The Claremont History and Philosophy of Mathematics Seminar meets on Mondays from 3:004:00 pm in Parsons 1289 on the Harvey Mudd College Campus. Click here for a map of the campus.
Parsons 1289 is on the first floor of Parsons. Parking is available in the lot off Foothill just east of Dartmouth. You may enter the Parsons door right off the parking lot by making the first right turn at the corridor and continuing on for about 50 feet. The room is on the right side of the corridor just before you leave the building by another exit/entrance.
Claremont Colleges Faculty can park in the HMC parking lot and visiting speakers will obtain a parking pass for the day.
For more information about the Seminar, or to suggest speakers, contact Jemma Lorenat,
Date  Speaker  Title and Abstract 
Monday September 10 3:00 pm 
Bogdan Suceava California State University Fullerton 
Followers of the Erlangen Program: The Birth of Affine Differential Geometry and the Development of the Bucharest School of Geometry ABSTRACT: In 1872, Felix Klein published a very important work, today known as the Erlangen Program, whose lasting impact influenced the modernist transformation of mathematics. A doctoral student of Gaston Darboux, Gheorghe Tzitzeica, introduced the first concepts of affine differential geometry in a series of papers published in 19071908. We discuss pertinent details of Tzitzeica's life and his work on affine differential geometry. We investigate his academic studies and the discovery of the centroaffine invariant that today bears his name  a concept that spawned research in affine differential geometry. Futhermore, Tzitzeica also discovered Dan Barbilian, a brilliant and original mathematician with inspiring contributions in algebraic geometry and metric geometry. He is remembered today, among other things, for Barbilian's metrization procedure, which can be used to generate Riemannian and Lagrange generalized metrics. 
Monday October 1 3:00 pm 
Reading Discussion 
Imre Lakatos, Proofs and Refutations (1963)

Monday October 15 3:00 pm 
Mordechai Feingold California Institute of Technology 
Teaching Mathematics (and Natural Philosophy) in Early Modern Universities ABSTRACT: In theory, mathematics was an integral part of the university curriculum during the early modern period. Attempts to assess the nature and quality of such instruction, however, has hitherto been based, almost exclusively, on statutory requirements, and on the examination of the assigned textbooks. My talk attempts to go beyond prescriptions; I shall attempt to imagine the venue in which instruction had taken place, and what actually transpired in the lecture hall. 
Monday November 5 3:00 pm 
Michael Barany University of Edinburgh 
Inclusion, Exclusion, and the Theory and Practice of "Truly International" Mathematics ABSTRACT: In theory, just about anyone can be a mathematician. Theorems and proofs, for the most part, don't discriminate based on race, class, gender, disability, national origin, or anything else, at least in principle. Historically, however, the field and profession has been open to very few. Mathematicians have grappled in many different ways with this gap between an ideal of openness and and a reality of exclusion and even outright discrimination. I will show how American mathematicians took leadership of the international mathematics community over the period between 1920, when they first proposed to host an International Congress of Mathematicians, and 1950, when they finally brought one to fruition. American mathematicians tried to reshape mathematics as a more interconnected and inclusive disciplineone they called "truly international"succeeding in some ways and failing in others. I will explain how the ambiguous and shifting meaning of that key phrase, "truly international," helped them navigate a wide range of political, financial, and other obstacles, while covering over persistent problems and blindspots. 
Monday November 19 3:00 pm 
Reviel Netz Stanford University 
Archimedes and Infinity One of the major themes of Archimedes' mathematics was the measurement of curvilinear objects. His approach to such measurements provided much of the inspiration for the modern rise of the calculus and it is natural to ask how Archimedes himself understood infinity and infinite divisibility. While Aristotle explicitly ruled out the existence of actual infinities, it is clear that Archimedes refers to them, but in ways which are always qualified. The talk presents the key cases, raises the question and offers a tentative conclusion for Archimedes use, and caution, regarding the infinite. 
Monday December 10th 3:00 pm 
Reading Discussion 
Karen Parshall, "Defining a mathematical research school: the case of algebra at the University of Chicago, 18921945" Historia Mathematica (2004)

Date  Speaker  Title and Abstract 
Monday January 29 3:00 pm 
Reading discussion on the Sleeping Beauty Controversy 
Peter Winkler, "The Sleeping Beauty Controversy" The American Mathematical Monthly (2017) 
Monday February 12 3:00 pm 
Marco Panza Chapman University and University of Paris 1 PanthéonSorbonne 
Universality in Euclid ABSTRACT: I shall address the classical problem of what in Euclid's arguments (proofs of theorems and solution of problems) can support the universality of his results, and try to provide an answer based on an original account of what universality might have meant in the context of Euclid's geometry. 
Monday March 5 3:00 pm 
Reading discussion on the philosophical implications of knot diagrams 
Silvia de Toffoli and Valeria Giardino, "Forms and Roles of Diagrams in Knot Theory" Erkenntnis (2014) 
Monday March 26 3:00 pm 
Niccolò Guicciardini University of Bergamo, Francis Bacon Professor in History at the California Institute of Technology 
Newton the mathematician ... and beyond ABSTRACT: This talk originates from the recent experiences (and troubles) I had in preparing for publication a short updated English version of an intellectual biography of Isaac Newton, which appeared in Italian some years ago. The magnitude and complexity of Newton's printed and manuscript works have allowed different interpretations of his personality and intellectual achievements. Since I am a historian of mathematics, I begin from the field that is most familiar to me. So, my opening question will be: What kind of mathematician was Isaac Newton? How should we interpret his mathematical works? I shall then move on to explore some territory that is foreign to me with a view to verifying whether the image of Newton the mathematician sketched in the first part of my talk can, at least in some degree, shed some light on the other intellectual endeavors to which he committed himself. 
Monday April 16 3:00 pm 
Karine Chemla Université Paris Diderot 
When Mathematicians' Philosophical Reflections Play a Key Part in the Advancement of Mathematics. The impact of Poncelet and Chasles' reflections on generality on Kummer's work with ideal divisors. ABSTRACT: Kummer's first public presentation of his "ideale complexe Zahlen" (first published in the 1846 issue of the Bericht über die zur Bekanntmachung geeigneten Verhandlungen der Kö nigl. Preufs. Akademie der Wissenschaften zu Berlin) draws a parallel between ideal elements in geometry and the "ideal prime factors" that he introduces in his study of cyclotomy. In this first publication of his theory, Kummer also drew other comparisons with the introduction of complex numbers into "algebra and analysis", and Gauss's work in number theory, whereas in later expanded presentations he established a parallel with chemistry. The historiography of Kummer's work in number theory has mainly dwelled on these other comparisons. However, the parallel with projective geometry has remained in the shadow. In this talk, I argue that Kummer's reflection on Poncelet's introduction of ideal relations in geometry, and the reconceptualization that Chasles offered for these "ideal elements" in his 1837 Aperçu historique played a key part in Kummer's introduction of "ideal complex numbers." This part is clearly perceptible in the structure of the 1846 publication, and I will explain how we can read the effect of Chasles' reconceptualization in the definitions that Kummer presents. I also argue the parallel between ideality in projective geometry and in Kummer's work on numbers helps us understand features of the "ideal complex numbers" that have puzzled historians. This episode is interesting at a higher level, since it suggests that the philosophical reflection on the value of generality that geometers like Poncelet and Chasles developed in the context of the shaping of projective geometry was instrumental as such in inspiring key developments in other domains of mathematics, and precisely in this case, the introduction of ideal elements more widely in mathematics. 
Monday April 30 3:00 pm 
Gizem Karaali Pomona College 
Defining Ada: On The Legacy of Augusta Ada Byron King Lovelace ABSTRACT: Augusta Ada, Countess of Lovelace, is today viewed as the first person to recognize the power of algorithmic machines and a pioneer in computer programming. Her biographers have often disagreed on her mathematical talents, her mathematical contributions, and her legacy. In this talk I explore the various approaches taken towards her, focusing explicitly on how the men in her life have been used to define her. I conclude with some thoughts on Adas impact and legacy. 
Date  Speaker  Title and Abstract 
Monday October 2 3:00 pm 
UC Riverside 
Richard Dedekind and the Structuralist Transformation of Mathematics. ABSTRACT: In recent history and philosophy of mathematics, “structuralism" has become an important theme. There are two different ways in which this theme is typically approached: (a) by focusing on a structuralist methodology for mathematics (the tools used in mathematics, the ways in which its various parts are organized, etc.); (b) in terms of a structuralist semantics and metaphysics for mathematics (conceptions of what we talk about when we study “the natural numbers”, “the real numbers”, “the cyclic group with five elements", etc.). In this talk, I will make a case that Richard Dedekind’s writings from the nineteenth century should be seen as a main historical sources for both strands. Moreover, for him the two strands are intimately connected; more specifically, methodological considerations lead naturally to semantic and metaphysical ones in his work. To make this evident will involve comparing his foundational work (on the natural and real numbers) with his contributions to algebra and number theory (his investigation of the notions of algebraic number, group, field, lattice, etc., including his famous theory of ideals). 
Monday October 23 3:00 pm 
University of San Francisco 
The Brouwer Fixed Point Theorem and Its History ABSTRACT: The Brouwer fixed point theorem first appeared in 1911, rather overshadowed by Brouwer's theorems on invariance of dimension and invariance of domain. It was revived, with a simpler proof based on Sperner's lemma, in 1928. But by then Brouwer himself had renounced the theorem, as nonconstructive and hence incompatible with his intuitionist principles. However, the fixed point theorem grew in importance, appearing in the textbook of Alexandroff and Hopf in 1935, and being used in applications such as the Nash equilibrium theorem. Recently the theorem has found a new role in reverse mathematics, a theory created by logicians over the last few decades. Reverse mathematics confirms that the Brouwer fixed point theorem is indeed nonconstructive, but it is "as close to constructive" as many basic theorems of analysis. 
Monday November 6 3:00 pm 

Reading Discussion: Dirk Schlimm, "Metaphors for Mathematics from Pasch to Hilbert'' Philosophia Mathematica, Vol. 24, No. 3 (2016) pp. 308329 
Monday November 20 3:00 pm 
Columbia University

The geometry of situation in seventeenthcentury England: Wallis, Barrow, Newton ABSTRACT: In 1685 John Wallis recalled his earlier study of conic sections: he had considered them "abstractly as Figures in plano, without the embranglings of the Cone." Around the same time, Isaac Newton criticized such an approach in extensive manuscript studies of ancient geometry. According to Newton, algebraic constructions were not only ugly; they could even be incomprehensible. In line with this judgment, most of the proofs in Newton's 1687 Principia mathematica were written in what Newton characterized as a traditional, geometric style. Scholarly literature on the period tends to oppose Wallis' preference for symbolic reasoning to geometry based on the generation of magnitudes; this was the philosophy of geometry advanced by Isaac Barrow, Newton's predecessor at Cambridge. By showing how Wallis' efforts to disembrangle the conics help explain Newton's preference for traditional geometry in the 1680s, this talk will argue that Wallis' 1655 On conic sections was an important part of the intellectual background of the mathematics of the Principia. Situation [situs] emerges as a key concept for explaining how Wallis, Barrow, and Newton understood the ontology of the diagram and its analysis by means of symbols. In English geometry, situation was not an alternative to measurement. Rather, as in the tradition of classical analysis, situation was an essential means for making measurements. 
Monday December 4 3:00 pm 
UC San Diego 
Algorithmic Thinking Before 1950 ABSTRACT: By looking past the ubiquitous narratives of electronic computing, I argue for an alternative, global genealogy of twentiethcentury algorithmic culture. I offer a foray into the design of mathematical technologies at the limits of human reason and how cultural discontents  anxiety, error, and doubt  stabilize technocratic control. Following the life of a single algorithm, the Confidence Interval parameter, I highlight a shift from its origins in an interwar practical philosophy movement to its sedimentation in the hegemonic modelbased planning rationality of the early Cold War era. I emphasize the interwar period as the critical moment in the shift from eighteenth and nineteenth century modes of statistical governance, including Victorian era political arithmetic and eugenics, to the rise of computerbased algorithmic cultures that remain the dominant order of today. 
Date  Speaker  Title and Abstract 
Monday January 23 3:00 pm 
Organizational Meeting 
Discussion of future speakers, readings, themes. 
Monday February 6 3:00 pm 
Reading Discussion

Frege Reading: Paolo Mancosu, ``Grundlagen, Section 64: Frege's Discussion of Definitions by Abstraction in Historical Context'' History and Philosophy of Logic, Vol. 36, No. 1 (2015) pp. 6289 
Monday March 6 
Janet Beery University of Redlands 
Navigating Between Triangular Numbers and Trigonometric Tables: How Thomas Harriot Developed His Interpolation Formulas By 1611, Thomas Harriot (15601621) was developing constant difference interpolation methods, work that culminated in 1618 or later in his unpublished treatise, "De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna," in which he derived symbolic interpolation formulas and showed how to use them to interpolate in tables. The interpolation formulas that appear in Harriot's surviving manuscript work vary in notation, structure, and method of application. In this presentation, we will use these largely undated manuscripts to show how Harriot may have developed and refined his methods over time. 
Monday March 20 
Brittany Carlson UC Riverside 
Victorian Puzzle Addiction: "The Final Problem" as a Mathematical Puzzle This talk examines the social conditions leading to the popularity of the Sherlock Holmes canon and the Victorian fascination with puzzles found in both detective fiction and recreational mathematics. This paper argues that Sir Arthur Conan Doyle's "The Final Problem," uniquely functions as both detective fiction and a mathematical puzzle, forcing its audience to think beyond the text to derive a solution to what game theoretical scholars term the "HolmesMoriarty Paradox." In "The Final Problem," Holmes and Moriarty allegedly arrive at their untimely deaths, with no witnesses, at Reichenbach Falls. The "HolmesMoriarty Paradox" arises out of this tension at Reichenbach Falls when the audience is confronted with the question of who will prevail and how: Professor Moriarty, who is an unstoppable evil genius, or Sherlock Holmes and his untouchable facilities of logic. Since many Holmes fans have become accustomed to observing his methods, they do not actively use them and generally react in one of two ways. Logicians, mathematicians, and puzzle enthusiasts can approach "The Final Problem" as a puzzle, similarly to those found in both mathematical and popular Victorian publications. However, the vast majority of the Victorian readership is distressed by Holmes's death, suffers media withdrawals from the Holmes canon, and do not make attempt to solve this puzzle. This paper also examines a game theoretical solution to "The Final Problem," which is statistically inconclusive; thus frustrating the majority its audience, since a clear and logical solution cannot be deduced even with the help of advanced statistical methods. This paper further asserts that although Conan Doyle attempts to transcend the bounds of the short story genre with a witty paradoxical puzzle to distract his fans from the loss of Holmes, it is a failure, forcing Conan Doyle to eventually revive him in "The Empty House." 
Monday April 3 
Reading and discussions

Reading: Eugene Wigner, ``The Unreasonable Effectiveness of Mathematics in the Natural Sciences'' Communications in Pure and Applied Mathematics, Vol. 13, No. 1 (1960) 
Monday April 17 
Audrey Lackner UC San Diego 
Visualizing Mathematics: Visual Aids in The Elements and the Hierarchy of Disciplines Visual aids to Euclid's Elements were considered so integral to the text that already in the first printed edition (1482) Erhard Ratholdt went to great lengths to include numerous diagrams despite the challenges of including illustrations in the early years of printing. Since diagrams were considered essential, over the years many publishers of The Elements relied on very similar images. Despite similarities between diagrams in various editions, the images in The Elements present arguments that go beyond the illustration of the geometric proofs. In the sixteenth century, visual aids in The Elements became part of arguments about the status of mathematics within the hierarchy of disciplines. In this talk, I will examine the treatment of diagrams in three versions of The Elements published in the 1570s to show how visuals created different visions of mathematics and its value. Sir Henry Billingsley, an English merchant who published a version of The Elements in 1570, used images to emphasize the physical nature of geometry by providing diagrams that were physical instances of the entities studied. He saw mathematics as a useful study of concrete bodies. Federico Commandino, an Italian humanist whose version of The Elements was published in 1570, emphasized the abstract by using diagrams to illustrate universal concepts and principles. He saw mathematics as a noble study of universal truths. Christopher Clavius, the mathematics teacher at the Jesuits' Collegio Romano, published his version in 1574 in which he used diagrams to draw connections between the physical world and the conceptual world. He treated mathematics as a bridge between concrete bodies and abstract ideas. 
Monday May 1 (in Parsons 1265) 
Norton Wise UCLA 
On the Narrative Form of Simulations

Date  Speaker  Title and Abstract 
Monday September 19 3:00 pm 
CalTech 
The Newtonian Origins of Experimental Error 
Monday October 10 3:00 pm 
Pitzer College 
Title: Charles Newton Little, America’s First Knot Theorist Abstract: The modern theory of knots, a subfield of topology, arose in the latter half of the 1800’s after Lord Kelvin proposed that atoms were “knotted vortices in the ether.” This led the Scottish physicist Peter Guthrie Tait to begin tabulating knots, a laborious task in which he was later joined by C.N. Little and Thomas P. Kirkman. Over a period of about 40 years, the three men created a list of all alternating knots with 11 or less crossings and all nonalternating knots to 10 crossings. While they could be sure that their tables listed, in theory, all possibilities, they had no proof whatsoever that their tables did not contain duplications. This would have to wait until well into the 20th century with the development of algebraic topology. In this talk I will review the early history of knot theory with a focus on the life and work of C.N. Little. 
Monday October 24 3:00 pm 
Reading discussion

Karine Chemla's Introduction, "Historiography and History of Mathematical Proof: A Research Programme," in her edited volume, The History of Mathematical Proof in Ancient Traditions, Cambridge University Press, Cambridge, 2012. 
Monday November 7 3:00 pm 
Jeremy Heis University of California Irvine 
Why Did Geometers Stop Using Diagrams? 
Monday November 21 3:00 pm 
Reading discussion

David Hilbert's
Grundlagen der Geometrie (The Foundations of Geometry)
trans. E. J. Townsend, Open Court Press, La Salle, 1950. 
Monday December 5 3:00 pm 
John Mumma Cal State San Bernardino 
Geometric diagrams and the logical form of the parallel postulate 
Date  Speaker  Title and Abstract 
Friday January 29 3:00 pm 
Organizational Meeting 
Discussion of future speakers, readings, themes. 
Friday February 12 3:00 pm 
Judy Grabiner Pitzer College 
Lagrange, Geometry, and Society For a long time, Euclid's geometry was seen as a model not only of how to reason, but of how to achieve certainty and truth. Is it? Were "Euclideans" like Newton, Leibniz, Euler, Lagrange, Voltaire, and Kant intellectual imperialists who misunderstood the nature of mathematics in general and space in particular? Did flaws in Book I of Euclid's Elements lead to General Relativity, surrealist art, conventionalism in mathematics, and multiculturalism? I'll describe how my own work on Lagrange unexpectedly forced me to grapple with these important questions. 
Friday February 19 
Kevin Lambert CSU Fullerton 
Counting on Power: George Peacock, Augustus De Morgan and the Circulation of Mathematical Knowledge between Britain and South Asia The object of this paper is to explore how English mathematician Augustus De Morgan's relationship to two books can tell us something about how both British and Indian mathematics was shaped as knowledge circulated between England and India during the late eighteenth and nineteenth centuries. The first book is Yesudas Ramchundra's, A Treatise on Problems of Maxima and Minima, Solved by Algebra, whose English publication De Morgan superintended. The other is a book De Morgan authored, Arithmetical Books from the Invention of Printing to the Present Time. The connection between them will be traced through George Peacock's investigation of the history of arithmetic in 1820s Cambridge. East India Company officers such as Henry Thomas Colebrooke provided important resources that Peacock would use to construct a world history of arithmetic that situated counting practices from a variety of times and places around the world into a progressive history culminating in European symbolic algebra. Peacock's subsequent philosophy of algebra, I will argue, directly informed the theoretical practices of a second generation of British mathematicians such as Augustus De Morgan, who in turn promoted the development of India mathematics. De Morgan's interest in the history and future of Indian mathematics, I hope to show, will bring us back to Ramchundra. 
Friday March 4 
Ted Porter UCLA 
Madness, Data, and Heredity Gregor Mendel's initial articulation of hybrid genetics cannot have been a purely empirical discovery. But most work on human heredity through about 1880 came down to data gathering, to which measures and indicators of association or correlation began later to be applied. When statisticians, biologists, and psychiatric researchers took up this study, they discovered that the institutions to which they looked for data were already at work trying to create a science from their numbers. That problem defines the historical topic or our discussion, the cultivation of data where explanatory hypotheses were generating more trouble than satisfaction. 
Friday April 1 
17th century readings and discussions part one readings will be posted 

Friday April 15 
17th century readings and discussions part two: Hobbes 
Reading: Douglas M. Jesseph, ``Of analytics and indivisibles: Hobbes on the methods of modern mathematics'' Revue d'histoire des sciences, Vol. 46, No. 2/3 (1993) pp. 153193 
Friday April 29 
Tom Archibald Simon Fraser University 
Abstract mathematics and axiomatization in mid20th c. America: Halmos, Dieudonné, and Measure.
Invented by Henri Lebesgue in 1901, the idea of measure and its application to integration was transformed over the next 50 years by many writers for a variety of purposes, and touches every corner of mathematical analysis. In keeping with the general trends toward axiomatics and abstraction, these new approaches aimed at theories that would have wide applicability and generality. But, as is often the case for mathematical tools, disagreement arose concerning the "best" definitions and the most important areas of application. In this paper, sparked by a reading of critical book reviews by Paul Halmos and Jean Dieudonné of work in this field, we examine divergent views on the nature and purpose of abstraction. Our examples will include work by Halmos' mentor John von Neumann, as well as the larger Bourbaki group of which Dieudonné was an important member. 
Friday May 6 
Jed Buchwald CalTech 
The Newtonian Origins of Experimental Error 