MATHEMATICS
In the Islamic perspective, mathematics is considered the access route that leads from the sensible to the intelligible world, the scale between the world of change and the sky of archetypes. Unity, the central idea of Islam, is an abstraction from the human point of view, even if in itself it is concrete. Compared to the sense world, mathematics is also an abstraction; but, considered from the point of view of the intelligible world, the "world of ideas" of Plato, is a guide to the eternal essences, which are themselves concrete. Like all the figures are generated from the point, and all the numbers from the unit, so the whole multiplicity comes from the Creator, who is One. Numbers and figures, if considered in the Pythagorean sense - that is, as ontological aspects of Unity, and not simply as pure quantity -, become vehicles for the expression of Unity in multiplicity. The Muslim mind has therefore always been attracted to mathematics, as can be seen not only in the great activity of Muslims in the mathematical sciences, but also in Islamic art.
The Pythagorean number, which is the traditional conception of numbers, is the projection of Unity, an aspect of Origin and of the Center which in a certain sense never leaves its source. In its quantitative aspect, a number can divide and separate; in its qualitative and symbolic aspect, however, it reintegrates multiplicity into Unity. It is also, by virtue of its close connection with geometric figures, a "personality": for example, the three corresponds to the triangle and symbolizes harmony, while the four, which is connected to the square, symbolizes stability. Considered in this perspective, the numbers are like many concentric circles, which echo, in many different ways, their common and immutable center. They do not "progress" externally, but remain united to their source thanks to the ontological relationship that they continue to maintain with unity. The same also applies to geometric figures, each of which symbolizes an aspect of Being. The majority of Muslim mathematicians, like the Pythagoreans, never cultivated mathematical science as a purely quantitative subject, nor did they ever separate numbers from geometrical figures, which conceptualize their "personality". They also knew too well that mathematics, by virtue of its internal polarity, was the "Jacob's ladder" which, under the guidance of metaphysics, could lead to the world of archetypes and to Being itself, but which would be separated from its source. instead it has become the means to descend into the world of quantity, to the pole which is always so far from the luminous source of all existence the more the conditions of cosmic manifestation allow it. There can be no "neutrality" on the part of man in relation to numbers: he rises to the world of Being through the knowledge of their qualitative and symbolic aspects, or descends through them, as mere numbers, to the world of quantity. When mathematics was studied in the Middle Ages, the first aspect was usually considered. The science of numbers was, as the Brothers of Purity wrote, "the first support of the soul by the Intellect, and the generous outpouring of the Intellect on the soul"; it was also considered "the language that speaks of Unity and transcendence".
The study of the mathematical sciences in Islam included almost the same topics as the Latin Quadrivium, with more in scope and a few other secondary topics. His main disciplines were - as in the Quadrivium - arithmetic, geometry, astronomy and music. Most Islamic scientists and philosophers were learned in all these sciences; some, like Avicenna, al-Fārābī and al-Ghazzālī, wrote important treatises on music and its effects on the soul.
Astronomy and sister astrology, with which it was almost always associated (in Arabic, as in Greek, the same word denotes both disciplines), were cultivated for a variety of reasons: there were chronological and calendar problems; the need to find the direction of Mecca and the time of day for daily prayers; the task of compiling horoscopes for princes and sovereigns, who almost always consulted an astrologer for their activities; and, of course, the desire to perfect the science of the motion of celestial bodies and to overcome its inconsistencies, so as to achieve the perfection of knowledge.
The main tradition of astronomy came to Muslims from the Greeks through Ptolemy's Almagest. However, there was also the Indian school, whose doctrines concerning astronomy, as well as arithmetic, algebra and geometry, were included in the Siddhānta translated from Sanskrit into Arabic. There were also some Chaldean and Persian texts, most of which originals were lost, as well as a pre-Islamic Arab astronomical tradition. Muslim astronomers, as we have already seen, made many observations, the results of which were recorded in numerous tables (zīj) larger than the old ones, and used until modern times. They also continued the school of Ptolemy's mathematical astronomy, applying their perfected science of spherical trigonometry to the most exact calculation of the motion of the heavens, in the context of the theory of epicycles. A geocentric theory usually followed, although it is aware, as al-Bīrūnī demonstrates, of the existence of the heliocentric system. And as al-Bīrūnī relates, Abū Sa'īd al-Sijzī even built an astrolabe based on the heliocentric theory.
The influence of Indian ideas would also have resulted in the development and systematization of algebra science. Although Muslims were familiar with Diophantus' work, there is little doubt that algebra, as cultivated by Muslims, has its roots in Indian mathematics, which they synthesized with Greek methods. The genius of the Greeks was highlighted in their expression of the finite order, of the cosmos, and therefore of numbers and figures; the perspective of Eastern wisdom is based on Infinity, whose "horizontal image" corresponds to the "undefined" character of mathematics. Algebra, which is integrally associated with this perspective based on Infinity, was born of Indian speculation and reached maturity in the Islamic world, where it was always connected to geometry and where it retained its metaphysical base. Together with the use of Indian numerals - known today as "Arabic numerals" -, algebra can be considered the most important science that Muslims added to the corpus of ancient mathematics. In Islam the traditions of Indian and Greek mathematics met and unified in a structure in which algebra, geometry and arithmetic would have possessed a contemplative, spiritual and intellectual aspect, in addition to that practical and purely rational aspect, which was the ¬ca part of medieval mathematics to be inherited and developed by the later known western science of the same name.
The history of mathematics in Islam begins with rigor with Muhammad ibn Mūsā al-Khwārazmī, in whose writings the Greek and Indian mathematical traditions were merged. This mathematician of the III / IX century left several works, among which the most important is the Compendium in the process of calculation by constraint and equation, which we will examine later. It was translated several times into Latin, with the title of Liber Algorismi, or "Book of al-Khwārazmī"; it became the root of the word "algorithm".
Al-Khwārazmī was followed in the same century by al-Kindī, the first famous Islamic philosopher who was also a mathematical expert, who wrote treatises on almost every subject of the discipline, and his disciple Ahmad al-Sarakhsī, best known for his works on geography, music and astrology. Also of this period was Māhānī, who continued the development of algebra and became particularly famous for studying the problem of Archimedes, and the three sons of Shākir ibn Mūsā - Muhammad, Ahmad and æasan -, who are also called the "Banu Mūsā ». They were all well known mathematicians, and Ahmad was also a physical expert.
The beginning of the IV / X century marks the appearance of various great translators, who were also mathematicians of money order. Particularly eminent among them was Thābit ibn Qurrah, who translated the Conics of Apollonius, various treatises of Archimedes and the Introduction to Nicomach's arithmetic, and he himself was one of the greatest Muslim mathematicians. He is credited with having calculated the volume of a paraboloid and of having given a geometric solution to some third degree equations. His contemporary Qusøā ibn Lūqā, who became famous in later Islamic history as a personification of the wisdom of the Ancients, was also a competent translator, and translated the works of Diophantus and Heron into Arabic.
Among the other scholarship mathematicians of the fourth / tenth century we must include Abū'l-Wafā 'al-Buzjānī, the commentator of the Compendium in the process of calculation for transport and equation, which solved the fourth degree equation x4 + px3 = q, by means of the intersection of a parabola and a hyperbola. To this century also belong Alhazen, of which we have already spoken, and the "Brothers of the Purità", that we will discuss in a moment. They were followed by Abū Sahl al-Kūhī, another of the most eminent Muslim algebraists and author of the Additions to the Book of Archimedes, who made a thorough study of the trinomial equation.
One could also mention Avicenna among the mathematicians active in this era, although his reputation is much greater as a philosopher and as a physician than as a mathematician. Avicenna, as before him al-Fārābī, elaborated the theory of Persian music of his time, a music that has survived as a living tradition to this day. It is not correct to say that their works are a contribution to the theory of "Arab music", since Persian music essentially belongs to a different musical family. It is very similar to the music of the ancient Greeks - to the music heard by Pythagoras and Plato - although it has had some influence on Arab music, as well as a strong influence on flamenco, and if it has also suffered the influence of rhythm and melody of Arab music. It was this tradition of Persian music that Avicenna, and before him al-Fārābī, theorized in the form of study then considered a branch of mathematics.
Avicenna was a contemporary of the famous al-Bīrūnī, who left us some of the most important mathematical and astronomical writings of the medieval period, and who led a special study of problems such as numerical series and the determination of the radius of the Earth. His contemporary Abū Bakr al-Karkhī also left two fundamental works of Islamic mathematics, the book dedicated to Fakhr al-Dīn on algebra and the requirements for arithmetic.
The fifth / eleventh century, which marks the coming to power of the Seljuks, was characterized by a certain lack of interest in mathematics in the official schools, although in this period numerous great mathematicians appeared. They were led by 'Umar Khayyām and a host of other astronomers and mathematicians who worked with him on the revision of the Persian calendar. The work of these mathematicians eventually led to the fruitful activity of the XNUMXth / XNUMXth century - when, following the Mongol invasion, the study of mathematical sciences was rejuvenated. The main figure of this period was Nasīr al-Dīn al-Tusī. Under his direction, as we have seen previously, many scientists, especially mathematicians, were gathered in the Maragha observatory.
Although, after the seventh and thirteenth centuries, interest in the study of mathematics gradually diminished, important mathematicians continued to flourish, and they solved new problems and discovered new methods and techniques. Ibn Bannā 'al-Marrākushī, in the VIII / XIV centuries, created a new approach to the study of numbers, followed a century later by Ghiyāth al-Dīn al-Kāshānī. The latter was the greatest Muslim mathematician in the field of calculation and number theory. He was the true discoverer of the decimal fractions and made an exact determination of the value of pi, and he also discovered many new methods and techniques for calculation. His is the Key to Arithmetic (Miftaá al-áisāb), which is the most fundamental work of this kind in Arabic. Meanwhile, a contemporary of al-Kāshānī, Abū'l-æasan al-Bustī, who lived in Morocco, on the other side of the Islamic world, was plotting new avenues in the field of the study of numbers, and the Egyptian Badr al- Dīn al-Māridīnī was composing important mathematical and astronomical treatises.
The Safavid revival in Persia marks the last period of relatively extensive activity in the field of mathematics, although little of it is known to the surrounding world. The architects of the beautiful mosques, schools and bridges of this era were all mathematicians. The most famous of these figures of the X / XVI century active in the field of mathematics was Bahā 'al-Dīn al-'Amilī. In the field of mathematics his writings were mostly a review and a compendium of the works of the previous masters; they became the standard texts in the various branches of this science from the time when, in the official schools, the study of mathematics was limited to a summary treatment, leaving the most serious study of individual initiative.
A contemporary of Bahā 'al-Dīn al-'Amilī, Mullā Muáammad Bāqir Yazdī, who flourished at the beginning of the 10th century, made some original mathematical studies. It has been claimed by some later mathematicians that he also made an autonomous discovery of the logarithm, but this statement has not yet been fully investigated and demonstrated. After Yazdī, mathematics remained mainly tied to the frame outlined by the medieval masters of this science. There were some occasional figures, such as the Narāqī family of Kashan, from the 12th / 18th century, whose members wrote various original treatises, or Mullā 'Alī Muhammad Isfahānī, which in the thirteenth / nineteenth century gave numerical solutions for third-degree equations. There were also some important Indian mathematicians. In general, however, the speculative force of Islamic society turned almost completely to questions of metaphysics and gnosis; mathematics, apart from its use in everyday life, essentially played the role of scale in the intelligible world of metaphysics. It thus fulfilled the function that the Brethren of Purity and many other earlier authors had considered its true raison d'être.
To sum up the results achieved by Islamic mathematics, we can say that Muslims first developed the theory of numbers in both its mathematical and metaphysical aspects. They generalized the concept of number beyond what was known to the Greeks. They also developed powerful new methods of numerical computation, which reached their culmination later with Ghiyāth al-Dīn al-Kāshānī in the 8th / 14th and 9th centuries. They also dealt with decimal fractions, numerical series, and related branches of mathematics related to numbers. They developed and systematized the science of algebra, while still preserving its link with geometry. The work of the Greeks continued in flat and solid geometry. Finally they developed trigonometry, both flat and solid, elaborating precise tables for functions and discovering many trigonometric relationships. Moreover, although this science was cultivated from the beginning in conjunction with astronomy, it was perfected and transformed for the first time in a science independent of Nasīr al-Dīn al-Tūsī in his famous work Figure of the secant, which represents one among the greatest achievements of medieval mathematics.
The Brethren of Purity, whose historical identity still remains doubtful, were a group of scholars, probably from Basra, who in the fourth / tenth century produced a compendium of the arts and sciences in 52 letters. There is also the Risālat al-jāmi'ah, which summarizes the teachings of the Epistles. Their clear style and the effective simplification of difficult ideas made their Epistles very popular, giving rise to so much interest in the philosophical and natural sciences. The sympathies of the Brethren of the Purity went decisively to the Pythagorean-Hermetic aspect of the Greek heritage, as is evident above all in their mathematical theories, which exerted a great influence in later centuries, particularly among the Shiite circles. Like the Pythagoreans, they emphasized the symbolic and metaphysical aspect of arithmetic and geometry, as can be inferred from the following selection of their writings.
It can be said that the algebra originated with the famous work of Muáammad ibn Mūsā al-Khwārazmī. A compendium in the process of calculation by constraint and equation (Kitāb al-mukhtaöar fī al-jabr wa'l-muqābalah), in which the Arabic word al-jabr was used for the first time, which means "constriction", and also "restoration". According to some authors, the word "algebra" derived from this word. Furthermore, al-Khwārazmī's book on arithmetic, which was later translated into Latin along with his work on algebra, contributed more than any other text to the spread of the Indian numbering system in both the Islamic world and the West.
The name of 'Umar Khayyām has become very familiar in the West thanks to the very beautiful, though sometimes free, English translation of his Rubā'īyāt or Quartine (Quatrains) by Fitzgerald [1859]. In his time Khayyam was known as a metaphysician and as a scientist rather than a poet, and today in Persia he is remembered above all for his mathematical works and for having participated with other astronomers in the elaboration of the solar calendar jalāli, which has been used since until today.
In his time he was known not only as a master of mathematical sciences and as a follower of Greek-inspired philosophy, and especially of the school of Avicenna, but also as a Sufi. Despite being attacked by certain religious authorities, and also by certain Sufis who wished to present Sufism under a more exoteric aspect, Khayyām must be considered a Gnostic, behind whose apparent skepticism there is the absolute certainty of intellectual intuition. His adherence to Sufism is demonstrated by the fact that he assigned to the Sufis the highest place in the hierarchy of knowledge holders.
In Khayyam, various perspectives of Islam are united. He was a Sufi and a poet, as well as a philosopher, an astronomer and a mathematician. Unfortunately, apparently he did not write much, and even a few works were lost. Nonetheless, the remaining works - which include, in addition to his poetry, treatises on existence, generation and corruption, physics, the totality of the sciences, the balance, metaphysics, and also mathematical works formed by research on the axioms of Euclid , on arithmetic and algebra - are sufficient proof of its universality. Khayyam's Algebra is among the most remarkable mathematical texts of the medieval period. It deals with cubic equations, which classifies and solves (usually geometrically), and always maintains the relationship between the unknowns, numbers and geometric shapes, thus maintaining the link between mathematics and the metaphysical meaning implicit in Euclidean geometry.