Apr 172010
 

benjamin_peirce_1857Benjamin Peirce, the father of Charles Sanders Peirce, taught mathe­matics and astronomy at Harvard from 1831 until his death in 1880. He was probably the leading American mathe­ma­ti­cian of his time. He is best known in the annals of mathe­ma­tics for his pioneering Linear Asso­ci­a­tive Alge­bra in 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 pub­lished over a dozen other mathe­matical works, in­clu­ding his well-known System of Ana­lytical Mechanics in 1855. He helped to cre­ate a modern science curri­culum at Harvard, and was an important force behind the profes­sion­al­iza­tion of mathe­matics and sci­ence educa­tion in America. [2]

Benjamin’s personality made 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 a 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, typically consisting of lectures that un­folded, in front of the students, his own remarkable intellectual life. Charles W. Eliot, another Harvard president, 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, Benjamin’s teaching inspired his students, and they reported an affection for him which bordered on reverence. Presi­dent Lowell’s explanation for this apparent paradox is as follows:

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]

And Byerly remarked 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 confined to students. Henry Cabot Lodge, in Early Memories, sketched 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]

During the 1860′s Harvard instituted a series of University Lectures which were open to the general public. President Eliot observed the impact of Benjamin’s lectures:

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 restricted to “Cambridge matrons” is shown by the report of Julius 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 remarked upon the surprise of various Cambridge friends — when Benjamin was ap­pointed superintendent of the United States Coast Survey — that the absorbed mathe­matician could be such 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’s reference to Benjamin’s “obvious disinterestedness,” and to his con­viction that the human story is fundamentally about knowledge, seems to capture the essential spirit of this man. A similar spirit animated the thought of Charles Peirce.

——————

There is an unmistakable platonism at the heart of Benjamin’s conception of mathe­ma­tics. Joseph Brent said that Benjamin “taught mathematics as a kind of Pytha­go­rean prayer”[12], and William Byerly described him as a “mathematician and mystic.” On a lecture by Benjamin on Hamilton’s quaternions, Byerly reported:

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 view — that the mind is adapted to understand nature — is a common resort of platonism. It reappears in Charles’ discussion of the logic of abduction, when he says 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.” [14]

 

In 1849 Benjamin read a brief note to the American Association for the Advancement of Science commenting upon a presentation by his colleague, the botanist Asa Gray. This note 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 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 same series can be detected in the relative rotation times of planets in the solar system. He concluded:

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 his 1854 address to the AAAS as retiring president, Benjamin again brought up the theme of a 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]

The idea that Benjamin defended in this address — that there is an inherent sympathy between science and religion — is not very popular today. Yet, for Benjamin, this idea was a reasonable inference from his experience as a mathematician and scientist.

Andrew Peabody, who taught mathematics at Harvard and often played chess with Benjamin in his early years, spoke of the lectures that Benjamin gave 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]

The Lowell lectures were published posthumously as Ideality in the Physical Sciences, and set out Benjamin’s understanding of the nature of science and mathe­matics.[22] In his first lecture Benjamin recounted Kepler’s attempts to account for the motion of the planets by the geometry of cycles and epicycles, and by using spheres in­scribed in, and cir­cumscribed about, the Platonic solids. Only when these approaches proved fruit­less, Benjamin pointed out, did Kepler resort to conic sections. Benjamin went on to praise Newton’s unification of Kepler’s laws, calling it “the sublime em­bod­i­ment of all-embracing thought.”  Benjamin then said:

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]

In this unapologetic platonism we can hear the scientific optimism of the nine­teenth cen­tury, along with an idealism nurtured by New England transcenden­­talism.[24] We also hear some of the intensity that must have transfixed Benjamin’s students.

Benjamin did not, in Ideality, systematically address the more difficult episte­mological issues of science. He 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]

And 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 preoccupation, though, was with the mathematical nature of the scien­tific enter­prise. Hence he observed that mathematicians have often anticipated 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]

Benjamin summarized his vision with a nice platonic metaphor:

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]

——————

In her discussion of Benjamin’s Linear Associative Algebra, Helena Pycior describes his achieve­ment in considering non-commutative algebras and, in parti­cular, permitting the coefficients to be complex rather than real.[29] Although he was criticized for having sacrificed the principle of permanence and the determinate­ness of division, Benjamin’s approach enabled him to identify idempotent and nilpotent elements in his algebras and thus to analyze their internal structure. Pycior notes the “freedom” of this ap­proach and remarks that 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]  Pycior concludes that “because of Linear Associative Algebra . . . Benjamin Peirce de­serves recognition, not only as a founding father of American mathematics, but also as a founding father of modern abstract algebra.” [31]

To the layperson, Benjamin’s Linear Associative Algebra is probably best known today for the famous defi­nition of mathe­matics with which he began the work:

Mathematics is the science which draws necessary conclusions. [32]

In the following passage, Benjamin described his definition as broadening the scope of mathematics, and as being essentially different from the Aristotelian definition:

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]

Notice that Benjamin did not attempt to specify mathe­matics by its subject matter, but by the character­istic activity of mathematicians: the drawing of necessary conclu­sions. This is why he called the definition ‘sub­jective’ rather than ‘objective’.

Benjamin also pointed out 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]

Benjamin’s definition has received much attention in the twentieth century. Google gives 13,000,000 hits for its full text.[35] His definition is clearly more inclusive than the Aristotelian definition; and it more readily accommodates such ­fields as abstract algebra, topology, combi­natorics and mathematical logic. It is both less rigid and more extensible than definitions which depend upon the delineation of content.

The best discussion of Benjamin’s definition occurs in the work of Charles Peirce, who was intimately involved in its formulation, and who also adopted it as an essential tenet of his own philosophy of mathematics. Thus, it is to the philosophy of Charles that one must turn for the full explication of Benjamin’s definition. I intend to discuss Charles’ view of mathematics in a future post; in the meantime I would recommend the excellent discussion by Shannon Dea in “’Merely a veil over the living thought’: Mathematics and Logic in Peirce’s Forgotten Spinoza Review”.[36].

Benjamin Peirce had an outsized influence both on his own students and on the course of American mathe­matics. Grattan-Guinness observes of Benjamin that:

“[he] 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.” [37]

–Paul


Notes:

1Benjamin 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, if there are any, 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. [↩]

12Joseph Brent, Charles Sanders Peirce: A Life (Bloomington and Indianapolis: Indiana University Press, 1993), 33. [↩]

13 Archibald, 6. [↩]

14 Collected Papers, 7.220. 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 An interesting recent reference to Benjamin’s definition is H. Berendregt and F. Wiedijk, “The Challenge of Computer Mathematics” , Philosophical Transactions of the Royal Society 363, no. 1835 (2005): 2351-2375. [↩]

36 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, 601. [↩]

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