction, we should give up the assumption of the eternity of the world, or so Philoponus thinks.

It must be noted that Philoponus does not talk merely about different infinite multitudes some of which are greater than others. He specifically talks about multiplying the infinite. He takes the infinite as something that (a) describes the size of an infinite multitude and (b) can be multiplied according to the examples he discussed. Thus, it seems that he treats the infinite as if it is a number. That is why I think Philoponus endorses NIM. Indeed, if NIM and EI are true, then there is a unique infinite number, say I, such that the number of the members of any infinite multitude is I. The reference of ‘the infinite’ in T4 seems to be such an I. T4 can be interpreted as arguing that the eternity of the past is incompatible with the conjunction of NIM and EI. The conjunction of NIM and EI implies that the size of every infinite multitude must be I. However, the eternity of the world implies that there can be infinite multitudes whose sizes are not just greater than I but even multiplications of I. More precisely, if the world has no beginning, the number of men living before Socrates is I. However, the number of men who have lived until the present time is greater than the number of men living before Socrates. As a result, there is something greater than I. Moreover, if the world is beginningless, not only the number of men who have lived until the present time but also the number of horses or dogs or members of any other species is I. This implies that the number of men and horses who have lived so far is 2I, and the number of men, horses, and dogs who have lived so far is 3I. Considering other species, we can see that I can be multiplied many more times. These results are incompatible with EI. Thus, we have to reject the eternity of the world.Footnote ¹⁷ If this reading of T4 is correct, then when Philoponus talks about the number of men, horses, dogs, and so on, he does not talk about the collection of those things. Rather, he talks about the number that describes the size of the collection in question.

Considering the arguments of Philoponus in T4, one might wonder how we can establish the premise that the number of men who have lived until the present time is greater than the number of men living before Socrates. Philoponus seems to take this premise for granted. He does not provide any explicit justification for it. Nevertheless, in later discussions of structurally similar arguments, the relevant counterparts of the aforementioned premise are justified by appealing to the fifth common notion from the first book of Euclid’s The Elements (Reference Euclid1908, vol. 1, p. 155). This common notion, which rejects the possibility of whole-part equality, goes as follows:

Here (and unless otherwise specified in what follows), ‘part’ should be understood as a proper part. This is because there are mereologies in which parthood is a reflexive relation, in the sense that everything is a part of itself (Sider Reference Sider2007, pp. 60 and 70), wherein every whole W includes a part equal to W, which falsifies CN5. Thus, to preserve the truth of CN5, we should take it as referring to the relation of proper parthood, which is, by definition, irreflexive. Nothing can be a proper part of itself. Understood as such, CN5 states that every whole is greater than any of its proper parts.

An immediate consequence of CN5 is that every magnitude is greater than any of its submagnitudes (i.e., the magnitudes obtained by removing a part of the initial magnitude). Moreover, since multitudes are interpreted as having a whole-part relationship with their submultitudes, CN5 implies that every multitude is greater than any of its proper submultitudes. Now, note that the multitude of men living before Socrates is a proper submultitude and, consequently, a part of the multitude of men living until the present time. Thus, according to CN5, the latter multitude must be greater than the former. This means that the number of men who have lived until the present time is greater than the number of men living before Socrates. Thus, the premise in question is established.Footnote ¹⁸

In a vague categorisation, I take any particular finitist argument which relies on the conjunction of CN5 and EI as a version of what I call ‘the Equality Argument’. The general aim of the Equality Argument for finitism is to establish the impossibility of the infinite wholes some of whose parts are also infinite.Footnote ¹⁹ Given CN5, the existence of such infinite wholes and parts implies that there are infinities some of which are greater than others. However, this contradicts EI. Accordingly, the infinite wholes in question must be regarded as impossible.

In another group of arguments, Philoponus argues against the eternity of the world by appealing to the Aristotelian idea of the untraversability of infinity. According to Aristotle (Physics VI.7, 238a20–31), no infinity can be traversed. This principle implies that (a) an infinite magnitude cannot be passed over, and (b) the members of an infinite multitude cannot be counted entirely. The core idea behind Philoponus’s finitist arguments from untraversability is that the world cannot be beginningless because otherwise some infinities would have been traversed until the present time. In one such argument, he argues that if the world has no beginning, there is an infinite collection of humans who have so far come into existence one after another, as if an infinity of humans has been counted out unit by unit. Philoponus takes this as that an infinity has been traversed. Since this contradicts the untraversability of infinity, the idea of the eternity of the world is untenable and must be rejected. Put differently, if the world has no temporal beginning, any individual X is the last ring of at least one infinite chain of human beings, each of whom is a child of the previous one. So, the existence of X in a world whose past is infinite means that an infinity has been traversed. However, if infinity is untraversable, the existence of X implies the finitude of any chain of its ancestors and, consequently, the finitude of the past. Developing such a line of argument, Philoponus (Reference Philoponus2004, Against Proclus, chapter 1, section 3, p. 24) contends that ‘the number of earlier individuals is not infinite. For [if it were] the generations of the race would not have reached down to each of us, for it is impossible to traverse the infinite’.Footnote ²⁰ In another line of argument, he contends that if the world is eternal, then the infinite has been traversed until the present time, not only once but many times. But if the infinite cannot be traversed once, then a fortiori cannot be traversed many times. This argument is presented as follows:

The argument explicitly mentioned in T5 is based on the idea of the untraversability of infinity. However, T5 can also inspire another argument against the eternity of the world, which is based on NIM and EI.Footnote ²² This argument that I call ‘the argument from the numbers of the revolutions of celestial bodies’ goes like this: recall the assumption that the number of the members of any infinite multitude is I. If the world has no beginning, then the number of revolutions of Saturn is I. However, during each of Saturn’s revolutions, Jupiter, the sun, and the moon respectively complete 3, 30, and 360 revolutions. This indicates that if the world is beginningless, the number of revolutions of Saturn, Jupiter, the sun, and the moon must respectively be I, 3I, 30I, and 360I. Since this outcome is incompatible with EI, the eternity of the world must be given up.

Medieval philosophers employed these ideas by Lucretius, Plutarch, and Philoponus in arguments for various purposes.Footnote ²³ Nevertheless, many of those philosophers did not know the exact origins of the employed ideas. In the early Arabic philosophy, a version of the Equality Argument was offered by a Mutazilite theologian, Ibrāhīm al-Naẓẓām (d. circa 845), in his criticism of the Dahrites’ account of the eternity of the world. Dahrites were a group of naturalists who denied the existence of God and believed that both the age and size of the world are infinite.Footnote ²⁴ Although al-Naẓẓām accepts the infinite divisibility of magnitudes, he denies that time and space are infinite (by addition).Footnote ²⁵ As reported by al-Khayyāt (d. circa 910) in The Book of Victory (Reference al-Khayyāt1957, chapter 20, pp. 34–35), al-Naẓẓām’s argument against the eternity of the world goes as follows: either all the stars (or celestial bodies) have the same velocity in their motions, or their velocities are different. If their velocities are different, the distances that they have traversed until now are different. Given EI, all these distances must be finite. This is because infinite distances must all be equal to each other. On the other hand, even if all the stars have the same velocity, the distance traversed by one star is much less than the total distance traversed by all the stars. This means again that all these distances must be finite. As it is implied by EI, infinite distances cannot be longer or shorter than each other. But if the distances traversed by the stars cannot be infinite, the world must have a beginning.

Interestingly, although al-Naẓẓām accepts the Equality Argument against the eternity of the world, he rejects its application against the anti-atomist view he endorsed. Among the early atomists of the Islamic world, a Plutarch–Lucretius-style argument was popular, which, roughly speaking, goes as follows: if atomism is false, then everything has an infinite number of parts. As a result, a mustard seed has the same number of parts as a mountain. But this is absurd. Therefore, atomism is true.Footnote ²⁶ According to al-Khayyāt’s report, al-Naẓẓām has rejected this argument by claiming that although the mountain has the same number of parts, each proportion of the mountain is bigger than the same proportion of the mustard seed. Apparently, al-Naẓẓām thinks that this would suffice to save his views from the aforementioned absurdity:

The Equality Argument is also mentioned in A Treatise on Infinity (1988) by Yaḥyā Ibn ʿAdī (d. 974):

Accepting both NIM and EI, Ibn ʿAdī rejects the existence of any actual infinity and, consequently, any infinite number. In the first sentence of T7, he says that the negation of EI is false. He then mentions some Philoponian examples to show that if the world has no beginning, then EI will be false. The obvious conclusion, which is not explicitly mentioned, is that the world has a beginning.

It is noteworthy that like many other figures in the Arabic tradition, Ibn ʿAdī uses different terms for referring to number and what is numbered (i.e., multitude, collection, or plurality). Referring to Aristotle’s Physics III.5, 204b6–7, Ibn ʿAdī (Reference Ibn ʿ Adī and Khalīfāt1988, A Treatise on Infinity, p. 137) says that neither ʿadad (number) nor maʿdūd (what is numbered) can be infinite. This should leave no doubt that when he talks about the number of years, days, or individuals of a species in T7, what he means is not merely the multitude of those things. Rather, he talks about the number that describes how many members that multitude has. This shows that Ibn ʿAdī accepts NIM. In other words, he accepts that the existence of an infinite multitude implies the existence of an infinite number. However, he rejects the existence of infinite numbers because he denies that the existence of any infinite multitude is possible.Footnote ²⁸

Versions of the Equality Argument are discussed by Ibn Sīnā in a still unpublished treatise, ‘On the Arguments of Those Who Maintain that the Past Has a Temporal Beginning’, in which he critically discusses the arguments for the finitude of the past.Footnote ²⁹ In particular, he analyses and rejects the following arguments in the fourth chapter of this treatise: (a) If the world has no beginning, both the number of things that have existed until the time of the Deluge and the number of things that have existed until our time are infinite, with the latter being greater than the former. Since this contradicts EI, the eternity of the world must be rejected. (b) If the world has no beginning, the existence of every individual human depends on the existence of an infinite number of their ancestors. However, something whose existence is dependent on an infinite number of things can never come into existence. So, if the world were beginningless, no individual would have existed. Since we exist now, the eternity of the world is false. More importantly, in the eighth chapter of this treatise, he argues that arguments from EI against the eternity of the world are unsound because EI applies only to the infinite multitudes or magnitudes all whose members or parts exist together. This condition of the applicability of EI is known as ‘the wholeness condition’ in the literature. A quantity S satisfies the wholeness condition if the following description is true of it:

According to Ibn Sīnā, it is impossible for infinities that satisfy WC – So sorry! It is not the most appropriate abbreviation – to be of different sizes. He thinks that no absurdity arises from the assumption that there are infinite multitudes or magnitudes of different sizes which fail to satisfy WC. Accordingly, no absurdity follows from that the multitude of things that have existed until our time is bigger than the multitude of things that have existed until the time of the Deluge, even if both multitudes are infinite. EI does not apply here. Thus, the infinitude of the past cannot be rejected through this line of argument.

Perhaps the most famous discussion of the Equality Argument in the Islamic tradition is offered by al-Ghazālī (d. 1111). In Discussion 1 of his The Incoherence of the Philosophers, he criticises the doctrine of the eternity of the world and offers several arguments for the finitude of the past. In particular, he presents a version of the argument from the number of revolutions of celestial bodies and then complements it with another argument which goes as follows:

Al-Ghazālī argues that if the world has no beginning, the number of revolutions of a celestial body is infinite and it is legitimate to ask about this number – as we can do it about any other number – whether it is even or odd. As a number, it must be either even or odd. But whether it is even or odd, only by adding one unit to it, its status changes from even to odd or vice versa. This means that there is always a lacking unit whose addition to the infinity in question can change the status of that infinity in terms of evenness and oddness. However, al-Ghazālī finds it implausible that an infinity lacks a single unit. He does not explain why he thinks so. But it is probably because of something like EI or the ideas that motivate it. An infinity must encompass everything, and nothing can be greater than it. Therefore, it should not be possible to change the status of an infinite multitude in terms of the evenness or oddness of the number of its members by just adding a new member to it. Accordingly, the number of the members of a multitude can be neither even nor odd. But this is not acceptable for al-Ghazālī because he thinks that every number must be either even or odd. Therefore, he denies that the number of revolutions of a celestial body can be infinite. Since the eternity of the world can grant the possibility of the infinitude of such a number, al-Ghazālī concludes that the world cannot be eternal and must have a temporal beginning.Footnote ³⁰

It is worth highlighting that he is so committed to NIM that he does not even allude to the possibility that a multitude might be infinite even if there is no infinite number that describes how many members this multitude has. Having said that, he considers the possibility that the evenness and oddness might be attributable only to finite numbers and not infinite ones. But he rejects this difference between finite numbers and the infinite number(s) in question. As it is stated in the premises of the argument from the numbers of the revolutions of celestial bodies, whether these numbers are finite or infinite, they can be multiplied and divided like other numbers. So, we can meaningfully talk about whether the number of revolutions of a star is divisible by six or ten regardless of whether or not that number is finite. Accordingly, it must also be possible to meaningfully talk about whether or not such a number is divisible by two. But this is exactly what establishes that assigning evenness and oddness to such numbers is plausible. As a result, al-Ghazālī’s argument cannot be refuted by contending that evenness and oddness cannot be attributed to the infinite number of revolutions of a celestial body, or so he thinks.

Moreover, al-Ghazālī provides two reasons why the above argument from evenness and oddness cannot be refuted by appealing to WC. First, he insists that we can meaningfully talk about the number of things even if they do not exist. More precisely, he thinks that we can legitimately attribute numbers to the multitudes some or even all of whose members are non-existents.Footnote ³¹ This implies that although the past revolutions of celestial bodies do not exist now, we can meaningfully talk about the number of such revolutions and their attributes and properties. Since the above argument is about the characteristics of such numbers rather than the revolutions themselves, the non-existence of the revolutions does not jeopardise the soundness of the argument. Second, al-Ghazālī argues that even if we cannot apply the above argument to the number of the past revolutions of celestial bodies because they fail to satisfy WC, we can apply this argument to the number of human souls who have been separated from their bodies until now. Ibn Sīnā (to whom al-Ghazālī is objecting in T8) believes that although the human soul has a temporal origination, it will never perish after coming into existence. Thus, the souls of all the humans who have passed away so far exist now altogether. Accordingly, if the world has no temporal beginning, the multitude of human souls who are now separated from the bodies they were attached to would be an infinite multitude that satisfies WC. Thus, if the satisfaction of WC is necessary for the success of the argument from evenness and oddness, we can apply this argument to the number of existing human souls instead of the number of revolutions of a celestial body. We can, therefore, successfully refute the eternity of the world by establishing the impossibility of the infinitude of the multitude of human souls that have passed away until now through the argument from evenness and oddness, or so al-Ghazālī believes. The problem of the infinite number of human souls is also linked to the Mapping Argument against the existence of actual infinities, which will be discussed in the next section.

Variations of the Equality Argument are also defended by another Muslim theologian, Muhammad Ibn ʿAbd al-Karīm al-Shahrastānī (d. 1153). In his Struggling with the Philosopher, he attempts to refute the main elements of Ibn Sīnā’s metaphysics by discussing five different issues, the fifth of which is the doctrine of the eternity of the world. Analysing the structure of the arguments for the finitude of the past, al-Shahrastānī identifies some of the primary principles that can be employed in such arguments. In particular, he contends that the mere fact that something is divisible to smaller parts or fractions shows that it cannot be infinite:

He then shows how these principles can be employed to develop variations of the Equality Argument, one of which goes as follows:

The primary aim of this argument is to show that the number of human souls cannot be infinite. But a corollary of it is that the past cannot be infinite either. The argument can be reconstructed as follows: if the number of human souls that exist on Sunday is infinite, it is, in principle, possible that this number increases on Monday by new souls that come into existence on Monday. But we know that infinity cannot increase. Thus, we can conclude, by a repetitive syllogism (namely here, a modus tollens), that the number of souls existing on Sunday cannot be infinite at all. Since the eternity of the world confirms the possibility of the existence of an infinite number of souls by any specific day, including that specific Sunday, we have to conclude that the eternity thesis must be rejected as well.

The last principle stated in T9 implies that the multitudes of revolutions of the celestial bodies cannot be infinite because we can talk about fractions of them. For example, we saw in T5 that the number of revolutions of Saturn is a third of Jupiter’s. Given this, the number of these revolutions must be finite because the infinite is indivisible and infractible, or so al-Shahrastānī thinks. Although this argument for finitism is not presented in Struggling with the Philosophers, a version of it can be found in al-Shahrastānī’s The End of Steps in Theology.Footnote ³²

As a proponent of the doctrine of the eternity of the world, Ibn Rushd (d. 1198) – who was referred to in the Latin tradition by ‘Averroes’ – does not find convincing either the argument from the number of revolutions of celestial bodies or the argument from the evenness and oddness of such a number. To respond to al-Ghazālī, Ibn Rushd appeals to the fact that the rotations of a celestial body form only a potentially infinite multitude. He believes that although it is meaningful to talk about proportions between the actual parts of potential infinities, it does not make any sense to talk about proportions between potential infinities themselves:

Ibn Rushd seems to believe that potential infinities are not comparable to each other. Imagine that there are two rope-making machines, M₁ and M₂, such that M₁ produces one metre of rope per minute, while M₂ produces 2 metres of rope per minute. Moreover, assume they both start working now and will never stop working. Can we say that the rope that will be produced by M₁ will be twice as long as that of M₂? Ibn Rushd’s answer to this question would be negative. Since the ropes that these machines will produce (if they never stop working) are potential infinities, they are not comparable to each other. These potential infinities will never have been actualised in the sense that there is no time up until which the processes of the production of such infinite ropes have been completed. This suffices to establish that the proportion between the length of the ropes that will be produced after a finite time by these two machines cannot be equal to the proportion between the total infinite ropes that these machines will produce. Indeed, such infinite ropes will never have been produced. Thus, it does not make any sense to talk about the ratio between them, or so Ibn Rushd would say. Although EI is, in a sense, violated here, this is not because one of these infinities is greater than the other. According to Ibn Rushd, EI is invalid in the case of potential infinities because they are not comparable to each other at all. Talking about the equality or inequality of potential infinities would involve a category mistake.

If the world is eternal, the multitudes of the revolutions of celestial bodies are merely potentially infinite. Thus, we cannot compare them to each other and meaningfully talk about their halves, thirds, or other fractions. Therefore, the finitist arguments from the revolutions of celestial bodies and from evenness and oddness fail. This seems to be what Ibn Rushd wants to convey by T11. Nevertheless, it is still unclear why Ibn Rushd assumes that potential infinities are not comparable to each other. One might think this is because potential infinities fail to satisfy WC. For example, one might read Ibn Rushd as saying that the infinite multitude of the revolutions of Saturn is incomparable to the infinite multitude of the revolutions of the sun because these multitudes fail to satisfy WC. However, if the comparability of two multitudes hinges on the satisfaction of WC, then even the finite fractions of those two multitudes are not comparable to each other. For example, WC is not satisfied either by the multitude of the revolutions of Saturn in the last three years or by the multitude of the revolutions of the sun in the same period. Thus, we should conclude that these finite multitudes are incomparable to each other because they fail to satisfy WC. Even worse, revolutions do not happen instantaneously. Thus, no revolution exists as a whole at any moment of (the past, present, or future) time. Accordingly, if we accept that only the present is real and only the present things exist, as Ibn Rushd and most of the medieval thinkers mentioned in this Element do, then no revolution—let alone any multitude of revolutions—satisfies WC. This implies that the multitudes of revolutions, whether finite or infinite, are not comparable to each other at all. But this does not seem a desirable consequence for Ibn Rushd. Unfortunately, he has not provided any more helpful explanation on how this undesirable result can be avoided.

A more abstract version of the Equality Argument goes as follows: assume that there is an infinite whole. Remove a finite part from it. What remains must be infinite. Otherwise, the initial whole is composed of two finite parts and must be finite. As a result, every infinite whole has infinite parts. However, given CN5, every whole is greater than its parts. Thus, if there are infinite wholes, there are infinities some of which are greater than others. This contradicts EI. Accordingly, the assumption of the existence of infinite wholes must be rejected. Arguments of this level of abstraction are presented by the Jewish philosopher, Bachya Ben Joseph Ibn Paquda (d. 1120) – whose Arabic name is ‘Baḥy Ibn Yūsuf Ibn Bāqūda – in his Duties of the Heart (1996, Gate 1, chapter 5, 82–85). A geometrical version of such arguments is presented by Ibn Bājja (d. 1138) – who was referred to in the Latin tradition by ‘Avempace’ – in his Commentary on Aristotle’s Physics (1991, Book III, p. 38): suppose that there is an infinite line AB that starts from A and extends infinitely in the direction of B. Moreover, suppose that AC is an initial final segment of AB. On the one hand, CB is a part of AB. Therefore, CN5 implies that AB is greater than CB. On the other hand, CB must be infinite. Otherwise, AB – which is composed of AC and CB – would also be finite. But if AB and CB are both infinite, then we have two infinities, one greater than the other. This contradicts EI. As a result, the initial assumption that AC is infinite must be rejected. There is no infinite magnitude (Fig. 2).Footnote ³⁵

Figure 2 The existence of infinite lines can be rejected by the conjunction of CN5 and EI.

Ibn Bājja (Reference Bājja and Fakhrī1991, Commentary on Aristotle’s Physics, Book III, p. 40) believes this argument only establishes the impossibility of the magnitudes that satisfy WC. That is why it cannot be employed to reject the infinitude of the past. Following Ibn Sīnā, Ibn Bājja contends that since the line of time fails to satisfy WC, the infinity of time is immune to the arguments for finitism.

Another interesting point that Ibn Bājja makes is about the compatibility of finitism with the methodology of mathematical sciences. Following Aristotle (Physics III.7, 207b28–38), Ibn Bājja says:

Ibn Bājja contends that actual infinities are not needed even for purely mathematical purposes. He seems to think that arguments for finitism apply to magnitudes and multitudes even if we take them to be merely mental (or mind-dependent) existents.Footnote ³⁶ He explicitly claims that ‘it is impossible for estimation (wahm) to present [the parts of an] infinity actually altogether’. Given that, in the context of the post-Avicennian Islamic philosophy, ‘estimation’ refers to the cognitive faculty which enables us to entertain fictional and imaginary objects and/or scenarios, Ibn Bājja must be interpreted as saying that the parts of an infinity cannot coexist even in mind and as purely mental existents. Put differently, Ibn Bājja believes that there is no infinity that satisfies WC even if the notion of EXISTENCE mentioned in WC is taken to be merely MENTAL EXISTENCE. His position must be understood in contrast with the views of people who believe that arguments for finitism establish the impossibility of an infinity only if it satisfies a condition that I call ‘the extramental existence condition’. A quantity S satisfies this condition if the following description is true of it:

Note that WC and EEC are independent of each other in the sense that a quantity might satisfy neither, both, or only one of these conditions. Ibn Bājja believes that the Equality Argument establishes the impossibility of any infinity that fulfils WC regardless of whether or not it also fulfils EEC. This condition also plays a crucial role in discussions of the Mapping Argument that is the subject of the next section.

The eminent Jewish philosopher, Moses Maimonides (d. 1204) – whose Arabic name is ‘Mūsa Ibn Maymūn’ – has mentioned the arguments from the revolutions of celestial bodies and from the number of human souls in his presentation of the anti-eternity views of the Muslim theologians in The Guide of the Perplexed (Reference Maimonides1967, I.74, p. 222). Although Maimonides thinks these arguments are not compelling, he does not specify where exactly they go wrong. But this does not undermine the significance of The Guide of the Perplexed in the history of the idea of infinities of different sizes. This book was translated first from Arabic into Hebrew and then from Hebrew into Latin in the thirteenth century. The Latin translation, which was entitled Dux neutrorum, was completed around 1220–1240. It might be through Dux neutrorum that the Equality Argument was transmitted into the Latin tradition. Dux neutrorum is likely to be one of the sources of William of Auvergne (d. 1249), the first Latin author (or, at least, one of the first ones) who offered arguments for the finitude of the past.Footnote ³⁷ These arguments are presented in his The Universe of Creatures, and some of them are in the style of the Equality Argument. According to one of these arguments, for each revolution of the sun, there are 360 degrees of the motion of the sun, and each degree of this motion is equal to one day. Therefore, for each revolution of the sun, there are 360 days. This establishes that if the world has no beginning, there are two infinities one of them 360 times greater than the other. Since this contradicts EI, the doctrine of eternity must be rejected (William of Auvergne Reference Teske1998, The Universe of Creatures, p. 137). William of Auvergne also presents another argument that is exclusively about the definite proportions between the revolutions of celestial bodies. Discussing the proportions between the revolutions of a few celestial bodies (e.g., Saturn, Jupiter, the sun, etc.), he concludes that ‘the revolutions of each of the planets will be according to astronomical calculation a definite part, that is, will have a definite proportion to all the revolutions of the heaven taken together, which are completed in the whole of time, which comes to an end in the present moment. It is impossible, however, that it be infinite since its parts are found to have a definite relation and proportion to it, etc.’ (William of Auvergne Reference Teske1998, The Universe of Creatures, p. 138). This means that if the world is eternal, then the multitude of the revolutions of each celestial body is a definite proportion of the multitude of all the revolutions happening in the heaven, while both of these multitudes are infinite. This, again, is incompatible with EI, given that the aforementioned proportion is not the ratio of 1 to 1.Footnote ³⁸ Thus, the world must have a beginning. Both of these arguments rely on observations about the actual state of the world. However, before these arguments, another argument is presented by William that appeals to our intuitions about counterfactual but still imaginable states of the world. That argument goes as follows:

Assume that the heaven is rotating with velocity v. Furthermore, assume that the world has no beginning and the number of the past revolutions of the heaven is I. However, the heaven could have a different velocity. But if the velocity of the motion of the heaven were different, the number of its revolutions would have been proportionally different. For example, if the velocity of this motion were v/2, then the number of revolutions would have been I/2. Generally, it is conceivable that the heaven has velocity v/2ⁿ for some n ≥1. But if the velocity were v/2ⁿ, the number of revolutions would have been I/2ⁿ. Thus, I has a half, a quarter, etc. This establishes that I cannot be infinite. This is because infinity does not have a half, a quarter, and so on. On the other hand, if I is a finite number, then the world must have a beginning. In the last two paragraphs of T13, the same argument is repeated by velocities 2ⁿv (n ≥1), which would change the number of revolutions of the heaven to 2ⁿI. As a result, I has also a double, a quadruple, and so on. The incompatibility of this result with EI affirms, once again, the finitude of I and the finitude of the past, or so William of Auvergne contends.Footnote ⁴⁰

The Equality Argument can be found in many later medieval Latin texts. For example, in a commentary on the Sentences of Peter Lombard (d. 1160), Bonaventure (d. 1274) mentioned a version of the argument from the number of revolutions of celestial bodies to reject the beginninglessness of the world (Reference Bonaventure1882, Commentary on the Sentences, Book II, Distinction 1, Part 1, Article 1, Question 2). Bonaventure (Reference Bonaventure1964, On the Eternity of the World, p. 108) makes it explicit that this argument is based on one of the ‘per se known propositions of reason and philosophy’ that states that ‘it is impossible to add to the infinite’. Murdoch (Reference Murdoch1981, p. 52) believes that it was probably the inclusion of this argument in Bonaventure’s ‘examination of the possibility of an eternal world that gave it the status of a sine qua non element in almost all subsequent discussions of this controversial problem’ (see also Murdoch Reference Murdoch, Kretzmann, Kenny and Pinborg1982, pp. 569–70).Footnote ⁴¹ According to Bonaventure (Reference Bonaventure1964, On the Eternity of the World, p. 109–10), another per se known proposition that can be taken as a ground for arguments against the eternity of the world states that ‘it is impossible that there be simultaneously an infinite number of things’. To use the terminology introduced earlier, this proposition can be read as saying that ‘no infinity that satisfies WC can be realised’. Bonaventure then argues that the eternity of the world is in tension with this principle because it implies the existence of an infinity of human souls that simultaneously exist.

As another example of the Latin discussions of the Equality Argument, it is interesting to see a slightly different construction that still relies on the principles mentioned in the previous versions we saw. Roger Bacon, in his Opus Tertium (Reference Bacon and Brewer2012, pp. 141–42), develops an argument whose core idea can be simplified as follows: consider a straight line that is infinitely extended in directions DC. Moreover, assume that A and B are two points existing on DC so that, compared to B, A is closer to C (Fig. 3).

Figure 3 A visualisation of Bacon’s version of the Equality Argument.

Now, consider the following line of argument:

1. (1) AD = AC. This is because they are both infinite and EI is true.

2. (2) BC = BD. This is because they are both infinite and EI is true.

3. (3) BC > AC. This is because AC is a part of BC and CN5 is true.

4. (4) BC > AD. This is true because of (1) and (3).

5. (5) BD > AD. This is true because of (2) and (4)

6. (6) BD < AD. This is because BD is a part of AD and CN5 is true.

7. (7) BD > AD and BD < AD. This is true because of (5) and (6).

But (7) expresses a contradiction. This means that the initial assumption of the existence of an infinite line extended in both directions is implausible. What Bacon concludes from this is that the size of the world cannot be infinite. The infinitude of the world’s size implies the existence of one-dimensional magnitudes infinitely extended in both directions. But the above proof reveals that the existence of such magnitudes is absurd.Footnote ⁴²

Versions of the Equality Argument have been criticised not only by philosophers who accepted the existence of actual infinities but also by philosophers who affirm that no actual infinity exists. As an example of the former group of philosophers, we can mention Henry of Harclay (d. 1317), who criticised the arguments for the finitude of the past from the number of revolutions of celestial bodies and defended the idea of the existence of actual infinities.Footnote ⁴³ By contrast, Naṣīr al-Dīn al-Ṭūsī (d. 1274) is one of those philosophers who cast doubt on the validity of such arguments against the eternity of the world without accepting the existence of actual infinities. He believes that although the past (or, more particularly, the multitude of the past revolutions of a celestial body) is infinite, its infinitude does not imply any absurdity. In his The Wrestlings Down of the Wrestler, al-Ṭūsī responds to al-Shahrastānī’s criticisms of Ibn Sīnā’s view regarding the eternity of the world by explicitly accepting that infinities which fail to satisfy WC can, in principle, be of different sizes. Such infinities can be divided into smaller parts that are themselves infinite. They can also be added to or multiplied. Put differently, al-Ṭūsī believes that EI does not apply to infinities whose parts fail to exist altogether. One such infinity can be greater or smaller than another, and this does not lead to any implausible consequences (al-Ṭūsī Reference al-Ṭūsī and Madelung2004, The Wrestlings Down of the Wrestler, pp. 183–85).

The Equality Argument has been addressed in many other texts that we cannot investigate here due to space constraints. Nevertheless, what we saw in this section from the long history of this argument and its miscellaneous versions suffices to reveal both its significance in the historical evolution of our conception of infinities of different sizes and its naivety and shortcomings. It was probably through the Equality Argument that the attention of the philosophers was drawn for the first time in history to the fact that the existence of an infinity could imply the existence of other infinities that might or might not be of the same size. Admittedly, no clear and explicit definition of size as it applies to infinities is offered in the context of discussions regarding this argument. The advocates of this argument take EI for granted without introducing any sufficient criterion for the equality of infinities. Nevertheless, it appears that they appeal to CN5 as a sufficient criterion for the inequality of infinities. Accordingly, the gist of the argument is to establish a tension between (a) the existence of unequal infinities, which follows from the conjunction of CN5 and the assumption of the existence of an infinity and (b) the equality of all infinities – that is, EI – that is taken for granted with no sufficient criterion for equality. One step forward towards a more accurate understanding of the possibility of the existence of infinities of different sizes is to introduce a sufficient criterion for the equality of infinities. This is precisely what has been done in the Mapping Argument.

4 The Mapping Argument

Another principle widely employed in medieval discussions of infinity is the fourth common notion from the first book of Euclid’s The Elements (Reference Euclid1908, vol. 1, p. 155). This common notion, usually known as the Axiom of Congruence, states that ‘things which coincide with one another are equal to one another’ (‘τὰ ἐφαρμόζοντα ἐπ ̓ ἀλλήλα ἴσα ἀλλήλοις ἐστίν’). The term ‘ἐφαρμόζοντα’ comes from the Greek verb ‘ἐφαρμόζειν’. Although the translation of ‘ἐφαρμόζειν’ to ‘to coincide’ (or ‘to correspond’) is not incorrect, it does not reflect the exact (or complete) meaning of this verb in the geometrical contexts in which Euclid employs the aforementioned common notion. He has appealed to this common notion in the proofs of some theorems (e.g., I.4, I.8, and III.24) to conclude certain equalities from coincidences or correspondences realised as a result of superposing or mapping some geometrical figures upon others. Thus, ‘ἐφαρμόζειν’ expresses a particular sort of coincidence or correspondence which occurs as a result of the application of a specific method of the comparison of quantities (i.e., the mapping or superposition technique). This sense of ἐφαρμόζειν is better captured in the later Arabic and Latin translations of The Element. For example, in the so-called Isḥāq-Thābit translation, this common notion is rendered as ‘things none of which exceeds another – whenever one of them is mapped [or superposed] on another – are equal’.Footnote ⁴⁴ This construal of the fourth Euclidean common notion, which can be expressed more simply as follows, plays a crucial role in the development of medieval theories of infinity:

CN4 has been used both in certain theories about infinitely large quantities originating from the Islamic–Arabic tradition and in certain theories about infinitely small quantities originating from the Christian–Latin tradition. The central notion in both groups of theories is the same notion that can be expressed by ‘ἐφαρμόζειν’ in Greek. This notion is associated with the Arabic ‘inṭibāq’ or ‘taṭbīq’ in the former theories about infinitely large quantities and with the Latin ‘superponere’ in the latter theories about infinitely small quantities. To highlight the different functions of the notions expressed by these terms, I prefer to translate them into different English terms. Accordingly, I translate ‘inṭibāq’ and ‘superponere’ as, respectively, ‘to map’ and ‘to superpose’ in English.Footnote ⁴⁵ However, it must be noted that in many (though not all) of the following discussions of finitism, these two verbs can be used interchangeably.

CN4 has been employed with CN5 in one of the most famous medieval arguments for finitism, usually called ‘the Mapping Argument’. The core idea behind this argument goes as follows: if an infinity exists, it can be shown, by appealing to CN4, that it would be equal to some of its parts. But this contradicts CN5. Therefore, no infinity exists. This general line of thought has been qualified in different versions of the Mapping Argument. Probably, the earliest version of this argument is presented by al-Kindī.Footnote ⁴⁶ He believes that this argument shows that every quantity must be finite. This implies that space (or, more specifically, the size of the world) and time (or, more specifically, the age of the world) must be finite. But this is compatible with the possibility of the expansion of the world with no limit over time and with the endlessness of the future. Thus, the size and age of the world can be potentially infinite but not actually infinite, or so al-Kindī (Reference al-Kindī2012, On First Philosophy, pp. 21–22) believes. Ibn Sīnā rehabilitated al-Kindī’s argument, and the Avicennian version of the Mapping Argument became popular among medieval thinkers. He has presented this argument with slight differences in several works.Footnote ⁴⁷ The version offered in The Physics of The Healing goes as follows:

Assume that there exists an infinite one-dimensional magnitude – that is, an infinite straight line – AB starting from point A and extending infinitely in the direction of B. Remove a finite part AC from its beginning, and map the remaining part – that is, CB or, more precisely, a copy of it, say C*B* – upon AC so that C* is positioned in front of A and the two lines are parallel to each other. If C*B* falls short of AC, then C*B* and, consequently, CB would be finite (Fig. 4a). Since AC is finite by assumption, AB, which is composed of two finite parts, must also be finite. However, this contradicts the initial assumption of the infinitude of AC. Thus, C*B* does not fall short of AC. But if the two lines correspond to (or coincide with) each other so that neither exceeds the other, then CN4 implies that they are equal to each other (Fig. 4b). This means that AB equals C*B* and, consequently, CB. But CB is a part of AB. Accordingly, CN5 implies that AB must be greater than (CB). They cannot be equal to each other. Thus, we have a contradiction. This establishes that we have to reject the initial assumption we started from, that is, the existence of an infinite one-dimensional magnitude.

Figure 4 A visualisation of the Mapping Argument against the existence of infinite magnitudes.

In this version of the Mapping Argument, the notion of MAPPING must be understood as the superposition of a geometrical figure upon another. If the two figures cover each other so that no part of any of them remains uncovered by the other, then the two figures coincide with and are equal to each other in the sense of CN4. Although this sense of mapping applies to magnitudes, it does not seem to apply to multitudes. That is why the version of the Mapping Argument presented in T14 appears to be inapplicable to numbers and multitudes of objects, even though T14 explicitly rejects the actual existence of not only infinite magnitudes but also infinite numbers and infinite multitudes of numbered things (albeit if they satisfy certain conditions). Of course, as shown in Books 7–9 of The Elements, numbers can be represented as magnitudes. As a result, any argument against the existence of infinite magnitudes can also be taken as an argument against the existence of infinite numbers.Footnote ⁴⁸ Nevertheless, the version of the Mapping Argument mentioned in T14 – which I call ‘the magnitude version’ – does not itself directly apply to discrete quantities and, in particular, to the multitudes of objects. Ibn Sīnā was apparently aware that there is another version of the Mapping Argument that directly applies to multitudes. Nonetheless, he never explicitly mentioned this version – which I call ‘the multitude version’ – in his works. The multitude version of the Mapping Argument is presented in the works of his commentators and critics. For example, al-Shahrastānī proposes an example of the multitude version of the Mapping Argument in his Struggling with the Philosopher to establish that the chain of the ancestors of a person cannot be infinite. Analysing the magnitude version of the Mapping Argument, he contends that, in the application of the mapping technique, we need to ‘map the point on the point and the line on the line’ (al-Shahrastānī, Reference al-Shahrastānī, Mayer and Madelung2001, Struggling with the Philosopher, p. 108). Apparently, what he has in mind here is the one-to-one correspondence of the points or equally long segments of the two lines, one of which is mapped upon the other. He believes that we can ‘apply this argument as it is to the numbers of human souls and the numbers of circular motion [of celestial bodies]’ (al-Shahrastānī, Reference al-Shahrastānī, Mayer and Madelung2001, Struggling with the Philosopher, p. 108). Regarding the former, he proposes the following argument for the finitude of the chain of the ancestors of any given human being:

The argument could have been presented in a more precise form. Nonetheless, the exact content that is supposed to be conveyed is not hard to guess: assume that the chain of the ancestors of any given human being is infinite. Therefore, the following multitudes are infinite: (a) the multitude of Zayd, his father, the father of his father, the father of the father of his father, and so on ad infinitum; and (b) the multitude of ʿAmr, his father, the father of his father, the father of the father of his father, and so on ad infinitum. If we assume that ʿAmr himself is one of the forefathers of Zayd and, accordingly, a member of (a), then (b) would be a part of (a). Moreover, we can suppose that the human beings in (a) are points with equal successive distances from each other on a straight line starting from a point corresponding to Zayd and extending infinitely. Part of this line is a line that starts from a point corresponding to ʿAmr and extends in the same manner as (a). Now, we can map the line corresponding to (b) on the line corresponding to (a) so that the point corresponding to ʿAmr is mapped upon the point corresponding to Zayd. To make this process of mapping more easily imaginable, al-Shahrastānī suggests that we can take ʿAmr as the twin of Zayd (and any ancestor of ʿAmr as the twin of the corresponding ancestor of Zayd). Of course, this does not mean that ʿAmr is both an ancestor and the twin of Zayd. Al-Shahrastānī just wants to reassure his readers that the mapping process can be completed and every member of (a) can be paired with a member of (a copy of) (b) in the same manner as twins (Fig. 5). If (a) is infinite, then (b) would be infinite too. Moreover, the completion of the mapping process establishes that (a) and (b) are equal due to CN4. However, this result is incompatible with CN5 because (b) is a part of (a). Thus, the infinitude of (a) and, consequently, (b) must be rejected.

Figure 5 An application of the multitude version of the Mapping Argument to show that the chain of the ancestors of a human being cannot be infinite.

A more abstract example of the application of the multitude version of the Mapping Argument is presented in Fakhr al-Dīn al-Rāzī’s commentary on Ibn Sīnā’s Fountains of Wisdom. In his commentary, after discussing a version of the Mapping Argument that is presented in the Physics part of Fountains of Wisdom, al-Rāzī anticipates twelve objections to this argument and tries to rebut them. In the eleventh objection, he says that one might reject the Mapping Argument because it implies the unacceptable consequence that even the multitude of all natural numbers must be finite:

In the multitude version of the Mapping Argument, mapping a multitude upon another is nothing but putting their members into one-to-one correspondence by pairing each member of one multitude with one and only one member of the other. According to this argument, some proper submultitudes of any infinite multitude can be mapped upon it so that no member of any of them remains unpaired. Coupling this with CN4, we can conclude that every infinite multitude is, in a sense, equal to some of its proper submultitude. But this is incompatible with CN5, according to which every multitude is greater than (and, therefore, not equal to) any of its proper submultitudes. Thus, the existence of infinite multitudes is impossible.

T16 objects to this line of reasoning. On the one hand, it is undeniable that the multitude of all natural numbers is infinite. On the other hand, if we apply the multitude version of the Mapping Argument to this multitude, we can show that it is equal to some of its proper submultitudes. For example, by employing CN4 and pairing 1 with 11, 2 with 12, 3 with 13, and n with n+10 in general, we can show that the multitude of all natural numbers is quantitatively equal (or, as it is usually said in modern set theory, equinumerous) to the multitude of the natural numbers greater than 10 (Fig. 6).Footnote ⁴⁹ But the latter multitude is a proper submultitude of the former. Thus, if we endorse CN5, a contradiction follows. Accordingly, the existence of the infinite multitude of natural numbers must be denied.

Figure 6 One-to-one correspondence between the multitude of all natural numbers and the multitude of all natural numbers greater than 10.

Before discussing al-Rāzī’s response to this objection, we should see what conditions Ibn Sīnā considers for the applicability of the Mapping Argument. In his presentations of the Mapping Argument in various places of his oeuvre, Ibn Sīnā consistently insists that this argument rejects the actual existence of the infinities that are or can be ordered. Put differently, the aim of the original version of the Mapping Argument presented by Ibn Sīnā is to establish the impossibility of any infinity that satisfies two conditions. One of these conditions concerns the actuality of the infinity in question, while the other concerns its orderedness. To be more specific, this argument cannot establish the impossibility of potential infinities. Putting together his discussions of this condition in various places of his works, we can conclude that what he means by the actual existence of an infinity is nothing but the fulfilment of WC – that is the simultaneous existence of all the parts of the infinity in question. In other words, he believes that the Mapping Argument cannot establish the impossibility of an infinity that fails to satisfy WC.

Ibn Sīnā does not explain why the satisfaction of WC is necessary for the applicability of the Mapping Argument. But what he had in mind is probably that if all the parts of S are not supposed to exist together simultaneously, then it does not make any sense to talk about whether or not S as a whole can be equal to some of its parts. On this interpretation, CN5 does not express a logical fact about any whole regardless of its nature. In particular, if it is legitimate to talk about wholes that are cross-temporal mereological sums of elements that do not exist simultaneously, then CN5 is not true of such wholes. As a result, the infinitude of such wholes cannot be rendered impossible by appealing to an argument which, like the Mapping Argument, relies on CN5. That is why, at the end of his discussion of the Mapping Argument in The Salvation (Reference Sīnā and Dānishpazhūh1985, chapter IV.2, pp. 245–46), Ibn Sīnā contends that this argument cannot establish the finitude of time. By contrast, al-Kindī did not believe that the applicability of (his own version of) this argument hinges on the satisfaction of a condition like WC. That is why he had no hesitation in rejecting the infinitude of the past by applying this argument to the past time.

In addition to WC, Ibn Sīnā considers another condition for the applicability of the Mapping Argument, which concerns the notion of ORDER. He believes that the mapping technique does not apply to an infinity whose parts are not ordered either in nature or in position (see, e.g., the first sentence of T14). Unfortunately, it is not crystal clear what he means by order in nature or position. Thus, looking at the origins of the discussion of order among the parts of a quantity might be beneficial. Aristotle contends that ‘some [quantities] are composed of parts which have position in relation to one another, others are not composed of parts which have position’ (Categories 6, 4b21–25). To provide examples of the former type of quantities, he mentions line and plane because each part of them is situated somewhere. These parts can be distinguished by where they are situated. Regarding these parts, ‘one could say where each is situated and which join on to one another’ (Categories 6, 5a21–22). By contrast, one cannot say regarding the parts of things like number and time that they ‘have some position in relation to one another or are situated somewhere, nor see which of the parts join on to one another’ (Categories 6, 5a24–27). Aristotle argues that the ontology of number and time is so that their parts cannot have a position. For example, regarding time, he says: ‘None of the parts of a time endures, and how could what is not enduring have any position?’ (Categories 6, 5a27–29) It is clear that he is talking about the category of position, and by ‘a position’, he means something like ‘a particular arrangement of the internal parts of a totality in relation to each other’. Such a position obtains when every part of that totality is situated somewhere in space (and therefore has a place). The parts of line and plane – taken as properties of physical entities – have a position in this sense. But the parts of number and time (i.e., individual numbers and moments of time) are not situated in space. They do not have a position obtained by their relational situations in space. Nevertheless, they can be distinguished by the order they have. The parts of time ‘have a certain order in that one part of a time is before and another after. Similarly with a number also, in that one is counted before two and two before three; in this way, they may have a certain order, but you would certainly not find position’ (Categories 6, 5a29–33). Thus, what Aristotle means by ‘order’ seems to be simply a beforeness-afterness relation by which the parts of the ordered thing can be distinguished from each other. In sum, according to Aristotle, some quantities have position and others order. He does not claim that the two types are mutually exclusive. It seems that the order between the members of a quantity might or might not be dependent (at least to some extent) on their position (if any). For example, if time has an order, a line in space is (or, at least, can be) ordered as well. Since they are topologically similar, it suffices to determine a direction for that line to attribute an order to the points of it. For example, if we assume that a line AB is extending from A to B, then for every couple of points C and D on it, C is before D if and only if AC is shorter than AD. Thus, a line in space can be considered having both position and order. Other quantities, for example an interval of time or a multitude of numbers, have order but not position. A fortiori, the order of such quantities does not depend on a position.

Given this construal of Aristotle’s view, some orders are positional, others not. But if a quantity has a positional order in the sense above, it has already satisfied WC. This is because such an order supervenes on the mutual or relational arrangement of the parts of that quantity in space. If the parts of a quantity fail to exist altogether, a fortiori, they fail to exist altogether in space. Therefore, talking about their mutual arrangement in space does not make sense. They do not have a position, nor do they have, hence, a positional order.

Al-Fārābī (d. 950), in his Bezels of Wisdom (Reference al-Fārābī and ĀL Yāsīn1985, p. 61, ll. 6–y), contends that the existence of infinities in ‘created things which has a place (makāna) and an order (rutbata)’ is impossible.Footnote ⁵⁰ If a quantity has a positional order in the Aristotelian sense delineated above, it satisfies both of the conditions proposed by al-Fārābī and must be finite. However, it must be noted that he does not propose these conditions in connection to the Mapping Argument. To my knowledge, he has never discussed this particular argument in any of his works. So, it is unclear what his justification is for the claim that having position and order implies finitude.

Ibn Sīnā has scrutinised the notions of ORDER and POSITION in connection to quantities in The Categories of The Healing (Reference Sīnā, Qanawātī, al-Khuḍayrī, al-Ihwānī and Zāyid1959, chapter IV.1, pp. 127–30). He explicitly links these notions to the Mapping Argument by claiming (in several places, though with different wordings) that this argument does not apply to quantities that are not ordered either in nature (fi al-ṭabʿ) or in position (fi al-waḍʿ). The contrast between ‘order in nature’ and ‘order in position’ in this context can be spelt out in at least two different ways. According to one interpretation, if something has a positional order, in the sense that it has an order imposed by the position of its parts, it has an order in position. By contrast, if something has a natural order, in the sense that it has an order imposed by the nature or essence of its parts, it has an order in nature. This interpretation better fits the historical background narrated above. Nevertheless, it overlooks the possibility of having quantities that are neither positionally nor naturally ordered but still can be conventionally ordered. If the Arabic term ‘waḍʿ’ (i.e., position) is understood in the more general sense of convention and supposition instead of the restricted sense of the category of position, the contrast between ‘order in nature’ and ‘order in position’ can be interpreted as the contrast between a natural order and a conventional order. For two reasons, the latter interpretation is preferable to the former. First, instead of ‘waḍʿ’, Ibn Sīnā has sometimes used ‘farḍ’, which less equivocally means supposition and has no relevance to the category of position.Footnote ⁵¹ Second, as we will shortly see, the applicability of the mapping technique to a quantity depends on whether or not it can be ordered; it does not matter whether this order is natural, positional (in the categorical sense), or conventional.

It is crucial to point out that having an order seems to be necessary only for the applicability of the multitude version of the Mapping Argument. It does not seem to play any significant role in the magnitude version of this argument. In the multitude version, to compare two multitudes A and B (or, more precisely, multitude A and its proper submultitudes B), we need to put them in a one-to-one correspondence by pairing each member of A with one and only one member of B. To achieve this goal, there must be procedures by which we can select the members of these multitudes one after another for pairing with each other: the first member of A with the first member of B, the second member of A with the second member of B, and so on. The mere possibility of mapping one multitude upon another in this way establishes the possibility of their being ordered. If the members of a multitude cannot be ordered, they cannot be paired with the members of another multitude one by one. Accordingly, the mapping technique cannot be employed to compare such a multitude with another to see whether or not they are equal. This means that we can appeal to CN4 to establish the equality of two multitudes only if they can be ordered.

Quite differently, to examine the equality of two magnitudes by employing the mapping technique, we only need to superpose one of them upon another and check if any part of any of them remains uncovered by the other. Having an order does not seem to play any crucial role in mapping a magnitude upon another. We can, for example, examine the equality of two triangular planes in this way without considering any specific order for the points on these planes (Fig. 7).

Figure 7 The two triangles are equal because when one is mapped upon another, no part of one remains uncovered by the other.

Putting all of the above observations together, it seems that, according to Ibn Sīnā, the Mapping Argument cannot establish the impossibility of an infinite multitude S unless it satisfies not only WC but also what is called ‘the ordering condition’. A quantity S satisfies this condition if the following description is true of it:

Thus, according to Ibn Sīnā, any infinite magnitude that satisfies WC and any infinite multitude that satisfies both WC and OC are impossible.Footnote ⁵² As an example of a multitude whose infinitude cannot be rejected by the Mapping Argument because it does not satisfy OC, Ibn Sīnā has mentioned the multitude of angels and devils. He has explicitly conceded, in The Salvation (Reference Sīnā and Dānishpazhūh1985, chapter IV.2, p. 246, ll. 5–7), that the multitude of angels and devils ‘is susceptible to increase, but this susceptibility does not make the [the application of] the mapping [technique] permissible; for what has no order in either nature or position is not susceptible to [the use] of the mapping [technique]’. Apparently, Ibn Sīnā believes that although the multitude of angels and devils satisfy WC, it fails to satisfy OC. Therefore, we cannot deny the possibility of its infinitude by appealing to the Mapping Argument. It is worth noting that Ibn Sīnā does not simply say that the mapping argument cannot reject the infinitude of the multitude of angels and devils because the members of this multitude are immaterial and, therefore, do not have a position (in its categorical sense). One might interpret this as a sign that, unlike some later medieval thinkers, Ibn Sīnā does not take materiality or having a position as one of the applicability conditions of the Mapping Argument. A quantity S satisfies what I call ‘the materiality condition’ if the following description is true of it:

Ibn Sīnā does not explicitly mention MTC as a necessary condition for the applicability of the Mapping Argument. However, the assumption that Ibn Sīnā does not take MTC to be necessary for the applicability of the Mapping Argument jeopardises the overall consistency of his philosophy. If MTC is not necessary for the applicability of the Mapping Argument, then it must be accepted that no multitude of immaterial entities that satisfy WC and OC can be infinite. This means that there can be no multitude of souls that are ordered and coexist together. Now, if one can show that other elements of Ibn Sīnā’s philosophy imply the possibility of the existence of such a multitude, then it will be established that his philosophy suffers from internal inconsistency. We will soon see that al-Ghazālī has actually established that Ibn Sīnā’s philosophy implies the possibility of the existence of an infinite multitude of souls that satisfies both WC and OC. That is why I have elsewhere (Zarepour Reference Zarepour2020, section 4.3) suggested that to preserve the internal consistency of Ibn Sīnā’s philosophy, we have to interpret him as endorsing MTC.

The Mapping Argument and its applicability conditions were the subjects of stimulating debates in various traditions of medieval philosophy.Footnote ⁵³ Ibn Sīnā’s pupil, Bahmanyār Ibn al-Marzubān (d. 1066), in The Attainment (1996, pp. 557–58), proposes the same applicability conditions for the Mapping Argument as his teacher. Al-Ghazālī was also well familiar with the Mapping Argument. He has presented a version of this argument in The Aims of the Philosophers (Reference al-Ghazālī and Bījū2000a, pp. 97–98). And, it was probably through the translation of this book in the third quarter of the twelfth century that the core idea of this argument was transmitted to the Latin tradition.Footnote ⁵⁴ In the Fourth Discussion of The Incoherence of the Philosophers, al-Ghazālī refers to the issue of the orderability of the multitude of human souls. He thinks that if the world has no temporal beginning, then it is, in principle, possible to have an infinite multitude of human souls that satisfies both WC and OC. The multitude he introduces can have a conventional order, which is not due to either the nature of human souls or their position.

Suppose the world has no temporal beginning, and every day and night, one and only one soul comes into existence. This implies that the multitude of human souls who have come into existence so far is infinite. Moreover, all the members of this multitude coexist together at present. This is because, according to Ibn Sīnā, who is the target of al-Ghazālī’s criticism, souls never perish. After coming into existence, a human soul will continue to exist forever, even when the body associated with that soul dies and perishes. This indicates that the multitude in question satisfies WC. Finally, by appealing to the natural order of days and nights, we can put a conventional order on the members of the multitude in question. This means that OC can be satisfied as well. In this imaginary scenario, we have an infinite multitude of souls that satisfy both WC and OC. But this contradicts the Mapping Argument. Accordingly, if we insist not to reject the soundness of this argument, we have to reject either the eternity of the world or the immortality of the soul. In any case, at least one of the doctrines endorsed by Ibn Sīnā must be rejected. As it is well known, al-Ghazālī himself denies the eternity of the world but endorses the soul’s immortality.Footnote ⁵⁵ However, it must be recalled that Al-Ghazālī’s line of reasoning is compelling only if MTC is not taken (by Ibn Sīnā) to be needed for the soundness of the Mapping Argument. If, for example, the mereology of the immaterial entities is so different from the material ones that whole-part equality (when equality is understood in the sense of correspondence as mentioned in CN4) is not necessarily absurd in the immaterial realm, then the infinitude of the multitude of immaterial entities like souls cannot be rejected by the Mapping Argument, regardless of whether or not the multitude in question satisfies WC and OC.

Al-Shahrastānī has presented the Mapping Argument in several works. In his Book of Religious and Philosophical Sects (Reference al-Shahrastānī and Cureton1846, vol. 2, p. 403; see in particular ll. 9 and 19–20), he discusses Ibn Sīnā’s account of the Mapping Argument and explicitly mentions that, according to Ibn Sīnā, the ordering condition is necessary for the applicability of the mapping technique. Moreover, he points out that the faculty of estimation is in charge of carrying out the mapping process in the case of infinite quantities. That is exactly why he thinks having a position – in its categorical sense – or, equivalently, the fulfilment of MTC is not necessary for the applicability of the Mapping Argument. Al-Shahrastānī contends, in his Struggling with the Philosophers (Reference al-Shahrastānī, Mayer and Madelung2001, p. 109), that when we argue against the infinitude of a body (which, being a body, necessarily has a position) by the Mapping Argument, we apply the mapping technique to an imaginary line existing in the body we have hypothesised. However, ‘what you suppose regarding an imaginary line is possible to be supposed regarding an imaginary number [i.e., multitude]’. And, this is valid regardless of whether or not the members of the multitude in question have a position. T15 is immediately mentioned after this line of argument to support the claim that MTC is not necessary for the Mapping Argument. Although al-Shahrastānī does not explicitly mention this, the same line of reasoning demonstrates that even EEC and WC are not necessary for the applicability of the Mapping Argument. By our imagination, we can apply the Mapping Argument to infinities that are supposed to exist only in our mind and to the cross-temporal mereological sum of things that do not coexist. That is why al-Shahrastānī (Reference al-Shahrastānī, Mayer and Madelung2001, Struggling with the Philosophers, p. 107–08) thinks the Mapping Argument applies to time and refutes the eternity of the world. By contrast with Ibn Sīnā, al-Shahrastānī believes that this argument establishes the finitude of the chain of the causal ancestors of an existent, regardless of whether the causes in question are efficient or material. For Ibn Sīnā, in contrast to material causation, which is a diachronic relation, efficient causation is a synchronic relation (Zarepour Reference Zarepour2022b, section 3.4). Accordingly, although everything exists simultaneously with its efficient cause, the material cause of that thing is temporally prior to it. This means that the chain of the causal ancestors of an existent fulfils WC only if the causes in the chain are efficient rather than material. Thus, Ibn Sīnā thinks that the Mapping Argument establishes the finitude of the chain of efficient causes but not the chain of material causes.Footnote ⁵⁶ Rejecting the necessity of WC for the applicability of the Mapping Argument, al-Shahrastānī (Reference al-Shahrastānī, Mayer and Madelung2001, Struggling with the Philosophers, p. 107–08) insists that the Mapping Argument equally applies to both of those chains and establishes that neither can be infinite. Moreover, he seems inclined to accept that every quantity can be ordered and, therefore, OC is not necessary for the applicability of the Mapping Argument (Lammer Reference Lammer and Al Ghouz2018, section 3).

Abu-l-Barakāt al-Baghdādī (d. 1165), an eminent Jewish philosopher who converted to Islam later in his life, has mentioned the Mapping Argument in his famous The Considered. But he thinks that this argument does not work:

Suppose we employ the mapping technique to show that the infinite line AB, which starts from A and extends infinitely in the direction of B, is equal to its proper part CB, which begins from C and extends infinitely in the direction of B. To complete the mapping process, we must move line CB so that C coincides with A. Al-Baghdādī believes that since CB is infinite, such a movement is impossible. Therefore, the mapping process cannot be completed, and the argument fails. He argues that CB does not move unless all parts of it move together. This means that the motion must happen not only for C but also in the direction of B. However, he thinks that no motion can happen in the direction of B unless we assume B as a limit. But this means that CB has two limits and is finite. We cannot even imagine CB in motion unless we conceive it as a finite line. Al-Baghdādī’s reasoning here does not make much sense from our modern perspective. Nevertheless, his discussion introduced another condition that he and many other medieval thinkers found necessary for the applicability of the Mapping Argument. A quantity S satisfies this condition – which I call ‘the movability condition’ – if the following description is true of it:

Al-Baghdādī believes that to map CB upon AB we need to move all the parts of CB together. However, this cannot happen, even in our imagination, unless we assume that CB is finite. Thus, if CB is infinite, it does not satisfy MVC, or so he suggests. As a result, from the outset, the mapping technique is not applicable to infinite quantities.Footnote ⁵⁷

Shihāb al-Dīn al-Suhrawardī (d. 1191) presents another version of the Mapping Argument in his The Philosophy of Illumination (Reference al-Suhrawardī, Walbridge and Ziai1999, section I.3, p. 44), which goes as follows: consider a series of objects that satisfies both WC and OC. Remove a finite number of these objects from somewhere in the middle of the series and connect the remaining parts to each other (or, more precisely, consider the two remaining subseries in continuity with each other as a new ordered series). Now, employing the mapping technique, compare the size of the new series with the initial series. If the new series is equal to the initial one, then CN5 is violated. Thus, the new series must fall short of the initial one. However, since the new series is obtained by removing only a finite part of the initial series, this indicates that the initial series must also be finite. Thus, the existence of an infinite series is impossible.

As I stated earlier, al-Rāzī has discussed and defended the Mapping Argument in many places in his oeuvre. For example, al-Rāzī presents this argument in Eastern Investigation (Reference al-Rāzī and al-Baghdādī1990, vol. 1, pp. 306–07) and Commentary on Fountains of Wisdom (Reference al-Rāzī, Hejazi and Saqa1994, vol. 2, pp. 49–50). Most of the objections against these arguments raised and evaluated in these two books are similar. He contends that MVC is not necessary for the applicability of this argument because the process of comparison through the mapping technique can be made entirely in the mind and by merely appealing to the intellectual ranks (al-marātib al-ʿaqliyya) that can be assigned to the parts of the infinities one of which is supposed to be mapped upon the other.Footnote ⁵⁸ By contrast, he thinks that OC is necessary for the applicability of the Mapping Argument. His best defence of this view can be found in a discussion of the applicability of the Mapping Argument to the multitude of human souls.Footnote ⁵⁹ As I mentioned earlier, philosophers like Ibn Sīnā believe that human souls have temporal origination, but they never perish after coming into existence. This implies that the multitude of human souls who existed one hundred years ago (call it M_–100) is smaller than the multitude of souls who exist now (call it M₀). Now, one might think that by applying the mapping technique, we can establish the finitude of M₀ (and, consequently, M_–100). Al-Rāzī reports that the sages (al-ḥukamā) do not accept this because these multitudes have no order either in nature or in position. He then states that some theologians find this manoeuvre extremely weak because they think that regardless of whether or not these multitudes are ordered, M_–100 is smaller than M₀, and this suffices to prove that M_–100 is finite. Coupling this with the fact that since a hundred years ago only a finite number of souls have come into existence, we can conclude that M₀ is finite too. However, al-Rāzī himself does not find such an approach plausible. He says that ‘the criteria mentioned by the sages is indeed valid’. Consider the following syllogism:

1. (8) Everything that falls short of another thing is finite.

2. (9) A falls short of B.

Therefore:

1. (10) A is finite.

Al-Rāzī focuses on the meaning of ‘finitude’ in this syllogism and says:

If we take the meaning of ‘finitude’ to be ‘falling short of another thing’, then (8) would be a self-evident (indeed, tautologously true) proposition. However, al-Rāzī seems to think that (8) is not a self-evident proposition and must be demonstrated. Apparently, he thinks that to draw the finitude of A from that A falls short of B, A must have a part that corresponds to part of B that has a finite rank.Footnote ⁶⁰ And he thinks that this cannot happen unless A and B are ordered. So, he thinks that OC is a necessary condition for the applicability of the Mapping Argument. He also thinks that WC is necessary for the applicability of this argument. That is why he thinks this argument cannot establish the impossibility of the multitude of past events. They fail to exist simultaneously. Thus, their infinitude is compatible with the soundness of the Mapping Argument, whose aim, according to al-Rāzī, is to establish the impossibility of infinities that satisfy WC, among other conditions.Footnote ⁶¹

It is worth mentioning that al-Rāzī does not think that the mapping technique applies to every ordered infinity, even if all its parts coexist. In particular, he does not think that the Mapping Argument can establish the finitude of the multitude of all natural numbers. In response to the objection described in T16, al-Rāzī (Reference al-Rāzī, Hejazi and Saqa1994, Commentary on Fountains of Wisdom, vol. 2, pp. 57) says that the Mapping Argument does not apply to numbers because they do not exist in the extramental world. So, it seems that he considers EEC necessary for the applicability of numbers. Apparently, he thinks that numbers are mind-dependent existents. Therefore, failing to satisfy EEC, the multitude of all numbers can, in principle, be actually infinite. However, as a matter of fact, this multitude does not form an actual infinity even in our minds. Al-Rāzī (Reference al-Rāzī, Hejazi and Saqa1994, Commentary on Fountains of Wisdom, vol. 2, p. 54) argues that the multitude of all numbers – as a whole – neither exists in the extramental realm nor even in the mind. He explicitly mentions that the mind is unable to present an infinite number of things all together at the same time.Footnote ⁶² Put differently, he thinks that the multitude of natural numbers do not satisfy WC either in the extramental world or even in our mind.

It seems that according to al-Rāzī, all numbers can coexist in an infinite mind like God’s.Footnote ⁶³ This does not jeopardise the soundness of the Mapping Argument because it does not apply to any multitude that fails to satisfy EEC. This implies that infinite sets of mind-dependent objects are not impossible. Given that al-Rāzī considers mathematical objects to be mind-dependent, he does not seem to have any problem with infinitism in mathematics. Endorsing a radically different ontology of mathematics, Ibn Sīnā seems to be a finitist about mathematics. He believes that mathematical objects are properties of physical objects. Since there cannot be any infinite body, there cannot be any infinite magnitude. In particular, there cannot be any infinite geometrical line. Geometrical entities do not exist independently from physical objects. Similarly, since (a) the multitude of all physical objects in the world is finite and (b) numbers are properties of the multitudes of objects, the multitude of all numbers existing in the world is finite. However, this multitude is potentially infinite because the number of existing physical objects can, in principle, increase with new objects coming into existence. Accordingly, the multitude of all numbers existing in the world can, in principle, become bigger boundlessly. Nevertheless, this multitude is never actually infinite. It might be possible (if the mind’s capacity allows) to imagine a number greater than the number of all the physical objects existing in the world or a geometrical shape bigger than the world. Nevertheless, such imaginations do not bring those mathematical objects into existence. Ibn Sīnā is a finitist, not only about physics but also about mathematics.Footnote ⁶⁴

Returning to al-Rāzī’s discussion of the Mapping Argument, a couple of other objections to this argument that he anticipates and rebuts are worth mentioning. For example, in the fifth objection to this argument in his Commentary on Fountains of Wisdom, al-Rāzī says:

The relevance of this objection to the Mapping Argument is not quite clear. However, one possible explanation goes as follows: assume that m and n are two infinite numbers that are made by doubling, respectively, a thousand and two thousand repeatedly and infinitely. Moreover, assume that M and N are the multitudes of all numbers from 1 to, respectively, m and n. M and N are two ordered multitudes whose sizes can be compared by the mapping technique. The objector insists that m is smaller than n. Accordingly, M is a proper submultitude of N. Thus, if we compare the two multitudes by the mapping technique, M falls short of N, and some of the numbers in N remain unpaired. This means that M and, consequently, m must be finite. But this is absurd because the number resulting from doubling one thousand repeatedly and infinitely is not finite. To show that this objection is untenable al-Rāzī says:

The first two sentences of this passage seem to imply that a description like ‘the number resulting from doubling a thousand repeatedly and infinitely’ is an empty description. It is made by putting the notions of DOUBLING and INFINITY together, but it does not refer to anything existing either in the extramental world or even in the mind. In particular, it does not refer to any number. According to this reading, the objection fails apparently because it talks about infinite numbers, while it does not make sense to treat infinity as a number. Put differently, the objection fails because it relies on the assumption of the existence of infinite numbers. However, in the last two sentences of the passage, the emphasis of the response shifts from the implausibility of the latter assumption to the fact that the infinities in question fail to satisfy EEC. I take these sentences as stating that even if it is legitimate to take the infinities resulting from doubling a thousand and two thousand repeatedly and infinitely as infinite numbers associated with infinite multitudes, those numbers and multitudes do not exist in the extramental world. These infinite things, as mathematical objects, would be merely mental existents. In particular, the infinite multitudes in question do not satisfy EEC. Thus, the Mapping Argument does not apply to them.

Another objection, whose relevance to the Mapping Argument is again not straightforwardly clear, goes as follows:

It seems that the rational numbers and the sums involved in this scenario are assumed to be represented by magnitudes or multitudes. Indeed, it is hard to see how this objection could be relevant to the mapping argument without this assumption. The passage can be interpreted in two different ways, making two different mathematical points. According to the first interpretation, the ratios the objector mentions in the first sentence of the passage are 1/2, 1/3, 1/4, and so on, and the sums that are compared are the following ones:

${\text{S}}_{\text{2}}=1/2+1/3+1/4+\dots$

${\text{S}}_{\text{12}}=1/12+1/13+1/14+\dots$

It is mathematically provable that none of the above sums converge to a finite number. Both of them are infinite. On the other hand, S₁₂ is the result of the deduction of the first ten ratios of S₂ from it. Thus, S₂ > S₁₂. Now, if we represent S₂ and S₁₂ by two magnitudes, one shorter than the other, we can apply the Mapping Argument to them and conclude that those magnitudes must be finite. But this is unacceptable. Therefore, the Mapping Argument is unsound, or so the objector might conclude.

However, I am not sure if the mathematical fact that the above series are divergent was known to al-Rāzī.Footnote ⁶⁶ Thus, an interpretation of T22 that does not appeal to the divergence of S₂ and S₁₂ might be preferable. According to such an interpretation, the ratios the objector talks about in the first sentence of T22 are 2/2, 3/3, 4/4, and so on, and the compared sums are the following ones:

${\text{S}}_{\text{2}}=2/2+3/3+4/4+\dots =1+1+1+\dots \text{.}$

${\text{S}}_{\text{12}}=12/12+13/13+14/14+\dots =1+1+1+\dots \text{.}$

Both sums are infinite because they are made of an infinite number of successive additions of 1. However, the number of 1s in S₁₂ is less than in S₂. Accordingly, S₂ > S₁₂.Footnote ⁶⁷ Now, if we represent these sums by two magnitudes and compare them by the mapping technique, we must conclude that the shorter magnitude, which is represented by S₁₂, must be finite. But this conclusion is absurd. In this interpretation, S₂ and S₁₂ can also be taken as the number of the elements of two infinite multitudes, one of which is a submultiple of the other. For example, S₂ and S₁₂ can be taken as the number of the members of the multitudes of all natural numbers greater than 1 and 11, respectively. By applying the mapping technique to these multitudes, we can conclude that the multitude associated with S₁₂ must be infinite. But this is absurd.

To block this objection, al-Rāzī (Reference al-Rāzī, Hejazi and Saqa1994, Commentary on Fountains of Wisdom, vol. 2, p. 57) again highlights the role of EEC and insists that the argument developed in T22 is flawed because the rational numbers mentioned in this argument do not exist in the extramental world. This implies that the magnitudes and multitudes associated with S₂ and S₁₂ do not satisfy EEC. Accordingly, their finitude cannot be established by the Mapping Argument. To summarise al-Rāzī’s view regarding the Mapping Argument, he finds it plausible and considers OC, WC, and EEC necessary for its application.

The Mapping Argument was widely discussed by many scholars in the later Islamic philosophy and theology. Many medieval Muslim theologians (particularly from the Sunni tradition) who addressed this argument believed that it could establish the impossibility of any infinity that satisfies EEC regardless of whether or not MTC and WC are satisfied.Footnote ⁶⁸ Things with OC are a bit more complicated. OC is usually considered to be necessary for the application of the mapping technique to multitudes. It was widely accepted that the members of two multitudes could be paired one by one only if their members could be ordered. Nevertheless, like al-Shahrastānī, many Muslim theologians believed that any multitude could be ordered. Put differently, although they took OC as a necessary condition for the applicability of the Mapping Argument to multitudes, they believed this condition would be trivial because every multitude satisfies it. On the other hand, OC was usually not considered necessary for the applicability of the Mapping Argument to magnitudes. Accordingly, most post-Avicennian Muslim theologians believed that this argument rejects the infinitude of any magnitude or multitude whose parts exist in the extramental world, whether these parts coexist simultaneously or exist at different times. As a result, these scholars believed that the Mapping Argument establishes the finitude of the size and age of the world, the multitude of all human souls, and any chain of causally connected entities (in any sense of causation). Nevertheless, since many such theologians take mathematical objects to be merely mental or estimative (wahmī) objects, they do not take the Mapping Argument as jeopardising mathematical infinitism.Footnote ⁶⁹

To provide more specific examples, we can consider a discussion of the Mapping Argument by Jalāl al-Dīn al-Dawānī (d. 1502) in his A Treatise on the Proof of the Necessary (Reference al-Dawānī and Ghūsh2013, pp. 139–56). He explicitly mentions that although this argument applies to the multitudes of events and human souls, it does not apply to the multitude of all numbers because they do not have extramental existence (al-Dawānī, Reference al-Dawānī and Ghūsh2013, A Treatise on the Proof of the Necessary, pp. 140–41). Moreover, he denies the necessity of OC by the following line of argument:

To establish that OC is unnecessary, al-Dawānī puts forward a dilemma. When we are applying the mapping technique to an infinite multitude, one of the following things must be the case: either (a) it is required that we consider all the members of that multitude one by one and individually, or (b) it is not. If (a), then the mapping process can never be completed regardless of whether or not the multitude in question is ordered. This is because our finite mind cannot entertain an infinite number of things one by one and individually. On the other hand, if we can apply the mapping technique to an infinite multitude just by thinking of it as a coarse whole without considering its members one by one and individually, this means that it does not matter whether the multitude in question is ordered or not. Therefore, the mapping technique can apply even to non-ordered multitudes, or so al-Dawānī concludes. However, it seems that his reasoning is flawed. What can be concluded from the fact that we can apply the mapping technique to a multitude without considering all its members individually is, at best, that, in the application of the mapping technique, it does not matter what the exact order of the members of the multitude in question is. Put differently, it does not matter which member of the multitude comes first and which next. This means that in the application of the mapping technique to a multitude, it does not matter what the exact order of its members is. Nevertheless, this does not imply that in such an application it does not matter whether or not the multitude in question can be ordered. Thus, it seems that al-Dawānī’s argument fails to show that OC is not necessary for the applicability of the mapping technique.

A more careful – though still implausible – claim regarding whether considering the members of a multitude one by one and individually is necessary for the application of the mapping technique to an infinite multitude is made by an Ottoman scholar, Khojazāda (d. 1488). He argues that in applying this technique to a multitude, we must consider all its members totally differentiated from each other and one by one only if they do not have an order either in nature or in position (in its categorical sense). On the one hand, he thinks that OC is indeed necessary for the applicability of the Mapping Argument. On the other hand, we cannot impose an arbitrary order on a multitude that lacks natural and positional order without considering its members one by one and individually. As a result, since our finite mind is unable to consider an infinite number of things, we cannot impose an order to an infinity that lacks natural and positional order. That is exactly why he thinks that the mapping technique is inapplicable to the multitude of human souls. Since they do not have natural and positional order, the possibility of their infinitude cannot be rejected by the Mapping Argument.Footnote ⁷⁰

Khojazāda develops the above argument against al-Ghazālī’s claim that the Mapping Argument establishes the finitude of the multitude of human souls. However, it is noteworthy that if we assume, as al-Ghazālī’s hypothetical scenario in T17 suggests, that in each day and night, only a single human soul comes into existence, then the natural order of days and nights is automatically transmitted to human souls. In this sense, we can say that the multitude of human souls has an order that supervenes on the natural order of day and night. So, even by Khojazāda’s criteria, the mapping technique would be applicable to the multitude of human souls.Footnote ⁷¹

As I mentioned earlier, the Mapping Argument was transmitted into both Jewish and Christian traditions. From at least the third quarter of the twelfth century (when al-Ghazālī’s The Aims of the Philosophers was translated into Latin) onwards, many scholars writing in Latin discussed this argument. For example, the magnitude version of the Mapping Argument is presented by Alexander Neckam (also recorded as ‘Nequam’, d. 1217) in his On the Nature of Things (Reference Neckam and Wright1863, p. 303).Footnote ⁷² William of Auvergne also presents the magnitude version of the Mapping Argument in his The Universe of Creatures. In this argument, he compares the length of two infinite lines by dragging one of them and mapping it upon another. Recall that the idea of mapping by moving was previously discussed by scholars like al-Baghdādī and al-Rāzī. William presents the mapping argument as follows: consider two parallel infinite lines. One of them starts from point A and extends infinitely in the direction of B, while the distance between A and B is one cubit. The other line starts from C and extends infinitely in the direction of D, while B corresponds to C (Fig. 8a). In the first step of the argument, William states that the infinite line starting from B must be equal to the infinite line starting from C. Otherwise – that is if one of the two lines falls short of the other – the shorter line has an endpoint and would be finite. If the line starting from B is finite, then the line starting from A would be finite, too. However, this contradicts the initial assumption. For the same reason, the line starting from C cannot be finite. This means that the lines starting from B and C correspond to each other and, given CN4, are equal. But the line starting from B is a part of the line starting from A. Thus, given CN5, the latter line is greater than the former. Putting these observations together, we can conclude that the line starting from A is greater than the line starting from C. In the second step of the argument, William says that by dragging the line starting from C for one cubit leftward, we can map the line starting from A upon the line starting from C (Fig. 8b). By a similar argument as what we saw in the first step, we can show that the two lines are equal to each other (Fig. 8c). But this contradicts, the result of the first step. So, the initial assumption of the existence of infinite lines must be rejected.Footnote ⁷³

Figure 8 Mapping one line upon another by dragging and moving.

William takes this argument to be applicable to time. This means that he does not assume that anything like WC is necessary for the Mapping Argument’s applicability. Peter John Olivi (d. 1298) has mentioned a version of the Mapping Argument that looks like al-Kindī’s version (see Murdoch Reference Murdoch1968, Reference Murdoch1969). Thomas Bradwardine (d. 1349), in his De causa Dei, has offered a version of the Mapping Argument against the eternity of the world, according to which the eternity of the world is unacceptable because if the eternity of the world implies the correspondence of the multitude of the souls of all the Popes to the multitude of all the human souls, which is absurd, or so Bradwardine thinks (Sylla Reference Sylla, Shapiro and Hellman2021, section 7). Finally, to mention an example of this argument in the Jewish tradition, we can refer to a discussion of this argument by Ḥasdai Crescas (d. 1410/11) in his Light of the Lord (Reference Crescas2018, pp. 37 and 73–74). He does not find this argument plausible because he thinks that infinity is not measurable. Nor can we describe them as being greater or lesser than each other. Therefore, it seems that he does not take CN5 to be applicable to infinities.

The cases we analysed in this section do not include all medieval engagements with the Mapping Argument. Nevertheless, they can sufficiently clarify the structure and subtleties of this argument and provide a reliable sample reflecting the general attitudes of medieval philosophers toward it. The application of the Mapping Argument to an infinite quantity S seems to have two crucial steps: (a) to compare the size of S with the size of a part of it by employing the mapping technique and establishing the equality of S to that specific part of it by appealing to CN4, and (b) to draw an absurdity from the whole-part equality established in the previous step by appealing to CN5. Interestingly, some conditions introduced for the Mapping Argument’s applicability are related to the first step and the others to the second one. Specifically, OC and MVC seem to be associated with the first step. On the other hand, WC, EEC, and MTC seem to be related to the second step. For example, the defenders of the necessity of OC believe that if a multitude is not ordered, it cannot be in one-to-one correspondence with another multitude. Accordingly, we cannot compare the size of such multitudes, let alone establish their equality by appealing to CN4. In this sense, OC is entirely different from a condition like WC. The latter condition does not seem to have anything to do with the first step of the Mapping Argument. To show this with a specific example, consider the application of the Mapping Argument to the multitude of the past days. If the world is eternal, the multitude of all the past days until today can be put in one-to-one correspondence with the multitude of all the days before two days ago, even though these multitudes do not satisfy WC. So, it does not seem implausible to conclude that these two multitudes are equal in the sense of equality mentioned in CN5. Nevertheless, those philosophers who think that WC is necessary for the soundness of the Mapping Argument seem to believe that the equality of an infinity that does not satisfy WC with a part of itself does not have any absurd consequences.

Now, we are in a position to see how the Mapping Argument has contributed to making our conception of infinity more sophisticated. In particular, we can compare the conception of infinity hidden in the discussions of the Mapping Argument with what is presumed in the Equality Argument, on the one hand, and our modern concept of infinity, on the other. I take up this comparison in the Conclusion, to which I now turn.

5 Conclusion

Throughout this Element, we have explored two early episodes of the history of the notion of INFINITIES OF DIFFERENT SIZES by analysing various versions of two medieval arguments for finitism, which somehow employ this notion. In the Equality Argument, EI – namely that all infinities are equal to each other – is taken for granted. However, no effective criterion is introduced for the equality of infinities. The medieval discussions of this argument do not tell us how one can confirm that two infinities are indeed equal. It is an assumption that must be accepted with no justification. Interestingly, although no sufficient criterion for equality of infinities is introduced, a criterion for their inequality is introduced and employed. Medieval philosophers took CN5 as implying that, like any other whole, an infinite whole is greater and, therefore, unequal to its parts, even if those parts are themselves infinite.

Taking one step forward, in the Mapping Argument, the assumption of EI is replaced by a sufficient criterion for the equality of infinities. The advocates of the Mapping Argument took CN4 as implying that one infinity is equal to another if one of them can be mapped upon the other and neither exceeds the other. In the case of infinite multitudes, we saw that, in practice, this criterion was often taken as a one-to-one correspondence between the members of the two infinities in question. If two infinite multitudes can be put in a one-to-one correspondence, they are equal. Instead of claiming that all infinite multitudes are equal to each other (in terms of their size), the medieval friends of the Mapping Argument seem to be open to the idea that if we cannot put two infinite multitudes in a one-to-one correspondence, we are not justified to insist that they are equal. This does not mean that all such philosophers believed that infinities that cannot be put in a one-to-one correspondence are indeed of different sizes. Rather, they merely believed that the sizes of such infinities are not comparable to each other. From this perspective, the medieval discussions of the Mapping Argument opened up a way for considering the possibility of infinities of different sizes by introducing a criterion for the equality of infinities that, in principle, may or may not be satisfied. Nevertheless, there was still a long way to go from those ideas to our current conceptions of infinite sets of different sizes.

The next step in the history of the notion of INFINITIES OF DIFFERENT SIZES was taken by those who did not find the Mapping Argument compelling. Assume that A is an infinite submultitude of B such that they can be put in one-to-one correspondence. Now consider the following premises:

1. (11) Since A corresponds to B and CN4 holds, they are equal.

2. (12) Since B is a part of A and CN5 holds, A is greater than B.

From these premises, we should conclude that such infinite multitudes are impossible. However, those who did not find this argument compelling believed that the notion of BEING GREATER THAN mentioned in CN5 and (12) is not necessarily incompatible with the notion of EQUALITY mentioned in CN4 and (11). This motivated some medieval philosophers to argue that the notion of BEING GREATER THAN can be understood in different senses, some of which are perfectly compatible with the notion of EQUALITY in the sense of correspondence. In particular, they tried to show that there is a legitimate notion of BEING GREATER THAN that allows A to be greater than B while they still correspond to each other. The attempts to provide clear definitions of such notions should be considered the next stage of the historical evolution of the notion of INFINITIES OF DIFFERENT SIZES.Footnote ⁷⁴ Having a more limited focus, I could not discuss that stage in this Element.

There are still a couple of other points that seem to be worth highlighting before closing this Element. First, the multitude version of the Mapping Argument clearly shows that medieval philosophers were aware of the characteristic feature of infinite sets that is highlighted in Dedekind’s definition of infinity. As is put by Reck (Reference Reck2023), a ‘set of objects is infinite – “Dedekind-infinite”, as we now say – if it can be mapped one-to-one onto a proper subset of itself’. Medieval philosophers’ notion of MULTITUDE might not completely fit the modern notion of SET. Nevertheless, it is striking that many medieval philosophers (who found the multitude version of the Mapping Argument compelling) believed that every infinite multitude can be in one-to-one correspondence with some of its proper submultitude. To my knowledge, unlike Dedekind, no medieval philosopher has mentioned this characteristic in a definition of infinite multitudes. Indeed, the defenders of the Mapping Argument somehow thought that this characteristic is precisely what makes the instantiation of infinite multitudes impossible. Nevertheless, the mere fact that they knew that infinite multitudes correspond with some of their submultitudes shows that medieval philosophers, compared to, for example, Ancient Greek philosophers, held a conception of infinity that more closely resembles the modern understanding of this notion.

Second, another striking fact about the medieval conception of infinity is that some medieval philosophers have taken OC as necessary for the applicability of the mapping technique. This means that we cannot talk about whether or not multitudes correspond to each other unless they are ordered. Borrowing the terminology of the modern set theory, they apparently believed that ordinality is prior to cardinality. Some philosophers, like Ibn Sīnā, thought that not every multitude can be ordered, while others, like al-Shahrastānī, believed that everything can be ordered.

This Element aimed to explore early medieval encounters with the notion of INFINITIES OF DIFFERENT SIZES through the lens of two medieval arguments for finitism. Admittedly, many other important and interesting things about the medieval encounters with this notion are yet to be explored in other works. Nevertheless, I hope the present study has shed some light on at least a few dark corners of the tantalising history of infinities of different sizes and encourages other scholars to explore other corners I could not explore.