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Medieval Finitism

This document discusses two medieval arguments, the Equality Argument and the Mapping Argument, which were initially proposed to argue against the possibility of infinity but have significantly influenced the understanding of infinities of different sizes. The author, Mohammad Saleh Zarepour, highlights the historical and philosophical connections of these arguments within medieval Jewish, Christian, and Islamic traditions, focusing primarily on the Islamic perspective. The document aims to explore the evolution of the concept of infinity and its implications in the philosophy of mathematics.

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Medieval Finitism

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Zarepour

Discussing various versions of two medieval arguments for the

impossibility of infinity, this Element sheds light on early stages
of the evolution of the notion of INFINITIES OF DIFFERENT
SIZES. The first argument is called ‘the Equality Argument’ and
relies on the premise that all infinities are equal. The second
argument is called ‘the Mapping Argument’ and relies on the The Philosophy of
assumption that if one thing is mapped/ superposed upon
another thing and neither exceeds the other, the two things are
Mathematics
equal to each other. Although these arguments were initially
proposed in the context of discussions against the possibility
of infinities, they have played pivotal roles in the historical
evolution of the notion of INFINITIES OF DIFFERENT SIZES.

Medieval Finitism
Medieval Finitism

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Stewart Shapiro
The Ohio State
Mohammad Saleh Zarepour
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their own take on what they believe are
the most significant debates influencing
research, drawing original conclusions.

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Elements in the Philosophy of Mathematics
edited by
Penelope Rush
University of Tasmania
Stewart Shapiro
The Ohio State University

MEDIEVAL FINITISM

Mohammad Saleh Zarepour

The University of Manchester

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Elements in the Philosophy of Mathematics

DOI: 10.1017/9781009047623
First published online: November 2024

Mohammad Saleh Zarepour

The University of Manchester
Author for correspondence: Mohammad Saleh Zarepour,
mohammadsaleh.zarepour@manchester.ac.uk

Abstract: Discussing various versions of two medieval arguments for

the impossibility of infinity, this Element sheds light on early stages of
the evolution of the notion of INFINITIES OF DIFFERENT SIZES. The first
argument is called ‘the Equality Argument’ and relies on the premise
that all infinities are equal. The second argument is called ‘the Mapping
Argument’ and relies on the assumption that if one thing is mapped/
superposed upon another thing and neither exceeds the other, the two
things are equal to each other. Although these arguments were initially
proposed in the context of discussions against the possibility of
infinities, they have played pivotal roles in the historical evolution of the
notion of INFINITIES OF DIFFERENT SIZES.

Keywords: inﬁnity, medieval philosophy, ﬁnitism, the mapping argument, the

equality argument

© Mohammad Saleh Zarepour 2024

ISBNs: 9781009618434 (HB), 9781009048507 (PB), 9781009047623 (OC)
ISSNs: 2399-2883 (online), 2514-3808 (print)

1 Introduction 1

2 Deﬁnition and Characteristics of Inﬁnity 3

3 The Equality Argument 10

4 The Mapping Argument 32

5 Conclusion 59

List of Abbreviations 62

List of Ancient and Medieval Thinkers Discussed

in This Element 63

References 65

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 philoso-
phy. 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 impossi-
bility 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 philo-
sophical 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.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.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,

1
For two seminal studies focused on historical engagements with the idea of inﬁnities of different
sizes, see Davenport (1999) and Mancosu (2009).
2
Such arguments are extensively discussed in Mancosu (forthcoming).

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 evolu-
tion 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 perspec-
tive, 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 math-
ematical 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 phil-
osophy, 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 counter-
parts 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:

T1. Al-Rāzī (1990, Eastern Investigations, vol. 1, p. 297)

What is literally said [to be limitless] is said either in the way of simple negation
(al-salb) or in the way of metathetic afﬁrmation (al-ʿudūl). As for what is said in
the way of simple negation, it [i.e., to say that it is limitless in the way of simple
negation] is to take away from that thing the meaning because of which it is
correct to describe that thing as having a limit. And that [meaning] is quantity.
This is like what is said of God Most High that He is limitless and of the point that
it is limitless. As for what is [said to be limitless] in the way of metathetic
afﬁrmation, there is something because of which it is [in principle] correct to
describe that thing as having a realised limit, but no limit is [in fact] realised.3

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.4 In our discussion of infinity, we are concerned with things that are
3
Unless otherwise mentioned, all the translations from Arabic and Persian are mine. Accordingly,
when I cite a work that includes both an original text in Arabic or Persian and its English
translation, the page numbers refer to the Arabic or Persian part of the cited work.
4
The origin of the distinction between simple negation and metathetic affirmation is Aristotle’s De
Interpretatione 10. To see how this distinction is usually understood in the context of the

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).5 Thus, roughly speaking, infinity must be
considered an attribute of magnitudes and multitudes.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 uni-
versally 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 (2009, 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’.7 Instead of appealing to
a repetitive process of taking from infinity, Ibn Sīnā defines infinity by

Aristotelian logic, consider a sentence ‘a is not F’. If this sentence is understood as expressing
a simple negation, then it says that it is not the case that a is F. Thus, the sentence in question can
be true regardless of whether a exists and whether it is capable of having F or not-F as a property.
On the other hand, if that sentence is taken as expressing a metathetic affirmation, then it says that
it is the case that a is not-F. Given the existential import of the affirmative claims, this sentence is
true only if all the following conditions are satisfied: (a) a exists, (b) F and not-F are in principle
attributable to a (or, equivalently, a is in principle capable of being F or not-F), and (c) as a matter
of fact, a does have the property of not-F. Regarding the engagements of the philosophers of the
classical period of Islamic philosophy with this distinction, see Thom (2008), Hodges (2012), and
Kaukua (2020). The first paper addresses the account of al-Fārābī (d. 950), and the latter two focus
on the view of Ibn Sīnā, which was the primary source for the majority of discussions concerning
this distinction in the postclassical Islamic philosophy.
5
All the translations of Aristotle’s terms and phrases are borrowed from Aristotle (1984, The Complete
Works of Aristotle). In this specific translation, ‘language’ is taken to be the translation of the Greek
term ‘λόγος’. Other translators have selected ‘speech’ as the translation of this term. In any case, as it
is explicitly mentioned in Categories 6, 4b32, what Aristotle here means by ‘λόγος’ is the spoken
language, which is constituted of a series of sounds and can be ‘measured by long and short syllables’.
So, it is comprised of distinct units that can be counted. This might explain why language is
considered a discrete quantity. Nevertheless, many scholars believe that it is not really clear why
language must be included in discrete quantities. This unclarity is intensified by the fact that there is
no reference to language in Aristotle’s discussion of categories in Metaphysics V.13. 1020a7–32. On
Aristotle’s account of quantity, see, among others, Studtmann (2004).
6
Hereafter, for the sake of simplicity and unless otherwise specified, by a ‘magnitude’, I mean
a straight line that represents a one-dimensional magnitude (e.g., weight or distance).
Accordingly, by ‘the length of a magnitude’, I mean the length of the line that represents that
magnitude. By setting a convention and taking a magnitude of a certain finite length as our
measuring unit, we can represent numbers by magnitudes: number n can be represented by
a magnitude of the length of n units. However, the possibility of making such conventions does
not undermine the fact that magnitudes in themselves are continuous quantities. I will later clarify
how a ‘multitude’ must be understood.
7
See also Ibn Sīnā (2009, The Physics of The Healing, chapter III.7, § 2 and chapter III.9, § 1). For
Ibn Sīnā’s definition of infinity, see McGinnis (2010, section 4) and Zarepour (2020, section 2).

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 (1990, 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 touch-
ing 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).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, CDn=2CDn+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

8
To be accurate, this form of referring to an infinite line is misleading and incompatible with the
standards of modern mathematics. This is because it leaves the impression that ‘B’ – in the same
manner as ‘A’ – refers to a point. However, this should not be the case because otherwise ‘AB’
refers to a finite line segment that is bounded with A and B. Nevertheless, this is how infinite lines
are referred to in many medieval texts. See, for example, T14. Thus, I remain faithful to their
reference style, hoping that the contexts of the following discussions of infinite lines will spare the
readers from potential misunderstanding caused by this style. In visualisation, the bounded side of
a line is represented by a bullet point and the infinitely extending side of it by an arrow point.

(a) A B

d d

(b) C D3 D2 D1 D

Figure 1 Inﬁnity by addition and inﬁnity by division.

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.9
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.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

9
However, it must be noted that if something is infinite by division, the number of the divisions
that can be made in that thing is infinite by addition. So, the notions of INFINITY BY DIVISION
and INFINITY BY ADDITION, though distinct, are related to each other. The relation between
these two conceptions of infinity is clearly visible in passages like T2 and T3.
10
Aristotle’s conception of infinity is studied, among others, by Hintikka (1966), Lear (1980),
Kouremenos (1995), Bowin (2007), Coope (2012), Nawar (2015), and Cooper (2016).

infinite totalities that are taken as actual infinities by some philosophers and as
potential infinities by others.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:

Equality of Infinities (EI): All comparable infinities are equal to each other. No
infinity is greater or lesser than another.

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 geom-
etry. 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 (2014, fr. 132, p. 144), it is ‘impossible that hanythingi should be
greater than the infinite, or that the infinite should be increased’.12 Also, EI might

11
An important medieval distinction that I do not touch on in this Element is the distinction
between the categorematic and syncategorematic senses of infinity, which is closely related to
the distinction between actual and potential infinities. On the origin of the distinction between
categorematic and syncategorematic infinities and its role in the medieval Latin discussions of
the theories of infinity and continuity, see, among others, Geach (1967), Kretzmann (1982),
Murdoch (1982, pp. 567–68), Duhem (1985, chapter 1), Uckelman (2015), and Moore (2019,
section 3.3).
12
The angle brackets are by the translator. The original text of Against Aristotle is lost.
Nevertheless, a large part of this treatise is now reconstructed based on the fragments quoted
in Greek, Arabic, and, in one case, Syriac sources. The most reliable fragments are those quoted
by Simplicius (d. 560), who had access to the original treatise, in his commentaries on Aristotle’s
Physics and On the Heavens. Fortunately, Simplicius’s quotes form the largest portion of the
reconstructed treatise.

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 back-
ground of remarkable ancient and medieval discussions of infinity, states that
infinity must be treated as a number:

Numericality of Inﬁnite Multitudes (NIM): For every multitude X, the

following claims are equivalent to each other: (a) X is an infinite multitude;
(b) the number of the members of X is infinite; and (c) there is an infinite number
describing how many members X has.

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.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

13
Apparently, some medieval philosophers were aware that a group of objects can be considered in
two different ways, one distributively and another collectively (something like the mereological
sum of certain elements). For example, such a distinction can be detected in Ibn Sīnā’s famous
Proof of the Sincere for the existence of God. See Świętorzecka et al. (forthcoming). The
difference between distributive and collective considerations of a multitude corresponds to the
Russellian distinction between class as one and class as many (if we interpret multitudes as
classes). See Russell (2010 [1903], section 70).

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 multi-
tudes 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.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 philo-
sophers 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

14
On deﬁning inﬁnite numbers with or without employing abstraction principles, see Mancosu
(2016).

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 investiga-
tions 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:

T2. Lucretius (2001, On the Nature of Things, Book I, 615–30, p. 58)

[I]f there is no smallest point, every minutest body will be composed of

an infinite number of parts (parvissima quaeque corpora constabunt ex
partibus infinitis), since a half of a half will always have a half and there
will be no limit to the possibility of division. If this is the case, what
will distinguish the whole universe from the smallest thing in it?
Nothing; for, no matter how fully infinite is the whole universe, the
minutest objects will equally be composed of an infinite number of parts
(parvissima quae sunt, ex infinitis constabunt partibus aeque). But since
sound judgment loudly protests against this conclusion and denies that
the mind can believe it, you must admit defeat and acknowledge the
existence of points that have no parts and are the smallest things; and
this being so, you must also acknowledge the existence of solid and
everlasting primary elements.

The most plausible reconstruction of the argument presented in T2 seems to be

something like the following:

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

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

Therefore:

(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:

T3. Plutarch (1976, Against the Stoics on Common Conceptions, 1079a, p. 813)

For it would not be possible to conceive one magnitude as greater or less

than another if for the parts of both alike it is characteristic to proceed to
infinity; but the nature of inequality is abolished, for, when things are
conceived as unequal, it is by the ultimate parts that the one leaves off
before the other and the other passes it by and is in excess of it. And, if
inequality does not exist, it follows that unevenness does not exist or
roughness of body either, for unevenness is inequality of a single surface
with itself and roughness is unevenness along with hardness, none of which
is left by those who bring no body to an end in an ultimate part but in number
of parts extend all bodies to infinity. Yet is it not completely clear that a man
consists of more parts than the man’s finger does and the universe again of
more parts than does the man? This all men know and have in mind if they
have not become Stoics; but, once they have become Stoics, their state-
ments and opinions are to the contrary effect that the man is not made up of
more parts than the finger is or the universe of more parts than the man, for
by division bodies are triturated to infinity and among infinites none is more
or less and none exceeds another in multitude at all or else the parts of the
one exceeded would stop being divided and making multitudes of
themselves.

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.15 Since EI
rules out the possibility of such inﬁnities, we have to reject the eternity of the

15
Throughout this Element, ‘eternity’ and ‘eternal’ must be taken as meaning, respectively,
temporal beginninglessness and temporally beginningless.

world, or so Philoponus contends. An argument developed based on this general

pattern goes as follows:

T4. Philoponus (2004, Against Proclus On the Eternity of the World, chapter
1, section 3, p. 24–25)

And if the world had no beginning and the number of men living before, say,
Socrates was infinite, and those living from Socrates until the present time have
been added to it, there will be something greater than the infinite, which is
impossible. And if the number of men that have lived is infinite, the number of
horses that have lived is certainly also infinite. [So] you will be doubling the
infinite again. [And] if you add to these the number of dogs as well, you will
triple the infinite, and if each of the other [species] is added it will be multiplied
many times over. And this is among the greatest of impossibilities, for it is not
possible to be greater than the infinite, not to mention many times greater.

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.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 multiplica-
tions 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
16
In Physics IV.11, 219b4–6, Aristotle distinguishes two senses of arithmos: what is counted (i.e.,
multitude or collection) and what by which we count (i.e., number). I am thankful to a reviewer
who drew my attention to Aristotle’s passage and raised the abovementioned objection.

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.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 (1908, vol. 1, p. 155). This common notion, which rejects
the possibility of whole-part equality, goes as follows:

Common Notion 5 (CN5): The whole is greater than the part.

Here (and unless otherwise specified in what follows), ‘part’ should be under-
stood 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 2007,
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

17
The arguments presented in T4 are based on the claim that the infinitude of the past implies the
beginninglessness of the generation of any species and the infinitude of the multitude of the
particulars of any species that have come into existence until any specific time. However, those
arguments can also be reconstructed based on the weaker claim that the possibility of the
infinitude of the past implies (a) the possibility of the beginninglessness of the generation of
such species and, consequently, (b) the possibility of the infinitude of those multitudes. To do
this, it suffices to appeal to a modal reading of EI, stating that it is impossible for infinities to be of
different sizes. On this interpretation, EI rejects not only the existence of infinities of different
sizes but also the possibility of the existence of such infinities. Accordingly, EI is incompatible
with even the possibility of the infinitude of the past, regardless of whether or not, as a matter of
fact, the infinitude of the past implies the existence of infinities of different sizes.

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, conse-
quently, 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.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.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

18
It seems that the claims (a) the minutest objects and the whole universe are not composed of an
equal number of parts (i.e., a premise of Lucretius’s argument in T2), and (b) the universe has
more parts than the man, and the man more than the man’s finger (i.e., a premise of Plutarch’s
argument in T3) must similarly be justified by appealing to CN5.
19
Does it make sense to talk about infinite wholes that do not have infinite parts? If it does not, the
Equality Argument aims to establish the impossibility of infinite wholes by appealing to CN5
and EI. However, following a conservative approach, I prefer not to take a position regarding the
answer to the aforementioned question. Therefore, I take the target of the attack of the Equality
Argument to be those infinite wholes some of whose parts are also infinite. Such a domain of
wholes might or might not include all infinite wholes, depending on the correct answer to that
question.

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 (2004, 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’.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:

T5. Philoponus (2014, Against Aristotle, fr. 132, p. 146)

[I]f the motion of the heavens is without a beginning, htheni it is necessary

that the sphere of Saturn has rotated with an infinite hnumber ofi revolutions,
but the hspherei of Jupiter with nearly three times more hrevolutionsi than
that. The hrevolutionsi of the sun will be thirty times greater hin numberi than
the ones of Saturn, the hrevolutionsi of the moon 360 times, and the
hrevolutionsi of the sphere of the fixed stars more than ten thousand times
greater. But how, if it is not heveni possible to traverse the infinite once, is it
not beyond all absurdity to assume ten thousand times the infinite, or rather
the infinite an infinite number of times? In consequence, it is necessary [ . . . ]
that the circular motion of the heavens did not exist before hbuti had
a beginning of existence.21

The argument explicitly mentioned in T5 is based on the idea of the untraver-

sability of inﬁnity. However, T5 can also inspire another argument against the
eternity of the world, which is based on NIM and EI.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,

20
The square brackets are by the translator.
21
The square brackets of this quote are mine. The angle brackets are by the translator.
22
In fact, I mentioned T5 just to discuss the argument that is based on EI, rather than the one that is
based on the untraversability of infinity. Medieval philosophers have developed a large number
of finitist arguments from untraversability. Nevertheless, since those arguments are rarely
concerned with the problem of infinities of different sizes, I have not touched on them in this
Element.

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.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.24 Although al-Naẓẓām accepts the infinite divisi-
bility of magnitudes, he denies that time and space are infinite (by addition).25
As reported by al-Khayyāt (d. circa 910) in The Book of Victory (1957, 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.26 According to al-Khayyāt’s report, al-Naẓẓām has

23
For the influence of Philoponus’s finitist arguments on the medieval philosophers, see, among
others, Davidson (1969, 1987, chapter IV), Pines (1972), Teske (1995), and Kohler (2006).
24
See van Ess (2017, p. 46, n. 40).
25
Some scholars have interpreted al-Naẓẓām’s view as implying that each body is actually
constituted of infinitely many parts. See Pines (1997, p. 14, n. 37).
26
In a slightly different version of this argument, it is mentioned that if atomism is false, the parts of
a mustard seed (whose number is infinite) can cover the whole earth. For various versions of
these arguments in the works of various Muslim thinkers, see, among others, Pines (1997,
pp. 15–17) and Dhanani (1994, pp. 15–17 and 2015).

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:

T6. Al-Khayyāt (1957, The Book of Victory, chapter 20, p. 34)

But as for his statement regarding the difference between the parts of
a mountain and a mustard seed, Ibrāhīm [al-Naẓẓām] claims that if the
mountain is halved into two halves and the mustard seed is halved into two
halves, then the halves of the mountain are larger than the halves of the
mustard seed. Similarly, if they are divided into quarters, fifths, or sixths, then
the quarters, fifths, and sixths of the mountain are larger than those of the
mustard seed. Likewise, every part of the mountain, whenever they are
divided in this way, is larger than every [corresponding] part of the mustard
seed, and all their parts are finite in extent and dimension.27

The Equality Argument is also mentioned in A Treatise on Inﬁnity (1988) by

Yaḥyā Ibn ʿAdī (d. 974):

T7. Ibn ʿAdī (1988, A Treatise on Inﬁnity, pp. 138–39)

The falsehood of the belief of who believes that there is an infinite thing greater
than another infinite thing is clear . . . [Nevertheless,] some people have
thought – based on their belief that the number of days is infinite and the number
of years is also infinite and it is obvious that the number of days is greater than
the number of years because every year is three hundred and sixty-five days and
a quarter [and is] nothing other than a fraction [of the number of days] – that they
have found an infinite number that is greater than another infinite number. Also,
since the individuals of any species are infinite in number, they have thought
that it is possible to add to their number an infinity, which is the number of
individuals of another species. So, the number of individuals of two species,
while it is infinite, becomes greater than the number of individuals of a single
species, which is also infinite. But it had become clear that there is no infinity in
the number. And when infinity is not in it, there is no number that is infinite.

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.,

27
ʿAbd al-Qāhir al-Baghdādī (d. 1037), in his The Principles of Religion (1981, p. 36), has
criticised al-Naẓẓām’s response to this argument.

multitude, collection, or plurality). Referring to Aristotle’s Physics III.5,

204b6–7, Ibn ʿAdī (1988, 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 individ-
uals 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.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.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:

28
It is worth noting that al-Rāzī’s view in T1 is somewhat anticipated by Ibn ʿAdī. He argues that
when we say that something is not white, our statement does not necessarily mean that that thing
has a colour other than white. Our statement is true even if that thing does not have any colour at
all. In the same manner, the statement that the number of something, say S, is not finite does not
necessarily mean that S has a number that is infinite. Even when S has no number at all – for
example, when S is not of the category of quantity – it is still true that the number of S is not
finite. In the latter case, S is not finite without having an infinite number. See Ibn ʿAdī (1988,
A Treatise on Infinity, p. 138). A very similar position is attributed to Moses Maimonides (d.
1024) by Ḥasdai Crescas (d. 1410/11). See Crescas (2018, Light of the Lord, Book I, Part I,
chapter I, class I, p. 32).
29
For the references to the various manuscripts of this treatise, see Gutas (2014, p. 445–46).
A summary of the content of this treatise is reported by Pines (1972, Appendix).

Wholeness Condition (WC): All the parts of S exist together.

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 absurd-
ity 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:

T8. Al-Ghazālī (2000b, The Incoherence of the Philosophers, pp. 18–19)

If one were to say: ‘Is the number of these revolutions even or odd, both even
and odd, or neither even nor odd?’, [what would you say?] If you were to say
‘both even and odd or neither even nor odd’, its falsity is necessarily known. If
you were to say ‘even’, then the even becomes odd by [the addition of] a unit.
How does the inﬁnite lack a unit? And if you were to say ‘odd’, then the odd
becomes even by [the addition of] a unit. How does that [inﬁnity] lack the unit
by the addition of which it becomes even? Thus, you have to say that it is neither
even nor odd.

If it is said that ‘it is only the ﬁnite that is described as even or odd and the
inﬁnite is not described as such’, we say:

[You claim that there is] a collection composed of units such that it has a sixth and
tenth – as previously mentioned – but it is not described as even or odd. The
falsity of this is necessarily known with no reﬂection. With what [justiﬁcation] do
you disassociate yourself from this? If it is said that ‘where your statement goes
wrong is [where you claim] that “the collection is composed of units. However,
these revolutions are non-existent. As for the past, it has ceased to exist, and as for
the future, it does not [yet] exist. But the collection [must] refer to present
existents. However, there is no existent here [in the collection]”’, we then say:

Number is divided into the even and the odd, and it is impossible for it to lie
outside this [division], regardless of whether the numbered remains or
perishes. If we assume a number of horses, we must believe that it is either
even or odd, regardless of whether we have supposed them to be existent or

non-existent. Even if they cease to exist after existing, this matter will not
change. We also have to say to them:

According to your principle, it is not impossible to have present existents that

are units varying in description while they are inﬁnite. And [an example of]
that is the human souls that are separated from bodies with death. There are
existents that cannot be described as even and odd . . . This view regarding the
souls is what Ibn Sīnā endorses, and it is perhaps Aristotle’s doctrine.

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.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

30
This argument is also presented in al-Ghazālī’s Moderation in Belief (2013, pp. 37–38).

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.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 argu-
ments for the finitude of the past, al-Shahrastānī identifies some of the primary

31
This can provide another piece of evidence that he endorses NIM. In the fourth paragraph of T8,
al-Ghazālī clearly distinguishes number from numbered and argues that we can talk about the
oddness and evenness of a number, regardless of whether or not the multitude of objects it has
numbered exist.

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:

T9. Al-Shahrastānī (2001, Struggling with the Philosophers, p. 105)

[T]he middle term in it [i.e., an argument for ﬁnitism] is primary matters,

amongst which is that the lesser in existing numbers is no equal to the greater;
and amongst which is that the lesser and the greater are only [found] in the
finite number, and the lesser and the greater are inconceivable to be [found] in
what is infinite; and amongst which is that a determinate part – such as the
half, the third, or the quarter – cannot be realised in the infinite.

He then shows how these principles can be employed to develop variations of

the Equality Argument, one of which goes as follows:

T10. Al-Shahrastānī (2001, Struggling with the Philosophers, pp. 105–06)

If inﬁnitely many human souls were to exist on Sunday, then it would not be
possible [for them] to increase by a number of souls on Monday. This is
because what is inﬁnite in number cannot increase by a number. But it has
increased. Thus, the repetition of the negation of the consequent implies the
negation of the antecedent.

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.32

32
The original Arabic title of al-Shahrastānī’s book has been read in two different ways: Nihāyat
al-aqdām fī ʿilm al-kalām (The End of Steps in Theology) and Nihāyat al-iqdām fī ʿilm al-kalām
(The Final Venture in Theology). But as it is explained by Monnot (1996, p. 215) it seems that the

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:

T11. Ibn Rushd (1998, The Incoherence of The Incoherence, pp. 124–25)

If you imagine two circular movements within the two limits of the same
[ﬁnite period of] time and consider a restricted part of each within the two
limits of the same [ﬁnite period of] time, then the proportion of one part to the
other is the same as the proportion of one whole to the other. For example,
since the revolution of Saturn in a period of time that we call a year is
a thirtieth of the revolution of the sun in that period, if we consider the totality
of the revolutions of the sun and the totality of the revolutions of Saturn that
have happened in the same period, then the proportion of the totality of the
revolutions of the former motion to the totality of the revolutions of the latter
motion must be the same as the proportion between their parts.

If, however, there is no proportion between the two total movements because
each of them is potential – i.e., they have neither a beginning nor an end – while
there is a proportion between the parts [i.e., one part of each totality] because
every one of them is actual, then the proportion between one whole to the other
is not necessarily the same as the proportion of one part to the other, as it is put
by the group of philosophers in their reasoning. This is because there is no
proportion between two magnitudes or multitudes each of which is assumed to
be inﬁnite. When the ancient assumed, for example, that the totality of the
motion of the sun has no beginning or end and that the same is true of Saturn,
there could be no proportion between them at all. This [is because it] (i.e., the
existence of a proportion between them) entails that the two totalities are ﬁnite,
as it is entailed regarding two parts of the totality.33 And this is self-evident.34

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

former reading is correct. The aforementioned version of the argument from the number of
revolutions is mentioned in al-Shahrastānī (1934, p. 29). Al-Shahrastānī’s view on the inﬁnitude
of the past is discussed, among others, by Mayer (2012) and Lammer (2018).
33
Given the context, the last word of this sentence should be ‘totalities’ rather than ‘totality’.
34
Another English translation can be found in Ibn Rushd, Averroes’ Tahfut al-Tahafut (1987, pp. 9–10).

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).35
Ibn Bājja (1991, 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:

T12. Ibn Bājja (1991, Commentary on Aristotle’s Physics, Book III, p. 40)

Mathematical sciences do not employ inﬁnity, except when an extending

magnitude is hypothesised [and] it is [assumed that it is] possible to take
a [magnitude] larger than it. They [i.e., mathematical sciences] do not need

A C B

Figure 2 The existence of inﬁnite lines can be rejected by the conjunction of

CN5 and EI.

35
Many other scholars have mentioned this geometrical argument in their works. Notably,
a discussion of it can be found in al-Shahrastānī’s The End of Steps in Theology (1934, p. 13).

the existence of the inﬁnite in actuality and perfection; they only need it in
potentiality and possibility.

Ibn Bājja contends that actual infinities are not needed even for purely mathem-
atical 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.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 under-
stood 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:

Extramental Existence Condition (EEC): All the parts of S exist

extramentally (or mind-independently).

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 (1967, 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

36
A criticism of this view is offered by Gersonides (d. 1344) in his discussion of Ibn Rushd’s
mathematical finitism. See Kohler (2006, section III). Endorsing this sort of mathematical finitism
was a prevalent position among those medieval philosophers who considered mathematical objects
as properties of existing physical objects. To give an example from the Latin tradition, Thomas
Aquinas (d. 1274) – who defends such a realist ontology of mathematics – argues, in his Summa
Theologiae (2006, Ia. 7, 3, p. 103), that ‘Geometers need not postulate lines which are actually
infinite, but lines from which they can cut off whatever length they require, and such are the lines
they call infinite’. For a recent work on Aquinas’s ontology of mathematics, see Rioux (2023).

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 trad-
ition. 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.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 1998, The
Universe of Creatures, p. 137). William of Auvergne also presents another
argument that is exclusively about the definite proportions between the revolu-
tions 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 1998, 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 1.38 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:

37
On the sources of William of Auvergne for the Equality Argument, see Mancosu (forthcoming,
section 3.1).
38
The fact that the aforementioned proportion is not the ratio of 1 to 1 can be concluded from CN5.
The multitude of the revolutions of each celestial body is a proper part of the multitude of all the
revolutions happening in the heaven. CN5 implies that the latter multitude is greater than the
former. Accordingly, the ratio between their sizes is not 1 to 1.

T13. William of Auvergne (1998, The Universe of Creatures, pp. 137)

But if he [i.e., Aristotle] says that the heaven completed an infinite number of
revolutions in the whole of past time, I shall imagine – for this imagining is
possible for the intellect – that the heaven moved in the whole of past time by
a velocity that was less by half. Because, then, the proportion of the one
motion to the other is the same as the proportion of amount traversed to the
other amount traversed, it is necessarily the case that in the same time it
completed only a half of the number of revolutions. The revolutions, then,
which it completed in the whole of past time, necessarily have a half, and in
the same way they have a quarter and an eighth and so on to infinity. But it is
evident that the infinite does not have a half.

Moreover, when the opposite is imagined, namely, that the motion of the whole
heaven in the whole time that has passed is twice as fast, then, for the same
reason the revolutions completed in the same time will be twice as many. But
the heaven will not complete more revolutions than those which have passed
and those which are in the future taken together, but it completed that number
by having revolved at twice the speed; hence, it has completed its motion.

Moreover, there is no doubt that it would have completed twice as many

revolutions if its velocity was doubled, and would have done so in the same
time. Hence, the number of revolutions already completed had a double, and
it is the half of some number. Such a number, therefore, is not inﬁnite, since it
has a half, as was said, and also a double.39

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 conceiv-
able 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.40

39
The square brackets are mine.
40
For more details about William of Auvergne’s arguments for that the universe has a beginning,
see Teske (1990, 1995, and 2000).

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
(1882, Commentary on the Sentences, Book II, Distinction 1, Part 1, Article 1,
Question 2). Bonaventure (1964, 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 (1981, p. 52) believes that it was probably the inclusion of this argu-
ment 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 1982, pp. 569–70).41 According to
Bonaventure (1964, 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 propos-
ition 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 prin-
ciples mentioned in the previous versions we saw. Roger Bacon, in his Opus
Tertium (2012, 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).

D B A C

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

41
In a brief note, Brown (1965) has interpreted Bonaventure’s argument as stating that if the past
was infinite, then the infinite set of the revolutions of the moon could be put in a one-to-one
correspondence with a proper subset of it. However, I think this interpretation is not tenable. The
discussion of one-to-one correspondence appears in a different type of medieval arguments for
finitism that will be discussed in the next section.

Now, consider the following line of argument:

(1) AD = AC. This is because they are both inﬁnite and EI is true.
(2) BC = BD. This is because they are both inﬁnite and EI is true.
(3) BC > AC. This is because AC is a part of BC and CN5 is true.
(4) BC > AD. This is true because of (1) and (3).
(5) BD > AD. This is true because of (2) and (4)
(6) BD < AD. This is because BD is a part of AD and CN5 is true.
(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 magni-
tudes infinitely extended in both directions. But the above proof reveals that the
existence of such magnitudes is absurd.42
Versions of the Equality Argument have been criticised not only by philo-
sophers 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.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
2004, 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

42
For a reconstruction of this argument that is more faithful to Bacon’s original presentation of it,
see Mancosu (forthcoming, section 4.1.1). In the same chapter, Mancosu discusses arguments of
the same spirit by John Peckham (d. 1292) and Peter John Olivi (d. 1298).
43
See, among others, Sylla (2021, p. 63).

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 discus-
sions regarding this argument. The advocates of this argument take EI for
granted without introducing any sufficient criterion for the equality of infin-
ities. 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 (1908,
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 ἐπ̓ ἀλλήλα ἴσα ἀλλήλοις ἐστίν’). The term
‘ἐφαρμόζοντα’ comes from the Greek verb ‘ἐφαρμόζειν’. Although the trans-
lation 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 correspond-
ences 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’.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:

Common Notion 4 (CN4): If one thing is mapped/superposed upon another

thing and neither exceeds the other, the two things are equal to each other.

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.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.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

44
The Isḥāq-Thābit translation is the second Arabic translation of The Elements. This translation
was made by Isḥāq Ibn Ḥunayn (d. circa 910) and revised by Thābit Ibn Qurra al-Ḥarrānī
(d. 901). A copy of this translation can be found in the codex 6581/1 of the Majlis Library in
Tehran. The fourth common notion is mentioned in folio 2a, ll. 4–5. The first Arabic translation
of The Element was made by al-Ḥajjāj Ibn Yūsuf Ibn Maṭar (d. 833) and was later revised by
himself. Neither of the two versions of al-Ḥajjāj’s translation is extant today. His translation was
the source of the first Latin translation of The Elements by Adelard of Bath (d. 1152?). The
formulation of the fourth Euclidean common notion in Adelard’s translation is almost the same
as in the Isḥāq-Thābit translation. Adelard has translated ‘ἐφαρμόζειν’ as ‘superponere’. See
Busard (1983, p. 33, ll. 80–81).
45
On the pivotal role of the notion of SUPERPOSITION in the medieval Latin theories of
infinitesimals and continuity, see Murdoch (1964).
46
This argument has been presented in al-Kindī’s On First Philosophy (2012, pp. 20–21) and three
short treatises on the finitude of the past (2012, pp. 60–73). Al-Kindī’s account of infinity has
been discussed, among others, by Rescher and Khatchadourian (1965), Shamsi (1975), Adamson
(2007, chapter 4), and Abdel Meguid (2018).

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 (2012, 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 pre-
sented this argument with slight differences in several works.47 The version
offered in The Physics of The Healing goes as follows:

T14. Ibn Sīnā (2009, The Physics of The Healing, chapter III.8, § 1)

It is impossible that a magnitude (miqdār), number, or [multitude of] numbered

things (maʿdūdāt) having an order (tartīb) either in nature (al-ṯabʿ) or in position
(al-waḍʿ) is instantiated as an actual existent with no end [i.e., as an infinite actual
existent]. That is because for every infinite magnitude and every infinite [multi-
tude of] numbered things possessing an order, their extension towards an actual
infinity is either in [all of] their directions or in a single direction. If [their
infinitude] is in all of their directions, then let us posit a limit within it – such as
a point in a line, or a line in a surface, or a surface in a body, or a unit in a numeric
totality – and make it a limit. So, we will talk about it inasmuch as we define it as
a limit. [Call this limiting point A.] Now, we take a finite part – say AC – from
AB, which is infinite in the direction of B. So, if [something] equal to CB were
mapped upon or laid parallel to AB or some [other appropriate] relation between
them is considered, then either it [i.e., CB] will extend infinitely in the same
direction as AB, or it [i.e., CB] will fall short of AB by an [amount] equal to AC.
Now, if AB corresponds (muṭābaqan) to CB [as they are extending] to infinity,
while CB is a part of AB, then the whole and the portion correspond [to each
other]. This is a contradiction. If CB falls short of AB in the direction of B and is
less than it, then CB is finite, and AB exceeds it by the finite [amount] AC.
Therefore, AB is finite. It was, however, [supposed to be] infinite. So, it becomes
evidently clear from this [discussion] that the existence of an actual infinity
among ordered numbers and magnitudes is impossible.

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

47
Zarepour (forthcoming) investigates the evolution of Ibn Sīnā’s view regarding the exact
structure, purpose, and applicability conditions of the Mapping Argument over the course of
his career.

(a) C* B*

A C B

(b) C* B*

A C B

Figure 4 A visualisation of the Mapping Argument against the existence

of inﬁnite magnitudes.

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.
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 multi-
tudes 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.48 Nevertheless,

48
Maimonides has listed twenty-ﬁve premises for the proof of the existence of God at the
beginning of The Guide of the Perplexed (1963, pp. 235–40). Muḥammad al-Tabrīzī (d. 13th
century), a thirteenth-century Persian Muslim scholar, has written a commentary in Arabic on
these premises. This work, which was probably the ﬁrst commentary on a part of The Guide, was

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, 2001, 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, 2001, 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:

T15. Al-Shahrastānī (2001, Struggling with the Philosophers, p. 109)

Hypothesise Zayd and consider him as a point; and hypothesise his fore-
fathers to inﬁnity [as points on] a straight line. Also, hypothesise ʿAmr and
consider him as a point falling short of Zayd with one forefather or two
forefathers or three; and consider his ancestors to inﬁnity as a line. Moreover,
suppose that Zayd and ʿAmr are twins in existence, and carry the argument to
its conclusion.

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,

translated into Hebrew by Isaac ben Nathan of Cordoba (d. 14th century) and used as a primary
source by Ḥasdai Crescas in many of the discussions of his Light of the Lord (2018). The first of
those premises rejects the existence of infinite magnitudes, and the second denies the existence of
infinite multitudes of bodies that satisfy WC. To justify the second premise, al-Tabrīzī appeals to
the first premise and argues that no such multitude exists because otherwise an infinite magnitude
will exist. More importantly, to justify the first premise, he presents a version of the mapping
argument in which the infinite magnitude in question is assumed to be constituted of consecutive
segments of equal lengths. See al-Tabrīzī (1981, Commentary on the Twenty-Five Premises for
the Proof of the Existence of God, pp. 7–8 and 17). For Crescas’s presentation of the Mapping
Argument, see Wolfson (1929, pp. 346–47) and Crescas (2018, p. 37).

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.
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

Amr B*

Zayd Amr

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.

Argument because it implies the unacceptable consequence that even the

multitude of all natural numbers must be ﬁnite:

T16. Al-Rāzī (1994, Commentary on Fountains of Wisdom, vol. 2, p. 53)

Let’s take the [multitude that includes every] number from one to what has no
end in terms of the ranks of additions together with another multitude [that
includes every number other than a few initial numbers]. We put the first rank
of this [latter] multitude in front of the first rank of that [former] multitude,
and the second of this in front of the second of that, and so on successively.
So, if the remainder does not appear [i.e., if by following this procedure
nothing of the latter multitudes remains unpaired], then the more is identical
to the less. And, if the remainder appears at the end of the ranks [of the former
multitude, and some ranks of the latter multitude remains unpaired], then it
entails the finitude of number in the direction of its increase; and it is self-
evident for the intellect that this is impossible.

In the multitude version of the Mapping Argument, mapping a multitude

upon another is nothing but putting their members into one-to-one correspond-
ence 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).49 But the latter multitude is a proper
49
Note a significant difference between the two versions of the Mapping Argument presented in
T15 and T16: in the former text, the members of the multitudes in question are imagined as points
with equal successive distances from each other on straight lines. The points on one line are then
paired with those of the other by superposing one of the two lines upon the other. However, the
argument developed in T16 is free from such an assumption. The numbers in one multitude are
paired with those of the other by successively putting them in front of each other. Nevertheless,
these numbers are not assumed to be lined up with equal distances from each other on straight
lines. The superposition of lines does not play any role in the latter version of the Mapping
Argument. In this sense, compared to T15, what we see in T16 is closer to a general notion of
ONE-TO-ONE CORRESPONDENCE.

11 12 13 14 15 16

1 2 3 4 5 6

Figure 6 One-to-one correspondence between the multitude of all natural

numbers and the multitude of all natural numbers greater than 10.

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.
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 condi-
tions concerns the actuality of the infinity in question, while the other concerns
its orderedness. To be more specific, this argument cannot establish the impos-
sibility 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 impossibil-
ity 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 (1985, 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 [quan-
tities] 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 distin-
guished 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 (1985, 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.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 (1959, 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

50
Kohler (2006, p. 97) claims that from the references to one of al-Fārābī’s lost treatises, that is, On
Changing Beings, by Ibn Rushd and Maimonides, it appears that al-Fārābī ‘has set two condi-
tions for the impossibility of an inﬁnite series or an actual inﬁnite number: the members of this
series must be ordered and they must exist simultaneously’. But it seems to me that what Ibn
Rushd and Maimonides have said about al-Fārābī’s view is not exactly what Kohler says.

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.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).

51
See Zarepour (forthcoming, section IV).

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:

Ordering Condition (OC): The members of S are (or, at least, can be) ordered.

Thus, according to Ibn Sīnā, any infinite magnitude that satisfies WC and any
infinite multitude that satisfies both WC and OC are impossible.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 (1985,
chapter IV.2, p. 246, ll. 5–7), that the multitude of angels and devils ‘is suscep-
tible 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:

52
For more details on why Ibn Sīnā must be interpreted as considering OC as a condition for the
applicability of only the multitude version of the Mapping Argument, despite that he has never
explicitly presented this version of the Mapping Argument, see Zarepour (2020, forthcoming).

Materiality Condition (MTC): All the parts of S exist in the material world.

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 2020,
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.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 (2000a,
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.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.

T17. Al-Ghazālī (2000b, The Incoherence of the Philosophers, p. 81)

The past days and nights are inﬁnite [if the world has no temporal beginning].
Thus, if we suppose the [coming into] existence of a single soul in each day
and night, what has come into existence up to the present would be inﬁnite,
being realised in an order of existence that is one [coming into existence] after
another.

53
For the evolution of Ibn Sīnā’s own views regarding this argument throughout his career, see
Zarepour (forthcoming).
54
This claim has extensively been discussed by Mancosu (forthcoming).

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.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 neces-
sarily 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 (1846, 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 neces-
sary 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 (2001, p. 109), that when we argue against the
infinitude of a body (which, being a body, necessarily has a position) by the

55
Al-Ghazālī, in his Moderation in Belief (2013, pp. 27–40), presents an argument for the existence
of God, one of whose premises is that the world has come into existence at a certain time and has
a temporal beginning. A reconstruction of this argument, which is now known as the Kalām
Cosmological Argument, is proposed by Craig (1979). On al-Ghazālī’s view regarding the
inﬁnite number of souls, see Marmura (1960) and Zarepour (2020, section 4).

Mapping Argument, we apply the mapping technique to an imaginary line

existing in the body we have hypothesised. However, ‘what you suppose regard-
ing 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 (2001, 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 2022b, section 3.4). Accordingly, although everything exists
simultaneously with its efficient cause, the material cause of that thing is tempor-
ally 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.56 Rejecting the necessity of
WC for the applicability of the Mapping Argument, al-Shahrastānī (2001,
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 2018, 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:
56
This position was later upheld by Thomas Aquinas. He believes motion and time can, in
principle, be infinite because they fail to satisfy WC. Therefore, if time is indeed infinite,
there can, again in principle, be a series of things that are casually connected to each other in
a diachronic sense of causation. See Aquinas (2006, Summa Theologiae, Ia. 7, 3, p. 105 and 4,
p. 107) and Cohoe (2013). Note that although Aquinas believes that time can be infinite, he does
not believe it is infinite. He thinks that the possibility of the infinity of time can be defended
philosophically. Nevertheless, the infinity of time – or, equivalently, the world’s eternity – can
neither be established nor rejected by philosophical arguments. On the subtleties of Aquinas’s
view regarding the eternity of the world, see İskenderoğlu (2002, chapter 3) and McGinnis
(2014).

T18. Al-Baghdādī (1938, The Considered, vol. 2, p. 85)

It is a fallacious argument because it is completed by moving the line and

dragging it inasmuch as it falls short until it coincides with the initial end.
However, motion is not conceivable for inﬁnity. If a motion is conceived for
it, its end will move along with the totality of it. If it rests with the extension
[it has] and does not move in the opposite direction, the statement [of the
argument] is useless. If [on the other hand] it is dragged from the opposite end
and moves, then it has an end and its [i.e., that end’s] place will be vacated so
that it [i.e., the moving line] falls short of the other [line]. But there is no end
or movement in length for it. The estimation cannot imagine the motion of the
cut end to correspond to the uncut end unless it imagines it as a ﬁnite [line] so
that the other limit moves, being dragged together with the initial limit.
Otherwise, it rests superseded [from the other line] while it is in its position
motionless. So, the argument is not established.

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:

Movability Condition (MVC): All the parts of S are movable simultaneously.

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.57
57
Apparently, al-Baghdādī thinks that MVC is necessary not only for the magnitude version of the
Mapping Argument but also for its multitude version. It is worth noting that some later Muslim
thinkers appealed to MVC to provide solutions for challenging theological problems. For
example, some philosophers and theologians try to show that although the Mapping Argument

Shihāb al-Dīn al-Suhrawardī (d. 1191) presents another version of the

Mapping Argument in his The Philosophy of Illumination (1999, 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 (1990, vol. 1, pp. 306–07) and Commentary on Fountains of Wisdom
(1994, 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.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.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, conse-
quently, 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

establishes things like the finitude of the past, it fails to establish the finitude of the multitude of
the objects of God’s knowledge and/or power. This is because the objects of God’s knowledge
and/or power do not satisfy MVC, or so those scholars suggest. See, for example, Beşikci
(forthcoming).
58
This objection – which argues that the Mapping Argument is unsound because (a) satisfying
MVC is essential for the mapping technique to be applicable and (b) infinite entities do not meet
MVC – is raised and responded to, among other places, in al-Rāzī (1994, Commentary on
Fountains of Wisdom, vol. 2, pp. 50 and 53–54).
59
This discussion is included in the second objection to the mapping argument that al-Rāzī
anticipates and blocks. See al-Rāzī (1994, Commentary on Fountains of Wisdom, vol. 2,
pp. 50–51 and 54–56).

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:

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

(9) A falls short of B.

Therefore:

(10) A is ﬁnite.

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

T19. Al-Rāzī (1994, Commentary on Fountains of Wisdom, vol. 2, pp. 55–56)

If we mean by its being ﬁnite that something is realised in others that is not
realised in it, then the meaning of it being ﬁnite is that it falls short of others.
Then, in this syllogism, the major [term] would be the same as the middle
[term]. And, if what we mean by that is the necessity of what falls short
ending up at a rank beyond which nothing else is left, [then] this can be
thought of [only] in what has an order in position or in nature. This meaning is
not realised in what is not like this [i.e., in non-ordered things]. If we intend
a third meaning, it would not be intelligible.

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.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.61

60
It is worth highlighting that he does not try to demonstrate (8) based on EI, even though it seems
that the former claim is a straightforward consequence of the latter.
61
See his discussion of the third and seventh possible objections to the Mapping Argument in (1994,
Commentary on Fountains of Wisdom, vol. 2, pp. 51–52 and 56–57). It must be noted, however,
that al-Rāzī does not take the inapplicability of this argument to the past events (or, more generally,

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 (1994,
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 (1994, 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.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.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

to time) as a justification for the eternity of the world. Al-Rāzī neither straightforwardly accepts nor
rejects the eternity of the world. To a large extent similar to Aquinas, he believes that both the
philosophical arguments for the eternity of the world and those for its temporal origination have
significant shortcomings, making them indecisive. Moreover, al-Rāzī argues that neither doctrine
is necessary for religious (and, in particular, Islamic) belief. For al-Rāzī’s views on the eternity of
the world and their philosophical affinities with Aquinas’s position, see İskenderoğlu (2002).
Refraining from a straightforward endorsement of either the eternity of the world or its temporal
origination was not a rare approach in medieval philosophy. Abu-l-Barakāt al-Baghdādī and
Maimonides are other well-known representatives of this approach.
62
See also al-Rāzī (1990, Eastern Investigation, vol. 1, p. 311).
63
I do not know if he has ever explicitly affirmed the existence of the infinite multitude of numbers
in God’s mind. Nevertheless, even if al-Rāzī denies this, it would not be because he thinks God
cannot grasp and entertain an infinite number of mental objects altogether. He believes that the
number of the objects of God’s knowledge is infinite. See, for example, al-Rāzī (1990, Eastern
Investigation, vol. 1, p. 310).

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 exist-
ence. Accordingly, the multitude of all numbers existing in the world can, in
principle, become bigger boundlessly. Nevertheless, this multitude is never actu-
ally 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 imagin-
ations do not bring those mathematical objects into existence. Ibn Sīnā is a finitist,
not only about physics but also about mathematics.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:

T20. al-Rāzī, Commentary on Fountains of Wisdom (1994, vol. 2, p. 51)

[The number resulting from] doubling a thousand repeatedly and inﬁnitely is

less than [the number resulting from] doubling two thousand repeatedly and
infinitely. And whatever is less than something other than itself is finite. So,
an infinite is finite. This is absurd.65

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:

T21. Al-Rāzī (1994, Commentary on Fountains of Wisdom, vol. 2, pp. 56–57)

These numbers do not exist in the extramental world or even in the mind. They
are present in the intellect by relating the meaning of inﬁnity to the meaning of
doubling. And this is nothing other than relating a meaning to a meaning. This
is contrary to the bodies and causes. They exist in the extramental world.

64
On this interpretation of Ibn Sīnā’s ontology of mathematics, see Zarepour (2016).
65
This objection is also mentioned in al-Rāzī (1990, Eastern Investigation, vol. 1, p. 310).

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:

T22. al-Rāzī, Commentary on Fountains of Wisdom (1994, vol. 2, p. 53)

One is a half of two, a third of three, a fourth of four, and so on ad inﬁnitum

(ilā mā lā nihāyah lah) in terms of ratios. Thus, we say: there is no doubt that
the sum of these ratios with the deduction of the first ten of them is less than
this sum without that deduction. This makes the sum of these ratios finite,
though we clarified that they are infinite. This is absurd.

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:

S2 ¼ 1=2 þ 1=3 þ 1=4 þ . . .

S12 ¼ 1=12 þ 1=13 þ 1=14 þ . . .

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.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:

S2 ¼ 2=2 þ 3=3 þ 4=4 þ . . . ¼ 1 þ 1 þ 1 þ . . . :

S12 ¼ 12=12 þ 13=13 þ 14=14 þ . . . ¼ 1 þ 1 þ 1 þ . . . :

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 > S12.67 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 (1994, 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 men-
tioned 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.

66
Nicole Oresme (d. 1382) is usually introduced as the ﬁrst person to prove the divergence of the
harmonic series (i.e., S1 = 1 + 1/2 + 1/3 + 1/4 + . . . ). If this is true, then al-Rāzī could not be aware
of the divergence of S2 and S12. For Oresme’s proof of the divergence of the harmonic series,
which was originally presented in his Questions on the Geometry of Euclid, see Grant (1974,
p. 135).
67
I am thankful to a reviewer who suggested this reading of the passage.

The Mapping Argument was widely discussed by many scholars in the later
Islamic philosophy and theology. Many medieval Muslim theologians (particu-
larly 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.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 multi-
tudes 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.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 (2013, 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, 2013, A Treatise on the Proof of the Necessary, pp. 140–41).
Moreover, he denies the necessity of OC by the following line of argument:

T23. al-Dawānī, A Treatise on the Proof of the Necessary (2013, p. 144).

Either [the applicability of] the mapping [technique] depends on considering

individuals as [totally] differentiated [from each other] (mufaṣṣalan) or it is
sufﬁcient to consider them [just] as a whole (mujmalan). In the ﬁrst case, the
mapping is impossible also for the ordered. In the second case, it applies also
to the non-ordered.

68
See Beşikci (2022, pp. 127–28).
69
On the view of Muslim theologians regarding the ontology of mathematics, see Zarepour (2022a,
section 1.2).

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 consider-
ing 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 possibil-
ity of their infinitude cannot be rejected by the Mapping Argument.70

70
See Shihadeh (2011, p. 151, section 7).

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.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 (1863, p. 303).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 pre-
sents 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

71
One might think that the Mapping Argument cannot reject the possibility that infinitely many
souls come into existence together simultaneously. However, this is not acceptable from the
point of view of Muslim philosophers because each soul must be associated with a body in its
origination and no body can be associated with more than one soul. But if one assumes – as the
majority of Muslim scholars do – that no multitude of coexisting bodies can be infinite, then there
can be no infinite multitude of souls that come into existence together simultaneously. Ibn Sīnā’s
example of the multitude of angels and devils does not have this shortcoming because angels and
devils are not assumed to be embodied at all.
72
A discussion of Neckam’s views about infinity can be found in Davenport (1999, chapter 1).

(a)

(b)

(c)

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

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.73
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 1968, 1969). 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 2021,
section 7). Finally, to mention an example of this argument in the Jewish

73
Mancosu (forthcoming, section 3.2) offers a detailed analysis of this argument.

tradition, we can refer to a discussion of this argument by Ḥasdai Crescas

(d. 1410/11) in his Light of the Lord (2018, pp. 37 and 73–74). He does not
find this argument plausible because he thinks that infinity is not measur-
able. 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 engage-
ments 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 applica-
tion 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 applic-
ability 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 discus-
sions 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 compel-
ling. 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:

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

(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.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 charac-
teristic feature of infinite sets that is highlighted in Dedekind’s definition of
infinity. As is put by Reck (2023), 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 corres-
pondence 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 submulti-
tudes 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

74
Such attempts can be seen in the works of Latin philosophers like Henry of Harclay, William of
Alnwick (d. 1333), and Gregory of Rimini (d. 1358). See Murdoch (1982, pp. 571–72).

prior to cardinality. Some philosophers, like Ibn Sīnā, thought that not every
multitude can be ordered, while others, like al-Shahrastānī, believed that every-
thing 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.

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Abbreviations
CN4 Common Notion 4
CN5 Common Notion 5
EI Equality of Inﬁnities
EEC Extramental Existence Condition
NIM Numericality of Inﬁnite Multitudes
MTC Materiality Condition
MVC Movability Condition
OC Ordering Condition
WC Wholeness Condition

Aristotle (d. 322 BCE)

Lucretius (d. circa 55 BCE)
Plutarch (d. circa 120)
Proclus (d. 485)
Simplicius (d. 560)
John Philoponus (d. 570)
al-Ḥajjāj Ibn Yūsuf Ibn Maṭar (d. 833)
Ibrāhīm al-Naẓẓām (d. circa 845)
al-Kindī (d. 870)
Thābit Ibn Qurra al-Ḥarrānī (d. 901)
Isḥāq Ibn Ḥunayn (d. circa 910)
al-Khayyāt (d. circa 910)
al-Fārābī (d. 950)
Yaḥyā Ibn ʿAdī (d. 974)
ʿAbd al-Qāhir al-Baghdādī (d. 1037)
Ibn Sīnā = Avicenna (d. 1037)
Bahmanyār Ibn al-Marzubān (d. 1066)
al-Ghazālī (d. 1111)
Bachya Ben Joseph Ibn Paquda = Baḥy Ibn Yūsuf Ibn Bāqūda (d. 1120)
Ibn Bājja = Avempace (d. 1138)
Adelard of Bath (d. 1152?)
Muhammad Ibn ʿAbd al-Karīm al-Shahrastānī (d. 1153)
Peter Lombard (d. 1160)
Abu-l-Barakāt al-Baghdādī (d. 1165)
Shihāb al-Dīn al-Suhrawardī (d. 1191)
Ibn Rushd = Averroes (d. 1198)
Moses Maimonides = Mūsa Ibn Maymūn (d. 1204)
Fakhr al-Dīn al-Rāzī (d. 1210)
Alexander Neckam = Alexander Nequam (d. 1217)
William of Auvergne (d. 1249)
Bonaventure (d. 1274)
Naṣīr al-Dīn al-Ṭūsī (d. 1274)
Thomas Aquinas (d. 1274)
Roger Bacon (d. 1292)

John Peckham (d. 1292)

Peter John Olivi (d. 1298)
Muḥammad al-Tabrīzī (d. 13th century)
Henry of Harclay (d. 1317)
William of Alnwick (d. 1333)
Gersonides (d. 1344)
Thomas Bradwardine (d. 1349)
Gregory of Rimini (d. 1358)
Nicole Oresme (d. 1382)
Isac ben Nathan of Cordoba (d. 14th century)
Ḥasdai Crescas (d. 1410/11)
Khojazāda (d. 1488)
Jalāl al-Dīn al-Dawānī (d. 1502)

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Acknowledgements
I owe a deep debt of gratitude to my wife, Samaneh, for her constant encour-
agement and love, without which I doubt I would have found the drive and focus
to pursue this project and probably many of my other projects.
I should also thank Amal Awad, Necmeddin Beşikci, Gholamreza Dadkhah,
Andreas Lammer, Fraser MacBride, Paolo Mancosu, Jon McGinnis, and Tony
Street for stimulating discussions on ideas presented in this Element. I am
particularly indebted to Paolo Mancosu and Necmeddin Beşikci for generously
sharing copies of their forthcoming works. My thanks also extend to Kaave
Lajevardi and two anonymous reviewers for their valuable feedback on an
earlier draft of this Element. This work would have never seen the light
of day without the patience and support of Stewart Shapiro, to whom I would
like to express my gratitude. I am grateful to the Leverhulme Trust for awarding
me a 2023 Philip Leverhulme Prize, which provided crucial support during the
ﬁnal stages of this project. Wal-ḥamdu li-l-lāhi awwalan wa ākhirā.

I want to dedicate this Element to my daughter, Sophia, who is a ﬁnitist by the

force of age! She can count only to ten and does not yet know anything about
inﬁnity. Nevertheless, she knows how to make her father inﬁnitely happy.

Penelope Rush
University of Tasmania
From the time Penny Rush completed her thesis in the philosophy of mathematics (2005),
she has worked continuously on themes around the realism/anti-realism divide and the
nature of mathematics. Her edited collection, The Metaphysics of Logic (Cambridge
University Press, 2014), and forthcoming essay ‘Metaphysical Optimism’ (Philosophy
Supplement), highlight a particular interest in the idea of reality itself and curiosity and
respect as important philosophical methodologies.

Stewart Shapiro
The Ohio State University
Stewart Shapiro is the O’Donnell Professor of Philosophy at The Ohio State University,
a Distinguished Visiting Professor at the University of Connecticut, and Professorial Fellow
at the University of Oslo. His major works include Foundations without Foundationalism
(1991), Philosophy of Mathematics: Structure and Ontology (1997), Vagueness in Context
(2006), and Varieties of Logic (2014). He has taught courses in logic, philosophy of
mathematics, metaphysics, epistemology, philosophy of religion, Jewish philosophy, social
and political philosophy, and medical ethics.

About the Series

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of mathematics in its many and varied forms. Distinguished authors will provide an
up-to-date summary of the results of current research in their fields and give their
own take on what they believe are the most significant debates influencing research,
drawing original conclusions.

Elements in the Series

Number Concepts: An Interdisciplinary Inquiry
Richard Samuels and Eric Snyder
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A. C. Paseau and Wesley Wrigley
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Gregory Lavers
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Dirk Schlimm
Elements of Purity
Andrew Arana
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