2. Mathematical Objects: A General Picture

Do mathematical objects exist? One may consider this question to be a paraphrased form of a more specific question: Are mathematical objects mind-independent substances? Nonetheless, from the perspective of Avicenna’s philosophy, we should distinguish these two questions. His answer to the former question, but not the latter, is trivially positive. In Avicenna’s philosophy, existence (wujūd) and thingness/objecthood (shay’iyya) are distinct but coextensive concepts. ^{Footnote 14} This justifies the interchangeable use of ‘existent’ and ‘object’ in the context of Avicenna’s philosophy. ^{Footnote 15} It also entails that not only mathematics but all sciences are pre-scientifically committed to the existence of their subject matters, i.e., their objects. Every science, inasmuch as it is a science, studies some things or objects. Moreover, thingness/objecthood and existence are coextensive, such that science studies some existents and, consequently, carries ontological commitments to its subject matter. Subject matters of all sciences exist, but that does not entail that they exist in the same way. Existence can be qualified in many different ways, and everything exists in a certain way. Avicenna believes that mathematical objects do exist, but not as mind-independent substances. Hence, his answer to the second question is negative.

Mathematical objects or subject matters of mathematical studies are quantities (kammiyyāt). They are either (a) continuous (muttaṣil) quantities or magnitudes (maqādīr), which are geometric objects (or shapes), or (b) discrete (munfaṣil) quantities or numbers (aʿdād), which are arithmetical objects. Both of these two groups of mathematical objects are accidents of material substances, ^{Footnote 16} which have mind-independent existence, but as accidents dependent on material substances, rather than as autonomous substances. Therefore, mathematical objects are not primarily mental constructions. However, we can separate them, in our minds, from the particular material substances to which they are attached in the extramental realm. Nonetheless, even in our minds, they have some sort of dependency on matter and materiality. A careful analysis of Avicenna’s writings on the classification of the sciences ^{Footnote 17} reveals that only subjects of metaphysical studies can be completely released from all sort of dependencies on matter and materiality.

According to Avicenna’s categorization of the sciences, two sciences are distinct either because they study objects with different natures or because they study objects with the same nature but from different aspects (ḥaythiyyāt). ^{Footnote 18} He believes that theoretical sciences are divided into natural sciences, mathematics, and metaphysics. Every object of a natural science is mixed with a specific kind of matter in both the external world and the mind. It may be possible to abstract this object, in the mind, from the specific kind of matter with which it is mixed. However, if we do so, the abstracted object cannot be the subject of theoretical studies in a natural science, ^{Footnote 19} but should instead be studied by mathematics or metaphysics. Every object of a natural science, inasmuch as it is the subject of theoretical studies in a natural science, is associated with a specific kind of matter. The objects of mathematics are similarly mixed with specific kinds of matter in the external world. Nevertheless, we can separate these objects, in our minds, from all particular kinds of matter. Nonetheless, this does not mean that mathematical objects are completely separated from materiality itself and that they have no dependency on matter. An object free from any kind of dependency on materiality cannot be the object of mathematical study; it should be studied in metaphysics. Given this classification, mathematical objects are separable from any specific kind of matter, but they still have some sort of dependency on materiality itself. The following passage supports this understanding:

According to this passage, the objects of natural science are mixed with specific kinds of matter in both the extramental world and the mind. Mathematical objects are similarly associated with specific kinds of matter in extramental reality, but they can be separated from all specific kinds of matter in the mind. This passage does not explicitly say whether mathematical objects still have some sort of materiality or dependency on materiality in the mind. However, there is a hint that this is the case. It seems that if we purify number of all characteristics of materiality, then the result is number inasmuch as it is number which, as Avicenna says in the above text, is the subject of metaphysical, not mathematical, studies. Admittedly, we need more persuasive evidence to support the dependency of mathematical objects on materiality in the mind. The nature of this dependency (if there is such) is itself unclear. So, it is better to address the subtleties of Avicenna’s view about geometric shapes and numbers. In the next section, I will discuss his views on geometric shapes.

3. Geometric Objects

Avicenna believes that geometric shapes, even in our minds, have some sort of necessary association with materiality. They are separable from all specific materials in our minds, but not from materiality itself. I will try to establish and expand this rendition of Avicenna by gleaning textual evidence for it from his various works. I start by analyzing a passage from the Isagoge:

If we consider what Avicenna says in this text about the quiddity (mahiyya) of squareness as his general view about quiddities of geometric shapes, then we should conclude that for him these quiddities have no existence unless undergoing admixture with motion and matter. ^{Footnote 22} In other words, it is impossible for them to be fully detached from materiality. Geometric shapes, inasmuch as they are geometric shapes, are necessarily mixed with materiality (because they lie under one of the subdivisions of the group (Ia) mentioned in the text). Nonetheless, according to this text, our estimation (wahm) has the ability to separate geometric shapes from all specific kinds of matter with which they may be mixed in the external world (because they lie under the group (Ia-2) of objects mentioned in the text). Geometric shapes, inasmuch as they are geometric shapes, are not necessarily associated with a specific kind of matter, though they are mixed with materiality. ^{Footnote 23} A square, inasmuch as it is square, is not necessarily mixed with gold, wood, or any other specific kind of matter, but it is necessarily associated with materiality. Therefore, contrary to the concepts of non-wooden triangle or non-golden triangle, which are easily intelligible, the concept of immaterial triangle is a self-contradictory and unintelligible concept, ^{Footnote 24} just as impossible as round square. Materiality is integrated with the quiddities of geometric shapes. Avicenna says that the core of the truth about geometric shapes, which Platonists and Pythagoreans have not ascertained, is that:

Here, Avicenna explicitly embraces the notion that association with materiality is a characteristic of not only the extramental existence of geometric shapes, but also of their definitions. This text explicitly shows that Ardeshir’s reading of Avicenna, according to which mathematical objects are “not combined with matter in definition but with matter in existence,” ^{Footnote 26} is misleading if not wrong. Geometric shapes, even in our minds, are connected to matter. However, it remains obscure how it is possible for a mental existent to be mixed with matter but not with a specific kind of matter. Obviously, when we consider a geometric shape in our minds as an object of our cognition, it is fully separated from the materiality that exists in the physical world. So, the materiality from which we cannot separate geometric shapes in our estimation is not the former kind of materiality existing in the extramental world. ^{Footnote 27} Geometric shapes are associated with some sort of estimative or, in Aristotelian terms, intelligible matter which may be considered as the cognitive counterpart to the perceptible materiality in the physical world. ^{Footnote 28} We can say, at least metaphorically, that geometric shapes are mixed with some sort of estimative or intelligible matter, which is neither any specific kind of matter we have in the external world, nor separable from geometric shapes. A significant consequence of this inseparability from intelligible matter is that geometric shapes are necessarily associated with material forms (ṣuwar māddiyya). Avicenna says:

Geometric objects, inasmuch as they are what they are, are necessarily attached to intelligible matters. They are always in the forms of material objects. ^{Footnote 30} So, they have some sort of ontological admixture and association with (or dependency on) materiality, or, more precisely, on material forms. ^{Footnote 31} In our estimation, we can separate them from all the particular matters mixed with which they may exist in the physical world; nevertheless, they remain attached to their material forms. ^{Footnote 32} It is impossible to conceive of geometric objects as being separated from their material forms. I shall now turn to an investigation of the nature of numbers.

4. Numbers

Avicenna’s position on the status of numbers differs slightly from his views on the nature of geometric objects. Numbers have no necessary association with material forms, but, inasmuch as they are the subject of arithmetical studies, they still have some sort of dependency on materiality. This long passage sets out the main characteristics of numbers:

This text reveals that numbers, inasmuch as they are numbers, are mixed with neither any specific kind of matter, nor even materiality itself. Therefore, they can be found and considered in three different forms, considered inasmuch as: (1) they are what they are, fully separated from materiality (i.e., category (a) of the text), (2) they are accidents of material things, associated with a specific kind of matter (i.e., category (b1) of the text), (3) they are accidents of material things, but dissociated from any specific kind of matter (i.e., category (b2) of the text). Subject matters of arithmetic are numbers only when they are considered in the latter form.

Numbers mixed with some specific kinds of matter should be studied in natural science. For example, the number four, inasmuch as it is accidentally true of the four elements, should be studied in natural science. On the other hand, numbers, inasmuch as they are what they are, fully separated from materiality, should be studied in metaphysics. To be the subject of arithmetical studies, numbers should be considered as accidents of material things. They are associated with materiality, but not necessarily with a specific kind of matter. The concept of immaterial number, contrary to the concept of immaterial triangle, is plausibly intelligible. ^{Footnote 34} When Avicenna discusses numbers in his metaphysics, he does indeed have a fully immaterial conception of numbers in mind. ^{Footnote 35}

This discussion shows that there is a dissimilarity between geometric shapes and numbers. Geometric shapes, inasmuch as they are what they are, are necessarily associated with materiality or material forms. As we saw in TEXT # 1, TEXT # 2, and TEXT # 3, Avicenna believes that inseparability from materiality is included in the definition of geometric shapes. Therefore, the dependency of geometric objects on materiality is an ontological dependency, in contrast to the dependency of numbers on materiality, which is only epistemological. Numbers, inasmuch as they are what they are, have no necessary accompaniment with materiality. But, inasmuch as they are the subject of arithmetical studies, we should consider or regard them as things dependent on materiality (i.e., accidents of material things). Therefore, numbers are not ontologically intertwined with materiality. It is only the consideration (naẓar) of the arithmetician that preserves numbers in association with materiality (not the ontological status of numbers inasmuch as they are numbers). Hence, numbers, inasmuch as they are the subject of arithmetical studies, have some sort of epistemological dependency on materiality. I will now show why we need to consider this dependency for numbers, and why numbers, inasmuch as they are numbers, cannot be the subject of mathematical studies:

This text contains at least three important points: First, it corroborates the view that number can be found in both inseparable and separable things. ^{Footnote 37} This means that number, inasmuch as it is number, is neutral with respect to materiality. As we saw, geometric shapes are not neutral in this respect. This alone, however, is not enough for us to conclude that numbers, inasmuch as they are the subject of arithmetical studies, are neutral with respect to materiality. Here we find the second point I want to make. Number, inasmuch as it is the subject matter of arithmetic, should be capable of participating in relations of decrease, increase, addition, subtraction, etc. And it is so only when it exists as accidents of material things. ^{Footnote 38} In our estimative faculty, we can separate numbers from all particular materials while preserving those aspects that they have acquired only after admixture with materiality. We separate numbers from materiality in our estimation, but consider them as if they had some material aspects and capabilities.

In another phrase, very similar to the view Avicenna expresses in the first sentence of TEXT # 6, he says that number “would apply to [both] sensible and non-sensible things. Thus, inasmuch as it is number, it is not attached to sensible things.” ^{Footnote 39} Ardeshir concludes from this that “discussion about number and its relations should be understood as abstracted from sensible objects, not when it may belong to sensible objects. So, discussion about numbers is not about sensible objects.” ^{Footnote 40} But this interpretation is misleading. As we saw, discussion of numbers, inasmuch as they are numbers, should be understood as abstracted from sensible objects, but discussion of numbers and their mathematical relations, inasmuch as they are the subject of arithmetical studies, are not completely independent of sensible or, more precisely, material objects.

The third interesting point in this text is that Avicenna does not say that numbers in our estimation are attached to (or associated with) materiality or nature; he says only that they have some sort of dependency (istinād) on nature, which seems weaker than an ontological association with materiality. That is what I call ‘epistemological’ dependency on matter.

What Avicenna says in TEXT # 6 is intended to refute the claim of a person who might say: “The purely mathematical things examined in arithmetic and geometry are also ‘prior to nature’—particularly number, for there is no dependency at all for its existence on nature because it cannot be found in nature.” ^{Footnote 41} Tahiri confusingly considers this passage to be something that Avicenna believed. He writes:

This interpretation is clearly false, and not just because it is anachronistic. According to Avicenna, as I have shown, arithmetic has an epistemological dependency upon nature. To summarize:

1. (I) Geometric shapes, inasmuch as they are geometric shapes, are mixed with estimative matter (or, consequently, with material forms). They have an ontological dependency on materiality.

2. (II) Numbers, inasmuch as they are numbers, are not mixed with materiality; but, inasmuch as they are the subject of arithmetical studies, they should be considered as mixed with materiality. They have an epistemological dependency on materiality.

Numbers, are, in a sense, more abstractable or more separable than geometric shapes. The subject matter of arithmetic is somehow closer to the subject matter of metaphysics. Hence, it is plausible to expect that the difference in the ontological status of numbers and geometric shapes would lead to further epistemological differences between geometry and arithmetic. However, I will refrain from engagement in this debate, which merits independent study.

In his discussion of the division of sciences, McGinnis says that, for Avicenna, “[t]hose existents that can be conceptualized without matter, even though they are necessarily mixed with motion and never subsist without matter, are the subject of mathematical sciences.” ^{Footnote 43} My analysis shows that this picture of the nature of mathematical objects suffers from imprecision. If by ‘matter’ McGinnis means particular kinds of matter existing in the physical world, then he is right. Mathematical objects can be conceptualized without any particular matter. However, if he intends to convey materiality qua materiality by ‘matter,’ then his interpretation is false. Mathematical objects, inasmuch as they are the subject of mathematical studies, cannot be abstracted from the materiality itself.

Unfortunately, we cannot arrive at a reliable understanding of Avicenna’s view on the subject matter of the theoretical sciences by consistently fixing one of these two meanings of ‘matter’ in McGinnis’s book. He writes:

As we saw, if by ‘matter’ he means materiality itself, then mathematical objects cannot be conceptualized without matter. On the other hand, if by ‘matter’ he means particular matters existing in the physical world, then subsisting and being conceptualized without matter (in this new sense) is not sufficient for being the subject of metaphysical investigation. Objects of metaphysics are separated from all material forms and from materiality itself, not merely from special kinds of matter. ^{Footnote 45}

5. Actual and Potential Perfect Objects

Mathematical objects, as we saw, are primarily accidents of material things in the external world. Mathematical enquiry is, therefore, primarily about accidents of material objects, not about independent immaterial entities. We can obtain a universalized conception of mathematical objects by abstracting them, via our faculty of estimation, from all particular materials with which they may be mixed in the extramental world. This purification procedure can, in principle, end in the production of some intelligible forms of exact and perfect mathematical objects that are not easily perceptible in tangible objects. ^{Footnote 46} But does this necessarily mean that there are no perfect mathematical objects in the physical world? Does it necessarily mean that perfect mathematical objects are merely mental constructions that have no counterpart in the external world? I will argue, in this section, that Avicenna endorses the existence of at least some perfect mathematical objects in the extramental world.

The most important evidence for Avicenna’s agreement with some sort of literalism is that, wherever he discusses the nature of mathematical objects in his writings, he affirms that mathematical objects exist in the external world in association with determinate kinds of matter (as accidents of specific material particulars). However, he simultaneously insists that mathematical forms are not sensible natural forms. ^{Footnote 47} Some might object that, although this provides strong evidence for the existence of mathematical objects in the extramental world (albeit not as independent substances), this cannot count as evidence that those objects are perfect. I think that this objection is untenable. If it stood, then Avicenna would need to distinguish between two sorts of perfect and imperfect mathematical objects, such that the latter could exist in the external world mixed with matter, but the former could exist only in the mind. He would need to say that, for example, quasi circular objects (which approximate the ideal shape of a circle but do not really satisfy the mathematical definition of a circle) could exist in the extramental world but quiet circular objects (which perfectly satisfy the mathematical definition of a circle) could exist only in the mind. This he does not do. ^{Footnote 48} As we will see, he denies that what we see in the external world are perfect mathematical objects, but this is so just because mathematical forms are not sensible (visible) forms, not because we see a mathematical form that is imperfect. Paying attention to the epistemological formalities that Avicenna proposes for grasping mathematical forms and producing mathematical concepts will show how mathematical literalism can, by and large, be compatible with Avicenna’s philosophy.

Interestingly, the mental faculty involved in apprehending mathematical concepts is estimation. Discussing the details of the role estimation plays in forming mathematical concepts and attaining mathematical knowledge is outside the scope of this paper. But consideration of some other objects of the estimative faculty may help us to reach a better understanding of the existential mode of mathematical objects in the extramental world. According to Avicenna’s epistemology, estimation is a bodily faculty with a distinct and unique cognitive power that lies between imagination and intellect in the hierarchy of cognitive faculties. Some of its activities are common to both humans and animals, while others are exclusively human. Looking at one of its activities will give us a better understanding of the role of estimation in perceiving mathematical objects in the external world; this activity is likened to incidental perception. ^{Footnote 49} When somebody perceives the sweetness of a yellow cake just by seeing it, she has an incidental perception. She has perceived a sensible form without employing the right perceptive faculty that we usually use to apprehend similar sensible forms. She has perceived the sweetness of the cake without tasting it. Such an apprehension, according to Avicenna, is feasible only because of our estimative faculty. It enables us to ‘see’ the sweetness of a yellow cake. Supposing that her apprehension is reliable and that the cake really is sweet, she has apprehended, by the aid of her estimation, something which really exists in the external world but is imperceptible by the sense she uses. ^{Footnote 50} It can be argued, analogously, that when we see a triangular wooden shape or a group of three balls we apprehend perfect mathematical objects (a perfect triangle or the number three ^{Footnote 51} ) that really exist in the extramental world as accidents of material objects; but they are not visible, or available, to our sensory faculties. ^{Footnote 52} Estimation, among its other roles, enables us to apprehend those things that actually exist in the extramental world but are invisible. ^{Footnote 53}

The existence of numbers as perfect mathematical objects in the external world seems more defensible than the existence of perfect geometric objects. The twoness of two tomatoes is as perfect as the twoness of two books or the twoness of one imaginary Santa Claus and one (hopefully real!) Christmas gift. They are different instantiations of the same universal concept, i.e., twoness. Twoness is not directly sensible, like whiteness or warmth, but we apprehend it thanks to the estimative faculty, and it is as real as the existence of any ordinary accidents that material objects may have. Hence, there is no serious hindrance to the attribution of arithmetic literalism to Avicenna.

The existence of perfect geometric objects, on the other hand, might seem more improbable. At first glance, it seems obvious that there is no perfect geometric object in the physical world. Since everything there has width, length, and depth, there is no perfect line with no width; hence, there is no perfect triangle. In fact, there are some passages from which someone may conclude that Avicenna endorses this view, i.e., anti-literalism. For example, he accepts that, when we want to prove a geometric theorem based on a composition of geometric shapes that we have drawn on a piece of paper, what we have drawn are not perfect geometric objects, and what we are trying to prove is not about those visible figures. He endorses a view that he attributes to Aristotle:

There is no doubt that perfect geometric objects cannot be drawn. Moreover, they are completely invisible. But this does not necessarily entail that there is no perfect geometric object in the external world. ^{Footnote 55} There is some evidence that supports the claim that Avicenna accepts the existence of perfect geometric objects in the external world, though not as sensible things. It can be argued, compatibly with Avicenna’s philosophy, that the role of estimation is not merely to construct perfect mental objects that have no counterpart in the extramental world. Estimation, at least in some cases, helps us to apprehend some sort of non-sensible perfection that really exists in the extramental world. There are some passages that support this construal. For example, Avicenna says that the ascertained doctrine is that:

From one perspective, this text is a criticism of Platonism. Avicenna believes that perfect geometric objects have no immaterial independent existence. From another perspective, it is a confirmation of the actual existence of perfect mathematical objects (e.g., point and line) in the extramental world. It is worth noting that the actual existence of point, line, and surface in the external world do not entail their separability from each other outside the mind. Avicenna emphasizes that we can separate point from line, line from surface, and surface from body (jism) only in our estimation. ^{Footnote 57} They exist in the external world, but they are not distinctly perceptible and cannot be separately predicated upon material particulars. Analogously, we can say that perfect triangles exist in the external world (e.g., in triangular bodies), but they are not distinctly perceptible and cannot be separately predicated upon material particulars. Given these considerations, the actual existence of at least some perfect geometric objects in the physical world seems, by and large, compatible with the tenor of Avicenna’s writings on the nature of mathematical objects. So, it is not incautious to say that he endorses some sort of geometric literalism.

Undoubtedly, a great deal of work is needed to establish literalism as a plausible view about the nature of mathematical and especially geometric objects. But my concern here is the compatibility and consonance of literalism (in the sense described) with Avicenna’s philosophy, rather than the plausibility of the view itself. My arguments show that he, by and large, endorses some sort of literalism. ^{Footnote 58}

Even if we accept the actual existence of some perfect mathematical objects in the physical world, it is undeniable that most mathematical objects do not exist in the physical world. If the number of material objects is finite, ^{Footnote 59} then there are some large numbers (larger than the number of all objects that we have in the extramental world) that are not accidents of any group of numbered material objects; so, those numbers do not literally exist. On the other hand, as Avicenna admits, there are many geometric objects that do not exist in the physical world. ^{Footnote 60} These objects, by the aid of the imaginative and estimative faculty, ^{Footnote 61} can, in principle, be constructed in the mind. But, obviously, there are infinitely many of these objects that have never been constructed. Consider a very large number (larger than the number of physical objects and larger than the largest number we have ever thought of), or a very strange geometric shape that nobody has never thought about. These objects do not exist, either as accidents of material objects in the physical world or as objects constructed by imagination and estimation in a human mind. Of course, if someone decides to construct such an object in her mind, she may succeed. But before that, these objects do not actually exist. Nevertheless, we can still attribute some sort of potential existence to these objects. We have the potentiality to create them in our minds, so they potentially exist. Therefore, Avicenna is somehow a potentialist: at least some mathematical objects only potentially exist. More precisely, some mathematical species exist only in a potential sense of existence. ^{Footnote 62}