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Jumat, 03 Mei 2013

Islamic Mathematics and Astronomy


All of the mathematical and astronomical work of this period was done by Muslims. It is interesting to recall that the mathematical work of the previous period had been done almost exclusively by Chinese. Some amount of stimulation had come from India. In addition to transmission of some Hindu mathematics.
Ibrahim al-Fazari is said to have been the first Muslim to construct astrolabes.

Ya'qub ibn Tariq and Muhammad, son of Ibrahim al-Fazari, are the first to be mentioned in connection with Hindu mathematics: Ya'qab met at the court of al-Mansur, a Hindu astronomer called Kankah (?), who acquainted him with the Siddhanta, and Muhammad was ordered to translate it. The physician al-Batriq translated Ptolemy's Quadripartitum. Two astrologers, one of them a Jew named Mashallah, the other a Persian called al-Naubakht, worked together to make the measurements necessary for the building of Bagdad. Al-Naubakht's son, al-Fadl, wrote astrological treatises and translations from the Persian into Arabic.

Ibrahim al-Fazari
Abu Ishaq Ibrahlm ibn Habib ibn Sulaiman ibn Samura ibn Jundab. Died c. 777.
Muslim astronomer. The first to construct astrolabes, he wa the author of a poem (qasida) on astrology and of various astronomical writings (on the astrolabe, on the armillary spheres, on the calendar). H. Suter: Die Mathematiker und Astronomer der Araber (3, 208, 1900)

Ya'qub Ibn Tariq
Probably of Persian origin, flourished in Baghdad, c.767-778 died c. 796. One of the greatest astronomers of his time. He probably met, c. 767, at the court of al-Mansur, the Hindu Kankah (or Mankah?), who had brought there the Siddhanta. He wrote memoirs on the sphere (c. 777), on the division of the kardaja; on the tables derived from the Siddhanta. H. Suter: Die Mathematiker und Astronomer der Araber (p. 4, 1900)

Muhammad Ibn Ibrahim Al-Fazari
Abu 'Abdallah Muhammad ibn Ibrahim al-Fazari. Son of the astronomer Ibrahim dealt with above, for whom he is sometimes mistaken (he may be the author of the astrological poem ascribed to his father). Died c. 796 to 806. Muslim scientist and astronomer. He was ordered by the Caliph al-Mansur in 772/3 to translate the Sanskrit astronomical work Siddhanta. This translation was possibly the vehicle by means of which the Hindu numerals were transmitted from India to Islam.
H. Suter: Die Mathematiker und Astronomen der Araber (p. 4,1900).
Cantor: Geschichte der Mathematik (I, 3rd ed., 698, 1907).
D. E. Smith and L. C. Karpinski: The Hindu-Arabic Numerals (p.92, Boston, 1911)

Mashallah
His real name was probably Manasseh (in Arabic, Misha). Latin translators named him Messahala (with many variants, as Macellama, Macelarma). Mashallah is a contraction of ma'aha Allah meaning "What wonders Allah has willed." (What hath God wrought.) Flourished under al-Mansur, died c. 815 or 820. One of the earliest astronomers and astrologers in Islam, himself an Egyptian (?) Jew. Only one of his writings is extant in Arabic, but there are many mediaeval Latin and Hebrew translations. The Arabic text extant deals with the prices of wares and is the earliest book of its kind in that language. He took part with the Persian astrologer al-Naubakht in the surveying preliminary to the foundation of Baghdad in 762-63. His most popular book in the Middle Ages was the 'De scientia motus orbis', translated by Gherardo Cremonese.

Text and Translation. The De scientia motus orbis is probably the treatise called in Arabic "the twenty-seventh;" printed in Nuremberg 1501, 1549. The second edition is entitled: 'De elementis et orbibus coelestibus', and contains 27 chapters. The De compositione et utilitate astrolabii was included in Gregor Reisch: Margarita phylosophica (ed. pr., Freiburg, 1503; Suter says the text is included in the Basel edition of 1583). Other astronomical and astrological writings are quoted by Suter and Steinsehneider.
An Irish astronomical tract based in part on a mediaeval Latin version of a world by Messahalah. Edited with preface, translation, and glossary, by Afaula Power (Irish Texts Society, vol. 14, 194 p., 1914. A relatively modern translation of the De scientia motus orbis, the preface is uncritical).

Rabu, 01 Mei 2013

Concept of Truth

The problem of truth was raised in medieval Islamic philosophy within the framework of discussions starting from the question of whether our knowledge corresponds to the "actuality of affairs." The notion of validity thus elaborated was comprehended as a quality of knowledge established through a comparison with "matters of fact." What was intended is not coincidence with what is and has existence. Existence (wujud) was generally understood in Islamic thought as one of the attributes (sifa) that a thing might or might not possess while still being "a thing" (shay'), and since our knowledge embraces things independently of their accidental attributes, the question about truth was placed on a wider footing. Validity, from that point of view, testifies that our knowledge conforms with reality in the immediate meaning of the term — thing-ness. This notion of reality(shayiyya) does not necessarily exclude Divinity, for God in Islamic sciences is often comprehended as The Thing, although different in every respect (except that of thing-ness) from all other things. The concept of "thing" serves to introduce some thing into the current of intellectual discourse rather than to state anything definite about it; to be a thing — that is, fixed and established — means to enter the field of discussion.
Validity as affirmation of conformity with reality was referred to as sidq (veracity, truth) or tasdiq (certification of truth). The "actuality of affairs" to which our knowledge conforms was comprehended also as a sort of "authenticity," and the corresponding term haqiqa may be rendered into English as "truth" as well. Thus verification is carried out by comparing our knowledge to the "truth of things," and if the result is positive, knowledge is "true (sadiq); if not, it is "false" (kadhib). Knowledge is valid by virtue of its coincidence with the truth of things, while the truth of the latter needs no verification. It follows from the fact of their "being affirmed": they just "are there" as "fixed" and "true." The ideas of truth, fixity and thing are closely linked in Arabic. The term "thing" (shay') is usually explained as "something that is established" (thabit), and the root h-q-q, from which "truth" (haqiqa) is derived, renders the same meaning. (For example, haqq means both "true" and "unshakable.")
The problem of truth was raised rather early in Islamic thought, and already the al-Rawafid discussed it. As al-Ash'ari informs us, most of them maintained that all human knowledge is "necessitated" (idtirar). From their point of view, a person is not free to acquire true knowledge or to reject the false; moreover, knowledge about the falsity or validity of our knowledge also cannot be obtained at our will. This [438] argument proceeded from the general assumption that all human deeds are "forced" (idtirar). At the same time, some of the al-Rawafid considered the human mind able to receive true knowledge independently, for example, to learn of God's unity (tawhid) before the prophets inform people of it. Knowledge gained independently, they argued, is obtained with the help of qiyas (literally "co-measuring"). The term designates rational epistemological procedures that produce new knowledge "by measure" of the old one and was used not only in Kalam, but in other sciences as well, denoting analogous judgment in fiqh (Islamic jurisprudence) and the syllogism in logic. However, al-Rawafid who affirmed the independent ability of reason to gain new knowledge were in the minority (al-Ash`ari, 1980, pp. 51-3).
The discussion of truth was deepened by the Mu'tazila. First, they were concerned with determining the types of propositions that can be true or false. These are statements containing "denial and affirmation" (al-nafy wa al-ithbat), "praise and reprobation" (al-madh wa al-dhamm) as well as "wonder" (ta`ajjub), while "question" (istifham), "order and interdiction" (al-'amr wa al-nahy),"regret" ('asaf), "hope" (tamanni) and "request" (mas'ala) are neither true nor false (al-Ash'ari, 1980, p. 444). The Mu`tazila seem to have been little occupied with how true knowledge is reached, and this is perhaps due to the fact that they discussed truth in connection with the reliability of prophetic sayings — which is not an art to be taught. The Mu"tazila had different opinions as to whether a proposition can be called true or false if its author was ignorant of the "actuality of affairs." (The question here is whether unintended deception can be called a lie, or whether a statement that incidentally happened to be exact can be called truth). When the relevant "actuality of affairs" does not exist (for example, if the event has not yet occurred) or is unknown to a person, the verification procedure that compares a proposition to the "truth of things" cannot be executed — for objective and subjective reasons respectively — and such a proposition is to be regarded as neither true nor false. This argument, however, was not generally accepted by the Mu"tazila.
As for Aristotelian logic, it took root in medieval Islamic thought above all due to peripatetism. This school gave much more sophistication to what the Mutakallimun said about truth and the possible ways of acquiring it. Many elements of Aristotelian logic introduced by the Islamic peripatetics became indisputable patterns of reasoning for Islamic thinkers, and no school of medieval philosophy seriously challenged the syllogism as a paradigm for the preservation of truth in argumentation. What wasdisputed was the sphere in which the syllogistic method is relevant. This method appears to have gained less favor among Islamic thinkers than it did among ancient or medieval Western thinkers, and in philosophy per se we find even among the peripatetics great reservations in this respect.
Elements of Aristotelian logic were rather well known to Islamic scholars from translations of Aristotle's works as well as from writings of his great commentators, among which must be mentioned in particular Porphyry's Eisagoge. There also existed quite a number of logical treatises of educational and propadeutical character composed in Arabic, many of which belong, or are ascribed to, al-Farabi.
According to the peripatetics, the purpose of logic is to gain true knowledge. Such knowledge is twofold, consisting of "notions" (tasawwur) and "certifications of truth" (tasdiq), which are both accessible only on the basis of some a priori know ledge. As for "notions" (that is, understanding what the thing is), this knowledge in the final analysis is based on the units of meaning that definitions, later used in arguments, are composed of. In "certifications of truth" this primary knowledge is represented by "principles of intellect" (awa'il al-`aql), that very intellect with the help of which, as al-Farabi interprets Aristotle, we perceive the "certainty (yaqin) of necessary and true general presuppositions" (al-Farabi, 1890, p. 40) with no prior investigation or argument.
This is how Ibn Sina expresses the point in his concise Book of Remarks and Admonitions:
The purpose of logic is to provide a canonical tool (ala qanuniyya) that prevents aberration of thought. By "thought" (fikr)I mean here what takes place when a person, having pulled himself up, passes from what is present in his mind, what he has a notion of or what he is certain of... to what is not [present] there. This transition has a certain order and figure that might be correct and might happen to be incorrect. The incorrect often looks correct or makes you believe that it is correct. So logic is a science that studies ways of transition from what is present in the human mind to what it acquires, ... the correct modes of ordering this transition and its figures, as well as the kinds of incorrect ones. (Ibn Sina, 1960. Pt I, pp. 167-78)
Atomic "individual meanings" (ma'ani mufrada), from which complex logical structures are produced by "ordering" (tartib) and "composing" (ta'lif), constitute the basis for all logical operations (Ibn Sina, 1960. Pt 1. pp. 179-80). These meanings entirely correspond to the things in question. The correspondence is based on what is established by the language-giver who assigns certain "meanings" (ma'ani) to certain "sounds" (alfaz): this correspondence is therefore called "established" (bi al-wad`). For example, the sound "human" corresponds to the meaning "animal endowed with speech." The sound and its meaning are the two elements that make up a "word" (kalima); the relation of "denotation" (dalala) exists between the first and the second. What is denoted by the "sound" is that very "meaning" that constitutes the "truth" (haqiqa) of things. Thus logic, dealing with sounds and their meanings, deals in fact with things — as long as the denotation originally established in language is preserved.
In order to acquire the correct notion of a thing, one must arrive at a "clarifying saying" (qawl sharih) about it. This can be achieved, first, in a "definition" (hadd) of the thing. The construction of definitions is described in every detail as a procedure of answering the question what is it? by providing its genus (jins) and specific difference (fasl) to produce a definition of the species(naw') that informs us of the quiddity (mahiya) of the thing in question. Besides a definition, a "description" (rasm) can also be given to clarify the notion of a thing, although this does not deal with thing's quiddity. A description has to be given to those tools that serve us in setting out definitions — that is, notions of genus, species and specific difference — as well as to the highest genera that have no genus above them (and, consequently, [440] for which no definition can be given). Second, a description may be given to the things that have quiddity; for example, "animal endowed with laughter" serves as a description for "human."
As for arguments, they are composed in the form of syllogisms. Aristotelian syllogistic doctrine was exposed in Islamic peripatetism in every detail, accompanied by the examination of possible errors, mistakes and sophisms. The validity of conclusions reached through syllogisms is based on the accuracy with which we establish true meanings in definitions.
A great project of the unification and hierarchization of sciences was advanced in Islamic peripatetism. The hierarchization was to be based on differences in the degrees of generality of the various sciences' subjects. What is proved in the more general sciences may serve as non-provable principles for the more particular ones. From that point of view, sciences form a pyramid of axiomatically subordinated branches of knowledge. AI-Farabi, in Kitab al-milla, al-Kindi, in Kitab al-falsafa al-'ula, Ibn Sina, inal-Burhan (part of Kitab al-shifa') (to give only examples, and not an exhaustive list) speak about such subordination of the more particular to the more general sciences. This structure of knowledge is conceived as corresponding to the universe, which is ordered along the same lines of generality-particularity.
Logic is an important instrument of cognition. This does not mean, however, that the peripatetics tend to exaggerate its significance. Besides knowledge acquired by means of logic, direct, intuitive (hadsiyy) knowledge is possible. This is granted as immediate manifestation, in which the thing unconditionally and completely expresses itself as such.
Knowledge of our ego serves as a paradigm of intuitive cognition, Ibn Sina introduces this thesis in his famous fragment about the "flying person" in his Book of Remarks and Admonitions:
Look at your soul and answer: when you are in sound health, or even not, but correctly perceive things, did it ever happen that you were ignorant of your self(dhat) or didn't ascertain your soul? ... Imagine that your self has just been created: assume that it is in its right mind and figure, sees none of its parts and its members don't feel each other, but it is spread and suspended at some moment in pure air. Then you will find that it notices nothing: however it observes fixity of its egoness (ana'iyya). (Ibn Sina. 1957, Pt 2, pp. 319-20)
The ego is always manifest to itself, and this manifestation is the primary fact of our consciousness. It depends on nothing and, furthermore, no sophisticated proof is needed to understand it: it suffices to imagine the situation described above for the fact of the ego's manifestation to itself to become clear.
Immediate manifestation can be considered a sort of completion for the logical form of cognition. This concluding step, however, already transcends the path that it completes and opens fundamentally new horizons. According to Ibn Sina, Ibn Tufayl and other authors, a person acquires complete and true knowledge through union with Active Intellect — the last of the Cosmic Intellects, repository of all forms and governor of the sublunar world. This full contact with the source of forms that [441] are the subject of logical inquiry no longer presupposes any necessity of transition from the already-present to the yet-unknown, and thus places one outside the framework of logical reasoning. Certainly, not everyone is able to achieve this union; only if the soul is pure, Ibn Sina argues, can it be inflamed by Active Intellect and directly imprinted with the forms of all possible knowledge. It is the same intention of achieving immediately manifest self-evidence that speaks for itself in these cases, as also when these authors abandon philosophical jargon and talk about directly witnessing the Divine world. It is also obvious that the patency of our ego for itself guarantees its ability to reach absolutely complete and true knowledge by witnessing the Divinity, for the two kinds of evidential witnessing differ with respect to their subject, rather than in their essence.
The exposition of the peripatetic doctrine of truth is in no way complete before Ibn Rushd's work Kitab fasl al-maqal wa taqrir ma bayna al-shari`a wa al-hikma min al-ittisal (or "Decisive Saying Establishing the Connection between Law and Wisdom") is mentioned. Despite its title, the chief idea of this little treatise is that the spheres of "wisdom" (that is, philosophy) and "Law" (the theoretical postulates followed in religious life as well as its practical prescriptions) may be separated. The work attempts to fix independent rights of reason for obtaining the truth that -within the limits defined for it — no one can violate. It is noteworthy that Ibn Rushd had predecessors among the Mutakallimun in the differentiation of what falls under the Law, which is established and can be revised under no circumstances, and what reason is permitted to discuss and decide. "What is known by reason and what is known only through Law," a chapter in Usul al-din ( Principles of Religion), a book by an Ash'arite author, Abu Mansur al-Baghdadi, bears a resemblance to Averroes's treatise not only by its title. Al-Baghdadi definitely states that only Divine prescriptions, either direct or transmitted through prophets, constitute the domain of Law, whereas problems of the world's origin and similar questions involve theories that human reason elaborates.
Isma`ilism may to a certain degree be regarded as a successor of peripatetism with respect to the theory of knowledge and truth. Isma`ili theoreticians, on the one hand, have no doubt concerning reason's capability of knowing the truth; more over, cognition of the truth is, in their view, indispensable for the person who wants to reach salvation. On the other hand, they give up the syllogistic method as the principal means of cognition. The Universe, in their estimation, is not a unified structure arranged in the hierarchical (generality-particularity) order that the peripatetics described. It is rather a system of structures that stand with respect to each other in relations of similarity, isomorphism and correspondence. This ontology presupposes a special method of cognition.
This is how Hamid al-Din al-Kirmani. the most prominent Isma`ili theoretician, expresses these theses. Any science, he argues, has its own "laws" (qawanin) — that is to say, criteria by which knowledge is tested in order to determine whether it really corresponds to the "true order" (nizam al-haqq) of the subject of study. And if the peripatetics strive to achieve their aim, that is, "knowledge of the meanings of existence" (ma`ani al-wujud), by means of logic. Isma`ili philosophers employ a different method.
Its basic premise can be expressed as follows: anything in the world belongs to some structure and may correctly be comprehended only within that structure, through its place in the overall framework and its structural role. Thus the preliminary step for cognition is to single out the universal structures that, being completely "balanced" (mutawazina) and "isomorphic"(mutashakila), form the created world. In their mutual conformity universal harmony is embodied, expressing the highest wisdom of their creation and giving evidence to the perfection of our world, which is the best of all possible worlds.
The principle of hierarchical harmony, penetrating the Universe, can be traced on different levels. As Isma`ili works show, this can be done with great accuracy and amazing sophistication. There are four basic structures to be identified; the metaphysical world, the religious community, the natural world, and the human being. It is of fundamental importance that knowledge of any of these structures allows us to know all of the others with the help of special rules of interstructural translation of meaning, since corresponding elements of different structures have a similar structural place, function and essence.
Had all the structures of the Universe been manifest to us, no special cognitive procedures would be necessary. However, universal structures fall into two classes. Some of them are "obvious" (zahir), while others are "latent" (batin). Isma`ili theory of knowledge proceeds from the premise that "latent" structure (as a whole, as well as any of its separate elements) can be known only through the "obvious." Since it is the structure of Isma`ili community (or as al-Kirmani prefers to say, the "world of religion") that is known to us in every detail, all new knowledge is acquired on this basis. This method is referred to as finding "balance"(muwazana) and "correspondence" (mutabaqa). Isma`ili community structure is harmoniously balanced with all other structures in the world (this is a postulate of Isma`ili philosophy, not a conclusion to be proved), and knowing it we can arrive at knowledge of anything. Besides, numeric structures are widely used in search of mutual structural correspondences. Using this method, al-Kirmani consistently and in every detail describes the metaphysical world (the hierarchy of Cosmic Intellects) as well as the natural world and the microcosm.
Structural correspondence is for al-Kirmani not only a method of finding new knowledge, but also a criterion for the verification of existing knowledge. Only that is valid which has a correct structure. "This criterion is such that what agrees with it. is true, and what disagrees, is false; it is this criterion that is so attractive for the intellect that seeks to know with its help what is given to it as well as what escapes it" (al-Kirmani, 1983, p. 236). In cognitive procedures the structure of the "religious world" (which means the Isma`ili community) is taken as a paradigm, but that structure too is verified by correspondence to "God's creation." The perfection of the manifest structure and its undoubted validity is proved for al-Kirmani by the fact that it disagrees with the Universe in no detail (al-Kirmani, 1983, p. 237).
Certainly, the person who would endeavor to apply this method of cognition in his own research will hardly succeed. This method serves well in the exposition and structuring of already acquired knowledge, but in spite of what al-Kirmani maintains, not in the search for new knowledge. The author of Rahat al-'aql leaves us [443] ignorant of the most interesting and important detail of his method — those inter structural semantic translation procedures that fill the unknown structure with new meanings so that it balances the structure manifest to us. In this respect what al-Farabi said on another occasion seems to be relevant. In this critique of astrologers, the "Second Teacher" argues that in the world one can single out diverse "sets" (kathra) of things, like animals' movements, the voices of birds, written signs, and so on, in order to put them in correspondence with the multitude of events that we experience; such a procedure, however, produces "only occasional, instead of necessary [truth] that reason should have accepted" (al-Farabi, 1890, p. III).
Illuminative philosophy is another successor to peripatetism with respect to the theory of truth. It is no exaggeration to say that Ibn Sina is the greatest authority for the most prominent representative of this school, Shihab al-Din Yahya al-Suhrawardi. The affinity of these two thinkers is surprising, in view of the disagreements between their teachings caused by al-Suhrawardi's adherence to a metaphysics of light and darkness; on the subject of the theory of truth, however, the disagreements between them are minimal.
Like Ibn Sina, al-Suhrawardi speaks about two kinds of true knowledge: immediate intuitive knowledge and logical knowledge. The first he also calls "truthful witnessing" (mushahada haqqiya), and the second "research" (bahth). Knowledge of the ego. or, as al-Suhrawardi himself calls it, ego-ness (ana'iyya), serves for him. as it did for Ibn Sina, as an archetype of the direct cognition of truth. But since the majority of people are unable to experience the completeness of truth immediately, they have to resort to indirect logical cognition, which starts with basic unquestionably valid premises and proceeds from them to the unknown (al-Suhrawardi, 1952, p. 18).
Al-Suhrawardi considers the elementary sensual perceptions "simple meanings." logical atoms from which the construction of concepts begins. These perceptions are simple, absolutely evident and self-identical; they are the principal elements known by anyone who has healthy organs of perception. Sensual perception is absolutely adequate, al-Suhrawardi argues: we perceive exactly what is there in the things perceived. Finally, basic sensual perceptions, being elements of knowledge, have no logical definition (al-Suhrawardi, 1952, p. 104). This sensualism of the celebrated mystic agrees well with his radical nominalism: according to al-Suhrawardi, no general concepts exist independently of our minds. On this basis he argues that quiddity is constituted not only by substantial features, as the peripatetics maintained, but also by accidental features. For the shape of a house, for example, is accidental with respect to the clay from which it is constructed, and nevertheless we say, in response to the question "what is it?" that it is a "house," rather than "clay" (al-Suhrawardi, 1952, pp. 85-6). Given Suhrawardi's metaphysics of light and darkness, he denies that the first matter is universal substance, and consequently is compelled to look for a different basis of individuation. For him it is not matter that is "responsible" for the multiplicity of individuals which all have the same quiddity and which therefore are, logically speaking, one and the same, but rather the degree of perfection (kamal) or degree of completeness by which this or that "universal meaning" is represented in the individual (al-Suhrawardi, 1952, [444] p. 87). This concept of individuality as a degree of perfection will later be elaborated in Sufism by Ibn 'Arabi.
As for the syllogistic method, the importance that al-Suhrawardi attaches to it is testified to by the fact that the first half of his chef-d'oeuvre, Hikmat al-ishraq (Wisdom of illumination), is solely devoted to its exposition. Al-Suhrawardi points out that it is a necessary propaedeutic for the second, metaphysical and mystical, part of his book. In his analysis of syllogisms accompanied by a detailed study of possible errors and sophisms, al-Suhrawardi strives to prove that all modes of syllogism can be reduced to a single positive categorical mode, which, in his estimation, makes knowledge of all the subtleties of the other modes superfluous.
As for the complete and perfect witnessing of truth, it is reached, according to al-Suhrawardi, in the state of "illumination"(ishraq). "Illumination" is the central concept of al-Suhrawardi's philosophy. It signifies direct irradiation of the soul by superior, metaphysical lights. The soul itself is a light that has descended from the world of light" into the "world of darkness" and is yet impotent to return to its original abode. This congeneity of human soul and the highest principles of being constitutes the ontological foundation for the possibility of such irradiation. Illumination discloses the truth (haqq) immediately and needs no verification (tasdiq). Logical instruments that verify the correctness of "transition" procedures are of no use when no such transition takes place.
The Sufi doctrine of the truth and the ways of acquiring it differs in its central point from any of the doctrines that we have hitherto discussed. No matter how truth is understood in Kalam, peripatetism, Isma`ilism, or the philosophy of illumination, all of these schools have in common an explicit or implicit understanding of true knowledge as something unhesitatingly established: the term "certainty" (yaqin) generally serves to express this fixity. Such certainty is understood as "quiescence" (itmi'nan), the same idea of quietude reached through complete and true knowledge is expressed by the title of al-Kirmani's magnum opus Rahat al-'aql — Peace of mind) on the basis of the generally accepted notion of the "perfect" (kamil) and "complete" (tamm) as immobile. The true. by virtue of its completeness, needs nothing external to be accomplished and, consequently, no movement is necessary for it. Against this understanding of truth as a state of clear certainty, Sufism opposes the doctrine of the truth being witnessed in its completeness in a state of "abashment" and "confusion" (hayra) that presupposes constant restlessness.
Although in this respect Sufism stands in opposition to other trends of medieval Islamic philosophy, there is doubtless continuity in the way Sufi theoreticians arrive at the above conclusion. Peripatetism, Isma`ilism and the philosophy of illumination understand the achievement of truth, at least in its logical form, as "transition" (intiqal) from what a person possesses as established truth to what he or she currently does not possess: as for mystical revelation, it also provides a sort of finally established and unequivocally valid knowledge. "The unknown — against the known," Ibn Sina writes (Ibn Sina, 1960. Pt 1. p. 181): all things are divided into two classes that stand to each other in a relation of exact mutual correspondence; everything is truly known after it has been unknown. Dividing things [445] into the "unseen" (batin) and the manifest" (zahir) was commonplace in medieval Islamic thought, and these concepts remain fundamental in Sufi epistemology as well, where cognition is often referred to as "making [the unseen] manifest" (izhar).
This substantial departure from traditional Islamic thinking in the final conclusion of Sufi epistemology (that is, the cognition of truth as "confusion" instead of as "fixed certainty") in spite of common intention of finding the solution (truth as unseen made manifest) is explained by a basic feature of this philosophy that may be defined as interiorization. Both Sufi ontology (seeCausality and Islamic thought) and epistemology are deeply marked by it. Just as the cause and effect relation is an inner division of the same essence rather than an external relation between two different essences, so the inner and the outside (the "hidden" and the "manifest") are not two different and definite aspects of things, but rather one and the same. What other schools of philosophy consider as occurring between, Sufi philosophy regards as taking place inside.
However, Sufi philosophy does not deny other points of view. As the doctrines of Ibn Sina and al-Suhrawardi demonstrated, logic may be regarded as an incomplete version of immediate and perfect truth-witnessing, rather than as its alternative. Ibn 'Arabi, the greatest of Sufi philosophers, adopts the same position. In this sense his theory of truth is inclusive rather than exclusive, for he regards non-Sufi ways of cognition as also true — within their limits, however, and not absolutely.
For example, the knowledge obtained through correct syllogisms is certainly true, and there is no doubt about its scientific results, like our knowledge of the sun's size or the rules of mathematics (Ibn 'Arabi, 1980. pp. 102-4). The intuitive "witnessing"(mushahada) gives true knowledge as well: the "inner sight" (basira) discovers immediately behind things their causes and thus discloses the inner essence of things hidden under their manifest outwardness. The cause of things and of their inner essence thus discovered is God, or The Truth (al-Haqq) — each time seen in one of His infinite aspects. However, the "witnessing" first brings into sight the thing, and then behind it, or inside it, discovers God. The two are still divided and differentiated, and the all-encompassing Truth that constitutes the core of everything is not found as the thing's outwardness. The highest stage of truth is to see things in God. to notice the sameness and equality of the different, to be unable to differentiate. This is the ability of the "heart" (qalb). Intellect, inner sight, and heart form an ascending hierarchy of organs with their corresponding methods of cognition.
Rational knowledge is acquired by moving "from" premises "to" a conclusion, by going along "the stretched path," as Ibn `Arabi puts it (Ibn 'Arabi, 1980, p. 73). The intuitive witnessing of God as the inner essence of things spheres this line. But only when the sphered line becomes equal to its center does "confusion" come, and the person sees the hidden as manifest and the manifest as hidden, sees God as His creation and the creation as God Himself. Total oneness and sameness, the transcendence of any differentiation and the non-fixity of any definiteness and any limit (the results of logical cognition included) — this is what such a way of seeing the truth boils down to. The Sufi [446] understanding of truth undermines well established stereotypes of dichotomizing divisions. The fundamental ontological sameness of God and His creation entails the sameness of any pair of opposed categories. The law of excluded middle is irrelevant for this point of view: what it points to is but a step that should inevitably be overcome. Truth turns out to be a transcendence of dichotomic divisions — a transcendence which, however, presupposes that each of them is fixed — but only as a step in an unceasing movement, equal to any other of its infinite steps.

Bibliography

Writings
AI-Ash'ari, Abu al-Hasan 1980: Maqalat al-islamiyyin wa ikhtilaf al-musallin [Sayings of Muslims and Differences between Those Who Pray] 3rd edn (Wiesbaden; Franz Steiner).
Ibn 'Arabi 1980: Fusus al-hikam [Bezels of Wisdom] 2nd edn (Beirut: Dar al-kitab al-'arabiyy).
Ibn Sina, Abu 'Ali 1957-60: Al-lsharat wa al-tanbihat, ma`a sharh Nasir al-Din al-Tusi [Book of Remarks and Admonitions, with the commentary of Nasir al-Din al-Tusi] ed. Suleyman Dunya, 4 parts (Cairo; Dar al-ma'arif).
AI-Kirmani, Hamid al-Din 1983: Rahat al-'aql [Peace of Mind] 2nd edn (Beirut: Dar al-andalus).
Al-Suhrawardi, Shihab al-Din Yahya 1952: "Hikmat al-ishraq," in Oeuvres philosophiques et mystiques de Shihabeddin Yahya Sohrawardi (1), ed. H. Corbin. Bibliotheque Iranienne, Vol. 2 (Teheran and Paris: Institut Franco-Iranien — Librairie d'Amerique et d'Orient), pp. 2-260.
References and further reading
Badawi, A. 1972: Histoire de la philosophie en Islam, 2 Vols (Paris: J. Vrin).
Al-Baghdadi, Abu Mansur 1981: 'Usul al-din, 3rd edn (Beirut: Dar al-kutub al-'ilmiyya).
Chittick, W. C. 1989: The Sufi Path of Knowledge: Ibn al-'Arabi's Metaphysics of Imagination (Albany: State University of New York Press).
Corbin. H. 1964: Histoire de la philosophie islamique (Paris: Gallimard).
Daftary, F. 1990: The Isma`ilis: Their History and Doctrines (Cambridge; Cambridge University Press).
Dunlop, D. M. 1956a: "Al-Farabi's Eisagoge." The Islamic Quarterly. A Review of Islamic Culture, 3, pp. 115-27.
—— 1956b: "Al-Farabi's Introductory Risalah on Logic," The Islamic Quarterly. A Review of Islamic Culture. 3. pp. 224-30.
—— 1956, 1959; "Al-Farabi's Paraphrase of the Categories of Aristotle," The Islamic Quarterly. A Review of Islamic Culture, 4 (1957), pp. 168-83: 5 (1959), pp. 21-37.
Fakhry, M. 1983: A History of Islamic Philosophy, 2nd edn (New York: Columbia University Press).
AI-Farabi, Abu Nasr 1890a: "Fi-ma yasihh wa ma la yasihh min ahkam al-nujum" [What Is Right and What Is Wrong in Astrology], in Al-thamra al-mardiyya fi ba`d al-risalat al-farabiyya [Longed-for Fruit of Some Treatises by al-Farabi] (Leiden: E.J. Brill), pp. 104-14.
—— 1890b: "Maqala fi ma'ani al-'aql" [On Meanings of "Intellect"], in Al-thamra al-mardiyya fi ba`d al-risalat al-farabiyya [Longed-for Fruit of Some Treatises by al-Farabi] (Leiden: E. J. Brill), pp. 39-48.
——1968: Kitab al-milla wa nusus ukhra [Book of Beliefs and Other Treatises], ed. M. Mahdi (Beyrouth: Dar al-mashriq).
Georr, Khalil 1948: Les categories d'Aristote dans leurs versions syro-arabes (Beyrouth).
Ibn 'Arabi 1859 and reprints: AI-Futuhat al-Makkiya [Revelations of Mecca]. 4 Vols (Cairo: Dar al-kutub al-'arabiyya al-kubra).
Ibn Sina, Abu 'Ali 1954: AI-Burhan min Kitab al-Shifa' [On Demonstration, from the Book of Healing] ed. Al-Badawi (Cairo: Maktabat al-nahda al-misriyya).
Ibn Rushd (Averroes) 1959: Kitab Fasl al-Maqal with its Appendix and an extract from Kitab al-kashf `an manahij al-adilla (Leiden: E. J. Brill).
Ibn Tufayl 1954: Qissat Hayy Ibn Yaqzan [Story of the Living, Son of the Wakeful] (Beirut: Dar al-Farabi).
Al-Kindi 1948: Kitab al-Kindi ila al-Mu`tasim bi-llah fi al-falsafa al-ula [Epistle of al-Kindi to al-Mu'tasim bi-llah in First Philosophy], ed. Ahmad al-Ahwani (Cairo).
Nasr. S. H. 1964: Three Muslim Sages: Avicenna, Suhrawardi, Ibn 'Arabi (Cambridge, Mass.: Harvard University Press).
Wolfson, H. A. 1976: The Philosophy of the Kalam (Cambridge, Mass.: Harvard University Press).

Minggu, 28 April 2013

a History of Ziro


by: J J O'Connor and E F Robertson
One of the commonest questions which the readers of this archive ask is: Who  discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as a empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name.

Neither of these uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.

One might think that once a place-value number system came into existence then the 0 as a empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.

We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.

Now the ancient Greeks began their contributions to mathematics around the time that zero as a empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle that the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.

Now there were exceptions to what we have just stated. The exceptions were the mathematicians who were involved in recording astronomical data. Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O. There are many theories why this particular notation was used. Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden". Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet). Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board. The suggestion here is that when a counter was removed to leave an empty column it left a depression in the sand which looked like O.

Ptolemy in the Almagest written around 130 AD uses the Babylonian sexagesimal system together with the empty place holder O. By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself. The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered as a sort of punctuation mark) makes its next appearance in Indian mathematics.

The scene now moves to India where it is fair to say the numerals and number system was born which have evolved into the highly sophisticated ones we use today. Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers. As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far. For example Mukherjee in [6] claims:-
... the mathematical conception of zero ... was also present in the spiritual form from 17 000 years back in India.

What is certain is that by around 650AD the use of zero as a number came into Indian mathematics. The Indians also used a place-value system and zero was used to denote an empty place. In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries. Let us examine this latter use first since it continues the development described above.

In around 500AD Aryabhata devised a number system which has no zero yet was a positional system. He used the word "kha" for position and it would be used later as the name for zero. There is evidence that a dot had been used in earlier Indian manuscripts to denote an empty place in positional notation. It is interesting that the same documents sometimes also used a dot to denote an unknown where we might use x. Later Indian mathematicians had names for zero in positional numbers yet had no symbol for it. The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
We now come to considering the first appearance of zero as a number. Let us first note that it is not in any sense a natural candidate for a number. From early times numbers are words which refer to collections of objects. Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects. Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division. In three important books the Indian mathematicians Brahmagupta, Mahavira and Bhaskara tried to answer these questions.
Brahmagupta attempted to give the rules for arithmetic involving zero and negative numbers in the seventh century. He explained that given a number then if you subtract it from itself you obtain zero. He gave the following rules for addition which involve zero:-

The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.

Subtraction is a little harder:-
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.

Brahmagupta then says that any number when multiplied by zero is zero but struggles when it comes to division:-
Positive or negative numbers when divided by zero is a fraction the zero as denominator. Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Really Brahmagupta is saying very little when he suggests that n divided by zero is n/0. Clearly he is struggling here. He is certainly wrong when he then claims that zero divided by zero is zero. However it is a brilliant attempt from the first person that we know to try to extend arithmetic to negative numbers and zero.
In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book. He correctly states that:-
... a number multiplied by zero is zero, and a number remains the same when zero is subtracted from it.

However his attempts to improve on Brahmagupta's statements on dividing by zero seem to lead him into error. He writes:-
A number remains unchanged when divided by zero.

Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing. The reason for this phrase is that some commentators on Mahavira have tried to find excuses for his incorrect statement.
Bhaskara wrote over 500 years after Brahmagupta. Despite the passage of time he is still struggling to explain division by zero. He writes:-
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

So Bhaskara tried to solve the problem by writing n/0 = infinity. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times infinitymust be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and sqrt0 = 0.

Perhaps we should note at this point that there was another civilisation which developed a place-value number system with a zero. This was the Maya people who lived in central America, occupying the area which today is southern Mexico, Guatemala, and northern Belize. This was an old civilisation but flourished particularly between 250 and 900. We know that by 665 they used a place-value number system to base 20 with a symbol for zero. However their use of zero goes back further than this and was in use before they introduced the place-valued number system. This is a remarkable achievement but sadly did not influence other peoples.

You can see a separate article about Mayan mathematics.

The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west. It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. This work was the first in what is now Iraq to use zero as a place holder in positional base notation. Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe. The Book of the Number describes the decimal system for integers with place values from left to right. In this work ibn Ezra uses zero which he calls galgal (meaning wheel or circle). Slightly later in the 12th century al-Samawal was writing:-
If we subtract a positive number from zero the same negative number remains. ... if we subtract a negative number from zero the same positive number remains.

The Indian ideas spread east to China as well as west to the Islamic countries. In 1247 the Chinese mathematician Ch'in Chiu-Shao wrote Mathematical treatise in nine sections which uses the symbol O for zero. A little later, in 1303, Chu Shih-Chieh wrote Jade mirror of the four elements which again uses the symbol O for zero.
Fibonacci was one of the main people to bring these new ideas about the number system to Europe. As the authors of [12] write:-
An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.

In Liber Abaci he described the nine Indian symbols together with the sign 0 for Europeans in around 1200 but it was not widely used for a long time after that. It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers. Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.

One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on. However, this was far from the case. Cardan solved cubic and quartic equations without using zero. He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics. By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.

Of course there are still signs of the problems caused by zero. Recently many people throughout the world celebrated the new millennium on 1 January 2000. Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified. Although one might forgive the original error, it is a little surprising that most people seemed unable to understand why the third millennium and the 21st century begin on 1 January 2001. Zero is still causing problems!

References

1.     R Calinger, A conceptual history of mathematics (Upper Straddle River, N. J., 1999).
2.     G Ifrah, From one to zero : A universal history of numbers (New York, 1987).
3.     G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
4.     G G Joseph, The crest of the peacock (London, 1991).
5.     R Kaplan, The nothing that is : a natural history of zero (London, 1999).
6.     R Mukherjee, Discovery of zero and its impact on Indian mathematics (Calcutta, 1991).
Articles:
7.     S Giuntini, A discussion concerning the nature of zero and the relation between imaginary and real numbers (Italian), Boll. Storia Sci. Mat. 4 (1) (1984), 25-63.
8.     R C Gupta, Who invented the zero?, Ganita-Bharati 17 (1-4) (1995), 45-61.
9.     P Mäder, "Wie die Puppe ein Adler sein wollte, der Esel ein Löwe, die äffin eine Königin - so wollte die Null eine Ziffer sein!" Ein überblick zur Geschichte der Zahl Null, in Jahrbuch überblicke Mathematik, 1995 (Braunschweig, 1995), 39-64.
10. R N Mukherjee, Background to the discovery of the symbol for zero, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, New Delhi, 1976, Indian J. Hist. Sci. 12 (2) (1977), 225-231.
11. K Muroi, The expressions of zero and of squaring in the Babylonian mathematical text VAT 7537, Historia Sci. (2) 1 (1) (1991), 59-62.
12. L Pogliani, M Randic and N Trinajstic, Much ado about nothing - an introductive inquiry about zero, Internat. J. Math. Ed. Sci. Tech. 29 (5) (1998),729--744.
13. S Ursini Legovich, The origin of the zero in Central American civilization. Comparative analysis with the Hindu case (Spanish), Mat. Ense nanza No. 13 (1980), 7-20.
14. M Ja Vygodskii, L'origine du signe de zéro dans la numération babylonienne (Russian), Istor.-Mat. Issled. 12 (1959), 393-420.

Jalaluddin Rumi, Penyair Sufi Terbesar dari Konya-Persia

          Dua orang bertengkar sengit di suatu jalan di Konya. Mereka saling memaki, “O, laknat, jika kau mengucapkan sepatah makian terh...