# Some of Carolyn Eisele's Papers

We give below a list of twelve of Carolyn Eisele's papers, most with a short extract from the Abstract and/or Introduction.

The Scientist-Philosopher C S Peirce at the Smithsonian

The Charles S Peirce-Simon Newcomb Correspondence

Charles S Peirce and the Problem of Map-Projection

The mathematics of Charles S Peirce

Salomon Bochner on Charles S Peirce

C S Peirce's Search for a Method in Mathematics and the History of Science

Mathematics as a Semiotic Factor in the Thought of C S Peirce

Mathematical methodology in the thought of Charles S Peirce

An introduction to Peirce's mathematical semiotic

Thomas S Fiske and Charles S Peirce

Fresh Light on Peirce's Theory of Signs from his Writings on Mathematics and the History of Science

Mathematics as a Key to Peirce's Semiotics

**Click on a link below to go to the information about that paper**The Scientist-Philosopher C S Peirce at the Smithsonian

The Charles S Peirce-Simon Newcomb Correspondence

Charles S Peirce and the Problem of Map-Projection

The mathematics of Charles S Peirce

Salomon Bochner on Charles S Peirce

C S Peirce's Search for a Method in Mathematics and the History of Science

Mathematics as a Semiotic Factor in the Thought of C S Peirce

Mathematical methodology in the thought of Charles S Peirce

An introduction to Peirce's mathematical semiotic

Thomas S Fiske and Charles S Peirce

Fresh Light on Peirce's Theory of Signs from his Writings on Mathematics and the History of Science

Mathematics as a Key to Peirce's Semiotics

**1.**Carolyn Eisele, The Scientist-Philosopher C S Peirce at the Smithsonian,

*Journal of the History of Ideas*

**18**(4) (1957), 537-547.

**1.1. Extract from the paper.**

A study in detail of Charles S Peirce's scientific achievements, which won him the respect and recognition of physicists and astronomers while he was still in the service of the United States Coast Survey, reveals much that is also significant for understanding his pre-eminence as America's greatest philosopher of science. However, the study of such detail is not the purpose of this short paper, in which only a chapter of the whole story will be told. An attempt will be made here merely to catch in Peirce's correspondence with Samuel P Langley glimpses of Peirce's scientific and philosophic thought, the thought of a practical scientist endowed with a deep capacity for philosophical reflection on the nature of knowledge and the means of acquiring it.

The "clarity of ideas" pursued by Peirce, the philosopher, was to be achieved, according to him, by the application of a procedure logically analogous to that which Peirce, the practical scientist, was accustomed to follow in his experimental work. This double aspect of Peirce's intellectual life has, of course, been generally recognised. It was first fully appreciated and aptly described by the late Morris R Cohen:

Charles Peirce of all American philosophers has shown the greatest insight into science. The son of a great mathematician and himself experienced in actual scientific work (geodetic survey), he understood what it was to engage in scientific measurements. With Chauncey Wright and other members of a philosophical club he worked out a substantial theory of science, analysing the nature of law, predictability, and other basic scientific concepts. Fundamentally, he regarded science as a method rather than a bundle of laws.Peirce's adult life extended over one of those exhilarating periods in the history of science when basic scientific assumptions were being re-examined. He worked and studied in an intellectual environment alive with a sense of impending discovery. Old questions in new guise plagued the intellectual world, and men of science like Mach, Pearson, and Poincaré thought that they had found answers to the classical questions on the nature of reality and truth and on fruitful methods of approach to all scientific knowledge. Peirce's practical experience as a scientist, his extraordinary intellectual endowments, and his unusual gift for lucid expression not men in matters of science and philosophy but rendered him also competent to criticise their philosophies.

**2.**Carolyn Eisele, The Charles S Peirce-Simon Newcomb Correspondence,

*Proceedings of the American Philosophical Society*

**101**(5) (1957), 409-433.

**2.1. Extract from the paper.**

Carefully tucked away in the files of the Manuscript Division of the Library of Congress and in the archives of Widener Library and Houghton Library at Harvard University are the two ends of a correspondence that stirs the imagination and quickens the pulse of the scientist or historian interested in scientific Americana of the late nineteenth century. The correspondents are two of the greatest intellects ever produced in America and their exchange of opinion regarding matters scientific and personal serves as an interesting personalised documentation of the scientific thought of their period. The renowned astronomer, Simon Newcomb, considered the sheaf of letters from Charles S Peirce important enough to file away with those of other important men in science whom he knew. Peirce, America's belatedly recognised giant in logic and philosophy, preserved a number of Newcomb's letters as well as the drafts of some of his letters to Newcomb of which we have no other record. The correspondence is being published herein for the first time.

The correspondence was discovered by the writer while doing research for a book on the activities of Charles S Peirce as a historian of science under a grant from the American Philosophical Society. It is being published with the permission of the heirs of both men and of the Philosophy Department of Harvard University. The writer is indebted to Dr Elizabeth McPherson of the Library of Congress who was first to suggest the possible existence of Peirce materials in the Newcomb Collection in the Manuscript Division.

It is not all continuous although most of it lies in the period 1889-1894. The fact that the two men were compatriots and contemporaries in science almost year for year in a time of unprecedented scientific inquiry and discovery suggests an inevitable personal and professional contact in the circumscribed society of American scientific life in those days.

**3.**Carolyn Eisele, Charles S Peirce and the Problem of Map-Projection,

*Proceedings of the American Philosophical Society*

**107**(4) (1963), 299-307.

**3.1. Extract from the paper.**

Among the official reports of the Assistants in the United States Coast and Geodetic Survey for the year 1877 is a letter from Charles S. Peirce to Superintendent Patterson that tells of Peirce's new experimental work on the flexure of the pendulum stand - work necessitated by his systematic attempt to compute the measure of gravity. For Peirce was a scientist in the employ of the United States government and had been charged in 1872 with the responsibilities of the investigation of gravity as well as the related one of determining the ellipticity of the earth. The letter contains, too, the following surprising remarks:

I have something else which you might possibly like to insert in the Coast Survey Report. When I was in Paris, I found that the best MS. of Ptolemy's catalogue of stars had never been properly transcribed. This I did and have set them down on a modern atlas and have the materials for new and improved identifications of them. I therefore propose to make a new edition of Ptolemy's catalogue with identifications and notes. Also with a planisphere showing the stars and the figures of the ancient constellations, which I carefully studied some years ago and laid down on my globe.Peirce's interest in map-making was further revealed in the report that he had "considered the projection of the sphere to be adopted and, setting out with the following conditions," he had invented two means of satisfying them. The conditions were:

1. The map must preserve the angles, like Mercator's and the Stereographic projection.

2. It must put the whole sphere on one finite map.

3. It must bring the sphere into an oblong shape, suitable for a page.

Peirce explained that both methods involved the use of "Elliptic Integrals" and he suggested that the Superintendent might be interested in having a brief description of these projections of the sphere as well as others he had invented, even though the Catalogue of Ptolemy were not desired.

**4.**Carolyn Eisele, The mathematics of Charles S Peirce, in

*Actes du Onzième Congrès International d'Histoire des Sciences, Varsovie-Cracovie, 1965, Sect. III: Histoire des Sciences Exactes*(Astronomie, Mathématiques, Physique) (Ossolineum, Wroclaw, 1968), 229-234.

**4.1. Review by: Koichi Inoue.**

*Mathematical Reviews*MR0260543

**(41 #5169)**.

The author presents an interesting report on Peirce's study in the field of mathematics on the basis of his letters and manuscripts which are now found in a collection at Harvard University.

C S Peirce, who had acquired "a reputation as a powerful logician" and was in a "deliberate posture as logician rather than mathematician", had, as the author points out, a "high status in the world of the professional mathematician of his time". "Peirce's needs in the logic of relatives led him to the investigation of another mathematical abstraction, that of 'collection' and its 'multitude', Cantor's

*Menge and Mächtigkeit*. Not only did he claim to have developed his theory independently of Cantor's early work, but he had come to his criterion for the infinite six years before Dedekind's famous work made its appearance."

**5.**Carolyn Eisele, Salomon Bochner on Charles S Peirce,

*The American Mathematical Monthly*

**82**(5) (1975), 477-478.

**5.1. Extract from the paper.**

My remarks at the Conference on the History of American Mathematics, relative to the paper written by and read for Salomon Bochner at Texas Tech University in May 1973, are still applicable to his

*Mathematical Reflections*in this Monthly,

**81**(1974) 827-853. For Professor Bochner's analysis of Charles S Peirce's mathematical treatment of the continuity concept in the overall framework of Peirce's philosophical system, reflects a lack of acquaintance with a large segment of basic material in that area of Peirce's writings. Those manuscripts lie unpublished in Peirce's handscript in Houghton Library at Harvard University, and are only gradually being made available to the scholarly world in new editions.

I am editing the forthcoming edition of

*The New Elements of Mathematics by Charles S Peirce*. Since this material is not readily accessible, Professor Bochner seems to have relied heavily on the

*Collected Papers of Charles Sanders Peirce*, vol. I-VI (1932-1935), probably in the belief that everything of value in Peirce's systematic thinking was included in those volumes. Peircean scholars, however, now speak of that collection as selected portions of selected papers where long mathematical passages of reference have been deleted and where purely mathematical papers have been totally omitted. It is unfortunate that Professor Bochner makes no mention of volumes VII and VIII of the

*Collected Papers*, edited by Arthur Burks and published in 1958, long after volumes I-VI.

Professor Bochner also cites as evidence of the validity of his assessment of Peirce as "a real American Tragedy" and "a great philosopher, and an even greater failure," negative conclusions found in parts of Murray Murphey's

*The Development of Peirce's Philosophy*(1961), written at a time when much of the unpublished manuscript material was still in a disordered state. Because of different depository arrangements at that time, Peirce papers were to be found in several different places in the Harvard libraries and some of them were restricted in use. It has taken years of dedicated work on the part of several scholars, notably Max and Ruth Fisch, to bring order into the collection. This was necessary before Professor Fisch could embark on the writing of the definitive biography of Peirce. Today the Peirce Collection is all of one piece and runs to some 1650 manuscripts and some 1600 correspondence folders, a vast monument to Peirce's genius.

**6.**Carolyn Eisele, C S Peirce's Search for a Method in Mathematics and the History of Science,

*Transactions of the Charles S Peirce Society*

**11**(3) (1975), 149-158.

**6.1. Extract from the paper.**

In an application to the Carnegie Institution in 1902 for aid in the production of a series of memoirs "forming a unitary system of logic in all its parts" Peirce wrote of having imbibed from his boyhood the spirit of positive science, especially of the exact sciences. Shortly after graduation from Harvard in 1859 he determined to devote his life to the study of logic and considered his appointment as an aid in the Coast and Geodetic Survey as an opportunity to learn about the methods of investigation in one science. He had come to treat logic as a science and made his studies "special, minute, exact, and checked by experience." The whole of Peirce's life was dedicated to a "Search for a Method," a title he gave in 1893 to the revision of a series of articles he had written in the period 1867-1893.

Granted that Peirce's primary objective lay in the development and refinement of the analysis of logical processes, two tools became increasingly indispensable to him in his search. The first of these was the method and content of the then contemporary corpus of mathematics; the second, the history of science and of mathematics, fields offering a rich harvest of episodes not only for his logic of science but for the refinement of the mechanics of his logic in a more general sense.

Peirce once described logic as the theory of reasoning whose "main business is to ascertain the conditions upon which the just strength of reasoning depends," and identified with that school of thought in which the rules of logical method were derived "by mathematical means from initial propositions which ordinary observation forces upon every man." He listed the principal achievements of that logical method as having been three in number. According to him the first was the doctrine of chances, which had become the logic of the physical sciences. It provided mathematically for the two conceivable certainties with respect to any hypothesis, the certainty of its truth and the certainty of its falsity for these extremes may be represented by 1 and 0 respectively and fractions between these values indicate the degree with which the evidence supports one or the other assumption. Thus, Peirce said, the general problem becomes that of determining the numerical probability of a possible fact from a given set of facts. The problem of probability thus is identified with the general problem of logic and the logic of probability is related to ordinary syllogistic, according to Peirce, as the quantitative to the qualitative branches of the same science. He wrote that this dichotomy resembles the distinction between projective geometry which asks only whether points coincide or not, and metric geometry which determines the numerical distance between them.

The logic of relatives was listed by Peirce as the second achievement of the logical methodology to which he adhered. De Morgan had first rendered it important in 1860 and Peirce held that it had reformed the conceptions of logic. Peirce regarded the third achievement to be Boole's algebraic procedure which he himself modified and developed in part. In Peirce's attempts in 1870 to extend the Boolian calculus to the logic of relatives, Peirce had come upon "a set of forms constituting a system virtually identical with the calculus of matrices," and consequently, with the subject of his father's Linear Associative Algebra. Peirce predicted that a fourth achievement was about to be witnessed with a "full elucidation of the logic of continuity," presumably by himself.

**7.**Carolyn Eisele, Mathematics as a Semiotic Factor in the Thought of C S Peirce,

*Semiotics*(1981), 417-422.

**7.1. Extract from the paper.**

I continue to live with the nagging conviction that a thorough understanding of any aspect of Peirce's tightly integrated thought can be attained only through an appreciation of the structure of the underlying mathematical framework on which that thought is built. Mathematics is a subject in closest alliance with semiotics, and Peirce has been a celebrated figure in this recently emerging academic discipline. It is natural, then, to examine the mathematical character of his thought in general and its effect on the nature of his semiotical procedure in particular.

In the celebration of Peirce as a philosopher and logician seldom is mention made of his thirty-one year career (1859-1891) in the service of the United States Coast and Geodetic Survey where he acquired an international reputation as geodesist and astronomer. Even less seldom is there any reference to his life-long exposure to the theoretical and applied forms of mathematics that were acquiring a new relevance in the mathematical and scientific revolution of the late 19th century. And yet he once wrote that his special business was to bring mathematical exactitude, and he meant modern mathematical exactitude into philosophy and to apply modern ideas of mathematics in philosophy. Moreover, "Philosophy requires exact thought, and all exact thought is mathematical thought." It must be bound down to a logic at least as rigid as that of Euclid, he claimed. Peirce believed that his own thought was so characterised, since he had been bred "in the lap of the exact sciences" and if he knew what mathematical exactitude is, "that is as far as he could see the character of" his philosophical reasoning. To examine this mathematical exactitude we shall turn to the methodology of the logic "at least as rigid as that of Euclid." Indeed we shall even stop to examine the pattern of treatment of a theorem in Euclid.

A basic assumption of this paper is that a mathematician by the very nature of his operations with a symbolic language devoid of any extraneous overtones is a practitioner of semiotic although he doesn't consciously sail under that flag. In his "search for a method" Peirce naturally adapted the symbolism and operational techniques of algebra to problems in logic. And his success in further developing the symbolic logic of Boole and De Morgan was well recognised in his time. But I turn here to the prescription given by Peirce for a valid procedure in logically examining the validity of a suspected "truth," like that of a theorem in geometry.

**8.**Carolyn Eisele, Mathematical methodology in the thought of Charles S Peirce, Historia Mathematica 9 (1982), 333-341.

**8.1. Extract from the paper.**

It is fitting that the celebration of Peirce's

*New Elements of Mathematics*should be taking place in New York City, where Peirce was often to be found attending mathematical meetings at Columbia University and where he consulted the resources of the old Astor Library for the production of many of his writings. This paper considers Peirce - a lifelong student of logic - as he examined scientific and mathematical methodology on all levels, in ages past as well as in the then-contemporary literature. Peirce hoped to create an exact philosophy by applying the ideas of modern mathematical exactitude. He developed a semiotic pattern of mathematical procedure with which to test validity in all areas of investigation.

After more than 30 years of "searching for the method" that underlies Peirce's investigation of problems in science, logic, and philosophy, I find the evidence becoming ever weightier that the procedure he prescribes is to be found in the gold mine of his mathematics that Peircean scholars have thus far failed to explore and exploit. Rich nuggets lie about waiting to be picked up. Digging brings even greater intellectual rewards. Perhaps when the four volumes (five books) of Peirce's New Elements of Mathematics become required reading for students in Peircean studies, we shall begin to see in depth why he became the greatest native philosopher of mathematics and science ever to appear on the American scene [Eisele 19761. In celebrating his thought on this occasion we do so in the New York academic community to which he was so greatly attracted at the end of the 19th century. We know of Peirce's early association with academe in Cambridge where his father was venerated as a leading American mathematician. We know of Peirce's 31-year association with outstanding mathematicians and scientists in Washington. We know a great deal about his mathematical ingenuity as the need arose in his astronomical and geodetic researches. We know of his 5-year lectureship in logic at The Johns Hopkins University when the celebrated Sylvester was Chairman of the Department of Mathematics and the equally famous Cayley became a visiting lecturer there during 1882. But it is not generally realised that after Peirce's retirement from the United States Coast and Geodetic Survey in 1891 he could often be found in the New York area where another great university and fine public library offered him the means of doing his "special great piece of work."

**8.2. Review by: Joseph W Dauben.**

*Mathematical Reviews*MR0667605

**(84c:01031)**.

This brief paper begins by emphasising Peirce's connections with New York City, where he often attended mathematical meetings at Columbia University, and where he consulted the Astor Library in producing many of his writings. The major point of the paper is concerned with Peirce's interest in scientific and mathematical methodology on all levels, both historically and as a subject of contemporary study. Above all, he developed a semiotic pattern of mathematical procedure with which to test validity in all areas of investigation. The author argues that Peirce hoped to create an exact philosophy by applying the ideas of modern mathematical exactitude.

**9.**Eisele, Carolyn, An introduction to Peirce's mathematical semiotic,

*The American Journal of Semiotics*

**2**(1/2) (1983), 45-54.

**9.1. Extract from the paper.**

In Peirce's logic of science the prescribed basic procedural pattern of investigation is termed

*theorematic reasoning*. Peirce's analysis of this semiotic device points to the mathematical grounding of his analytical thought in every field. With the exception of application to problems in pure mathematics, use of this semiotic machine entails an accompanying recognition and evaluation of the mathematical element of probability.

When, in my teens, I was first reading the masterpieces of Kant, Hobbes and other great thinkers, my father, who was a mathematician, and who, if not an analyst of thought, at least never failed to draw the correct conclusion from given premises, unless by a mere slip, would induce me to repeat to him the demonstrations of the philosophers, and in a very few words would usually rip them up and show them empty. In that way, the bad habits of thinking that would otherwise have been indelibly impressed upon me by those almighty powers, were, I hope, in some measure, overcome. Certainly, I believe the best thing for a fledgling philosopher is a close companionship with a stalwart practical reasoner.In those terms Charles Peirce, often referred to as the father of semiotic, described the early intellectual influence that eventually mathematicised all his thought.

A basic assumption of this paper is that a mathematician, by the very nature of his operations with a symbolic language devoid of emotional overtones, is a practitioner of semiotic although he is not generally aware of it. In personal direction of Charles's mathematical education, his father, famous for his work in mathematics and astronomy, unwittingly made a formal semiotician of his son.

**10.**Carolyn Eisele, Thomas S Fiske and Charles S Peirce, in

*Peter Duren (ed.), A Century of Mathematics in America, Part I*(American Mathematical Society, 1988), 41-55.

**10.1. Extract from the paper.**

The

*Bulletin of the American Mathematical Society*, volume XI, February 1905 carried the Presidential address of Thomas Scott Fiske telling of "Mathematical Progress in America." It had been delivered before the Society at its eleventh annual meeting on December 29, 1904. Fiske spoke of three periods during which one might consider the development of pure mathematics on the American scene. He pointed in the first period to the work of Adrain and of Bowditch and of Benjamin Peirce whose

*Linear Associative Algebra*was the first important research to come out of America in the field of pure mathematics. The second period, Fiske said, began with the arrival in 1876 of James Joseph Sylvester (1814-1897) to occupy the Chair of Mathematics at the newly founded Johns Hopkins University in Baltimore where his first class consisted of G B Halsted. C S Peirce, son of Benjamin, once described Sylvester as "perhaps the mind most exuberant in ideas of pure mathematics of any since Gauss." Shortly after Sylvester's arrival in Baltimore, he established the

*American Journal of Mathematics*(AJM), a great stimulant to original mathematical research, which ultimately included some thirty papers by himself. In 1883 he returned to England to become Savilian Professor of Geometry at Oxford.

The first ten volumes of the

*AJM*contained papers by about ninety different writers and covered the years 1878 to 1888. About thirty were mathematicians of foreign countries; another twenty were pupils of Professor Sylvester; others had come under the influence of Benjamin Peirce; some had been students at German Universities; some were in large degree self-trained. Several of them had already sent papers abroad for publication in foreign journals. Among the contributors to the early volumes of the

*American Journal of Mathematics*special mention must be made of Newcomb, Hill, Gibbs, C S Peirce, McClintock, Johnson, Story, Stringham, Craig, and Franklin.

Reminiscing in 1939, Fiske recalled in the

*Bulletin*that while a graduate student in the Department of Mathematics at Columbia University, he had been urged in 1887 by Professor Van Amringe to visit Cambridge, England. There he attended all the mathematical lectures in which he was interested and met Cayley, On November 24, 1888, after his return to New York, Fiske interested two fellow students, Jacoby and Stabler, in the idea of creating a local mathematical society which was to meet monthly for the purpose of discussing mathematical topics. Thus began the New York Mathematical Society and the third period in the Fiske account of American mathematical progress.

By December 1890, the idea of the Society publishing a

*Bulletin*, as did the London group, was suggested and led to the decision to have the American Society ape them. The first number appeared in 1891 with Fiske as editor-in-chief. By 1899 arrangements were made for the publication of a

*Transactions*and Fiske again became a member of the editorial staff (1899-1905).

As secretary of the Society, Fiske corresponded with a few of the top mathematicians of this country to determine if their assistance could be counted on for the new enterprise. Cooperating at once were Newcomb, Johnson, Craig, and Fine.

Among those invited in 1891 were two unusual individuals: Charles P Steinmetz and C S Peirce. Fiske relates that Steinmetz who was born in Breslau on April 9, 1865, was "a student at the University of Breslau where he had been the ablest pupil of Professor Heinrich Schroeter. In the spring of 1888 he was about to receive the degree of Ph.D., but in order to escape arrest as a socialist he was compelled to flee to Switzerland. Thence he made his way to America, arriving in New York June I, 1889." Fiske continued, "About a year later my attention was attracted to an article of sixty pages or more in the

*Zeitschrift für Mathematik und Physik*on involutory correspondences defined by a three-dimensional linear system of surfaces of the $n$th order by Charles Steinmetz of New York. This was the doctor's dissertation."

Fiske tells of having invited him to Columbia University and of having offered help with the English in his papers. He also invited him to become a member of the New York Mathematical Society where he presented a number of papers, two of which were published in the

*American Journal of Mathematics*. Although soon gaining fame as one of America's leading electrical engineers, he remained a member until his death on October 26, 1923.

The other addition to the Society in 1891 is of particular interest to us. "Charles S Peirce, B.Sc., M.A., member of the National Academy of Sciences" was elected at the November meeting. It is of interest to learn that Van Amringe had proposed Peirce's name for membership and that Harold Jacoby had sent the invitation. It is to be noted that Peirce was already a member of the London Mathematical Society. The Fiske

*Bulletin*account of 1939 tells the story this way:

Conspicuous among those who in the early nineties attended the monthly meetings in Professor Van Amringe's lecture room was the famous logician, Charles S Peirce. His dramatic manner, his reckless disregard of accuracy in what he termed "unimportant details," his clever newspaper articles describing the meetings of our young Society interested and amused us all. He was advisor of the New York Public Library for the purchase of scientific books and writer of the mathematical definitions in the Century Dictionary. He was always hard up, living partly on what he could borrow from friends, and partly from odd jobs such as writing book reviews for the 'Nation' and the 'Evening Post' .... At one meeting of the Society, in an eloquent outburst on the nature of mathematics C S Peirce proclaimed that the intellectual powers essential to the mathematician were "concentration, imagination, and generalisation." Then, after a dramatic pause, he cried: "Did I hear someone say demonstration? Why, my friends," he continued, "demonstration is merely the pavement upon which the chariot of the mathematician rolls."The last sentence in the Fiske account causes one to pause momentarily and observe that Fiske was apparently more sensitive to the nature of Peirce's vast outpourings in those days than most of their contemporaries in mathematics could be. Truly Peirce had elected to trace the taming of the human mind in its ordered discovery of the patterns of nature's design to the development of logical restraint. Peirce's probes for materials in his analytical development of such themes led to massive descriptions of events in the history of mathematics and science. And his attempts to systematise those historical reflections permeate Peirce's writings generally - philosophical, mathematical, scientific.

**10.2. Review by: Joseph W Dauben.**

*Mathematical Reviews*MR1003161

**(90g:01022)**.

This paper, in one of the centenary volumes of the American Mathematical Society, is devoted to the subject of American mathematics at the turn of the century, and the extent to which an indigenous community of mathematicians was working here at a high scholarly level comparable to what European mathematicians at the same time were accomplishing. Here the author focuses her attention upon the period beginning in 1888, when the New York Mathematical Society was founded, along with its Bulletin, in 1891 with T. S. Fiske as editor in chief. Among mathematicians of this era whose work is discussed are Newcomb, Johnson, Craig, Fine, Steinmetz and C. S. Peirce. She devotes the majority of her article, in fact, to Peirce, who contributed seven pieces (between 1878-1885) to the American Journal of Mathematics (Sylvester's journal). Peirce, who at the time was working for the U.S. Coast and Geodetic Survey, also taught logic for a time at Johns Hopkins. Among his various mathematical projects were theoretical studies of map projections, the four-colour problem, graph theory, non-Euclidean geometry, gravity research and pendulum studies, articles for the Century dictionary (1889), and studies of nomenclature and the history of mathematics. He even published one paper on "logical machines" in 1887. Ultimately, the author's paper is designed to demonstrate through such specific examples the quality and vitality of American mathematics at the turn of the century.

**11.**Carolyn Eisele, Fresh Light on Peirce's Theory of Signs from his Writings on Mathematics and the History of Science, in

*Tasso Borbé (ed.), Semiotics Unfolding: Proceedings of the Second Congress of the International Association for Semiotic Studies Vienna, July 1979*(De Gruyter Mouton, Berlin, Boston, 2015), 91-98.

**11.1. Extract from the paper.**

To this writer the most pressing need in Peircean scholarship today is a confirmation in detail of the assertion that the many facets of Peirce's logical and philosophical thought are grounded in the mathematics revolution of the late nineteenth century. It becomes ever more apparent that a working knowledge of the then newly designed basic concepts of the mathematics and its methodology underlying scientific research in that period is necessary for any deep-rooted understanding of his general thought. Therein can be discerned numerous elements in the development of his semiotic stance. Pure mathematics in itself in modern dress is one of the purest forms of semiotics with its working tools all set up in symbolic or iconic form. And the names associated with its progress are similarly symbols of what that progress consisted in. Fortunately Peirce's writings are rich in reference to his scientific and mathematical forebears and contemporaries so that one can probe the source of his innovations and determine the cast of his approach to the subject matter of logic and philosophy.

In the language of semiotics, Peirce's constant reference to Fermat, Kepler, Bolzano, Cauchy, Cantor, Dedekind and to Grassmann, Riemann, Boltzman, Hamilton, Mach, Cayley etc., etc., makes indices of their names pointing to specific areas of his intellectual concern. Although some of these persons had life spans that overlapped his, their writings appeared in the records of Peirce's yesterday and were available to him. He was as familiar with them as he was with those of past ages. In all of them he searched relentlessly for evidences of reliable methodology in the quest for a knowledge of the 'truth'. He was particularly interested in episodes that would reveal step by step the secret of human ingenuity and the growth of the human mind. He was equally on the hunt for currently developing logical procedures that might lead one to an understanding of the real world.

In the introduction to a series of lectures on the history of science in 1892 Peirce mentioned the motivations for such studies and his own interest in discovering how such 'truth' was revealed. He was ever on the alert for illustrations in his own analysis of reasoning. In this paper an attempt will be made to associate some of the elements of that analysis with the logical practice of some of the great innovators in mathematics and science.

**12.**Carolyn Eisele, Mathematics as a Key to Peirce's Semiotics, in

*Annemarie Lange-Seidl (ed.), Zeichenkonstitution. Akten des 2. Semiotischen Kolloquiums Regensburg 1978*(De Gruyter, Berlin, Boston, 2017, 123-128.

**12.1. Extract from the paper.**

With the recent publication of

*The New Elements of Mathematics by Charles S. Peirce*in 4 volumes and 5 books, it becomes ever more evident that an understanding of the foundations and range of his thought depends on an understanding of the mathematics that he embraced during the wave of revolution that swept over that field during his lifetime. Since mathematics is the greatest exponent of language that is totally symbolic and since Peirce had been trained to think and to speak in that language from early child-hood, it is no surprise to have him maintain repeatedly that all thought is carried on in signs in a way analogous to that in which mathematical theory develops.

In a letter to Mr Kehler in 1911 he wrote:

It was about 1870, I don't think it could have been as late as 1872, that I invented the word "pragmatism" to mean that way of thinking that regards thinking as consisting not necessarily in talking to oneself because an algebraist like Boole plainly thought in algebraic symbols; and so did I, until at great pains, I learned to think in diagrams which is a much superior method. . . By pragmatism I meant a philosophy which should regard thinking as manipulating signs so as to consider questions. But attention whether voluntary or not is always an act-, and a general conception is a habit, and believing, real genuine belief, consists in a habit with which one is contented and which one recognises... [as] consisting in the general fact that under certain circumstances one would act in a definite way, and would be content to do so... Out of such considerations which turn, as if upon a pivot, about the idea that a thought is nothing but a habit connected with a sign one can build up quite a little philosophy which is what I meant by "Pragmatism". I think the idea was suggested to me by Berkeley's two little books about vision; and while the idea was a fresh one for me in the early seventies, I used to talk my friends to death about "pragmatism".By 1878 Peirce had written of pragmatism as a doctrine that the meaning or ultimate translation of a conceptual sign, that is of a general sign, lies in purposive action. One will learn in time from the intellectual biography of Peirce that is being prepared by Max Fisch just when it was that Peirce adopted this semiotic stance, one that permeates his thought and writings in the years of his greatest intellectual power.

The writer's sympathy for Peirce's position is readily comprehended in terms of her long association with the world of mathematics where one automatically becomes a semiotician in professional communication. Peirce appreciated the art of transcribing verbal signs into lean mechanistic symbols that are transformed into other signs under precise rules of operation already agreed upon. This procedure leads to the "truth" in the long run, by the deductive verification of a statement that had been but a mere assumption.

Last Updated December 2021