托马斯·布拉德华Thomas Bradwardine

托马斯·布拉德华的出生日期并不确定。在[1]中，日期被给出为1290年至1300年之间，在[2]中，日期被给出为大约1290年，而例如在[15]中，则被给出为大约1300年。这些仅仅是基于我们所知的布拉德华的第一个确切日期——1321年8月，当时他成为牛津大学默顿学院的会士——所做的猜测。1321年8月是布拉德华被授予学士学位的日期。我们只是取了各种建议出生日期的平均值！

我们将布拉德华的出生地定为Chichester，但这同样基于很少的证据。我们从布拉德华的著作中得知，他的父亲在写作时住在Chichester，但由于这一定是在布拉德华出生二十多年后写的，因此假设他的父亲没有搬家有些异想天开。

虽然不确定，但有强有力的证据表明布拉德华一直留在默顿学院直到1335年。在此期间，他担任了包括监考在内的各种大学职务，并获得了多个学位，如1323年的文学硕士学位和1333年之前的某个时间的神学学士学位。正是在牛津的这段时期，他几乎所有关于逻辑、数学和哲学的著作都写成了。

1333年，布拉德华被任命为林肯的教士，然后在1335年，他加入了围绕达勒姆主教理查德·德·伯里的人群。两年后，当他于1337年9月19日被任命为圣保罗大教堂的牧师时，他搬到了伦敦。此后不久，他成为国王爱德华三世的 chaplain。

国王爱德华三世于1340年1月声称拥有法国国王的头衔。他的主张并非完全没有理由，因为通过他的母亲，他与卡佩王朝的最后一位统治者的关系比法国国王菲利普六世更密切。爱德华于1340年6月在佛兰德斯城市斯鲁伊斯附近赢得了一场海战，但缺乏资源推进迫使他达成休战。1346年7月，爱德华在诺曼底登陆， accompanied by his eldest son Prince Edward (known as the Black Prince)。布拉德华肯定是爱德华入侵部队的一部分。

英国入侵部队于1346年8月26日在克雷西对法国人取得了决定性胜利。然后他们在1346年9月围攻加来，并于1347年8月投降。苏格兰的弗洛伦斯·南丁格尔·大卫二世是法国的盟友，意识到爱德华正忙于围攻加来，于1346年10月入侵英格兰北部。爱德华预料到了这一点，并在英格兰北部留下了一支强大的部队来对付苏格兰人。大卫二世在内维尔十字战役（靠近达勒姆）中被击败。在法国举行的庆祝克雷西和内维尔十字战役胜利的胜利庆典上，布拉德华在爱德华三世面前发表了Sermo epinicius，其文本保存了下来。

爱德华于1347年10月返回英格兰。布拉德华于1348年8月31日被选为坎特伯雷大主教，但相当奇怪的是爱德华取消了这一任命。1349年6月4日，布拉德华第二次被选为坎特伯雷大主教，似乎没有爱德华的任何反对。他于1349年7月10日在阿维尼翁被祝圣，但在他返回伦敦履行职责后不久就死于瘟疫。黑死病于1348年蔓延到英格兰和法国，大约在布拉德华的同时，伦敦可能有三分之一的人口因此死亡。

布拉德华是一位著名的数学家和神学家，被称为“深邃的博士”。他在论文De proportionibus velocitatum in motibus（1328）中研究了匀速运动的物体和速度比。这部作品在支持与批评亚里士多德的物理学之间采取了一条相当奇怪的路线。也许这并不那么奇怪，因为亚里士多德的观点对当时的学习如此根本，以至于对布拉德华所能期望的也许只是对亚里士多德关于运动物体和作用在其上的力的观点的重新解释。很可能他的意图不是批评亚里士多德，而是从数学上证明对亚里士多德陈述的重新解释是合理的。

亚里士多德声称，只有当作用在物体上的力超过阻力时，运动才可能发生。尽管他没有用这些术语来表达，但从亚里士多德的Physics中也可以推导出，物体的速度与作用在它上面的力除以阻力成正比。布拉德华用数学论证表明这两者是不一致的。他通过假设一个初始的力和阻力，然后假设阻力加倍，再加倍等等，同时保持力不变。布拉德华论证说，在某一点上，阻力将超过力，因此物体不能运动。但是，根据第二条规则给出的速度，不可能为零。

布拉德华接着声称，速度的算术增加对应于力与阻力原始比率的几何增加。这巧妙地消除了矛盾，但当然是不正确的。然而，他的观点非常有影响力，并且被接受为力学定律长达一百多年。例如，尼克尔·奥里斯姆遵循了布拉德华的力学思想。16中的Takahashi对布拉德华的比率概念给出了一个有趣的新解释。

布拉德华在试图用几何学反驳原子论时，提出了另一个有趣但错误的论证。然而，他质疑自己论证的逻辑，因为他觉得也许几何学的存在已经假设了原子论是错误的。

布拉德华是第一位研究“星形多边形”的数学家。后来约翰内斯·开普勒对它们进行了更彻底的研究。布拉德华的其他著作包括以下内容。

On insolubles讨论了诸如“我在说谎”这样的逻辑问题。

On "it begins" and "It ceases"论证了一个时间间隔有一个第一瞬间但没有最后瞬间。该间隔的结束以其不存在的第一瞬间为标志。

Speculative geometry包含初等几何，但并非全部基于欧几里得。例如，上文提到的“星形多边形”在此出现，关于通过相接触的多面体填充空间问题的讨论也在此出现。这部著作分为四节，在[12]中有详细讨论。

Speculative arithmetic基于波爱修斯的文本。历史学家之间对于哪部算术文本是布拉德华所著存在争论。这在[8]中有所讨论，并提出了一种理论。

On the continuum讨论了原子理论。它相当紧密地遵循了亚里士多德的观点。在其中，布拉德华陈述道：-

没有连续统是由原子组成的，因为每个连续统都是由无限多个同类的连续统组成的。

这项工作在[14]中进行了讨论，作者指出布拉德华的这项工作是将数学应用于自然哲学的一个很好的早期例子。

On future contingents和In defence of God against the Pelagians and on the power of causes是布拉德华的两部主要神学著作。它们试图反对自由意志，支持神圣意志。

Thomas Bradwardine's date of birth is not known with any certainty. In [1] the date is given as between 1290 and 1300, in [2] the date is given as about 1290 while, for example, in [15] it is given as about 1300. These are merely guesses based on the first definite date we know for Bradwardine and that is August 1321 when he became a Fellow of Merton College, Oxford. August 1321 is the date on which Bradwardine was awarded his B.A. We have merely taken an average of the various birth dates suggested!

We have given Bradwardine's birthplace as Chichester, but again this is based on little evidence. We know from Bradwardine's writings that his father, at the time of writing, was living in Chichester but since this must have been written more than twenty years after Bradwardine was born, it is somewhat fanciful to assume his father had not moved.

Although it is not certain, there is strong evidence to suggest that Bradwardine remained at Merton College until 1335. He held various university posts during this time including proctor and he received a number of degrees such as M.A. in 1323 and B.Th. some time before 1333. It is during this period at Oxford that almost all of his works on logic, mathematics, and philosophy were written.

In 1333 Bradwardine was appointed canon of Lincoln then, in 1335, he joined the group of men surrounding Richard de Bury who was Bishop of Durham. He moved to London two years later when he was made chancellor of St Paul's Cathedral on 19 September 1337. Soon after this he became chaplain to King Edward III.

King Edward III claimed the title of king of France in January 1340. His claim was not totally without justification since through his mother he was more closely related to the last ruler of the Capetian dynasty than was the French King Philip VI. Edward won a naval victory off the Flemish city of Sluis in June 1340 but lack of resources to press forward forced him to make a truce. In July 1346 Edward landed in Normandy accompanied by his eldest son Prince Edward (known as the Black Prince). Bradwardine was certainly part of Edward's invasion force.

The English invading force won a decisive victory over the French at Crécy on 26 August 1346. They then laid siege to Calais in September 1346 and it surrendered in August 1347. David II of Scotland was an ally of the French and realising that Edward was occupied in the siege of Calais, invaded the north of England in October 1346. Edward had expected this and left a strong force on the north of England to deal with the Scots. David II was defeated at the Battle of Neville's Cross (near Durham). At a victory celebration in France to celebrate the victories at both Crécy and Neville's Cross, Bradwardine gave as address the Sermo epinicius in the presence of Edward III, the text of which has survived.

Edward returned to England in October 1347. Bradwardine was elected Archbishop of Canterbury on 31 August 1348, but rather strangely Edward annulled the appointment. On 4 June 1349 Bradwardine was elected Archbishop of Canterbury for the second time, without it appears any objections from Edward. He was consecrated at Avignon on 10 July 1349 but died of the plague soon after he returned to London to take up his duties. The Black Death had spread to England and France in 1348 and perhaps a third of the population of London died as a result at around the same time as Bradwardine.

Bradwardine was a noted mathematician as well as theologian and was known as 'the profound doctor'. He studied bodies in uniform motion and ratios of speed in the treatise De proportionibus velocitatum in motibus (1328). This work takes a rather strange line between supporting and criticising Aristotle's physics. Perhaps it is not really so strange because Aristotle views were so fundamental to learning at that time that perhaps all that one could expect of Bradwardine was the reinterpretation of Aristotle's views on bodies in motion and forces acting on them. It is likely that his intention was not to criticise Aristotle but rather to justify mathematically a reinterpretation of Aristotle's statements.

Aristotle claimed that motion was only possible when the force acting on a body exceeded the resistance. Although he did not express it in these terms, it had also been deduced from Aristotle's Physics that the velocity of a body was proportional to the force acting on it divided by the resistance. Bradwardine used a mathematical argument to show that these two were inconsistent. He did this by assuming an initial force and resistance, then supposed that the resistance doubled, doubled again etc keeping the force constant. At some point, argues Bradwardine, the resistance will exceed the force so the body cannot move. But the velocity, given by the second rule, could not be zero.

Bradwardine then claims that an arithmetic increase in velocity corresponds with a geometric increase in the original ratio of force to resistance. This cleverly removes the contradiction, but of course is incorrect. His view, however, was very influential and it was accepted as a law of mechanics for over a hundred years. For example Oresme followed Bradwardine's ideas of mechanics. Takahashi in [16] gives an interesting new interpretation of Bradwardine notion of ratios.

Another interesting, but fallacious, argument was produced by Bradwardine when he tried to disprove atomism using geometry. However, he questioned the logic of his own arguments as he felt perhaps the existence of geometry already assumes that atomism is false.

Bradwardine was the first mathematician to study "star polygons". They were later investigated more thoroughly by Kepler. Other works by Bradwardine include the following.

On insolubles which discusses logical problems such as "I am telling a lie".

On "it begins" and "It ceases" which argues that a temporal interval has a first instant but no last instant. The end of the interval is marked by the first instant of its non-existence.

Speculative geometry contains elementary geometry which is not all based on Euclid. For example the "star polygons" referred to above appear here as does a discussion of the problem of the filling of space by touching polyhedra. This work, in four sections, is discussed in detail in [12].

Speculative arithmetic is based on a text by Boethius. There is argument between historians about which arithmetic text is the one due to Bradwardine. This is discussed, and a theory proposed, in [8].

On the continuum discusses the atomic theory. It follows Aristotle's views fairly closely. In it Bradwardine states:-

No continuum is made up of atoms, since every continuum is composed of an infinite number of continua of the same species.

This work is discussed in [14] where the author notes that this work by Bradwardine is a good early example of the application of mathematics to natural philosophy.

On future contingents and In defence of God against the Pelagians and on the power of causes are Bradwardine's two major theological works. They attempt to argue against free will and in favour of Divine Will.