Abstract
Ibn Sīnā (980–1037), one of the most influential philosopher-scientists who was known to the West by the Latinised name Avicenna, has introduced a major shift in the philosophy of mathematics. His conception of number is structured into 5 main conceptual developments: (1) the recognition of mathematical objects as intentional entities and the acknowledgment that this amounts to provide an intentional notion of existence; (2) the link between the intentional act of apprehending unity and the generation of numbers by means of a specific act of repetition made possible by memory; (3) the identification of a specific intentional act that explains how the repetition operator can be performed by an epistemic agent; (4) the development of a notion of aggregate (or constructive set) that assumes an inductive operation for the generation of its elements and an underlying notion of equivalence relation; (5) the claim that plurality and unity should be understood interdependently (we grasp plurality by grasping it as instantiating an invariant).
There is no knowledge of philosophy without knowledge of the mathematical sciences… hence he who wants to acquire knowledge of philosophy should first begin studying the books of mathematics according to the order I have established.
Al-Kindī in Rasā’il al-Kindī al-falsafiyya, p. 378.
This paper was part of a wider investigation that was recently published under the title Mathematics and the Mind. An introduction into Ibn Sīnā’s Theory of Knowledge, Springer Brief Series, 2016.
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Notes
- 1.
The Arabic-Islamic tradition is heavily under-researched because of the structure of modern scientific research that reflects a peculiar perception of the development of the history of science. Little efforts have been made to update the prevailing historical knowledge despite, as Rashed points out, “the accomplishments of the last five decades that outweigh everything we owe to the last two centuries.” (Rashed p. 3, 2015) As a a result, many modern scholars continue to skip over the Arabic-Islamic period as if it has never existed! For more on this topic, see Tahiri (2018).
- 2.
Unless indicated otherwise, all Ibn Sīnā’s quotations are taken from his al-Ilāhiyāt, the number of pages and paragraphs refers to the bilingual edition of the book which was translated by Marmura under the title “The Metaphysics of The Healing”.
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It is this grasping of elementary arithmetic processes like sum as operators which will be crucial to his account for repetition by remarkably making it the most basic mathematical operator. It is important to point out that Ibn Sīnā does not conduct this linguistic analysis by chance for he explicitly mentions that the analysis of language is among the tasks of the logician: “because there is some link between word and meaning, the modes of the word can influence the modes of meaning. That’s why the logician should also consider the absolute semantic aspect of the word inasmuch as it is not restricted by the language of a certain people to the exclusion of other, except in rare cases” (Ibn Sīnā 1983, p. 131). In Manṭiq al-Mashriqiyyīn, he becomes closer to the pragmatist position of the linguists for he begins with this topic in which logical analysis is tightly linked to the discussion of the meaning of sentences. A prominent example is his analysis of Arabic sentences, he explains their structure by admitting the existence of a second type of propositions which are made of just two components and he points out that the copula is not needed to link the subject to the predicate:
- 8.
Here is one of the passages in which Frege argues against the view that number is a property of things; “It marks, therefore, an important difference between colour and Number, that a colour such as blue belongs to a surface independently of any choice of ours. […] The Number 1, on the other hand, or 100 or any other Number, cannot be said to belong to the pile of playing cards in its own right, but at most to belong to it in view of the way in which we have chosen to regard it; and even then not in such a way that we can simply assign the Number to it as a predicate.” Frege (1884, p. 29, §22).
- 9.
This is strikingly similar to Frege’s objection that will be discussed later.
- 10.
In his Commentary on al-Ilahiyāt, al-Shirāzī better known as Mulla Ṣadrā (1572–1640) further illuminates what Ibn Sīnā has in mind by this distinction: “the unity which is a principle of the mathematical numbers is other than the unity which can be found in the separate substances for the separate substances do not possess a quantitative number generated by repetition of similar units.” (in Avicenne 1978, p. 302).
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But time seems to be essential not to counting as such but to the actual recognition of identity for it looks as if the mind needs time to be able to effectively objectify a phenomenon. Further discussion of this topic is out of the scope of this paper.
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This general concept of al-qiyās, which can be translated as co-relational inference (Young 2016), comes from al-fiqh or Islamic jurisprudence, it is an iterative constructive procedure by which the jurists infer a general law of which particular legal judgements appear as instantiations, the same concept is used in similar way by al-Khwārizmī in his landmark book Algebra and al-Muqābala to capture the relationship between particular equations and the six canonical equations of which they appear as instantiations (Al-Khwārizmī 2007, p. 51). It is important to point out that Ibn Sīnā, who belongs to the ḥanafite juridical school like al-Khwārizmī , describes the relation of a universal to individuals as one-to-many, he further makes sure to distinguish it from the psychological conception making it in effect a logical-epistemic relation, that is why his construction of the natural numbers can be called a logical-epistemic construction. This is the relation that he captured by “falling under a concept” in his later work Pointers and Remainders.
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This principle is clearly expressed by Ibn Sīnā’s predecessor al-Fārābī in the following passage:
And we can also say in each of the two things in virtue of each one of them leads to the same purpose that they are all the same. We therefore say regarding their plurality use whatever you wish since they are one and the same. (Al-Fārābī 1989, §3 p. 38; my emphasis)
Al-Fārābī distinguishes between the two kind of substitutions: the substitution of elements of an equivalence class and the substitution of names that refer to the same object, he calls this identity relation “wāhid bi al-‘adad i.e. one in number” or more significantly “wāhid bi‘aynihi i.e. one in itself.” One of the interesting examples he provides is the customary use of name and kunya to refer to the same person (ibid., p. 41 §6). Ibn Sīnā provides the following specific example in his al-Ilāhiyāt: “Zayd and Ibn ‘Abdallah (i.e. ‘Abdallah’s son) are one” (p. 74, §2); many of the chapters of book 3 in which Ibn Sīnā discusses, among other things, the one and many cannot be very well understood without reading al-Fārābī’s distinctive Kitāb al-wāḥid wa al-waḥda (On One and Unity). Ibn Sīnā has not only explicitly acknowledged his debts to al-Fārābī in his autobiography (in Gohlman 1974, pp. 32–34), the latter seems to be the only philosopher that he praised and appreciated among all his predecessors as he declared in the following passage:
As for Abū Naṣr al-Fārābī, we must have a very high opinion of him and he should not be put on the same group of people
for he is all but the most excellent of our predecessors 
May God facilitate the meeting with him, so it shall be useful and beneficial. (in Badawi 1978, p. 122) - 20.
One is astounded by those who define number and say, “number is a plurality composed of units or of ones,” when plurality is the same as number
and the reality of plurality consists in that it is composed of units. Hence, their statement, “plurality is composed of units,” is like their saying, “plurality is plurality 
.” For plurality is nothing but a name for that which is composed of units 
. (p. 80, §5; my emphasis) - 21.
Ibn Sīnā explicitly states here an equality relation between the two members, hence we can substitute whatever term we like in the first proposition. A similar proof of a circular definition using substitution is expressed in the following passage:
If you say ‘The thing is that about which
it is valid to give an informative statement 
,’ it is as if you have said, ‘The thing is the thing about which 
it is valid to give an informative statement 
because the meaning of ‘whatever’ 
, ‘that which’ 
and the ‘thing’ 
is one and the same. You would have then included ‘the thing’ in the definition of ‘the thing’. (p. 24, §7) - 22.
Hodges and Moktefi (2013, p. 79).
- 23.
Ibid., p. 95.
- 24.
It is interesting to point out that al-Jurjānī (1340-1413) knows the transitivity relation that he calls the rule of equivalence (qiyās al-musāwāt). In his famous Mu‘jam al-ta‘rīfāt or Dictionary of Definitions, he provides two kinds of example to illustrate his definition. The first represents the class of relations which are transitive like equality, and the second those relations which are not transitive like “to be half (niṣf)” as he explains: “A is half of B and B is half of C, it is not true (falā yaṣduqu) to infer that A is half of C since half of half is not a half but a quarter.” (Al-Jurjānī 2004, p. 153).
- 25.
Al-Qiyās II. 4 in Hodges and Moktefi (2013, p. 36).
for he is all but the most excellent of our predecessors 
May God facilitate the meeting with him, so it shall be useful and beneficial. (in Badawi 
and the reality of plurality consists in that it is composed of units. Hence, their statement, “plurality is composed of units,” is like their saying, “plurality is plurality 
.” For plurality is nothing but a name for that which is composed of units 
. (p. 80, §5; my emphasis)
it is valid to give an informative statement 
,’ it is as if you have said, ‘The thing is the thing about which 
it is valid to give an informative statement 
because the meaning of ‘whatever’ 
, ‘that which’ 
and the ‘thing’ 
is one and the same. You would have then included ‘the thing’ in the definition of ‘the thing’. (p. 24, §7)References
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Tahiri, H. (2018). The Foundations of Arithmetic in Ibn Sīnā. In: Tahiri, H. (eds) The Philosophers and Mathematics. Logic, Epistemology, and the Unity of Science, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-93733-5_13
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