数学家传记
Ibn Sina或伊本·西那是所有伊斯兰哲学家-科学家中最有影响力的一位。他不仅撰写了医学著作,还涉及几何学、天文学、算术和音乐。
伊本·西那常以其拉丁名伊本·西那为人所知,尽管今天大多数提及他的地方都已恢复使用正确的 ibn Sina 这一形式。我们了解他生平的许多细节,因为他写了一部自传,而这部自传又由他的一位学生所写的传记材料加以补充。这部自传并不仅仅是对其生平的叙述,而是为了阐明他关于达到终极真理的思想而写的,因此必须谨慎地加以解读。这部自传的一个有用的校勘本见于[7],而一个新的译本见于[9]。
伊本·西那的一生为他所经历的那个政治极度动荡的时期所主导。萨曼王朝是穆斯林阿拉伯征服之后在伊朗兴起的第一个本土王朝,大约从900年起控制着河中地区与呼罗珊。布哈拉是他们的首都,它与撒马尔罕同为帝国的文化中心。然而,从10世纪中叶起,萨曼王朝的势力开始削弱。到伊本·西那出生时,Nuh ibn 阿布·纳斯尔·曼苏尔是布哈拉的苏丹,但他正竭力维持对帝国的控制。
伊本·西那的父亲是Nuh ibn 曼苏尔一处地产中一个村庄的治理者。他由父亲教育,其家是该地区学者聚会的场所。伊本·西那确实是个非凡的孩子,其记忆力和学习能力令在他父亲家中聚会的学者们惊叹。到十岁时,他已背下《古兰经》以及他读过的大部分阿拉伯诗歌。伊本·西那十三岁时开始学医,到十六岁开始诊治病人时已精通这门学科。他还学习了逻辑学和形而上学,接受当时一些最好的教师的指导,但在所有领域他都继续自学。在其自传中(见[7]或[9]),伊本·西那强调他或多或少是自学成才的,但在他一生中的关键时刻曾得到过帮助。
正是他在医学上的技能对伊本·西那来说具有重大价值,因为正是通过他在这一领域的声誉,萨曼王朝统治者Nuh ibn 曼苏尔才听说了他。伊本·西那治好萨曼王朝统治者的一场病之后,作为奖赏,他获准使用萨曼王朝的皇家图书馆,这对伊本·西那在整个学术领域的发展都极为重要。
如果萨曼王朝统治者的命运有所好转,伊本·西那的一生将会截然不同。努赫·伊本·曼苏尔为了保住权力,曾任命前突厥奴隶塞布克提金为加兹尼的统治者,并任命其子马哈茂德为呼罗珊总督。然而,已控制大部分河中地区的突厥喀喇汗人联合马哈茂德,着手推翻萨曼王朝。在夺得呼罗珊之后,他们于999年占领了布哈拉。随后五年间,萨曼王朝试图重新夺回控制权,但他们的权力时代已经结束。正如[2]中所述:-
命运将[伊本·西那]抛入了伊朗历史上一个动荡时期,当时新的突厥势力正在中亚取代伊朗人的统治,而当地伊朗王朝正试图从巴格达(位于现代伊拉克)的阿拔斯哈里发政权中获得政治独立。
萨曼王朝的失败以及另一件创伤性事件——他父亲的去世——彻底改变了伊本·西那的生活。在没有赞助人或父亲支持的情况下,他开始了在呼罗珊各城镇漂泊的生活,白天行医和担任行政官员,每晚则召集学生围绕他进行哲学和科学讨论。他曾在古尔甘吉担任法学家,到过花剌子模,之后在古尔甘任教,接着在雷伊担任行政官员。也许最引人注目的是,尽管生活动荡,他仍继续产出顶尖水平的学术成果。因为[2]:-
……[伊本·西那]的专注力和智力才能如此出众,以至于他能够以非凡的连贯性和持续性继续其智力工作,完全不受外界干扰的影响。
经过这段漂泊时期,伊本·西那前往伊朗中西部的哈马丹。他在这里定居了一段时间,成为宫廷医生。统治的布韦希王子沙姆斯·道拉两次任命他为维齐尔。当时的政治并不容易,伊本·西那被他的政治对手迫使躲藏了一段时间,他也在监狱中作为政治犯度过了一些时光[26]
……但他伪装成苏菲派逃到了伊斯法罕,并加入了阿拉·道拉。
伊本·西那最重要的两部著作是The Book of Healing和The Canon of Medicine。第一部是一部科学百科全书,涵盖逻辑学、自然科学、心理学、几何学、天文学、算术和音乐。第二部是医学史上最著名的单部著作。这些著作是在他于哈马丹期间开始撰写的。
被囚禁后,伊本·西那于1022年在他所侍奉的白益王朝王子去世时决定离开哈马丹,前往伊斯法罕。在这里,他进入了当地王子的宫廷,并在相对平静中度过了生命的最后几年。在伊斯法罕,他完成了在哈马丹开始的主要著作,还写了许多关于哲学、医学和阿拉伯语的其他著作。
在军事战役期间,伊本·西那被期望陪同他的赞助人,他的许多作品都是在这样的战役中创作的。正是在这样一次军事战役中,他病倒了,尽管试图将自己的医疗技能应用于自身,还是去世了[1]:-
……死于一种神秘的疾病,显然是一种治疗不当的绞痛;然而,他可能是被他的一个仆人毒死的。
伊本·西那写了大约450部作品,其中约240部留存下来。在留存的作品中,150部关于哲学,而40部致力于医学,这是他贡献最大的两个领域。他还写了心理学、地质学、数学、天文学和逻辑学方面的作品。然而,就数学而言,他最重要的著作是他的巨著百科全书式的作品,即Kitab al-Shifa'Ⓣ(《治疗之书》)。这部作品的四个部分之一致力于数学,伊本·西那在百科全书中将天文学和音乐列为数学的分支。事实上,他将数学分为四个分支:几何学、天文学、算术和音乐,然后他又细分了每个主题。他将几何学细分为大地测量学、静力学、运动学、流体静力学和光学;将天文学细分为天文和地理表格以及历法;将算术细分为代数学以及印度加法和减法;将音乐细分为乐器。
毫不奇怪,百科全书的几何部分是基于欧几里得的Elements。伊本·西那给出了证明,但表述缺乏欧几里得所采用的严谨性。事实上,伊本·西那在这部作品中并没有将几何学作为从公理出发的演绎系统来呈现。然而,我们应该注意到,这是伊本·西那选择在百科全书中呈现该主题的方式。在其他关于几何学的著作中,他像许多穆斯林科学家一样,试图给出欧几里得第五公设的证明。百科全书中几何部分涉及的主题有:线、角和平面;平行线;三角形;用尺规作图;平行四边形和三角形的面积;几何代数;圆的性质;不提及无理数数的比例;与多边形面积有关的比例;圆的面积;正多边形;以及多面体和球的体积。完整细节见[17]。
伊本·西那进行了天文观测,我们知道有些是在伊斯法罕进行的,有些是在哈马丹进行的。他从观测中做出了几个正确的推论。例如,他观察到金星是太阳表面的一个斑点,并正确推断出金星必须比太阳更接近地球。这一观察以及伊本·西那的其他相关工作在[53]中讨论。伊本·西那发明了一种观测恒星坐标的仪器。该仪器有两条腿,一端枢接;下腿绕水平量角器旋转,从而显示方位角,而上腿标有刻度并带有观测瞄准器,在与下腿垂直的平面内抬起以给出恒星的高度。伊本·西那对天文学的另一个贡献是他试图通过在古尔甘观测月球的子午中天来计算巴格达和古尔甘之间的经度差。他还正确地指出光速是有限的,尽管很难看出其依据是什么。
由于伊本·西那将音乐视为数学的分支之一,简要说明他在这一主题上的工作(主要涉及音程、节奏模式和乐器)是合适的。一些专家声称,伊本·西那对大三度协和音的推广导致了纯律的使用,而不是与毕达哥拉斯相关的音准。更多信息包含在T S Vyzgo的论文《论伊本·西那对音乐学的贡献》中,见[5]。
力学是伊本·西那归类在数学下的一个主题。在他的著作Mi'yar al-'aqul中,伊本·西那定义了简单机械及其组合,涉及滚轮、杠杆、绞盘、滑轮等许多。尽管这些材料是众所周知的,当然不是原创的,然而伊本·西那对机械的分类超越了海伦的分类,具有高度的原创性。
由于伊本·西那的主要贡献在哲学,我们至少应提及他在这一领域的工作,尽管我们肯定不会给予这项工作应有的篇幅。他讨论了理性与实在,声称神是纯粹理智,知识在于心灵把握可理解之物。要把握可理解之物,理性和逻辑都是必需的。但伊本·西那声称26:-
……获得知识是重要的。对可理解之物的把握决定了理性灵魂在来世的命运,因此对人类活动至关重要。
伊本·西那给出了一种知识理论,描述的是感知一个对象时的抽象,而非对象本身的具体形式。在形而上学中,伊本·西那考察了存在。他考虑关于世界的科学和数学理论以及神作为终极因。他的目标在1中被描述如下:-
伊本·西那试图在一个宏大的形而上学视野中整合科学与宗教的所有方面。凭借这一视野,他试图解释宇宙的形成,并阐明恶、祈祷、天命、预言、奇迹和奇观等问题。其范围还包括与依照宗教法律组织国家有关的问题以及人的终极命运问题。
已知伊本·西那曾与比鲁尼通信。在10中,给出了伊本·西那寄给比鲁尼的十八封信,以回答后者提出的问题。这些信件涵盖哲学、天文学和物理学等主题。伊本·西那还有其他保存下来的通信,已在文章31中作了综述。这些信件的主题包括对神学家和自称拥有魔力者的论辩,以及对那些对某一知识分支只有肤浅兴趣者观点的驳斥。伊本·西那就哲学中的某些主题写作,并写信给学生,这些学生必定曾请求他解释他们在某部经典文本中遇到的困难。31的作者们认为伊本·西那在促进自然科学,并论辩反对那些试图遮蔽真理的宗教人士。
Ibn Sina is often known by his Latin name of Avicenna, although most references to him today have reverted to using the correct version of ibn Sina. We know many details of his life for he wrote an autobiography which has been supplemented with material from a biography written by one of his students. The autobiography is not simply an account of his life, but rather it is written to illustrate his ideas of reaching the ultimate truth, so it must be carefully interpreted. A useful critical edition of this autobiography appears in [7] while a new translation appears in [9].
The course of ibn Sina's life was dominated by the period of great political instability through which he lived. The Samanid dynasty, the first native dynasty to arise in Iran after the Muslim Arab conquest, controlled Transoxania and Khorasan from about 900. Bukhara was their capital and it, together with Samarkand, were the cultural centres of the empire. However, from the middle of the 10th century, the power of the Samanid's began to weaken. By the time ibn Sina was born, Nuh ibn Mansur was the Sultan in Bukhara but he was struggling to retain control of the empire.
Ibn Sina's father was the governor of a village in one of Nuh ibn Mansur's estates. He was educated by his father, whose home was a meeting place for men of learning in the area. Certainly ibn Sina was a remarkable child, with a memory and an ability to learn which amazed the scholars who met in his father's home. By the age of ten he had memorised the Qur'an and most of the Arabic poetry which he had read. When ibn Sina reached the age of thirteen he began to study medicine and he had mastered that subject by the age of sixteen when he began to treat patients. He also studied logic and metaphysics, receiving instruction from some of the best teachers of his day, but in all areas he continued his studies on his own. In his autobiography (see [7] or [9]) ibn Sina stresses that he was more or less self-taught but that at crucial times in his life he received help.
It was his skill in medicine that was to prove of great value to ibn Sina for it was through his reputation in that area that the Samanid ruler Nuh ibn Mansur came to hear of him. After ibn Sina had cured the Samanid ruler of an illness, as a reward, he was allowed to use the Royal Library of the Samanids which proved important for ibn Sina's development in the whole range of scholarship.
If the fortunes of the Samanid rulers had taken a turn for the better, ibn Sina's life would have been very different. Nuh ibn Mansur, in an attempt to keep in power, had put Sebüktigin, a former Turkish slave, as the ruler of Ghazna and appointed his son Mahmud as governor of Khorasan. However the Turkish Qarakhanids, already in control of most of Transoxania, joined with Mahmud and moved to depose the Samanids. After gaining Khorasan they took Bukhara in 999. There followed a period of five years in which the Samanids tried to regain control but their period of power was over. As recounted in [2]:-
Destiny had plunged [ibn Sina] into one of the tumultuous periods of Iranian history, when new Turkish elements were replacing Iranian domination in Central Asia and local Iranian dynasties were trying to gain political independence from the 'Abbasid caliphate in Baghdad (in modern Iraq).
The defeat of the Samanids and another traumatic event, the death of his father, changed ibn Sina's life completely. Without the support of a patron or his father, he began a life of wandering round different towns of Khorasan, acting as a physician and administrator by day while every evening he gathered students round him for philosophical and scientific discussion. He served as a jurist in Gurganj, was in Khwarazm, then was a teacher in Gurgan and next an administrator in Rayy. Perhaps most remarkable is the fact that he continued to produce top quality scholarship despite his chaotic life style. For [2]:-
... the power of concentration and the intellectual prowess of [ibn Sina] was such that he was able to continue his intellectual work with remarkable consistency and continuity and was not at all influenced by the outward disturbances.
After this period of wandering, ibn Sina went to Hamadan in west-central Iran. Here he settled for a while becoming court physician. The ruling Buyid prince, Shams ad-Dawlah, twice appointed him vizier. Politics was not easy at that time and ibn Sina was forced into hiding for a while by his political opponents and he also spent some time as a political prisoner in prison [26]
... but he escaped to Isafan, disguised as a Sufi, and joined Ala al-Dwla.
Ibn Sina's two most important works are The Book of Healing and The Canon of Medicine. The first is a scientific encyclopaedia covering logic, natural sciences, psychology, geometry, astronomy, arithmetic and music. The second is the most famous single book in the history of medicine. These works were begun while he was in Hamadan.
After being imprisoned, ibn Sina decided to leave Hamadan in 1022 on the death of the Buyid prince whom he was serving, and he travelled to Isfahan. Here he entered the court of the local prince and spent the last years of his life in comparative peace. At Isfahan he completed his major works begun at Hamadan and also wrote many other works on philosophy, medicine and the Arabic language.
During military campaigns ibn Sina was expected to accompany his patron and many of his works were composed on such campaigns. It was on one such military campaign that he took ill and, despite attempting to apply his medical skills to himself, died [1]:-
... of a mysterious illness, apparently a colic that was badly treated; he may, however, have been poisoned by one of his servants.
Ibn Sina's wrote about 450 works, of which around 240 have survived. Of the surviving works, 150 are on philosophy while 40 are devoted to medicine, the two fields in which he contributed most. He also wrote on psychology, geology, mathematics, astronomy, and logic. His most important work as far as mathematics is concerned, however, is his immense encyclopaedic work, the Kitab al-Shifa' Ⓣ. One of the four parts of this work is devoted to mathematics and ibn Sina includes astronomy and music as branches of mathematics within the encyclopaedia. In fact he divided mathematics into four branches, geometry, astronomy, arithmetic, and music, and he then subdivided each of these topics. Geometry he subdivided into geodesy, statics, kinematics, hydrostatics, and optics; astronomy he subdivided into astronomical and geographical tables, and the calendar; arithmetic he subdivided into algebra, and Indian addition and subtraction; music he subdivided into musical instruments.
The geometric section of the encyclopaedia is, not surprisingly, based on Euclid's Elements. Ibn Sina gives proofs but the presentation lacks the rigour adopted by Euclid. In fact ibn Sina does not present geometry as a deductive system from axioms in this work. We should note, however, that this was the way that ibn Sina chose to present the topic in the encyclopaedia. In other writings on geometry he, like many Muslim scientists, attempted to give a proof of Euclid's fifth postulate. The topics dealt with in the geometry section of the encyclopaedia are: lines, angles, and planes; parallels; triangles; constructions with ruler and compass; areas of parallelograms and triangles; geometric algebra; properties of circles; proportions without mentioning irrational numbers; proportions relating to areas of polygons; areas of circles; regular polygons; and volumes of polyhedra and the sphere. Full details are given in [17].
Ibn Sina made astronomical observations and we know that some were made at Isfahan and some at Hamadan. He made several correct deductions from his observations. For example he observed Venus as a spot against the surface of the Sun and correctly deduced that Venus must be closer to the Earth than the Sun. This observation, and other related work by ibn Sina, is discussed in [53]. Ibn Sina invented an instrument for observing the coordinates of a star. The instrument had two legs pivoted at one end; the lower leg rotated about a horizontal protractor, thus showing the azimuth, while the upper leg marked with a scale and having observing sights, was raised in the plane vertical to the lower leg to give the star's altitude. Another of ibn Sina's contributions to astronomy was his attempt to calculate the difference in longitude between Baghdad and Gurgan by observing a meridian transit of the moon at Gurgan. He also correctly stated, with what justification it is hard to see, that the velocity of light is finite.
As ibn Sina considered music as one of the branches of mathematics it is fitting to give a brief indication of his work on this topic which was mainly on tonic intervals, rhythmic patterns, and musical instruments. Some experts claim that ibn Sina's promotion of the consonance of the major third led to the use of just intonation rather than the intonation associated with Pythagoras. More information is contained in T S Vyzgo's paper "On Ibn Sina's contribution to musicology" in [5].
Mechanics was a topic which ibn Sina classified under mathematics. In his work Mi'yar al-'aqul ibn Sina defines simple machines and combinations of them which involve rollers, levers, windlasses, pulleys, and many others. Although the material was well-known and certainly not original, nevertheless ibn Sina's classification of mechanisms, which goes beyond that of Heron, is highly original.
Since ibn Sina's major contributions are in philosophy, we should at least mention his work in this area, although we shall certainly not devote the space to it that this work deserves. He discussed reason and reality, claiming that God is pure intellect and that knowledge consists of the mind grasping the intelligible. To grasp the intelligible both reason and logic are required. But, claims ibn Sina [26]:-
... it is important to gain knowledge. Grasp of the intelligibles determines the fate of the rational soul in the hereafter, and therefore is crucial to human activity.
Ibn Sina gives a theory of knowledge, describing the abstraction in perceiving an object rather than the concrete form of the object itself. In metaphysics ibn Sina examined existence. He considers the scientific and mathematical theory of the world and ultimate causation by God. His aims are described in [1] as follows:-
Ibn Sina sought to integrate all aspects of science and religion in a grand metaphysical vision. With this vision he attempted to explain the formation of the universe as well as to elucidate the problems of evil, prayer, providence, prophecies, miracles, and marvels. also within its scope fall problems relating to the organisation of the state in accord with religious law and the question of the ultimate destiny of man.
Ibn Sina is known to have corresponded with al-Biruni. In [10], eighteen letters which ibn Sina sent to al-Biruni in answer to questions that he had posed are given. These letters cover topics such as philosophy, astronomy and physics. There is other correspondence from ibn Sina which has been preserved which has been surveyed in the article [31]. The topics of these letters include arguments against theologians and those professing magical powers, and refutation of the opinions of those having a superficial interest in a branch of knowledge. Ibn Sina writes on certain topics in philosophy, and writes letters to students who must have asked him to explain difficulties they have encountered in some classic text. The authors of [31] see ibn Sina as promoting natural science and arguing against religious men who attempt to obscure the truth.
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