Abstract
One decisive point in Ibn Sīnā’s strategy is to acknowledge that numbers are objects. Furthermore, if they are objects they possess (some kind of) reality.
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Notes
- 1.
This text appears to make irrational numbers nonintentional objects which assumes that they are part of the nonexistents, which is not the case for the following reason: “reference to the nonexistent that has no concept in any respect at all in the mind is impossible” (p. 25, §13). Ibn Sīnā tells us here why nonexistents are not intentional objects: the words used seem to refer to something while in fact they do not refer at all that is why it is impossible for the mind to use means by which to have access to what they refer; an example of such nonexistents is the squared circle or the set of all sets that are not members of themselves. To distinguish irrational numbers and nonexistents, Ibn Sīnā uses one of his key epistemic concepts, the notion of accessibility relation, in this significant passage: “there are, here, relationships in irrational roots and in numerical relations that are easily accessible (قريبة المنال) to the soul.” Unlike nonexistents, irrational numbers, though they fail to refer to external things, are within reach of the mind for it can actually “see” the limit of their infinite development. As a result of the apprehension of their convergence, irrational numbers can be accessed and manipulated by the mind through symbols and numerical relations. Al-Manāl or accessibility, in the sense of within reach, is thus a specific mental act by which the mind can grasp and refer to a determined ongoing open process that does not need to be (assumed) actually realised. It is not though clear what Ibn Sīnā astonishingly means by “there are here.” The next sentence appears to say that what he has in mind is the manipulation of irrational numbers due to the successful extension of arithmetical operations to them by al-Khwārizmī. If it is the case, the statement can then be seen as the conclusion drawn by Ibn Sīnā in which he seems to adopt some kind of structural epistemic realism in physics according to which the human mind has access to the external world through mathematical structures by constructing relations that are satisfied by some class of objects, as suggested by the following passage: “knowledge (معرفة) of the order of the arrangement of the spheres can only be arrived at through astronomy (علم الهيئة); and astronomy is only arrived at through the science of arithmetic and geometry.” (p. 15, §6).
- 2.
The theory of intentionality in the Arabic tradition is another chapter of research yet to be investigated. The jurists are behind its development since questions about the meaning of legal rules and their application to individual cases are an essential part of the discussion between the parties in al-jadal (Miller 1985, Chap. I, p. 15). For example, in his discussion of one of the specific technical attacks, al-Juwainī (1028–1085) points out that for the jurists “the meaning is what is intended by i.e. the reference of the verbal sentence (المعنى هو المقصود باللفظ)” (al-Juwainī 1979, p. 212). This is the usual meaning of ma‘nā defined by the lexicographers such as Ibn Manẓour in his Lisān al-‘arab:
i.e. “the ma‘nā of any discourse is its intended meaning, and ‘anaytu by the saying this’ is ‘what I meant and intended’”. Abū ‘Abd Allah ibn Ziād better known as Ibn al-A‘rābī (767–845), one of the early leading linguists, is more precise when he defines ma‘nā as: “the intention (القصد) which comes out and appears in the thing when it was searched for” (in Ibn Fāris 1984, Vol. IV, pp. 148–149). It is al-Juwainī’s disciple al-Ghazālī who fixed the meaning of intentionality as qasd and showed its basic epistemic function in any discourse through his popular books like Maqāsid al-falāsifa or The Intentions of the Philosophers. He also establishes its use in law and jurisprudence making it a general methodological interpretation of legal texts famously known as maqāsid al-sharī‘a. As for Ibn Sīnā, he not only identifies the mind with intentionality by declaring: “every intention tends towards something (إن كل قصد فله مقصود)” (Ibn Sīnā, book 9, Chap. 3 §7, p. 320), repeated almost exactly in another passage: “every intention is designed for the intended object (كل قصد يكون من أجل المقصود)” (ibid. §5, p. 319). He further specifies that meaning is not transparent to the mind, it is rather determined by the will of the enunciator to convey his intention: “the verbal word by itself has no meaning… it only acquires its meaning by the will of the enunciator (اللفظ بنفسه لا يدل ألبتة إنما يدل بإرادة اللافظ)” (Ibn Sīnā 1952, p. 25). Though intentionality is not Ibn Sīnā’s innovation, it is how he integrates it in his epistemology by using it in correlation with abstraction that should be appreciated. For it is particularly remarkable how he swiftly gets to intentionality through his conception of abstraction. As it will be explained in the next section, his brilliant idea is to conceive of abstraction as that mental operation which makes the mind an object of study for it is the locus of complex mental activities whose analysis can lead to the distinction of those intentional acts involved in the construction of the concept of number. The scholastics translation of Arabic words (ma‘nā, murād, qaṣd) as intentio seems to only stress the presence in the mind aspect as suggested by Husserl’s master Brentano in his Psychology from an Empirical Standpoint’s famous passage:Every mental phenomenon is characterized by what the Scholastics of the Middle Ages called the intentional (or mental) inexistence [i.e. existence in intentio] of an object, and what we might call, though not wholly unambiguously, reference to a content, direction toward an object (which is not to be understood here as meaning a thing), or immanent objectivity. Every mental phenomenon includes something as object within itself, although they do not all do so in the same way.” (Brentano 1995, p. 88)
- 3.
This section can be seen as an indication of the kind of philosophy that Ibn Sīnā describes “as it is in nature” which can best be illuminated by “intuitive” as used in the computer software sense, i.e. easy to understand and use. Hence by intuitive theory of knowledge we mean a concrete descriptive account as experienced by the epistemic subject.
- 4.
The substitution of elements of an equivalence class is clearly expressed by 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 both one and the same استعمل أيهما شئت فكلهما واحد. (Al-Fārābī 1989, p. 38 §3; my emphasis)
He also examines the case of 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” (Ibn Sīnā 2005, p. 74 §2), we will come back to this in the last section of this chapter.
- 5.
- 6.
The term “regions” seems most appropriate as “stages” or “levels” are inappropriate for they connote some kind of ladder-like hierarchical structure of the mind making difficult to speak of their interaction. This way of presenting this distinction is to signal three kinds of object correspond to the three main regions of the mind: memory, imagination and the intellect.
- 7.
التحليل تمييز أشياء صح وجودها في المجتمع و لكنها
مختلطة عند العقل، فيفصل بعضها عن بعض بقوته و بحده. أو
يكون بعضها يدل على وجود الآخر، فإذا تأمل حال بعضها
انتقل منه إلى الآخر.
- 8.
“الخيال قوة تحفظ مُثَلَ المحسوسات بعد الغيبوبة مجتمعة فيها” (Ibn Sīnā: 1992, p. 377); ḥafiẓa, which means to retain something in memory so that you no longer need the thing itself hence memorise, also involves the idea of preserving which suggests that the objects of memory are kept intact i.e. clearly distinguished from objects of imagination which in turn are distinguished from those of the intellect.
- 9.
ibid., p. 379.
- 10.
The role of memory in the foundation and activity of science has been little recognised by contemporary philosophers. Epistemologically, memory represents that vital region of the mind by which the epistemic subject is aware of its objectivity since its objects provide the evidence that his past experiences are not and cannot be of his own making but acquired through his constant interaction with the outside world.
- 11.
Ibn Sīnā has discussed in more detail the ontological and epistemic status of fictions such as phoenix (‘anqā’ mughrib) in his Risālah fī al-nafs or Letter on the Soul. He is particularly very critical of imagination and recommends it be used with great care in the sciences other than mathematics.
Representation in imagination, except in mathematics, is often misleading (مضل), and it is a guide and heuristic (هاد و مرشد) in mathematics (Ibn Sīnā: 1956, p. 197).
It is because the intended meaning of geometric expressions (معاني ألفاظ الهندسيات) is unequivocally known by the realisation (بالتحصيل) of its relation to its objects, and as a result no object can be imagined other than the one intended by the meaning (فلا توهم غير المعنى المقصود به). Rather each expression has an understood meaning according to the intention (بل لكل لفظ منها معنى مفهوم بحسب الغرض). (ibid., p. 196).
The active nature of imagination tends thus to confuse the mind because of the polysemy of language and its ability to create its own objects i.e. fictions such as a flying horse or a phoenix. And it seems to him that anything that can be imagined can be posed as a possible existent for possibility assumes consistency and consistency should be proved, since from imagined entities the mind can indeed be led to believe to their existence in the external world. To prevent such unwarranted and misleading inference, he requires a proof of existence from any claim that tends to confuse real with imaginative objects or any attitude that tends to substitute the latter for the former. And for the realm of natural beings instantiation seems to be the only way to prove consistency. In his Maqāṣid or The Intentions of the Philosophers, al-Ghazālī further refines Ibn Sīnā’s theory of knowledge by describing how knowledge turns mental into actual existence.
When it happens to us to imagine something we desire, from this imagination results the power of desire. If this desire is intense and perfect and our judgment that it should be is added to it, from this comes a virtue which runs through the muscles, then it moves the tendons and results in the movement of the limbs that serve us as instruments; and from there comes the desired action. Similarly, when we imagine a line that we want to draw and we judge what it should be, from the desire that that line should be comes the power to make it; the power of desire moves therefore the hand and the pen, and from them the line will result as we had imagined it. When we said that “it should be”, we meant that we know or we believe that this existence will be, for us, useful, pleasant or good. The movement of the hand comes thus from the power of desire, and the movement which is the power of desire comes from the imagination and knowledge that the thing should be. We therefore find in our knowledge of a thing the beginning of its execution فقد وجدنا العلم فينا مبدأ لحصول شيء (Al-Ghazālī 1961, p. 236).
- 12.
و أما الخيال الباطن فيخيله مع تلك العوارض، لا يقدر
على تجريده المطلق عنها، لكنه يجرده عن تلك العلاقة
الذكورة التي تعلق بها الحس، فهو يتمثل صورته مع غيبوبة
حاملها. أما العقل فيقتدر على تجريد الماهية المكنوفة
باللواحق الغريبة المشخصة، مستثبتا إياها كأنه عمل
بالمحسوس عملا جعله معقولا. و أما ما هو في ذاته بريء عن
الشوائب المادية، واللواحق الغريبة التى لا تلزم ماهيته
عن ماهيته ، فهو معقول لذاته ، ليس يحتاج إلى عمل يعمل
به بعده لأن يعقله ما من شأنه أن يعقله، بل لعله من جانب
ما من شأنه أن يعقله
- 13.
It is remarkable that Ibn Sīnā’s successors like the great scientist al-Ṭūsī for example, in his extensive comment of al-Ishārāt, has perfectly grasped his basic conception of knowledge by elaborating: “Intellection is the apprehension of the thing inasmuch as it is and nothing else
(التعقل إدراك الشيء من حيث هو هو فقط); not inasmuch as it is another thing, either considered alone or with its other attributes that are apprehended by this kind of apprehension.” (Ibn Sīnā: 1992, pp. 367–368)
- 14.
The support of memory is equally necessary for the activity of pure reason.
- 15.
ونقول إنه إنما تكتسب تصور المعقولات بتوسط الحس على
وجه واحد ، وهو أن الحس يأخذ صور المحسوسات ويسلمها الى
القوة الخيالية فتصير تلك الصور موضوعات لفعل العقل
النظري الذي لنا، فتكون هناك صور كثيرة مأخوذة من الناس
المحسوسين، فيجدها العقل متخالفة بعوارض مثل ما نجد
زيدا مختصا بلون وسحنة وهيئة أعضاء، وتجد عمرا مختصا
بأخرى غيرتلك. فيقبل على هذه العوارض فينزعها فيكون
كأنه يقشرهذه العوارض منها ويطرحها من جانب حتى يتوصل
الى المعنى الذى يشترك فيه و لا يختلف به، فيحصلها و
يتصورها. و أول ما يفتش عن الخلط الذي في الخيال فإنه
يجد عوارض وذاتيات ، ومن العوارض لازمة وغيرلازمة ، فيفرد
معنى معنى من الكثرة المجتمعة في الخيال ويأخذها الى
ذاته.
- 16.
The famous mathematician, physicist and philosopher Ibn al-Haytham, who is contemporary with our philosopher, too founds science on the apprehension of an invariant by the epistemic subject:
Science is an opinion which does not change, and the opinion is a belief in a certain notion. Science is therefore a belief in a certain notion, as it is, and it is moreover an invariable belief… But there is no belief without a believer and a believed notion.” (in Rashed 2002, p. 445)
i.e. “the ma‘nā of any discourse is its intended meaning, and ‘anaytu by the saying this’ is ‘what I meant and intended’”. Abū ‘Abd Allah ibn Ziād better known as Ibn al-A‘rābī (767–845), one of the early leading linguists, is more precise when he defines ma‘nā as: “the intention (القصد) which comes out and appears in the thing when it was searched for” (in Ibn Fāris 1984, Vol. IV, pp. 148–149). It is al-Juwainī’s disciple al-Ghazālī who fixed the meaning of intentionality as qasd and showed its basic epistemic function in any discourse through his popular books like Maqāsid al-falāsifa or The Intentions of the Philosophers. He also establishes its use in law and jurisprudence making it a general methodological interpretation of legal texts famously known as maqāsid al-sharī‘a. As for Ibn Sīnā, he not only identifies the mind with intentionality by declaring: “every intention tends towards something (إن كل قصد فله مقصود)” (Ibn Sīnā, book 9, Chap. 3 §7, p. 320), repeated almost exactly in another passage: “every intention is designed for the intended object (كل قصد يكون من أجل المقصود)” (ibid. §5, p. 319). He further specifies that meaning is not transparent to the mind, it is rather determined by the will of the enunciator to convey his intention: “the verbal word by itself has no meaning… it only acquires its meaning by the will of the enunciator (اللفظ بنفسه لا يدل ألبتة إنما يدل بإرادة اللافظ)” (Ibn Sīnā 1952, p. 25). Though intentionality is not Ibn Sīnā’s innovation, it is how he integrates it in his epistemology by using it in correlation with abstraction that should be appreciated. For it is particularly remarkable how he swiftly gets to intentionality through his conception of abstraction. As it will be explained in the next section, his brilliant idea is to conceive of abstraction as that mental operation which makes the mind an object of study for it is the locus of complex mental activities whose analysis can lead to the distinction of those intentional acts involved in the construction of the concept of number. The scholastics translation of Arabic words (ma‘nā, murād, qaṣd) as intentio seems to only stress the presence in the mind aspect as suggested by Husserl’s master Brentano in his Psychology from an Empirical Standpoint’s famous passage:
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Tahiri, H. (2016). Ibn Sīnā’s Basic Theory of Knowledge. In: Mathematics and the Mind. SpringerBriefs in Philosophy. Springer, Cham. https://doi.org/10.1007/978-3-319-25238-4_4
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