Benjamin Peirce
Benjamin Peirce, the father of Charles Sanders Peirce, taught mathematics and astronomy at Harvard from 1831 until his death in 1880. and was probably the leading American mathematician of his time. He is best known, in the history of mathematics, for his Linear Associative Algebra of 1870, and for his proof, as a young man, that there is no odd perfect number with fewer than four distinct prime factors.[1] Benjamin also published over a dozen other mathematical texts and treatises, including his well known A System of Analytical Mechanics in 1855. He helped to create a modern science curriculum at Harvard, and was an importent force in the professionalization of mathematics and science education across America. [2]
Benjamin’s personality left a powerful impression on those who encountered him. Lawrence Lowell, president of Harvard from 1909 to 1933, described him as follows:
Looking back over the space of fifty years since I entered Harvard College, Benjamin Peirce still impresses me as having the most massive intellect with which I have ever come into close contact, and as being the most profoundly inspiring teacher that I ever had. His personal appearance, his powerful frame, and his majestic head seemed in harmony with his brain. [3]
William E. Byerly, the first Harvard Ph.D. in mathematics in 1871, and professor of mathematics at Cornell and Harvard, provided this sketch of Benjamin:
The appearance of Professor Benjamin Peirce, whose long gray hair, straggling grizzled beard and unusually bright eyes sparkling under a soft felt hat, as he walked briskly but rather ungracefully across the college yard, fitted very well with the opinion current among us that we were looking upon a real live genius, who had a touch of the prophet in his make-up. [4]
Benjamin’s classes were unpredictable; they were lectures, without much interaction, and he seems to have mainly unfolded his own remarkable mathematical and intellectual life in front of his students. Charles W. Eliot, another president of Harvard, recounted one particularly excruciating example of this:
I remember that this great master began one day with unusual promptness to put on the slates a series of calculations and formulae in which he seemed to be much interested. He said but little; but wrote diligently with the chalk, stopping now and then to examine his work and to rub out some of it, but only to resume it, and go on eagerly. The class before him said not a word, took notes as well as they could of what he wrote on the slates, and watched him. Suddenly near the end of the hour the worker looked despairingly at the contents of the last slate he had filled, turned to the class, and remarked, “there is an error somewhere in this work, but I cannot see where it is. This last line — the conclusion — is obviously wrong.” Whereupon he seized the rubber and rapidly rubbed out everything he had put on the slates. Professor Peirce sat down in his armchair visibly fatigued. The class slowly folded their notebooks and departed without a word, even to each other. I, for one, have always remembered vividly that hour’s spectacle. [5]
Despite such disasters — or perhaps because of them — Benjamin’s teaching inspired students and they reported a strong affection, bordering on reverence, for him. President Lowell attempted to explain this paradox:
Described in this way it may seem strange that such a method of teaching should be inspiring; yet to us it was so in the highest degree. We were carried along by the rush of his thought, by the ease and grasp of his intellectual movement. The inspiration came, I think, partly from his treating us as highly competent pupils, capable of following his line of thought even through errors in transformations; partly from his rapid and graceful methods of proof, which reached a result with the least number of steps in the process, attaining thereby an artistic or literary character; and partly from the quality of his mind which tended to regard any mathematical theorem as a particular case of some more comprehensive one, so that we were led onward to constantly enlarging truths. [6]
Byerly noted that “in his personal relations with his students he was always courteous, kind, and helpful, if rather prone to overrate their ability and promise, and they loved him.” [7]
The force of Benjamin’s personality was not restricted to students. Henry Cabot Lodge, in Early Memories, described a child’s impression of Benjamin:
Altogether he had a fascination which even a child felt, and all the more because he was full of humor, with an abounding love of nonsense, one of the best of human possessions in this vale of tears. I know that I was always delighted to see him, because he was so gentle, so kind, so full of jokes with me, and so ‘funny.’ As time went on I came as a man to know him well and to value him more justly, but the love of the child, and the sense of fascination which the child felt, only grew with the years. [8]
President Eliot noted the impact of his University Lectures upon the general public:
Benjamin Peirce’s lectures dealt, to be sure, with the higher mathematics, but also with theories of the universe and the infinities in nature, and with man’s power to deal with infinities and infinitesimals alike. His University Lectures were many a time way over the heads of his audience, but his aspect, his manner, and his whole personality held and delighted them. An intelligent Cambridge matron who had just come home from one of Professor Peirce’s lectures was asked by her wondering family what she had got out of the lecture. “I could not understand much that he said; but it was splendid. The only thing I now remember in the whole lecture is this — ‘Incline the mind to an angle of 45°, and periodicity becomes non-periodicity and the ideal becomes real.’” [9]
That this effect was not confined to “Cambridge matrons” is shown by the report of Julius Erasmus Hilgard, in 1870, on a meeting of the National Academy of the Sciences:
Prof. Peirce at one of the sessions of the Academy had just finished, with inspired eye and prophetic manner, delivering to a confounded audience one of the most abstruse portions of his researches, and a respectful silence had ensued when Agassiz rose and said: “I have listened to my friend with great attention and have failed to comprehend a single word of what he has said. If I did not know him to be a man of great mind, if I had not had frequent occasion to feel his power, to admire his judgement and discrimination on ground where our several lines of study touch, in organic morphology, in physics of the globe, I could have imagined that I was listening to the vagaries of a madman, or at best to empty and baseless literal dialectics. But knowing my friend to be not only profound but fruitful in all matters on which I have any judgment, I am forced to the conclusion that there are modes of thought familiar to him, which are inaccessible to me, and I accept in faith not only the logical truth of his investigations, but also their value as means of opening to our comprehension the laws of the universe.” Upon this epilogue there was what the French call a sensation in the hall, and the audience recovered their cheerfulness. [10]
Eliot also noted the surprise of various Cambridge friends — after Benjamin was appointed superintendent of the United States Coast Survey — that the absorbed mathematician could be an able administrator:
Those of us who had long known Professor Peirce heard of this action with amazement. We had never supposed that he had any business faculty whatever, or any liking for administrative work. A very important part of the Superintendent’s function was to procure from Committees of Congress appropriations adequate to support the varied activities of the Survey on sea and land. Within a few months it appeared that Benjamin Peirce persuaded Congressmen and Congressional Committees to vote much more money to the Coast Survey than they had ever voted before. This was a legitimate effect of Benjamin Peirce’s personality, of his aspect, his speech, his obvious disinterestedness, and his conviction that the true greatness of nations grew out of their fostering of education, science, and art. [11]
Eliot was percipient, I think, in identifying Benjamin’s “obvious disinterestedness,” and his conviction that the human story is about knowledge. A similar spirit animated the thought of his son, Charles.
There was an unmistakable platonism at the heart of Benjamin’s conception of mathematics. Joseph Brent remarks that he “taught mathematics as a kind of Pythagorean prayer.”[12] William Byerly called Benjamin a “mathematician and mystic,” describing one of his lectures on Hamilton’s quaternions as follows:
He must have been working recently on his “Linear Algebras” for he said that “of possible quadruple algebras the one that had seemed to him by far the most beautiful and remarkable was practically identical with quaternions, and that he thought it most interesting that a calculus which so strongly appealed to the human mind by its intrinsic beauty and symmetry should prove to be especially adapted to the study of natural phenomena. The mind of man and that of Nature’s God must work in the same channels.” [13]
This supposition — that the mind is somehow adapted to understand nature — has been a frequent resort of platonism, and reappears in Charles’ discussion of the logic of abduction. [14]
Benjamin read a note to the American Association for the Advancement of Science in 1849, commenting upon a presentation by his colleague, the botanist Asa Gray, which began by observing the wide scope of geometry in nature:
The Association may wonder what a mathematician can have to do with Botany, and what right he has to discuss such a subject as vegetable morphology. But let me assure you that the geometer is somewhat omnivorous in intellect, and although he has lived and thriven for centuries upon the sun and moon, the planets and comets, and other such inorganic food, he is already aspiring to a vegetable diet, and may ere long be whetting his teeth for flesh and blood. . . [15]
Based upon the observations of Gray, Benjamin described the laws for the spacing of phytons, (the successive angles of leaves around a stalk) as continued Fibonacci fractions, i.e., as fractions of the form: 1/(n + a), where n is a small integer and a is the continued fraction.[16] He asked why this particular series should appear in plant morphology — instead of, say, the iteration of 7/19 which would also provide an adequate spacing of phytons — and conjectured that the series can be detected in the relative rotation times of planets in the solar system, concluding:
May I close with the remark, that the object of geometry in all its measuring and computing, is to ascertain with exactness the plan of the great Geometer, to penetrate the veil of material forms, and disclose the thoughts which lie beneath them? When our researches are successful, and when a generous and heaven-eyed inspiration has elevated us above humanity, and raised us triumphantly into the very presence, as it were, of the divine intellect, how instantly and entirely are human pride and vanity repressed, and by a single glance at the glories of the infinite mind, we are humbled to the very dust. [17]
Asa Gray eventually communicated these results to his friend, Charles Darwin, who wrote back to him in May of 1863 as follows:
Your little discussion on Angles of Divergence of leaves in a Spire has almost driven me mad. . . I have been drawing all the real angles & unreal angles on a spire, & I see the angles which do not occur in nature, are just as symmetrical in position as the real angles. If you wish to save me from a miserable death, do tell me why the angles of 1/2 1/3 2/5 3/8 etc. series occur, & no other angles. –It is enough to drive the quietest man mad. . . . Did you & some mathematician publish some paper on [the] subject; Hooker says you did. Where is it? . . . Do you know of any plant in which [the] angle is fluctuating or variable?[18]
Gray replied to Darwin that he had “no notion in the world” why the angular divergence should reflect this sequence rather than some other sequence. [19]
In 1854, in his address to the AAAS as retiring president, Benjamin revisited the theme of a real connection between mathematical thought and nature,
. . . the true thought of the created mind must have had its origin from the Creator; but with him thought is reality. It must then be that the loftiest conceptions of transcendental mathematics have been outwardly formed, in their complete expression and manifestation, in some region or other of the physical world; and that there must always be interwoven with the discoveries of observation these striking coincidences of human thought and nature’s law. They are the reflections of the divine image of man’s spirit from the clear surface of the eternal fountain of truth. [20]
explicitly arguing for the underlying unity of science and religion.
Andrew Peabody, who taught mathematics and played chess with Benjamin in his earlier years at Harvard, recounted his lectures for the Lowell Institute in 1879:
In these lectures, he showed, as he always felt with adoring awe, that the mathematician enters, as none else can, into the intimate thought of God. This was his pervading consciousness, and it gave tone to his life-work and to his whole life. [21]
These lectures, published posthumously as Ideality in the Physical Sciences, provide Benjamin’s most extensive discussion of his scientific and mathematical platonism. [22] In the first lecture he described Kepler’s attempt to account for the motion of the planets by the geometry of cycles and epicycles, and by considering spheres inscribed and circumscribed in Platonic solids. Only when these approaches proved fruitless, he said, did Kepler resort to conic sections. Benjamin praised the Newtonian unification of Kepler’s laws as a “sublime embodiment of all-embracing thought,” and then observed:
The whole domain of physical science is equally permeated with ideality. You cannot escape from it if you would. It illumines the remotest star and the first-born of the nebulae. . . . Ascend from the infinitesimal to the infinite; pass from the elementary particle to the universal cosmos. With the increased grandeur of dimension, the intellectual utterance is not enfeebled. There is everywhere in Nature a voice audible to human ears, and a speech intelligible to human understanding. It is the truth of science, the beauty of poetry, the logic of philosophy, and the faith of religion. Ignorance cannot hide it, nor deformity degrade it, nor superstition corrupt it, nor skepticism conceal it. It vibrates in every soul; it is the consolation of the slave, and the conscience of the king. It is the corner-stone upon which sound government is built, and the fulcrum by which eloquent speech moves the world. [23]
This is unabashed platonism. It expresses the scientific optimism of the nineteenth century and reflects an idealism which was nurtured by New England transcendentalism. [24] It also shows the intensity that must have transfixed Benjamin’s students. Although he did not address the difficult epistemological issues surrounding science, Benjamin recognized its empirical character:
The wisest physical philosophers have ever been the most rigid observers; they have penetrated through fact to the inmost soul of Nature; and their proudest discoveries have invariably been vast intellectual conceptions exhumed from the recesses of the material world. [25]
His rejection of relativism was unequivocal:
When the sculptor develops his Apollo or his Venus from the quarried marble, it is his own creation, and has his image stamped upon it; but the truth which the man of science extracts has an absolute character of its own, which no power of genius can transform, and which is neither attributable to accident nor born of human parentage. It pervades the meanest chips of stone, which the artist rejects as superfluous. [26]
Benjamin’s overriding interest, however, was always in the mathematical character of science. He observed that mathematicians frequently anticipate the course of empirical science:
The dreams of Pythagoras and Plato upon the mysteries of number have been surpassed in the numerical relations discovered by modern science. The doctrine of the polyhedrons, which Kepler did not find in the system of the planets, has as real a relation to Nature as it had to his generous mind. It is found to be an essential feature of the modern theory of crystallization, just as he recognized in the paths of the planets and comets, marked out by the Creator, the same conic sections which were but ideal existences with Euclid and Apollonius. The imaginary square root of algebra, from which the puzzled analyst could not escape, has become the simplest reality of Quaternions, which is the true algebra of space, and clearly elucidates some of the darkest intricacies of mechanical and physical philosophy. The highest researches undertaken by the mathematicians of each successive age have been especially transcendental, in that they have passed the actual bounds of contemporaneous physical inquiry. But the time has ever arrived, sooner or later, when the progress of observation has justified the prophetic inspiration of the geometers, and identified their curious speculations with the actual workings of Nature. [27]
The culmination of Benjamin’s platonic vision is that:
The divine image, photographed upon the soul of man from the centre of light, is everywhere reflected from the works of creation. The origin is as distinctly imprinted upon the records of philosophy and the laws of Nature as are the lines of the sun upon every solar spectrum. [28]
——————
Helena Pycior documents the remarkable achievement of Benjamin’s Linear Associative Algebra in accepting non-commutative algebras and, especially, in permitting their coefficients to be complex rather than real.[29] Despite criticism for sacrificing the principle of permanence and the determinateness of division, this leap enabled Benjamin to abstractly identify idempotent and nilpotent elements in his algebras, and thus to analyze their internal structure. Pycior notes the “freedom” of this approach, and how Benjamin showed a “peculiar ability to invent all sorts of strange algebras without much apparent concern for their justification through mathematical use or specific scientific application.” She attributes this freedom to his “religious belief in a correspondence between the minds of God and man,”[30] and gives the assessment: “because of Linear Associative Algebra, therefore, Benjamin Peirce deserves recognition, not only as a founding father of American mathematics, but also as a founding father of modern abstract algebra.”[31]
Linear Associative Algebra is best known, however, for the definition of mathematics with which Benjamin began the work:
Mathematics is the science which draws necessary conclusions. [32]
The discussion immediately following shows that this definition was intended, in part, to accommodate higher levels of abstraction:
This definition of mathematics is wider than that which is ordinarily given, and by which its range is limited to quantitative research. The ordinary definition, like those of other sciences, is objective; whereas this is subjective. Recent investigations, of which quaternions is the most noteworthy instance, make it manifest that the old definition is too restricted. The sphere of mathematics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dogmatic teaching. Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims’ and neither law can rule nor theory explain without the sanction of mathematics. . . [33]
Benjamin also remarks that “mathematics, under this definition, belongs to every enquiry, moral as well as physical”:
The branches of mathematics are as various as the sciences to which they belong, and each subject of physical enquiry has its appropriate mathematics. In every form of material manifestation, there is a corresponding form of human thought, so that the human mind is as wide in its range of thought as the physical universe in which it thinks. The two are wonderfully matched. [34]
Much has been written about Benjamin’s definition of mathematics.[35] But the most important analysis of it was likely by Benjamin’s son, Charles, whose conception of mathematics had more in common with his father’s platonism than has often been recognized. For Charles, his father’s definition of mathematics was the focal point of an understanding of mathematics which, in retrospect, seems to obviate the foundational disputes of the twentieth century.[36] Charles’ understanding of mathematics, in turn, formed the basis for his scientific realism, and extended into many other areas of his thought. The inspiration that Benjamin provided the young man who was to become America’s greatest philosopher might be seen as yet another testament to his vision and character — it was certainly not the least of his accomplishments.[37]
–Paul
Notes:
1 Benjamin Peirce, Linear Associative Algebra, (private printing, 1870); edited with notes by C. S. Peirce in American Journal of Mathematics 4 (1881): 97–229; reprinted (New York: Van Nostrand, 1982). Benjamin Peirce, “On perfect numbers”, New York Math. Diary, 2 XIII (1832): 267-277. Lebesgue proved this result 12 years later, and it was reproven by J. J. Sylvester, a pallbearer at Benjamin’s funeral, in 1888. It is not known if there actually are any odd perfect numbers — which is one of the oldest open problems in mathematics — but the lower limit on the number of distinct prime factors of such numbers currently stands at 9. See Steven Gimbel and John Jaroma, “Sylvester: Ushering in the Modern Era of Research on Odd Perfect Numbers”, Integers. Electronic Journal of Combinatorial Number Theory 3, (2003). [↩]
2 Raymond Clare Archibald, “Benjamin Peirce: 1809-1880”, American Mathematical Monthly 32 (1925): 1-30, is the indispensable source for first-hand accounts. Benjamin helped found the Lawrence Scientific School and the Harvard Observatory; in 1853 he was elected president of the American Association for the Advancement of Science, and helped to organize the Smithsonian Institute; with Louis Agassiz and others, he founded the National Academy of Sciences. Between 1867 and 1874, Benjamin served as superintendent of the United States Coast Survey. [↩]
3 Archibald, 4. [↩]
4 Archibald, 5. [↩]
5 Archibald, 3. Another story, told by Robert Rantoul, is that “The famous experiment of the pendulum hung inside of Bunker Hill Monument from the top, to demonstrate the rotation of the earth, was all the rage in my day in College. We thought we had arrived at an explanation of it, which we discussed together with much enthusiasm, until Professor Peirce volunteered one day to explain it. After that nobody thought he understood it at all.” Edward Waldo Emerson, The Early Years of the Saturday Club, 1855-1870 (Boston: Houghton Mifflin, 1918), 99. [↩]
6 Archibald, 4-5. [↩]
7 Archibald, 7. [↩]
8 Henry Cabot Lodge, Early Memories (New York: Scribners, 1913) : 55-56. [↩]
9 Archibald, 3. [↩]
10 Cited in Ivor Grattan-Guinness, “Benjamin Peirce’s Linear Associative Algebra (1870): New Light on its Preparation and ‘Publication’”, Annals of Science 54 (1997): 604. This letter was copied by Thomas Hill, president of Harvard from 1862 to 1868, and kept in his personal copy of Linear Associative Algebra. [↩]
11 Archibald, 3-4. [↩]
12 Joseph Brent, Charles Sanders Peirce: A Life (Bloomington and Indianapolis: Indiana University Press, 1993), 33. [↩]
13 Archibald, 6. [↩]
14 E.g., 7.220. “I have already pointed out that it is a primary hypothesis underlying all abduction that the human mind is akin to the truth in the sense that in a finite number of guesses it will light upon the correct hypothesis.” Also, cf. 5.173, 6.530. [↩]
15 Benjamin Peirce, “Mathematical Investigation of the Fractions which occur in Phyllotaxis,” Proceedings of the American Association for the Advancement of Science 2 (1849) : 444-447. Asa Gray, “On the Composition of the Plant by Phytons, and some Applications of Phyllotaxis,” Proceedings of the AAAS 2 (1849) : 438-444. [↩]
16 Hence, when n = 2 this produces the sequence 1/3, 2/5, 3/8, 5/13. . . which is the sequence most often occurring in nature, although n = 3, 4 and 5 also occur. [↩]
17 Benjamin Peirce, “Phyllotaxis”, 447. [↩]
18 Letter from Charles Darwin to Asa Gray, May 11, 1863. Darwin Correspondence Project Database, letter # 4153. http://www.darwinproject.ac.uk/entry-4153 [↩]
19 Letter from Asa Gray to Charles Darwin, May 25, 1863. Darwin Correspondence Project Database, letter # 4186. http://www.darwinproject.ac.uk/entry-4186 [↩]
20Benjamin Peirce, “Address of Professor Benjamin Peirce, President of the American Association for the year 1853, on Retiring from the Duties of President,” in Cornell University Library, Historical Math Monographs, (1853): 14. Permanent link at:
http://projecteuclid.org/euclid.chmm/1263240516 [↩]
21 Andrew Preston Peabody, Harvard Reminiscences (Boston: Ticknor, 1988), 186. [↩]
22 Benjamin Peirce, Ideality in the Physical Sciences (Boston: Little, Brown, 1881), edited by James Mill Peirce. [↩]
23 Benjamin Peirce, Ideality, 18-19. [↩]
24 Auguste Comte was an influence on Ideality: cf., 11-12, 33, 35, passim. The Saturday Club, to which Benjamin belonged, included: Louis Aggasiz, James Russell Lowell, Ralph Waldo Emerson, Charles Eliot Norton, Oliver Wendell Holmes, John Greenleaf Whittier, Henry James, and others. See Emerson, The Early Years. [↩]
25 Benjamin Peirce, Ideality, 25. [↩]
26 Benjamin Peirce, Ideality, 26. [↩]
27 Benjamin Peirce, Ideality, 28-29. [↩]
28 Benjamin Peirce, Ideality, 31. [↩]
29 Helen Pycior, “Benjamin Peirce’s Linear Associative Algebra,” Isis, 70, no. 4 (Dec. 1979): 537-571. [↩]
30 Pycior, 549. Benjamin invites Pycior’s interpretation, writing in his preface: “This work has been the pleasantest mathematical effort of my life. In no other have I seemed to myself to have received so full a reward for my mental labor in the novelty and breadth of the results. I presume that to the uninitiated the formulae will appear cold and cheerless; but let it be remembered that, like other mathematical formulae, they find their origin in the divine source of all geometry.” Benjamin Peirce, Linear Associative Algebra (1882), front. I would hesitate to call this belief theological, although it was certainly religious and Christian–in a vague Unitarian sort of way. [↩]
31 Pycior, 551. [↩]
32 The drafts of this definition, “Mathematics is the science that draws inferences” and “Mathematics is the science that draws consequences,” are discarded before the 1870 manuscript. See Grattan-Guinness, 602. [↩]
33 Benjamin Peirce, Linear Associative Algebra (1882), 1. [↩]
34 Benjamin Peirce, Linear Associative Algebra (1882), 2. [↩]
35 Google gives 2650 hits for the full text of the definition, and about twice that many for the partial “science which draws necessary conclusions”. Benjamin’s definition begins the Wikipedia article on “Mathematics”. An interesting recent reference, for example, is H. Berendregt and F. Wiedijk, “The Challenge of Computer Mathematics” , Philosophical Transactions of the Royal Society 363, no. 1835 (2005): 2351-2375. [↩]
36 An excellent discussion of Charles on mathematics is Shannon Dea, “‘Merely a veil over the living thought’: Mathematics and Logic in Peirce’s Forgotten Spinoza Review”, Transactions of the Charles S. Peirce Society 42, no. 4 (2006): 501-517. [↩]
37 Grattan-Guinness observes of Benjamin: “Peirce was a major figure in nineteenth-century American science, to an extent which renders astonishing the lack of any substantial historical work on his achievements when so many lesser figures have been treated extensively; possibly his focus upon mathematics has granted him historical leperhood. But this leaves unknown not only an impressive body of publication and teaching and much participation in the professionalization of American science but also a Nachlass, hardly known or used by historians. Grattan-Guinness, 601. [↩]