Avicenna on Mathematical Infinity

3.1 The Collimation Argument

There are some places in which Avicenna provides an indirect discussion of the problem of infinity. For example, in his discussions of the void in The Physics of the Healing^[25] and The Physics of the Salvation,^[26] he appeals to the impossibility of circular motion in an infinite void as one of his premises in arguing for the impossibility of the void. To justify this premise, he proposes an auxiliary argument that is known as The Collimation Argument (burhān al-musāmita) or The Parallelism Argument (burhān al-muwāzāt). This argument likely originates in Aristotle’s De Caelo.^[27] Aristotle’s original argument was proposed to show that the infinite “cannot revolve in a circle; nor could the world, if it were infinite.” Avicenna extensively revised this argument to show, primarily, that circular motion in an infinite void is impossible. Coupling this result with the claim that the void, if it exists, cannot be finite, Avicenna concludes that the void does not exist.^[28] But in other places such as The Physics of Fountains of Wisdom,^[29] he also proposed this argument as an independent argument against the actual infinitude of intervals (ab‘ād). The argument goes as follows:

Consider the line L which is infinite in one direction; it starts from the center O of a finite circle C, intersects the circumference of the circle, and extends infinitely. Consider, moreover, another line L′ which is parallel to but distinct from L, and extends infinitely in both directions. Now, suppose that the circle C together with L start to rotate around O, while L′ remains motionless and fixed. As a result of this circular motion, these two lines intersect. Therefore, there is a moment of time in which these lines are parallel and there is a moment of time in which they intersect with each other. From this fact, Avicenna concludes that there should be a moment of time T and, accordingly, a point P on L′ in which these lines intersect each other for the first time (after the beginning of the circular motion). But there is obviously no such point. For every point P which we consider as the first intersection point of these lines, there are infinitely many points on L′ prior to P which would have been passed and intersected by L (Fig. 1). Since Avicenna believes that circular motion undeniably can happen, he concludes that what should be rejected is the existence of infinite lines and intervals.^[30]

Fig. 1

Although this argument is intuitively powerful, it is not convincing. The argument is not only invalid from the point of view of modern mathematics, but is also incompatible with some of Avicenna’s own views. From the fact that there is a moment of time in which L and L′ are parallel and there is another moment in which they intersect, we cannot conclude that there is a moment in which these lines intersect with each other for the first time, nor that there is a point on L in which these lines intersect for the first time. Here Avicenna seems to suppose that the set of all temporal moments in which L and L′ are intersected has a least element (with respect to the natural order and succession we consider for temporal moments). Correspondingly, there is a first point of intersection on these lines.^[31] However, this supposition is false. It is not the case that every subset of temporal moments or every subset of the set of points on a straight line has a least or first element.^[32]

Interestingly, while Avicenna endorses the above supposition in the context of The Collimation Argument, he seems to reject it in his discussion of time and linear motions. If the above supposition were true, we could argue similarly against the possibility of even linear motions on finite spatial intervals. Consider a finite interval AB. Suppose that an object X is first motionless on A, then it moves and goes without stopping from A to B, and finally stops on B. Therefore, there is a time in which X is at rest and there is a time in which X is in motion. Now, if every subset of the points on a line (a spatial or temporal interval) had a least member, then there would be a first intermediary point of AB on which X does not stop (i. e., it merely passes that point). Correspondingly, there would be a moment in which X is on an intermediary point for the first time. But, given the continuity of time, space and motion – which Avicenna holds – there is neither such point nor such moment.^[33] Therefore, if the supposition under discussion were true, we should conclude, in a way parallel to what we had in The Collimation Argument, that it is impossible for X to move from A to B. It means that not only is circular motion in an infinite space impossible, but linear motion on a finite magnitude is impossible as well. However, Avicenna accepts, not surprisingly of course, that continuous linear motion is possible. It indicates that he should reject the supposition that every subset of temporal moments or every subset of points on a line has a least or first element. He does so, indeed. McGinnis has elegantly shown that, according to Avicenna’s theory of time, “if one takes some instant t as a limit, then for any other instant t´, no matter how close one wants to take t´ to t, then there is another instant t´´ that is not identical with t, but is closer to t than t´. Since this same analysis will be true of t´´, t´´´ and so on, one can get indefinitely close to t without actually being at t.”^[34] It means that the set of all temporal moments between t and t´ has no least number. So the aforementioned supposition is rejected.^[35] But rejecting this supposition renders The Collimation Argument invalid. It seems therefore that this argument is controversial even with respect to Avicenna’s own philosophical framework. At least, Avicenna owes us an explanation of why this argument is based on a supposition that he rejects in another context.

Avicenna could however slightly revise this argument such that it validly entails the impossibility of the actual infinity, at least in an Aristotelian framework which would be acceptable for him. Avicenna accepts, following Aristotle, that the infinite cannot be traversed in a finite time.^[36] Therefore, he could argue that if the circle C is finite, then it takes a finite time for it to rotate once around O. Accordingly, it takes a finite time for L to rotate once around O. But in each round of rotating around O, L traverses the whole line L′. This means that the infinite length of L′ can be traversed in a finite time (equal to half the time of L’s rotating around O). This argument shows that the conjunction of (a) the principle that the infinite cannot be traversed in a finite time, (b) the possibility of having infinite intervals or lines, and (c) the possibility of circular motion, entails a contradiction. Avicenna could, therefore, argue that since (a) and (c) are obviously true (according to him), (b) must be rejected; there is no actually infinite interval. This argument seems perfectly sound, at least in an Aristotelian framework. However, Avicenna did not propose such a revised version of The Collimation Argument, and, as I mentioned, his own original version is problematic.

A more careful inspection of the original version of The Collimation Argument (especially an inspection of how this argument is related to the argument I proposed against the impossibility of linear motion) reveals some interesting aspects of Avicenna’s understanding of the notion of continuity; but discussing this issue would take us too far from our main concerns.^[37] I turn, therefore, to another argument against the actuality of infinite magnitudes.

3.2 The Ladder Argument

The idea of motion plays an important role in The Collimation Argument; that argument should therefore be categorized as a physical argument in the aforementioned sense. Now we briefly review a mathematical argument against the actuality of infinite intervals that does not appeal to such physical notions. This argument, known as The Ladder Argument (burhān al-sullam), is first proposed in The Physics of the Healing^[38] as a potential rehabilitation of one of Aristotle’s physical arguments in De Caelo.^[39]The Ladder Argument is also Avicenna’s only argument against the actuality of infinity in Pointers and Reminders.^[40] The argument, as appeared in The Healing, goes as follows:

TEXT # 1: Let us posit a certain interval between two opposite points on two lines extending infinitely. Now, let us connect the [points] by a line that is a chord of the intersecting angle. So, because the extension of the two lines, which is infinite, is proportional to the increase of the interval [that is, the length of the chord], the increases to that interval are infinite. [Those increases] can also exist together equally, because the increases that are below will actually be joined to those that are above. For instance, the [amount that] the second increases the first will belong to the third, together with any other increase. So the infinite increases must actually exist in one of the intervals, and that is because the increases actually exist, and every actual increase will exist and so will belong to a certain one [of the intervals]. In that case, it necessarily follows that some interval will exist in which there is an actual infinity of equal increases. So that interval would increase the first finite [interval] by an infinite [amount], in which case there would be an infinite interval […]. This infinite can exist only between two lines, in which case it is finite and infinite, which is absurd.^[41]

To have a more diagrammatic understanding of this argument, suppose that L and L′ are two distinct lines that start at the same point A and extend infinitely to make an acute angle with infinite sides. Now, consider two arbitrary points D₁ and E₁ on L and L′ respectively. As a result, D₁ E₁ is an interval which lies between L and L′. Furthermore, consider all intervals D₂ E₂ , D₃ E₃ , D₄ E₄ , etc., parallel to D₁ E₁, such that D_i and E_i (for every natural number 1≤i) lie respectively on L and L′ and the difference between the lengths of every two consecutive intervals is constant. If we suppose that this difference is d, then for every natural number 1≤i, D_i+1 E_i+1 – D_i E_i= d. So we have a hierarchy of intervals in which every interval is formed by adding an interval of the length d to the previous interval.^[42] For short, every interval is formed by an increase to the previous interval. Therefore, every interval is formed by a number of increases to D₁ E₁ . Moreover, if an increase belongs to an interval, then all the previous increases belong to the same interval too. As a result, every interval somehow includes all the previous intervals. For instance both the first and the second increases belong to D₃ E₃ and, in a sense, it includes both D₁ E₁ and D₂ E₂ , i. e.,

D₃ E₃ = D₂ E₂ + d = D₁ E₁ + d + d.

The number of the increases is infinite, and all of them actually exist, since otherwise there is an upper bound for the lengths of D_i E_i and, as a result, L and L′ would be finite. From these premises, Avicenna concludes that there should be an interval BC (in such a way that B and C lie respectively on L and L′) which includes all of the infinitely many increases. BC is, therefore, larger than any D_i E_i we consider. It should be itself infinite. However, BC is restricted to L and L′, and terminates at them, which indicates that it is finite (Fig. 2). Consequently, it is both finite and infinite. Contradiction. Ergo, there are no infinite lines such as L and L′.

Fig. 2

This argument again seems to be controversial. It is based on three premises:

1. Every increase belongs to an interval. In other words, for every increase there is a D_i E_i to which the increase belongs.

2. If an increase belongs to an interval, then all of the previous increases also belong to the same interval. In other words, if the n^th increase belong to D_i E_i , then all the first, the second, […], and the (n-1)^th increases also belong to the same interval.

3. All of the infinitely many increases actually exist.

From these premises, which seem to be uncontroversial, Avicenna then concludes that:

1. The infinite increases all together must actually exist in one of the intervals. In other words, some interval will exist in which there is an actual infinity of equal increases.

It is, however, far from clear how (4) can be validly entailed from premises (1)–(3). These premises indicate that there exists an actually infinite hierarchy of finite intervals with increasing lengths. Each interval is longer than the previous by the amount d, and each interval is formed by a finite number of increases to D₁ E₁ . Nonetheless, these facts do not seem to imply that there is an actually infinite interval which includes all the increases.

As I mentioned earlier, The Ladder Argument is discussed in The Physics of the Healing as a rehabilitation of one of Aristotle’s physical arguments which Avicenna finds weak. However, it seems that The Ladder Argument is still vulnerable to one of the very objections that he himself puts forward against Aristotle’s argument. Roughly speaking, Aristotle’s argument goes as follows:^[43] Consider a finite circle in an infinite space and suppose that it takes a finite time for it to rotate once around its center. Suppose, moreover, that two distinct radii of this circle are extended infinitely. Now, Aristotle argues that there should be an infinite interval – e. g., an infinite arc parallel to the circumference of the circle – between these two radii. But if so, this infinite interval would be traversed in a finite time by one of the two infinitely extended radii during the rotation of the circle. This however leads to a contradiction, since the infinite is not traversable in a finite time. Given the assumption that circular motion is possible, Aristotle concludes that space cannot be actually infinite. Consequently, there is no magnitude. In criticizing Aristotle’s argument, Avicenna says:

TEXT # 2: It is not the case that an infinite interval [[e. g., an infinite arc]] must occur [[between two infinitely extended radii of a circle]]; rather, the increase [[of the lengths of the intervals between those radii]] will proceed infinitely, where every increase will involve one finite [amount] being added to another, in which case the interval will be finite. This is just like what you learned concerning number – namely, that [number] is susceptible to infinite addition, and yet, any number that occurs is finite without some number actually being infinite, since any given number in an infinite sequence exceeds some earlier number [in that sequence] only by some finite [number].^[44]

I think, however, that the same point can be made concerning The Ladder Argument. By borrowing Avicenna’s own wording, we can say: The increase of the length of the intervals D_iE_i will proceed infinitely, where every increase will involve one finite interval of the length d being added to another finite interval, in which case the outcome will be finite. Therefore, although we have actually an infinite hierarchy of finite intervals with increasing lengths, it is not the case that there is an actually infinite interval terminated at L and L′ which includes all of the intervals D_iE_i. This is exactly what happens in the case of numbers. The supposition that all numbers actually exist does not imply that there actually exist a number which is infinite. Even if the chain of consecutive numbers extends infinitely and all numbers actually exist, this does not entail that infinity itself is a number and occupies a place in the chain of numbers. Similarly, the actual existence of infinitely many intervals terminated at L and L′ does not necessarily entail the actual existence of an infinite interval terminated at these two lines.

It is not clear how Avicenna intended to save The Ladder Argument from the above objection. This is why the soundness of this argument was the subject of a longstanding discussion in the post-Avicennan Islamic philosophy.^[45] But regardless of this historical debate, the argument does not seem to be convincing for modern readers. From the perspective of contemporary mathematics, the above objection is quite compelling and reveals a fatal flaw in The Ladder Argument. Avicenna seem to believe that the existence of an infinite plane entails the existence of an infinite triangle ABC whose sides are infinite. He argues, then, that this is impossible. On the one hand, all three sides of this triangle should be infinite; so, BC is infinite. On the other hand, BC is restricted to AB and AC, since it terminates at B and C; so, BC is finite. Consequently, BC is both finite and infinite. Contradiction. Ergo, there is no infinite plane and no infinite interval at all. However, this is unsound. The existence of an infinite plane does not imply the existence of an infinite triangle, or any other infinite shape with a closed boundary on a plane. Therefore, even this mathematical argument – which, unlike The Collimation Argument, does not appeal to physical notions such as motion – does not work, at least from our modern point of view. I now turn to Avicenna’s main argument for the non-actuality of mathematical infinity.

4.1 The Structure of the Mapping Argument

Consider the straight line AB which starts from the point A and extends infinitely in the direction of B. AB represents a one-dimensional magnitude infinitely extended in one direction, or an infinite set of numbered objects possessing an order, the first element of which is placed on A while the other elements are successively lined up on some discrete points on the rest of AB.^[59] Suppose that we take the finite part AC from AB. Avicenna argues that:

TEXT # 5: [I]f some amount equal to CB were mapped on or parallel to AB (or you were to consider some other analogous relation between them), then either [CB] will proceed infinitely in the way AB does, or it will fall short of AB by an amount equal to AC. If, on the one hand, AB corresponds with CB [in proceeding] infinitely, and CB is a part or portion of AB, then the part and the whole correspond [with one another], which is a contradiction. If, on the other hand, 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, in which case AB is finite; but it was infinite. So it becomes evidently clear from this that the existence of an actual infinite in magnitudes and ordered numbers is impossible.^[60]

Here, Avicenna argues that after taking the finite part AC from AB, we can compare the sizes of CB and AB by mapping (aṭbaqa) something equal (musāw) to the former on the latter, i. e., by mapping a copy of CB on AB in a way that the first point of the copy of CB corresponds with A. Avicenna uses the term ‘CB’ (jīm bā’) equivocally in referring to both CB and its copy (i. e., the thing equal to CB). For clarity, we use the term C*B* to refer to the copy of CB. Therefore, Avicenna believes that we can compare the sizes of CB and AB, by mapping C*B* onto AB in a way that C* corresponds with A. After this mapping, either C*B* extends infinitely in the direction AB does (Fig. 3a) or C*B* falls short of AB (Fig. 3b). In the former case, C*B* corresponds with AB. But C*B* is equal to CB. Therefore, CB corresponds with AB. This entails that a whole (i. e., AB) totally corresponds with its part (i. e., CB). Avicenna believes that this is a contradiction. Now we should check the other horn. Suppose that C*B* falls short of AB. This means that C*B* corresponds with a part of AB which starts at A and terminates at a determinate point on AB. Call this latter point D (Fig. 3b). AD is an interval which terminates at two determinate points A and D; therefore, it has determinate limits and is finite. Now, since B*C* corresponds with AD, B*C* is finite too. On the other hand, the amount by which C*B* falls short of AB is equal to AC, for the fact that C*B*=CB implies that AB–C*B* = AB–CB = AC. This means that the sum of C*B* and AC is equal to AB. But C*B* and AC are both finite. Therefore, their sum, which is equal to AB, is finite. As a result, we should accept that AB is finite. This contradicts our first supposition. Avicenna concludes that an infinity like AB cannot actually exist.

This clarification of The Mapping Argument, which brings some geometrical diagrams (such as Fig. 3) to mind, shows how this argument is intended to be applied to magnitudes. But the last sentence of TEXT # 5, in addition to the aforementioned evidence, confirms that Avicenna believes that this argument can also work perfectly against numerical infinity and show “that the existence of an actual infinite in […] ordered numbers is impossible.” Therefore, one might ask how this argument works in the case of numbers or numbered things. To the best of my knowledge, Avicenna has not explicitly replied to this question, at least not in his major works. He has astutely realized that the mapping technique can be employed against the actual infinity of numbers and numbered things, but it seems that his own explanations of the application of this technique are more perfectly matched to the case of magnitudes, rather than that of numbers. Fortunately, it is not very difficult to guess what he had in mind for the case of numbers.

If we consider AB as an infinite set of numbers or numbered objects possessing an order the first element of which is placed on A and the other elements are successively lined up on some discrete points of the rest of AB, then AC can be considered as the finite subset of the initial elements of AB placed from A to C. Accordingly, mapping C*B* (i. e., a copy of CB) onto AB pairs the first element of C*B* with the first element of AB, the second element of C*B* with the second element of AB, and so on (i. e., for every natural number n, pairing the n^th element of C*B* with the n^th element of AB). If this pairing procedure ends at some finite stage by pairing the last element of C*B* with an element of AB, this means that C*B* and consequently CB are finite. Therefore AB, which is the union of AC and CB, would be finite too (Fig. 4b). On the other hand, if the elements of C*B* extend infinitely, then it is possible to set a one-to-one correspondence between C*B* and AB by pairing every n^th element of the former with the n^th element of the latter. Therefore, AB corresponds with B*C* and, consequently, with BC (Fig. 4a). This means that a whole (i. e., AB) corresponds with one of its proper parts (i. e., BC). Since Avicenna sees the correspondence between a whole and its proper part as a contradiction, this line of argument can establish for him that there cannot be any actually infinite set of numbers or numbered things possessing an order.

However, as far as I know, Avicenna himself has nowhere clarified the details of the application of The Mapping Argument to the case of numerical infinity. His explanations, as I discussed, highlight only the application of this argument to the case of geometrical infinity. He clearly claims that this argument rejects the actuality of numerical infinity, but he does not say how it really works in that case. Nonetheless, it does not seem implausible to accept that Avicenna had in mind something very similar to what is set out in the above paragraph. At least four different considerations support this claim. The first consideration is that the only natural way to develop the notion of mapping from the context of continuous geometrical magnitudes to the context of sets containing discrete elements seems to be interpreting mapping between these sets as one-to-one correspondence between their elements. Since Avicenna believes that the mapping technique can successfully work in the case of numbers and numbered things, it is highly probable that he has such a natural understanding of the notion of mapping in the context of numbers. The second consideration comes from an abstruse passage in The Discussions presenting a brief version of The Mapping Argument in response to a question probably raised by Ibn Zayla (c. 983–1048). It is argued there that it is possible to have two potential infinities of different sizes (i. e., one of them is bigger than the other). But it is impossible to have such infinities in actuality, because:

TEXT # 6: When it [i. e., an infinity] became concurrent with and parallel to it [i. e., another infinity] in terms of connectedness or in terms of orderedness, or when it [i. e., one of those infinities] became a part of the other, then one of them would come to an end in one side and a remnant of one of them would remain at the other side. Therefore, the finitude of [… the shorter infinity] is necessary [and this is absurd].^[61]

The passage seems to be saying that after mapping one of those infinities on the other, they become parallel to each other in terms of connectedness or in terms of orderedness (muḥādhāt fī ittiṣāl aw fī tartīb). This disjunctive phrase suggests that being parallel in terms of connectedness/continuity differs from being parallel in terms of order/orderedness. If the mapping technique was applicable only to continuous magnitudes, then we would have only one kind of parallelism. All continuous magnitudes are isomorphic to each other. Consequently, their being parallel to each other cannot be of more than one kind. Even if Avicenna thought that all parallelisms of magnitudes are of both mentioned kinds, he would have to use a conjunctive phrase rather than a disjunctive one. This he does not do. It indicates that the mapping technique is applicable to not only continuous magnitudes but also numerical infinities. The distinction between these two kinds of parallelism brings to mind the two different ways of the application of the mapping technique I explained above. By applying this technique to infinite magnitudes they become parallel to each other in terms of connectedness and continuity. By contrast, the application of this technique to the case of numerical infinities makes them parallel to each other in terms of order and orderedness. Therefore, ‘parallelism in terms of orderedness’ could be interpreted as Avicenna’s term for the notion of one-to-one correspondence.

The third consideration supporting that Avicenna was aware of the application of the mapping technique to the case of numerical infinities is this: even early commentators on Avicenna (who are obviously closer than we are to the historical context of Avicenna’s discussions) have had the same interpretation of the intended function of Avicenna’s mapping argument in the case of numbers. For example, Fakhr Al-Din Al-Rāzī, in his commentary on Avicenna’s Fountains of Wisdom, explains this function of The Mapping Argument along exactly the same lines as the interpretation mentioned above. Consider two sets of numbers: (1) A proper subset of natural numbers, for example, the set of natural numbers bigger than 10, and (2) the set of all natural numbers from one to infinity. Al-Rāzī writes that one can argue, based on the mapping technique, that:

TEXT # 7: We put the first position of this [latter] set in front of the first position of that [former] set, 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 set remains unpaired], then the more is identical to the less. And, if the remainder appears at the end of the positions [of the former set, and some positions of the latter set 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.^[62]

The text evidently shows that Al-Rāzī understands Avicenna’s mapping notion to be one-to-one correspondence in the case of numbers and numbered things.^[63] The plausibility of the attribution of this position to Avicenna is reinforced by a fourth consideration: the explanatory power of this interpretation as to why Avicenna emphasizes that possessing an order, in either nature or position, is a necessary condition for the applicability of The Mapping Argument in the case of numbers and numbered things. If we understand Avicenna as believing that correspondence between sets of discrete elements is one-to-one correspondence between their elements, then we have a very convincing explanation of why non-ordered sets of discrete objects cannot be the subject of The Mapping Argument. I will come back to this issue in section 4.3 below.^[64] Now it is time to discuss the significance of the insights behind the surface structure of The Mapping Argument.

4.2 Insightful Ideas behind the Surface Structure

An exhaustive analysis of Avicenna’s explanation of The Mapping Argument reveals some remarkably interesting aspects of his treatment of the concept of infinity. The general structure of this argument is as follows: Avicenna argues that the actual existence of an infinity entails the equality of a whole to some of its parts. He denies the equality of whole and part, at least for physical objects and mathematical objects which are, as he believes, properties of physical objects. He concludes therefore that the infinite cannot exist in actuality. Before Avicenna, some other philosophers had correctly noticed that the existence of an actual infinity entails the equality of a whole to some of its parts;^[65] and indeed, some of them had appealed to this fact to reject the actuality of infinity. For example, Al-Kindī followed such a line of reasoning in parts of his previously mentioned argument against infinity, from which The Mapping Argument was inspired.^[66] Both Al-Kindī and Avicenna employed the mapping technique to compare the sizes of infinities. They however had two different understandings of the notion of equality (tasāwī). According to Al-Kindī, things with an equal distance between their limits (ḥudūd), or things whose dimensions between their limits are the same are equal. This understanding is however controversial for three reasons. First, Al-Kindī defines equality in terms of itself. It seems therefore that the definition is question-begging and somehow uninformative. One can take the question of equality one step further and legitimately ask how the equality of the dimensions between the limits of things can be investigated. Al-Kindī seems not to have a non-circular convincing answer for this question. Second, it is not clear how this notion of equality can be applied to infinite magnitudes. If a magnitude is infinite, then it has no limit, at least in one direction. Therefore, the sizes of infinite magnitudes cannot be compared based on the equality of the dimensions between their limits. Third, even if we accept that the sizes of infinite magnitudes can be compared based on this understanding of the notion of equality, it is far from clear how this understanding can be applied to numerical infinities. Talking about the dimensions between the limits of infinite sets of numbers or discrete objects does not seem to be meaningful.^[67] The latter point is one of the reasons why Al-Kindī’s argument, despite having the same general structure as that of Avicenna, should not be interpreted as directly applicable to the case of numerical infinities.

Avicenna, on the other hand, employs the notion of mapping or correspondence (taṭābuq) to compare the sizes of infinities. The idea of understanding the notion of equality in terms of a mapping or correspondence relation goes back to Euclid. According to his fourth common notion in the first book of TheElements, “things which fit onto/coincide/correspond with (ta epharmozonta) one another are equal to one another.”^[68] This common notion is translated into Arabic in the so-called Isḥaq–Thābit version as “allatī lā yufaḍḍal ’aḥaduha ‘alā al-ākhar idha inṭabaq ba‘ḍuhā ‘alā ba‘ḍ fahīya mutasāwīya.”^[69] “Things which none of them exceeds another, when some of them are mapped on some others, are equal.” Avicenna appeals to this simple principle to compare the sizes of different infinite magnitudes or different infinite sets of numbers or numbered things.^[70] He elegantly develops the application of this principle to the context of infinities. The significance of this strategy becomes more evident when we consider that most of the tools we usually use to compare the sizes of finite things are ineffective in the case of infinities.^[71]

The equality of the sizes of two different sets of objects is usually understood as the equality of the numbers of their elements. Since these sets are finite, we can count and determine the number of their elements. If these numbers, which are obtained by two distinct counting procedures, are equal to each other, then we can say that those sets are of the same size; otherwise, they are of different sizes and the larger set is the one whose number of elements is larger. Similarly, we can measure the lengths of two different finite magnitudes separately and one by one, then we can compare the numbers obtained to decide whether they are equal or not. But it is obviously impossible to follow this approach in the case of infinities. Enumerating the elements of an infinite set or measuring the length of an infinite magnitude is impossible.^[72] By definition, the sizes of infinities cannot be described by a finite number. Therefore, comparison between the sizes of different infinities must be grounded on something else: Avicenna’s creative suggestion is returning to Euclid and borrowing his notion of correspondence. Avicenna shows that appropriate interpretations of this notion can be successfully applied to the case of infinities. I will return to this issue in section 4.3.

The Mapping Argument indicates, therefore, that the existence of an actual infinity entails the correspondence between that infinity and some of its infinite parts – which is, according to Avicenna, simply contradictory.^[73] A careful investigation of the notion of correspondence, especially when it applies to the infinite sets of numbers or numbered things, makes it clear that Avicenna’s comprehension of the notion of infinity is much more modern than we might expect. In the previous section I argued that what Avicenna means by correspondence between two sets of numbers or numbered things is one-to-one correspondence between their elements, in such a way that every element of each of those sets corresponds with one and only one element of the other set. So, if I am right, Avicenna believes that on the one hand, (1) correspondence between sets is correspondence between their elements, and on the other hand, (2) the existence of an infinite set entails its correspondence with some of its proper subsets. The moral is that Avicenna sees infinite sets of numbers or numbered things as sets which are in one-to-one correspondence with some of their proper subsets. But this is exactly what Dedekind proposed in 1888 as a definition for infinite sets. Erich Reck’s reading of Dedekind’s definition is as follows:

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. (A set can then be defined to be finite if it is not infinite in this sense.)^[74]

It is definitely surprising that more than eight centuries earlier, Avicenna had had a similar conception of the notion of infinity. This does not mean, however, that the above definition is Avicenna’s own definition of infinity. He was aware, if I am right, that every infinite set of numbers or numbered things has the property of being in one-to-one correspondence with some of its proper subsets (borrowing the first letters of the names of Avicenna and Dedekind, I would like to call this property ‘AD-property’), but he never explicitly proposed having this property as a definition for being infinite.^[75] Another important dissimilarity between Avicenna and Dedekind’s views concerning infinity is that Dedekind, contrary to Avicenna, does accept the actual existence of infinite sets.^[76] In other words, for Avicenna no actual thing can instantiate or exemplify the AD-property.^[77] So, I do not want to exaggerate the commonalities between Avicenna’s view and our post-Dedekindian, post-Cantorian understanding of infinity. My claim is merely that Avicenna had correctly noticed that every infinite set of objects has the AD-property, and this is enough to show that there is a strong connection between his views about infinity and ours. I do not know of any philosopher before Avicenna who was aware of this property of infinite sets.^[78]

Jon McGinnis believes that, before Avicenna, the Ṣābian mathematician Thābit Ibn Qurra al-Ḥarrānī (d. 901) recognized the “modern definition of infinity as a set capable of being put into one-to-one correspondence with a proper subset of itself.”^[79] I disagree. To justify his position, McGinnis refers to two sections of a series of questions addressed by Abū Mūsā ʿĪsā Ibn Usayyid to Thābit Ibn Qurra in which Thābit argues that there is an infinite number of different sizes that an infinite set may have.^[80] I believe, however, that a careful analysis of the relevant passages to which McGinnis has referred reveals nothing confirming that Thābit recognized the idea of one-to-one correspondence and the AD-property of infinite sets. This quotation expresses the core of Thābit’s claim and clarifies the structure of his argument:

TEXT # 8: We [i. e., Ibn Usayyid and his friends] questioned him [i. e., Thābit] regarding a proposition put into service by many revered commentators, namely that an infinite cannot be greater than an infinite. – He pointed out to us the falsity of this (proposition) also by reference to numbers. For (the totality of) numbers itself is infinite, and the even numbers alone are infinite, and so are the odd numbers, and these two classes are equal, and each is half the totality of numbers. That they are equal is manifest from the fact that in every two consecutive numbers one will be even and the other odd; that the (totality of) numbers is twice each of the two [other classes] is due to their equality and the fact that they (together) exhaust (that totality), leaving out no other division in it, and therefore each of them is half (the totality) of numbers. – It is also clear that an infinite is one third, or a quarter, or a fifth, or any assumed part of one and the same (totality of) numbers. For the numbers divisible by three are infinite, and they are one third of the totality of numbers; […] and so on for other parts of (the totality of numbers). For we find in every three consecutive numbers one that is divisible by three, […] and in every multitude of consecutive numbers, whatever the multitude’s number, one number that has a part named after this multitude’s number.^[81]

Here, Thābit is providing an elegant argument that there are infinities of different sizes. He argues that from every two consecutive numbers one is even and the other odd. Therefore, the totality of even numbers is equal to the totality of odd numbers; since the union of these totalities is equal to the totality of numbers, each of the totalities of even or odd numbers is equal to a half of the totality of numbers. By similar lines of argument, Thābit shows that the totality of numbers divisible by three is equal to one third the totality of numbers, and the totality of numbers divisible by four is equal to a quarter the totality of numbers, and so on. But he does not say anything confirming that any of those proper subsets of numbers are equal to the totality of numbers; nor does he mention the idea of one-to-one correspondence. Even his argument for the equality of the set of even numbers to the set of odd numbers is not grounded on the one-to-one correspondence of the elements of these two sets. Paolo Mancosu is right to say:

When ibn Qurra states that odd numbers and even numbers have the same size one should be careful not to immediately read his argument as being the standard one based on one-to-one correspondence, for the motivation adduced does not generalize to other arbitrary infinite sets. Rather, it would seem that some informal notion of frequency (how often do even numbers (respectively odd numbers) show up?) is in the background of ibn Qurra’s conception of infinite sizes (“we find in every three consecutive numbers one that is divisible by 3”).

What would he have replied to the possible objection that there are as many even numbers as natural numbers based on a one-to-one correspondence between the two collections? The text is silent on this issue.^[82]

Without doubt, Thābit’s discussion has genuinely innovative aspects.^[83] But he aims only at proving the existence of different sizes of infinite sets, not at showing the equality of infinite sets to some of their proper subsets by appealing to the notion of one-to-one correspondence. We do not have enough evidence, therefore, to claim that Thābit had recognized that infinite sets of numbers have the AD-property.^[84] At least, the passages cited by McGinnis do not provide such evidence.

It is worth mentioning that neither Thābit nor Avicenna nor any other scholar before Georg Cantor recognized the (either actual or potential) existence of infinite sets of different cardinalities. From the standpoint of modern mathematics, two sets are of the same cardinality if there is a one-to-one correspondence between their elements. Therefore, the existence of infinite sets of different cardinalities means the existence of infinite sets which cannot be put in one-to-one correspondence with each other. Avicenna, as I argued, was aware that infinite sets of numbers can be put in one-to-one correspondence with some of their proper subsets, but he did not know that there are some infinite sets that cannot be put in one-to-one correspondence with the set of natural numbers.^[85] So he did not know about infinite sets of different cardinalities. A fortiori, Thābit (who, as I showed, was not familiar with the notion of one-to-one correspondence) did not know about the different cardinalities of infinite sets. This means that Thābit’s claim that the totality of even numbers is equal to the half of the totality of all numbers should not be understood as the claim that the cardinality of the former set is less than the latter’s.^[86]

In sum, Avicenna employs the notion of mapping or correspondence as a tool for the comparison of the different sizes of infinite magnitudes or infinite sets of numbers or numbered things possessing an order. By The Mapping Argument he shows that an infinite magnitude is necessarily in correspondence with some of its proper parts. He believes and indeed explicitly states that the argument is appropriate to the case of numerical infinities, but he himself does not discuss the details of this application. By (1) appealing to the most natural development of the notion of correspondence to the case of numbers and numbered objects, and (2) relying on some of Avicenna’s own texts and some early commentaries on his works which confirm that Avicenna had such a development in mind, we can say that, according to Avicenna, correspondence between two sets of numbers or numbered things means nothing other than one-to-one correspondence between the elements of these sets. Coupling this fact with Avicenna’s insistence on the applicability of The Mapping Argument to the case of numbers and ordered objects, we can conclude that he was aware, like Dedekind, that every infinite set of objects possessing an order has the AD-property. However, by contrast to Dedekind, he did not propose the having of this property as a definition for being infinite. Moreover, he did not know anything about infinities with different cardinalities, in the modern sense of the term ‘cardinality’. Finally, contrary to Dedekind and most other modern set-theorists, Avicenna rejects the actual existence of infinite sets of numbers or numbered objects. He believes that, on the one hand, (I) every mathematical infinity corresponds with some of its proper subsets, and on the other hand, (II) it is impossible for a set of actually existent mathematical objects to correspond with some of its proper subsets. Therefore, he concludes that it is impossible for a magnitude or a set of discrete mathematical objects to be infinite. His justification for (I) comes from The Mapping Argument, but he himself does not provide any justification for (II); he simply accepts it. I will argue, in the next two sections, that Avicenna’s position regarding the ontology of mathematics justifies (II).

There is still another problem that we have not yet touched on. According to TEXT # 3 and TEXT # 4, The Mapping Argument works against the actual existence of infinite magnitudes and infinite sets of numbers or numbered things having an order. But we have not yet clarified why having an order is a necessary condition for the elements of an infinite set of objects to be the subject of The Mapping Argument. In the following section, I discuss this issue and show how it is related to the possibility of having an actual infinity of immaterial objects.

4.3 Non-Ordered Can Be Actually Infinite

In the following passage, from The Physics of the Salvation, Avicenna explicitly says that The Mapping Argument does not work against infinities such that either their parts do not exist totally together at the same time (e. g., the infinite time line and infinite motions) or they cannot be ordered (e. g., immaterial objects such as angels and devils):

TEXT # 9: [There are two situations in which we may have infinites:] either when [(1) the totality of] the parts are infinite and do not exist all together – therefore, it is not impossible for them to exist one before or after another, but not all together – or when [(2)] the number [i. e., the set of numbered things] itself is not ordered in either nature or position – therefore, there is nothing preventing it [i. e., the non-ordered set of numbered things] from existing [with] all [its members] together. There is no demonstration for its impossibility, rather there is a demonstration for its [actual] existence. As for the first kind [of these infinites], time and motion are proven to be such. As for the second kind, [the existence of] a multiplicity of angels and devils – that are infinite with respect to number – is proven to us, as will become clear to you [too]. All of this [i. e., the set of angels and devils] is susceptible to increase, but this susceptibility does not make the [the application of] the mapping [technique] permissible; for what has no order in either nature or position is not susceptible to [the use] of the mapping [technique]; and [the use of this technique] in the case of what does not exist all together is even more impermissible.^[87]

In this passage Avicenna discusses two conditions for the applicability of The Mapping Argument. Following McGinnis, I call these conditions respectively (1) the ‘wholeness condition’ and (2) the ‘ordering condition’. According to the wholeness condition, The Mapping Argument is applicable to a set or totality only if its elements exist all together at the same time. According to the ordering condition, the argument is applicable to a set or totality only if its elements have an order in either nature or position. Avicenna seems to believe that both magnitudes and sets of numbers (or numbered things) must satisfy the wholeness condition to be the subject of The Mapping Argument. For him it is nonsense to speak about the (non-)equality of things – whether magnitudes or numbers or numbered things – that do not actually exist. This is why time cannot be the subject of the mapping argument. Time does not satisfy the wholeness condition because temporal moments do not exist simultaneously and all together. In each moment there actually exists only one point of the temporal line. Therefore, the infinity of the temporal line and the eternity of the world do not entail the actual existence of an infinity. Since temporal moments do not exist all together at the same time, it is nonsense for Avicenna to say that the whole of the temporal line corresponds with – and is therefore equal to – some of its parts. As a result, The Mapping Argument is not applicable to time and cannot reject its potential infinity.^[88]

The case of the ordering condition is more complicated. Is having an order a necessary condition for both numbers and magnitudes to be the subject of The Mapping Argument? In his discussions of The Mapping Argument Avicenna repeatedly says that this argument rejects the actual infinity of magnitudes and numbers or numbered things having an order in either nature or position (tartīb fi al-ṭabʿ aw al-waḍʿ). Accordingly, in the above passage, he says that no argument, including The Mapping Argument, can reject the actual existence of an infinite set of non-ordered numbered things. It is therefore undisputable that, for Avicenna, the ordering condition is a necessary condition for numbers and numbered things to be the subject of the mapping argument. Is it also a necessary condition for the case of magnitudes? McGinnis argues for a positive answer to this question. He believes that (1) there is a sense in which magnitudes are (or at least can be) ordered and (2) having an order is a necessary condition for magnitudes to be the subject of The Mapping Argument.^[89] The former claim seems to be defensible. However, I think that there are reasons to be suspicious of the latter. To justify these claims, we should first have a clear understanding of what exactly Avicenna means by having an order. In the introduction to his discussion of The Mapping Argument in the Metaphysics part of Alā’ī Encyclopedia Avicenna writes:

TEXT # 10: Beforeness and afterness (pīshī wa sipasī) is either by nature – as in number (shumār) – or by supposition (farḍ) – as in measures (andāza-hā) [i. e., one-dimensional magnitudes] – that you can start from any direction you may want. And, everything that either there is beforeness and afterness in it by nature or it is a magnitude (miqdār) which has parts that exist all together is finite.^[90]

A comparison of this passage with TEXT # 3 and TEXT # 4 shows that Avicenna uses the Arabic phrase ‘tartīb’ as an equivalent for the Persian phrase ‘pīshī wa sipasī’. It indicates that, for Avicenna, a set or totality of things have an order only if for every two members of this totality one of them is, in a sense, before (or after) the other. As Avicenna explicitly confirms in the above passage, numbers have a natural and essential order. This is because for every two numbers one of them is before or less than the other. One-dimensional magnitudes (i. e., lines), on the other hand, do not have a natural order. This is because they lack essential directionality. They can be traversed in two opposite directions. But as soon as we fix one of these directions, we have a beforeness/afterness (or priority/posteriority) relation between the parts of this one-dimensional magnitude. Therefore, every one-dimensional magnitude can in principle have two different suppositional orderings, depending on the direction we consider for it. This demonstrate (1); at least in the case of one-dimensional magnitudes. Nonetheless, it does not show that satisfaction of this condition is necessary for magnitudes to be the subject of The Mapping Argument. Indeed, although they are ordered, it is not in virtue of this orderedness that The Mapping Argument is applicable to them. Let me explain why.

According to Avicenna, correspondence between two geometrical magnitudes L and L′ is nothing more than projection and coverage: L corresponds with L′ if and only if it is possible to map L onto L′ in such a way that no part of either L or L′ remains uncovered. It seems, therefore, that the notion of correspondence could be understood by appealing merely to our geometrical intuitions without the aid of the notion of ordering (or, equivalently, beforeness/afterness). It is in virtue of the continuity and connectedness of geometrical magnitudes – as suggested by TEXT # 6 – that the mapping technique and the notion of correspondence in the sense of coverage and projection are applicable to them. Considering the case of two-dimensional magnitudes shows us that the application of the mapping technique to the case of magnitudes is not in virtue of their orderedness. The sizes of two-dimensional figures can be compared by defining equality in terms of coverage. Consider two triangles drawn on a paper. By mapping one of them on the other and checking whether or not they completely cover each other, we can compare their areas. But there is no obvious/natural order among the points of two-dimensional magnitudes/shapes. More precisely: the natural positions of the points on a two-dimensional magnitude do not impose a beforeness/afterness relation on them. Those points are not ordered by their natural positions. It shows that the applicability of the mapping technique in the sense of coverage and projection is something completely independent from orderedness in the sense of having beforeness/afterness. That is why Avicenna mentions the ordering condition only for the case of numbers. Orderedness does not play any crucial role in the applicability of The Mapping Argument to the case of magnitudes.

Quite differently, correspondence between two sets of discrete objects A and B is one-to-one correspondence between their elements.^[91] But the notion of one-to-one correspondence cannot be reduced to geometrical projection and coverage; for the elements of A and B may be of different sizes, different shapes and have different places (if they have these properties at all). If A and B are ordered then we can compare their sizes by pairing the first element of A with the first element of B, the second element of A with the second element of B, and so on (i. e., for every natural number n, pairing the n^th element of A with the n^th element of B). In fact, Avicenna seems to believe that the possibility of being ordered is a necessary condition for the possibility of being put in a one-to-one correspondence. He thinks that if a set of objects cannot be ordered, it cannot be put in a one-to-one correspondence with other sets.^[92] In other words, the possibility of picking elements of a set one-by-one to put them in a one-to-one correspondence with the elements of another set is equivalent to the possibility of putting an order on the elements of that set. Therefore, if it is impossible for a set to be ordered, then it is equivalently impossible for it to be put in a one-to-one correspondence. Accordingly, the size of such a set – that cannot be ordered – cannot be compared to the sizes of other sets, including its own proper subsets. As a result, it cannot be shown that that set corresponds with some of its subsets. So, no contradiction arises and The Mapping Argument does not work in such a case.^[93]

Apparently, it is by following this line of argument that Avicenna claims that The Mapping Argument is not applicable to the case of immaterial objects such as angels and devils; for he believes that the set of those objects cannot be ordered. More precisely, they do not have a natural and essential order, nor it is possible to posit a conventional order for them (at least it seems inconceivable how they might have such an order). The actual infinity of such sets cannot be rejected by appealing to The Mapping Argument. Avicenna seems to believe that (1) other arguments against the actual infinity are of no use in the case of sets of discrete elements, and (2) there are some arguments for the actual infinity of the set of angels and demons. Consequently, he believes that such infinite sets are uncontroversially actual.^[94]

But what will happen if – by contrast to the case of angels and devils – it is possible to put a conventional order on a set of immaterial objects? Can we still argue, based on the mapping technique, that such a set cannot be actually infinite?

For example, Al-Ghazālī (1058–1111) argued that the eternity of the world implies the eternity of the species human being. This means that every moment of time is preceded by an infinite number of people who have died. Therefore every moment of time is preceded by an infinity of human souls who have been separated from matter but still actually exist. Moreover, Al-Ghazālī believes that we can put a conventional order on (at least a subset of) the set of the souls who have been separated from matter until now.^[95] Suppose that the first element of the set A is the last soul who was separated from her body before the present time, the second element of A is the last soul who was separated from her body before the end of yesterday, the third one is the soul who was separated from its body before the end of the day before yesterday, and so on.^[96] The eternity of the world and human beings implies the actual existence of and the infinity of the set A. But, since A is ordered, its actual infinity can be rejected by The Mapping Argument. This is simply a contradiction. Based on this line of argument, Al-Ghazālī concludes that Avicenna’s doctrine of the eternity of the world and human being contradicts his rejection of the actual infinity.^[97]

I think that we have a strong strategy to rebut this objection on behalf of Avicenna. Having an order guarantees that we can employ the mapping technique to argue that an infinite set of discrete objects corresponds with some of its proper subsets. But it cannot itself guarantee that such a correspondence is a contradictory conclusion. Avicenna can say that although the correspondence of a whole to its proper parts is a contradiction in the case of material objects and their properties (e. g., mathematical objects), such a correspondence is not controversial in the case of immaterial objects. In other words, the equality of the whole and its part is unacceptable for physical and mathematical objects, but it is acceptable for immaterial objects, because the mereology of the realm of materiality is not similar to that of the realm of immateriality. The principle of the impossibility of the equality of the whole and its proper part is not necessarily valid in the latter realm. Therefore, it is in principle possible for some ordered infinite set of objects to exist actually; but they cannot be material or dependent on materiality.^[98]

According to this solution, there are two crucial steps in the application of The Mapping Argument: (1) employing the mapping technique to show that a whole corresponds with some of its proper subsets, and (2) extracting a contradiction from such a correspondence. Having an order, either natural or conventional, guarantees that the first goal can be accomplished. The second one depends on the ontology of the objects under discussion. If they are material, such a correspondence is absolutely unacceptable. But if they are immaterial, it may be justified. It is worth emphasizing, however, that I do not claim that there is no such immaterial whole-part correspondence that is contradictory.

In sum, Avicenna believes that The Mapping Argument (his only argument against the actuality of discrete infinity) does not work in the case of those sets of immaterial objects which cannot be ordered. It is therefore possible, in principle, that some such sets are actually infinite. However, such an infinite immaterial numerosity cannot be the subject of mathematical studies, because Avicenna believes that numerosity completely separated from matter should be studied by metaphysics, not mathematics. Moreover, it is not conceivable, at least from Avicenna’s standpoint, that one could undertake mathematical studies on a set that is not ordered and cannot be put in a simple one-to-one correspondence. It is also worth mentioning that there is no immaterial infinite continuity; there is no immaterial infinite magnitude. This is because magnitudes, according to Avicenna’s ontology of mathematics, by contrast with numbers, cannot be separated from matter. They have an ontological dependency on matter. Therefore, there is neither finite nor infinite immaterial continuity.^[99]