数学家传记
罗吉尔·培根是一位英国数学家。他最重要的数学贡献是将几何学应用于光学。
罗吉尔·培根的父母是富裕的地主。然而,他的父亲必定高度重视教育,因为他的两个儿子成了学者,而且他愿意在经济上支持他们。在给出已知的关于他早年岁月的少数细节之前,我们应当稍微谈谈我们为培根给出的出生日期。已知的全部情况是,1267年培根写下了以下内容:-
我在科学和语言上勤奋钻研,自我最初学习字母表以来,四十年已经过去。我一直勤学不辍,在这四十年中,除了两年之外,我一直处于学习之中。
1214年这个日期是这样得出的:假定这段话的意思是,从他开始大学学习以来已经过去了四十年。这些学习始于十三岁,所以从1267年减去53年,就得出1214年作为他的出生年份。然而,还有另一种解释。这就是把他在1227年最初学习字母表当作字面事实来接受。如果是这样,那么也许他出生于1222年前后,但培根会声称自己从五岁起就一直在学习吗?对于这个时期地主的儿子来说,这似乎不太可能。
培根并不是他父母最年长的儿子,因此他本不会被期望继承家族的产业和财富。他的父母很可能期望他成为一名神职人员,这是像他们这样地位的家庭中,除长子以外的儿子通常走的道路。尽管没有记录显示培根在进入牛津大学之前的教育情况,但他很可能由当地神职人员教授拉丁语和算术,为他上大学做准备(当时所有教学都用拉丁语进行)。他十三岁时进入牛津大学,父亲为他支付了食宿、生活费和学费。
当然,我们不能以今天三四年制的大学课程来看待培根的大学课程。十三岁进入大学意味着大学同时提供今天会被视为中学和大学的教育,因此培根在牛津度过了多年的学习时光。他最初的学习涵盖了三艺:语法、逻辑和修辞。随后他进入四艺阶段,学习几何、算术、音乐和天文。他从牛津大学获得硕士学位,并留在那里任教直到约1241年。
由于亚里士多德不是基督徒,亚里士多德的学说曾在巴黎大学被禁多年。然而在1240年代初,巴黎将亚里士多德的学说重新引入课程。他们指望年轻的讲师培根来巴黎大学讲授亚里士多德的思想,因为他在牛津已成为亚里士多德方面的专家,其学说在牛津构成了课程材料的主要部分。他加入了巴黎的文学院,该院分为四个行政单位,三个是法国人的,一个——培根加入的那个——是英国人的。这仅仅是行政设置,绝不表示教学语言,与牛津一样,教学语言是拉丁语。今天觉得上午9点的课早得不可思议的大学生应当注意,在培根时代,巴黎的课从早上6点开始。
我们应当注意,直到这时,培根对科学表现出很少的兴趣。正是在巴黎期间,他遇到了彼得·佩雷格里努斯,后者对他产生了重大影响。培根后来写道:-
[Peregrinus]通过实验获得了关于自然、医学和炼金术的知识,以及天上和地下的一切知识。……此外,他还考察了老巫婆的实验和算命,以及她们的咒语和所有魔法师的咒语。同样也考察了所有魔术师的幻象和诡计;这样做的目的是为了使任何应当知道的事情都不被他遗漏,并且使他能够看出在多大程度上谴责一切虚假和魔法之事。
Peregrinus后来在安茹的查理一世的军队中担任工程师时,写了一篇关于磁石的著名论文。他的杰作是在1269年8月,军队在教皇批准的十字军东征中围攻意大利的卢切拉时写成的。
培根对数学和自然哲学的兴趣,很可能是由佩雷格里努斯激发的,在他1247年回到牛津后占据了他的生活。这成了一种激情,他倾注了家族的全部财富。他购买书籍、设备、仪器和数学表。这些都极其昂贵,因为所有书籍都是手稿,每一卷都必须手工抄写。他深受罗伯特·格罗斯泰斯特著作的影响,开始了对语言、数学、光学和科学的深入研究。他学习如此刻苦,以至于没有时间结交朋友或参与周围的大学生活。他全神贯注于学习,以至于周围的人惊叹他的健康竟能承受他花费的漫长时光。
他最重要的数学贡献是将几何学应用于光学。他说:-
数学是科学的门和钥匙。
培根读过海什木的Optics,这使他意识到数学应用于现实问题的重要性,见[26]。他追随罗伯特·格罗斯泰斯特,强调使用透镜放大以辅助自然视觉。他用透镜和镜子进行了一些系统的观察。他似乎以一种非常现代的科学方法计划和解释这些实验。然而,他的著作中描述了许多他从未在实践中进行过的实验。Lindberg [23]说培根的实验包括:-
……设计的或人为的实验、偶然的日常经验、其他观察者的经验报告、神圣启示的精神体验,以及我[Lindberg]所称的几何实验。
在De mirabile potestate artis et naturae中,这基本上是一封写于约1250年的信,培根描述了他的科学思想,特别是他对机械装置的设想以及一些光学成就。
培根于1251年访问巴黎,但后来离开牛津大学,加入了小兄弟会,即牛津的方济各会修道院。他做出这一举动的原因尚不清楚。当然,他是一位虔诚的基督徒,相信他的科学工作有助于理解世界,从而通过理解上帝的创造来理解上帝。圣奥古斯丁曾鼓励基督徒学习异教哲学家的教导并加以基督教化的利用。培根坚信圣奥古斯丁的这一教导,并研读了他所能得到的所有希腊和阿拉伯著作。然而,这并非教会中许多人的观点,他们认为研究圣经是通往知识的唯一途径,不赞成研究非基督教哲学家。加入方济各会——该会有学术传统——可能是一个举动,以便为培根提供一些保护,免受那些反对他观点的人的攻击。也可能仅仅是因为此时他在科学研究上的巨额支出已耗尽了他家的全部资金,他被迫寻找一种维持生计的手段。一些历史学家认为,这一举动很大程度上是由于健康不佳,可能是过度工作所致。
在牛津修道院,他继续对科学保持兴趣。爱德华·斯图迪是修士生活的重要组成部分,尽管培根的实验科学会是一种独特的形式。修道院有一个极好的图书馆,所以尽管按照修会规则他现在不允许拥有个人财产,他仍然可以继续他的科学追求。然而,1256年,康沃尔的理查德成为英格兰方济各会学术方面的负责人。这对培根来说是不幸的,因为他对理查德的思想非常批评。因此,也许不足为奇的是,理查德被任命后不久,培根就被迫结束在牛津修道院的学术研究,并被送到巴黎的一所修道院。他写道:-
他们以难以言表的暴力强迫我服从他们的意志。
大约10年间,培根与外界没有个人接触,尽管他能够通过信件通信。在此期间他能够做的少数学术任务之一是历法改革的工作。他向教皇提出的改革历法的请求没有被听取,而当300年后教会确实按照培根建议的思路改革了历法时,他并没有因为早期的提议而获得赞誉。培根在巴黎修道院期间也能够教授数学,所以尽管看起来意图是阻止他进行教会不批准的研究,某种生活仍然是可能的。
培根在1264年联系了富尔克枢机主教,提议写一本对教会有益的科学书籍。似乎有一些误解,因为富尔克枢机主教以为这本书已经写好了,并要求看看。培根将此视为他重新开始科学研究并摆脱巴黎修道院的唯一机会。然而,由于没有钱,而且此时他在英格兰的家族因选择了失败的政治派别而破产,他几乎没有手段来写出所需的文本。然而,1265年,富尔克枢机主教成为教皇克雷芒四世,培根现在得到了教皇的支持。他联系了教皇,教皇在1266年6月22日的一封信中回复,告诉培根秘密地写他的作品,这样他的上级就不会知道他违反了修会规则。这对培根来说似乎是不可能的,所以他去找他的上级,向他们展示了教皇的要求。他被允许继续。
到1267年,培根已经写出了看起来与今天数学家或科学家可能提出的资助申请非常相似的东西。他的提议是由一个合作者团队在一家教会机构的协调下编写一部涵盖所有科学的百科全书。他的提议包含在迅速写成的Opus maius(包含840页的伟大著作)、Opus minus(较小著作)和Opus tertium(第三著作)中。培根旨在向教皇表明科学在大学课程中具有正当角色,并且对教会很重要。他在Opus maius中写下了一系列令人震惊的想法,例如他提出了望远镜的设想:-
因为我们能够如此塑造透明物体,并相对于我们的视觉和视觉对象以这样的方式安排它们,使得光线会按照我们期望的任何方向反射和弯曲,并且在我们希望的任何角度下,我们都可以看到近处或远处的物体……因此我们也可能使太阳、月亮和星星在外观上降临到下面……
这部著作还给出了彩虹最大高度为42°,这比之前给出的任何值都更精确,培根必定是通过实验发现的。文章[16]出色地叙述了培根关于逻辑理论的思想在其职业生涯中如何演变。从他早期的工作开始,他就在语义理论中引入了创新,并在Opus maius中扩展了这些思想,将它们应用于神学和哲学问题。
Opus maius由信使弗瑞兹·约翰送给教皇,弗瑞兹·约翰是培根最喜爱的学生。我们知道手稿到达了罗马,但教皇克莱门特四世在见到它之前就去世了,培根使其伟大计划得以实现的希望破灭了。
然而大约在这个时候,培根得以离开巴黎的修道院返回英格兰。在那里,他开始着手他向教皇提出的宏大计划,开始撰写Communia naturalium(自然哲学通论)和Communia mathematica(数学科学通论)。这最后一部著作包含了对量和比例理论的研究,培根打算将其作为他知识综合体系第二卷的一部分。只有部分内容曾经出版,很可能大部分从未写出,但其中再次包含了一些关于天文学和历法改革的非凡见解,这些是培根在进行观测后形成的。据报道,培根
……有时会在夜间登上这个地方(他在福利桥上的书房,位于泰晤士河中央的一个小岛上),四周环水,在那里测量恒星的高度和距离,并为自己方便而加以利用……
培根相信地球是一个球体,人们可以环绕它航行。他估计到恒星的距离,得出的答案是1.3亿英里。
大约在1278年,培根被他的方济各会同伴关进了意大利安科纳的修道院,罪名是在教学中suspected novelties。在这里,那些观点与上级不一致的修士被单独监禁,甚至不允许与看守说话,因为害怕他们的观点会产生影响。他们被拒绝忏悔,也不被赦罪,因此,他们的上级相信(被囚禁的修士也相信),他们将永远下地狱。然而,1290年方济各会领导层发生变化,瓜弗雷迪的雷蒙德控制了修会,遵循圣方济各的爱的教义,释放了安科纳的囚犯。虽然我们没有明确证据表明培根也在这些人当中,但他很可能在其中,而且他一定尽快回到了英格兰。
从他的著作中可以清楚地看出,培根一直为他所相信的而辩护,反对他认为错误的东西。即使在安科纳的监狱中受了大约12年的苦之后,他仍然继续陈述自己的观点。这些观点在他1293年的最后著作Compendium studii theologiae中,与他一生中任何时候一样激烈地表达出来。
Roger Bacon's parents were well-off landowners. His father, however, must have had a high regard for education since two of his sons became academics and he was prepared to support them financially. Before giving the few details of his early years that are known, we should say a little about the date of birth we have given for Roger. All that is known is that in 1267 Bacon wrote the following:-
I have laboured diligently in sciences and languages, and forty years have passed since I first learned the alphabet. I have always been studious and for all but two of these forty years I have been in study.
The date of 1214 is arrived at by assuming that the passage means that forty years have passed since he began his university studies. These began at the age of thirteen, so 53 from 1267 gives the date of 1214 as the year of his birth. There is, however, another interpretation. This is to accept as a literal fact that he first learned the alphabet in 1227. If so then perhaps he was born around 1222 but would Bacon claim to have been in study since the age of five? This would seem unlikely for the son of landowners of this period.
Roger was not the oldest of his parent's sons so he would not have been expected to inherit the family estates and wealth. It is likely that his parents would have expected him to have become a priest which was the usual route for a son, other than the first, of a family in their position. Although there is no record of Roger's education before he entered Oxford University it is likely that he would have been taught Latin and arithmetic by the local priest to prepare him for university studies (where all teaching was carried out in Latin). At the age of thirteen he entered Oxford University, his father putting up the money for his board, subsistence, and tuition.
Of course we must not think of Bacon's university course in terms of the three or four year university course of today. Entering university at the age of thirteen meant that the university was providing both what would be considered today as a secondary and tertiary education, so Bacon would have spent many years of study at Oxford. His initial studies covered the trivium of grammar, logic, and rhetoric. He then progressed to the quadrivium, studying geometry, arithmetic, music and astronomy. He received Master's Degree from the University of Oxford and remained there teaching until around 1241.
The teachings of Aristotle had been banned at the University of Paris for several years on the grounds that Aristotle was not a Christian. However in the early 1240s Paris reintroduced the teachings of Aristotle into their courses. They looked to the young lecturer Bacon, who had become an expert on Aristotle at Oxford where his teachings formed a major part of the course material, to lecture at the University of Paris on Aristotle's ideas. He joined the Faculty of Arts in Paris, which was divided into four administrative units, three being French and one, which Bacon joined, being English. This was only an administrative set-up and in no way indicated the language of teaching which, like Oxford, was Latin. Students who find 9 a.m. lectures impossibly early in universities today should note that lectures in Paris in Bacon's time began at 6 a.m.
We should note that up to this time Bacon showed little interest in science. It was while at Paris that he met Peter Peregrinus who was to be a major influence on him. Bacon later wrote:-
[Peregrinus] gains knowledge of matters of nature, medicine, and alchemy through experiment, and all that is in the heaven and in the earth beneath. ... Moreover, he has considered the experiments and the fortune-telling of the old witches, and their spells and those of all magicians. And so too the illusions and wiles of all conjurors; and this so that nothing may escape him which ought to be known, and that he may perceive how far to reprove all that is false and magical.
Peregrinus later wrote a famous treatise on magnets while serving as an engineer in the army of Charles I of Anjou. His masterpiece was written while the army was besieging Lucera in Italy in August 1269 while on a crusade approved by the Pope.
Bacon's interest in mathematics and natural philosophy, probably aroused by Peregrinus, took over his life in Oxford after he returned there in 1247. It became a passion into which he poured all his family's wealth. He bought books, equipment, instruments, and mathematical tables. These were all very expensive for all books were in manuscript and each volume had to be copied by hand. He was much influenced by the writings of Grosseteste and he embarked on a deep study of languages, mathematics, optics and sciences. So hard did he study that he had no time for friends or the university life around him. He was so fully occupied with his studies that those around him marvelled that his health stood up to the long hours he spent.
His most important mathematical contribution is the application of geometry to optics. He said:-
Mathematics is the door and the key to the sciences.
Bacon had read al-Haytham's Optics and this made him realise the importance of the applications of mathematics to real word problems, see [26]. He followed Grosseteste in emphasising the use of lenses for magnification to aid natural vision. He carried out some systematic observations with lenses and mirrors. He seems to have planned and interpreted these experiments with a remarkably modern scientific approach. However many experiments are described in his writings which he never carried out in practice. Lindberg [23] says Bacon's experiments included:-
... contrived or artificial experiments, casual everyday experience, reports of the experiences of other observers, the spiritual experience of divine illumination, and what I [Lindberg] call geometrical experiments.
In De mirabile potestate artis et naturae, which is essentially a letter written around 1250, Bacon described his scientific ideas, in particular his ideas for mechanical devices and some of his optical achievements.
Bacon visited Paris in 1251 but later left the University of Oxford and entered the Order of Friars Minor, the Franciscan Friary in Oxford. It is unclear exactly what his reasons were for this move. Certainly he was a devout Christian who believed that his scientific work would aid an understanding of the world, and so of God through understanding His creation. St Augustine had encouraged Christians to learn from and make Christian use of the teachings of pagan philosophers. Bacon strongly believed in this teaching by St Augustine and studied all the Greek and Arabic works he could lay his hands on. This was not the view of many in the Church, however, who believed that a study of the scriptures was the only path to knowledge and disapproved of the study of non-Christian philosophers. Joining the Franciscans, who had a tradition of scholarship, may have been a move to give Bacon some protection from those opposed to his views. It may simply be that by this time his enormous expenditure on scientific study had used all his family's funds, he was forced to find a means to provide his keep. Some historians suggest that the move was largely due to ill health, perhaps caused by over-work.
In the Oxford friary he continued his interest in the sciences. Study was a major part of the life of the friars, although Bacon's experimental science would have been a unique form. The friary had an excellent library so, although he was now not allowed personal possessions by the rules of the Order, he could still continue his scientific pursuits. However, in 1256 Richard of Cornwell became head of the academic side of the Franciscan Order in England. This was unfortunate for Bacon who was very critical of Richard's ideas. Perhaps, therefore, it is not surprising that soon after Richard was appointed, Bacon was forced to end his academic studies at the Oxford friary and was sent to a friary in Paris. He wrote:-
They forced me with unspeakable violence to obey their will.
For about 10 years Bacon had no personal contact with the outside world although he was able to correspond by letter. One of the few academic tasks he was able to do during this time was work on calendar reform. His plea to the Pope to reform the calendar was not listened to and when 300 years later the Church did reform the calendar along the lines suggested by Bacon, he received no credit for his early proposals. Bacon was also able to teach mathematics while in the Paris friary, so although it appears that the intention was to prevent him from undertaking research which the Church did not approve, life of a sort was still possible.
Bacon contacted Cardinal de Foulques in 1264 proposing to write a book on science which would be of benefit of the Church. There seemed to be some misunderstanding, for Cardinal de Foulques got the idea that the book was already written and asked to see it. Bacon saw this as his only chance to restart his scientific studies and be free from the Paris friary. However with no money, and by now his family in England were ruined through choosing the losing political side, he had little means to produce the required text. However, in 1265 Cardinal de Foulques became Pope Clement IV and Bacon now had the support of the Pope. He contacted the Pope who replied in a letter written on 22 June 1266 telling Bacon to write his work in secret so that his superiors would not know that he was breaking the rules of his Order. This seemed impossible to Bacon, so he approached his superiors showing them the Pope's request. He was allowed to proceed.
By 1267 Bacon had written what looks remarkably similar to a grant proposal that a mathematician or scientist might make today. His proposal was for an encyclopaedia of all the sciences worked on by a team of collaborators, coordinated by a body in the Church. His proposal was contained in the rapidly composed Opus maius (Great Work consisting of 840 pages), Opus minus (Smaller Work) and the Opus tertium (Third Work). Bacon was aiming to show the Pope that sciences had a rightful role in the university curriculum and were important to the Church. He wrote down in Opus maius an astounding collection of ideas, for example he gives a proposal for a telescope:-
For we can so shape transparent bodies, and arrange them in such a way with respect to our sight and objects of vision, that the rays will be reflected and bent in any direction we desire, and under any angle we wish, we may see the object near or at a distance ... So we might also cause the Sun, Moon and stars in appearance to descend here below...
This work also gives 42 ° for the maximum altitude of the rainbow, a more accurate value than any previously given and one which Bacon must have discovered by experiment. The article [16] is an excellent account of how Bacon's ideas on logical theory evolved during his career. From his early work he introduced innovations in semantic theory and in Opus maius he extended these ideas applying them to problems of theology and philosophy.
The Opus maius was sent to the Pope by a courier John, who was Bacon's favourite pupil. We know that the manuscript reached Rome but Pope Clement IV died before seeing it and Bacon's chances of having his great project come to fruition vanished.
About this time, however, Bacon was able to leave the friary in Paris and return to England. There he embarked on the great project he had proposed to the Pope, starting to write the Communia naturalium (General Principles of Natural Philosophy) and the Communia mathematica (General Principles of Mathematical Science). This last work contains a study of quantity and of the theory of proportions which Bacon intended as part of volume II of his synthesis of knowledge. Only parts were ever published, probably most was never written, but again there was some remarkable insights on astronomy and calendar reform which Bacon had formed after making observations. It was reported that Bacon
... did sometimes use in the night season to ascend this place (his study on Folly Bridge, on an eyot midstream in the Thames) environed with waters and there to take the altitude and distance of stars and make use of it for his own convenience...
Bacon believed that the Earth was a sphere and that one could sail round it. He estimated the distance to the stars coming up with the answer 130 million miles.
Around 1278 Bacon was put in prison in the convent in Ancona in Italy by his fellow Franciscans, the charge being of suspected novelties in his teaching. Here friars who had views with which their superiors disagreed were put in solitary confinement and not allowed to speak even to their guards for fear their views would have influence. They were refused confession and denied absolution so, their superiors believed (and so did the imprisoned friars), they would go to hell for all eternity. A change in the Franciscan leadership in 1290, however, saw Raymond of Guafredi take control of the Order and following the teachings of love by St Francis, released the prisoners in Ancona. Although we have no explicit evidence that Bacon was among these men, it seems very likely that he was and he must have returned to England as soon as he could.
From his writings it is clear that Bacon had always argued for what he believed and against those he believed to be wrong. He continued to state his views even after suffering the prison at Ancona for around 12 years. They were as aggressively stated in Compendium studii theologiae, his last writings of 1293, as at any time in his life.
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