本杰明·皮尔斯Benjamin Peirce

本杰明·皮尔斯的父母是Lydia Ropes Nichols和皮尔斯。因此，本传记的主人公皮尔斯与他的父亲同名，他的父亲既是马萨诸塞州的州议员，也是哈佛学院的图书馆员。我们注意到这个姓氏读作“Purse”。尽管在皮尔斯出生时哈佛已经建立了大约170年，但它仍处于相当严格的政治控制之下，并不是一所真正具有全国地位的机构。随着皮尔斯终身与哈佛的联系及其影响力稳步增长，这种情况将会改变。皮尔斯是拉斐尔·塞勒姆私立文法学校的学生，在那里他与Henry Ingersoll 纳撒尼尔·鲍迪奇成为朋友，后者是纳撒尼尔·鲍迪奇及其妻子Mary Ingersoll的八个孩子之一。

皮尔斯于1825年进入哈佛学院，四年后毕业。在这些年里，他协助他的朋友Ingersoll 鲍迪奇的父亲。纳撒尼尔·鲍迪奇翻译的皮埃尔·西蒙·拉普拉斯的Traite de mécanique célesteⓉ（《天体力学论》）前四卷到1818年已经完成，但在皮尔斯还是本科生期间，他仍未将其出版。几乎可以肯定，出版费用导致了延迟，但纳撒尼尔·鲍迪奇在1818年之后并没有把这项工作搁置一旁，在随后的几年里仍在继续改进它。在皮尔斯还是本科生时，他参与了纳撒尼尔·鲍迪奇的项目，通读文本进行校对并提出改进建议。

皮尔斯于1829年至1831年在马萨诸塞州北安普顿的George Bancroft的Round Hill学校任教，然后于1831年被任命为哈佛学院的导师，此时距他毕业已有两年。他于1833年获得哈佛大学硕士学位，同年与参议员Elijah Hunt Mills的女儿Sarah Hunt Mills结婚。皮尔斯和Sarah Peirce有四个儿子和一个女儿，其中包括查尔斯·桑德斯·皮尔士（本档案中也有他的传记）和James Mills 皮尔斯，后者曾在哈佛数学系任教，然后从1890年到1895年担任哈佛研究生院院长，此后担任文理学院院长。他们的另外两个儿子也事业有成，皮尔斯 Mills 皮尔斯成为采矿工程师，Herbert Henry Davis 皮尔斯成为外交官。

正是在哈佛，皮尔斯从1831年起度过了他的整个职业生涯。他于1833年被任命为那里的数学与自然哲学教授，担任此职位九年，直到数学与天文学教授职位空缺时，他被调任该讲席。他的第一个讲席不是捐赠设立的，但他在1842年被任命的讲席是捐赠设立的Perkins讲席，他继续担任此职直到去世。

在他职业生涯的早期，皮尔斯 出版了许多教科书。例如 An Elementary Treatise on Plane Trigonometry（1835年）、First Part of an Elementary Treatise on Spherical Trigonometry（1836年）、An Elementary Treatise on Sound（1836年）、An Elementary Treatise on Algebra : To which are added Exponential Equations and Logarithms（1837年）、An Elementary Treatise on Plane and Solid Geometry（1837年）、An Elementary Treatise on Plane and Spherical Trigonometry（1840年）以及 An Elementary Treatise on Curves, Functions, and Forces 第1卷（1841年）、第2卷（1846年）。这些书具有原创性且在数学上优雅，但对当时的美国学生来说要求过高。学生们觉得它们过于简练，而且除了最优秀的学生之外，其他人也觉得他的讲课风格非常难懂 [7]：-

他讲课时不停下来回答问题，黑板上写满了大量潦草的字迹。然而，最优秀的学生能够欣赏他的怪癖，并被他对数学的热情所激励。

在 [3] 中，Archibald 引用了 Lawrence Lowell 的观点，他是 皮尔斯 教过的最优秀的学生之一：-

他一完成题目或写满黑板，就会把所有内容擦掉，然后重新开始。他对细节缺乏耐心，有时结果会不对；但他不是检查自己的工作以找出错误，而是把它擦掉，说他在某个地方弄错了一个符号，我们复习笔记时就会发现。这样描述起来，这种教学方法竟然具有启发性，可能显得奇怪；然而对我们来说，它在最高程度上确实如此。我们被他思想的奔涌、他智力活动的从容与把握所带动。

如果普通学生无法应对皮尔斯向他们呈现的数学，那么他认为最好只让更有天赋和专注的学生在一年级之后继续学习数学。他提出了三条不同的路径供学生选择。一种选择是为期一年的实践课程，第二种是主要为学校教师设计的一年理论课程，第三种选择是为期三年的课程，旨在培养未来的数学家。1847年哈佛大学劳伦斯科学学院成立后，皮尔斯得以在美国首次教授研究生水平的数学。他设置的课程令人印象深刻，包括学习西尔维斯特·佛朗索瓦·拉克鲁瓦、奥古斯丁·路易·柯西、加斯帕尔·蒙日、让-巴蒂斯特·毕奥、威廉·哈密顿、皮埃尔·西蒙·拉普拉斯、西莫恩·德尼·泊松、卡尔·弗里德里希·高斯、于尔班·勒维耶、弗里德里希·威廉·贝塞尔、约翰·柯西·亚当斯、乔治·比德尔·艾里、詹姆斯·麦古拉和恩斯特·弗朗茨·诺伊曼的著作。这是一门真正雄心勃勃的课程，但有些超前于时代，他每年只教大约两名学生。

皮尔斯 在广泛的数学主题上进行了研究，从应用方面的天体力学和大地测量学到纯数学方面的线性结合代数和 数论。在一篇早期的数论论文中，他证明了不存在少于四个不同 prime 因子的奇 完全数。他还修订并撰写了关于 纳撒尼尔·鲍迪奇 翻译的 皮埃尔·西蒙·拉普拉斯 的 Traité de mécanique céleste Ⓣ（《天体力学专论》）前四卷的评论，他在本科时曾亲自校对过这部译作。这些卷中只有三卷在 纳撒尼尔·鲍迪奇 生前出版，而 皮尔斯 本人完成了第四卷的编辑并使其出版。

皮尔斯 帮助确定了海王星的轨道（1846年发现），并计算了海王星对天王星轨道和其他行星产生的摄动。事实上，他对此问题持有有争议的观点，因为他声称 于尔班·勒维耶 和 约翰·柯西·亚当斯 对扰动天王星的行星位置所做的预测与 Galle 发现的海王星不符。他还声称海王星没有遵循 于尔班·勒维耶 和 约翰·柯西·亚当斯 预测的轨道。

1852年，皮尔斯 将方法引入应用于观测的误差理论，允许剔除有缺陷的观测。这产生了一个有趣的后果，即他在1856年的一桩法庭案件中被传唤为专家证人（见 [14]）。该案件涉及在一份有争议的遗嘱上伪造 Sylvia Ann Howland 的签名。皮尔斯 检查了签名中的三十条向下笔画，并测量了每条的角度。然后他声称，将这些角度与 Sylvia Ann Howland 另一个签名的角度进行比较，差异如此之大，以至于所谓的伪造签名必定是伪造的，而且必定是描摹的。为了支持他的主张，他对42个真实的 Sylvia Ann Howland 签名中向下笔画的角度进行了统计分析。皮尔斯 的统计论证存在弱点，[14] 中指出了这些弱点，特别是关于他所考虑的概率的独立性。也许幸运的是，该案件是基于法律形式问题判决的，皮尔斯 的证据在决定案件结果中并非关键。[14] 的一个好处是作者给出了 皮尔斯 研究过的所有签名，所以我们都可以自己扮演专家证人！

1870年，皮尔斯自费出版了Linear Associative Algebra，对所有维数小于七的复结合代数进行了分类。他使用如今熟悉的幂等和幂零元素（由皮尔斯创造的术语）工具，为线性结合代数的一般理论奠定了基础，并在书中给出了150多个新代数的乘法表。他深受威廉·哈密顿关于四元数的工作影响，从而发展了这条特定的代数思考路线。Pycior在[19]中讨论了这项工作，她声称：-

皮尔斯值得认可，不仅作为美国数学的奠基人，而且作为现代抽象代数的奠基人。

皮尔斯于1870年在华盛顿向国家科学院宣读了该文本，然后分发了100份石印副本[10]：-

皮尔斯从1867年起向国家科学院提交了他的一些成果，四年前他被任命为该院的创始成员；但他们无力印刷。因此，在海岸调查工作人员的倡议下，找到了一位没有数学训练但字迹优美的女士，她既能读懂他潦草的手稿，又能每次12页地将全文写在石印石上。

他的儿子查尔斯·桑德斯·皮尔士后来编辑了这部作品，于1881年在American Journal of Mathematics中出版。

皮尔斯的职业生涯还有另一个方面我们仍需提及。他以重要方式参与了美国海岸调查局，1852年担任经度测定主任，随后在1867年时任主任Alexander Dallas Bache去世后，他被任命为美国海岸调查局局长。他继续担任局长直到1874年，除了监督美国地图的制作外，他还组织了调查局前往西西里、长崎、查塔姆群岛和阿拉斯加的探险，以观测天文事件。他在1853年美国协会主席任期结束时的演讲中说道[17]：-

在天文学中……必须承认，土星和木星、火星和金星仍受未解释的运动不规则性影响；小行星理论尚未超越算术的最早阶段；土星环与其主星通过一种[神秘的]……力相连；潮汐遵循太阳和月亮引力的规律还很不发达。海岸调查局在这一主题上所做的卓越研究已经确立，这里仍有一个待征服的世界……

在[17]中，皮尔斯表明他坚信数学：-

……它是伟大的万能钥匙，开启知识的每一扇门，没有它，任何发现——任何值得称为发现的、是规律而非孤立事实的发现——都不曾或永远无法做出。

然而，他警告不要为纯数学而纯数学：-

被其对称性所吸引，几何学家有时可能过于专注于他的科学，忘记了它的应用；他可能将其提升为偶像并崇拜它；他可能将其贬低为玩具，幼稚地自娱于它可能呈现的奇异形状，而本应努力工作，将其用于人类的利益……

鉴于这一说法，皮尔斯被称为美国的“纯粹数学之父”就略显奇怪了（例如见[4]）。

皮尔斯因其对美国数学的杰出贡献而获得了许多荣誉。他致力于建立美国科学促进会（1847年成立）和国家科学院（美国）（1863年成立）。皮尔斯是美国艺术与科学院于1847年成立的组织史密森学会的五人委员会成员之一。他还被任命为三人科学委员会成员，该委员会于1955年至1858年在纽约奥尔巴尼组织了达德利天文台。皮尔斯当选为美国哲学会会士（1842年）、Royal Astronomical Society会士（1850年）和皇家学会会士（1852年）。

最后让我们指出，皮尔斯是一位虔诚的基督徒，并认为[10]：-

……数学是上帝造物对上帝作品的研究。

皮尔斯在[17]中详细阐述了这一点。我们给出简短引文：-

每一门科学都提供了足够的证据，但几何学提供的证据最多，证明我们被分配到的这个世界特别适合我们的心智，并且极好地促进了我们的智力进步。造物主的计划中包含这一点是无可置疑的。整个秩序本来可以多么轻易地被颠倒！我们本来可以多么轻易地被安排到某个我们微弱而有限的能力无法解开的复杂体系中！

Benjamin Peirce's parents were Lydia Ropes Nichols and Benjamin Peirce. Benjamin, the subject of this biography, therefore had the same name as his father who was both a state legislator in Massachusetts and also a librarian at Harvard College. We note that this surname is pronounced "Purse". Although Harvard had been established for around 170 years at the time of Benjamin's birth, it was still under fairly tight political control and was not an institution of true national standing. This would change through Peirce's life long association with Harvard as its influence steadily increased. Benjamin was a pupil at Salem Private Grammar School and there he became friends with Henry Ingersoll Bowditch who was one of the eight children of Nathaniel Bowditch and his wife Mary Ingersoll.

Peirce entered Harvard College in 1825 and graduated four years later. During these years he assisted his friend Ingersoll Bowditch's father. Bowditch's translation of the first four volumes of Laplace's Traite de mécanique céleste Ⓣ had been completed by 1818 but he had still not published it during the years that Peirce was an undergraduate. Almost certainly the cost of publication caused the delay, but Bowditch had not put the work on one side after 1818, still continuing to improve it over the succeeding years. While he was an undergraduate, Peirce became involved in Bowditch's project and worked through the text doing proof-reading and suggesting improvements.

Peirce taught at George Bancroft's Round Hill School in Northampton, Massachusetts, from 1829 to 1831, and then was appointed as a tutor at Harvard College in 1831, two years after graduating. He was awarded a Master's Degree by Harvard in 1833 and in the same year he married Sarah Hunt Mills, the daughter of Senator Elijah Hunt Mills. Benjamin and Sarah Peirce had four sons and one daughter, among them Charles Peirce (who also has a biography in this archive) and James Mills Peirce who taught in the Mathematics Department at Harvard, and then from 1890 to 1895 served as Dean of the Graduate School at Harvard and, after that, as Dean of the Faculty of Arts and Sciences. Their other two sons also enjoyed successful careers, Benjamin Mills Peirce as a mining engineer and Herbert Henry Davis Peirce as a diplomat.

It was at Harvard that Peirce spent the whole of his career from 1831. Appointed professor of Mathematics and Natural Philosophy there in 1833 he held this position for nine years until the professorship of Mathematics and Astronomy became vacant when he was moved to that chair. His first chair was not endowed, but the chair he was appointed to in 1842 was the endowed Perkins Professorship and he went on to hold this until his death.

In the early part of his career Peirce published a number of textbooks. For example An Elementary Treatise on Plane Trigonometry (1835), First Part of an Elementary Treatise on Spherical Trigonometry (1836), An Elementary Treatise on Sound (1836), An Elementary Treatise on Algebra : To which are added Exponential Equations and Logarithms (1837), An Elementary Treatise on Plane and Solid Geometry (1837), An Elementary Treatise on Plane and Spherical Trigonometry (1840), and An Elementary Treatise on Curves, Functions, and Forces Vol 1 (1841), Vol 2 (1846). These books were original and mathematically elegant but were rather too demanding for the American students of the time. The students found them too concise and all but the very best students also found his lecturing style very difficult [7]:-

He lectured without stopping for questions and filled the blackboard with a mass of scribblings. The best students, however, were able to appreciate his quirks and were inspired by his enthusiasm for mathematics.

In [3] Archibald quotes the opinion of Lawrence Lowell, one of the best students that Peirce taught:-

As soon as he had finished the problem or filled the blackboard he would rub everything out and begin again. He was impatient of detail, and sometimes the result would not come out right; but instead of going over his work to find the error, he would rub it out, saying that he had made a mistake in a sign somewhere, and that we should find it when we went over our notes. Described in this way it may seem strange that such a method of teaching should be inspiring; yet to us it was so to the highest degree. We were carried along by the rush of his thought, by the ease and grasp of his intellectual movement.

If the average student could not cope with the mathematics that Peirce was presenting them with, then he decided it was better he saw only the more talented and dedicated students continue with mathematics beyond their first year. He proposed three different tracks from which the students could make a choice. One option was a one year practical course, the second was a one year theoretical course designed primarily for school teachers, and the third option was a three year course which would train mathematicians of the future. After the Lawrence Scientific School was founded at Harvard in 1847, Peirce was able to teach graduate level mathematics for the first time in the United States. The course he set up was impressive, including the study of works of Lacroix, Cauchy, Monge, Biot, Hamilton, Laplace, Poisson, Gauss, Le Verrier, Bessel, Adams, Airy, MacCullagh and Franz Neumann. It was a truly ambition course but it was somewhat ahead of its time and he taught it to only about two students per year.

Peirce undertook research on a wide range of mathematical topics from celestial mechanics and geodesy on the applied side to linear associative algebra and number theory on the pure side. In an early number theory paper he proved that there is no odd perfect number with fewer than four distinct prime factors. He also revised and wrote a commentary on Bowditch's translation of the first four volumes of Laplace's Traité de mécanique céleste Ⓣ which he had himself had proof-read as an undergraduate student. Only three of these volumes appeared in Bowditch's lifetime, and Peirce himself completed editing the fourth volume and saw it through publishing.

Peirce helped determine the orbit of Neptune (discovered in 1846) and calculated the perturbations produced by Neptune on the orbit of Uranus and on the other planets. In fact he held controversial views on this topic for he claimed that the predictions for the position of the planet perturbing Uranus as made by Le Verrier and Adams did not coincide with the planet Neptune which Galle discovered. He also claimed that Neptune was not following the orbit predicted by Le Verrier and Adams.

In 1852, Peirce introduced methods into the theory of errors applied to observations which would allow faulty observations to be discarded. There was an interesting consequence of this, namely that he was called as an expert witness in a court case in 1856 (see [14]). The case concerned the forging of the signature of Sylvia Ann Howland on a contested will. Peirce examined the thirty downward lines in the signature and measured the angle of each. He then claimed that the comparison of these angles with those of another of Sylvia Ann Howland's signatures was so great that the alleged forged signature must indeed be a forgery and must have been traced. To support his claims he did a statistical analysis of the angles of the downstrokes in 42 genuine Sylvia Ann Howland signatures. There are weaknesses in Peirce's statistical arguments which are pointed out in [14], particularly concerning the independence of the probabilities he considered. Perhaps it was fortunate that the case was decided on the basis of a legal formality and Peirce's evidence was not crucial in determining the outcome of the case. One of the nice things about [14] is that the authors give all the signatures that Peirce worked with, so we can all play at being expert witnesses ourselves!

In 1870 Peirce published, at his own expense, Linear Associative Algebra a classification of all complex associative algebras of dimension less than seven. He used the, now familiar, tools of idempotent and nilpotent elements (terms invented by Peirce) to establish the foundations of a general theory of linear associative algebra and he presented multiplication tables for over 150 new algebras in the book. He had been much influenced by Hamilton's work on the quaternions to develop this particular line of algebraic thinking. Pycior discusses this work in [19] where she claims:-

Benjamin Peirce deserves recognition, not only as a founding father of American mathematics, but also as a founding father of modern abstract algebra.

Peirce read the text before the National Academy of Sciences in Washington in 1870 before circulating 100 lithographed copies [10]:-

Peirce had presented some of his results from 1867 onwards to the National Academy of Sciences, of which he had been appointed a founder member four years earlier; but they could not afford to print it. Thus, in an initiative taken by Coast Survey staff, a lady without mathematical training but possessing a fine hand was found who could both read his ghastly script and write out the entire text 12 pages at a time on lithograph stones.

His son Charles Peirce later edited the work for publication in the American Journal of Mathematics in 1881.

There is another side to Peirce's career which we still have to mention. He was involved in a major way in the United States Coast Survey, as director of the longitude determinations in 1852 then he as made director of the United States Coast Survey in 1867 on the death of the then director Alexander Dallas Bache. He continued as director until 1874 and as well as overseeing the production of a map of the United States, he organised expeditions by the Survey to Sicily, Nagasaki, the Chatham Islands, and Alaska to observe astronomical events. He said in his address as the end of his presidency of the American Association for the Year 1853 [17]:-

In astronomy ... it must be conceded that Saturn and Jupiter, Mars and Venus, are yet subject to unexplained irregularities of motion; that the theory of the asteroids has not advanced beyond the earliest stage of arithmetic; that the rings of Saturn are connected with their primary by a [mysterious]... force; and the laws under which the tides obey the attractions of the sun and moon are quite undeveloped. The remarkable researches upon this subject made in the Coast Survey, have established that here still remains another world to be conquered ...

In [17] Peirce shows that he strongly believes in mathematics:-

... it is the great master-key, which unlocks every door of knowledge, and without which no discovery - no discovery which deserves the name, which is law and not isolated fact - has been or ever can be made.

However, he warns against pure mathematics for its own sake:-

Fascinated by its symmetry, the geometer may, at times, have been too exclusively engrossed with his science, forgetful of its applications; he may have exalted it into his idol, and worshipped it; he may have degraded it into his toy, and childishly amused himself with the singular shapes which it would assume, when he should have been hard at work with it, using it for the benefit of mankind ...

Given this statement it is slightly strange that Peirce has been called the "Father of pure mathematics" in America (see for example [4]).

Benjamin Peirce received many honours for his outstanding contributions to American mathematics. He worked for the establishment of the American Association for the Advancement of Science (founded 1847) and the National Academy of Sciences (United States) (founded 1863). Peirce was one of a committee of five set up by the American Academy of Arts and Sciences in 1847 to organise the Smithsonian Institution. He was also appointed to a scientific council of three who organised the Dudley Observatory at Albany, New York, from 1955 to 1858. Peirce was elected to the American Philosophical Society (1842), the Royal Astronomical Society (1850), and the Royal Society (1852).

Let us finally note that Peirce was a devout Christian and saw [10]:-

... mathematics as study of God's work by God's creatures.

Peirce expands on this at length in [17]. We give a short quote:-

There is proof enough furnished by every science, but by none more than geometry, that the world to which we have been allotted is peculiarly adapted to our minds, and admirably fitted to promote our intellectual progress. There can be no reasonable doubt that it was part of the Creator's plan. How easily might the whole order have been transposed! How readily might we have been assigned to some complicated system which our feeble and finite powers could not have unravelled!