The Claremont History and Philosophy Seminar will be meeting online on Friday mornings for the entirety of the "virtual" year. For more information about the Seminar, to suggest speakers, or to obtain the relevant zoom information, please contact Jemma Lorenat,
Date  Speaker  Title and Abstract 
Friday January 22 10:00 am 
Paola Cantu AixMarseille University and CNRS 
Peano's formalization and semantics ABSTRACT: This talk develops one issue contained in a paper that investigates Peano's philosophical views through a detailed analysis of several mathematical practices that can be fruitfully investigated to compare Peano's philosophical views with logicism and structuralism: the link between functions and relations, the role of metatheoretical investigations, the kind of semantics, the use of definitions by abstraction, and the foundational or nonfoundational value of axiomatics. In the paper Peano's view is characterized as a form of structural algebraism, which differs from both the algebra of logic tradition using mathematical symbols to express logical calculi, and from Frege's logical investigation centered on the effort to understand the functional nature of predication. This talk will focus on Peano's semantics, which radically differs both from Frege's and from Padoa's semantics, as described in the 1902 paper on formal deductive systems. The influence of Leibniz can be seen both in the development of a kind of semantics that is neither representational nor conceptualist (symbols mean linguistic items of ordinary language), and in the consideration of symbolic signs as mirroring what they stand for. 
Friday January 29 9:15 am 
Matthew Jones Columbia University 
Learning to Love Opacity: Decision Trees and the Genealogy of the Algorithmic Black Box ABSTRACT: My talk concerns the genesis and the development of one of the foremost kinds of algorithms for supervised learning: decision trees. A series of researchers, each slightly askew to the dominant practices and epistemic virtues of their fields, came obliquely to trees in the 1970s: a datadriven statistician, a machine learning expert focused on large data sets, social scientists unhappy with multivariate statistics, a physicist interested mostly in computers who eventually was tenured in a statistics department. In case after case, the creators of different forms of trees deployed "applied" philosophies of science in critiquing contemporary practices, epistemic criteria and even promotion practices in academic disciplines. Faced with increasing amounts of highdimensional data, these authors time and again advocated a datafocused positivism. The history of trees does not cleanly divide into a theoretical and an applied stage; an academic and a commercial phase; a statistical and a computational stage; or even an algorithm design and an implementation stage. This history is iterative: the implementation of algorithms on actually existing computers with various limitations drives the development and transformation of the techniques. Before the very recent renaissance and current triumph of neural networks, decision trees were central to the transformation of artificial intelligence and machine learning of recent years: the shift in the central goal to a focus on prediction at the expense of concerns with human intelligibility, and of a shift from symbolic interpretation to potent but inscrutable blackboxes. Trees exploded in the late 1980s and 1990s as paragons of interpretable algorithms but developed in the late 1990s into a key example of powerful but opaque ensemble models, predictive but almost unknowable. We need to explain, rather than take as given, the shift in values to predictionto an instrumentalismcentral to the ethos and practice of the contemporary data sciences. Opacity needs its historyjust as transparency does. 
Friday February 19 9:15 am 
Vincent Peluce City University of New York 
The Perception of Time in Intuitionistic Arithmetic ABSTRACT: In L.E.J. Brouwer's first act of intuitionism, the subject's perception of time is put forth as the foundation on which arithmetic will be built. According to Brouwer, proper intuitionistic arithmetic, as with the rest of intuitionistic mathematics, is not tied to any particular formal system. When we try to axiomatically approximate an intuitionistic arithmetical system, we are faced with the problem of incorporating the subject and their perception into the axiom system itself. We discuss some unsatisfactory responses to this problem and then offer a solution. 
Friday February 26 10:00 am 
Laura Turner Monmouth University 
Women as data and as individuals: public dialogues on sexism in mathematics during the 1970s, 1980s, and 1990s ABSTRACT: The Association for Women in Mathematics (AWM) came of age in the 1980s, emerging by the early 1990s as a serious mathematics organization engaged in improving the status of women in mathematics. The same period also saw resistance to amelioratory measures and highprofile claims of sexism in mathematical practice. Through the public dialogues surrounding these episodes, in this talk we explore aspects of how the AWM and its members addressed the complex issue of sexism in mathematics in the contexts of reactions to affirmative action mandates; sexlinked theories of mathematical ability; antiscience gender essentialism; the Jenny Harrison tenure dispute; and the myth of objectivity in mathematical practice. As we will see, the deeplyheld and widespread belief in the inability of women to do mathematics, exacerbated by the myth of the canonical feminine woman and the understanding that mathematics as a discipline was essentially unbiased, simultaneously emphasized the importance of treating each woman mathematician as an individual, instead of as a representative of her sex, and left her vulnerable to the same discrimination the AWM sought to combat. In particular, while women as data held the power for reform, the perspectives of individual women were vulnerable to controversy. 
Friday April 2 10:00 am 
Dirk Schlimm McGill University 
'Calculus' as Method or 'Calculus' as Rules? Boole and Frege on the aims of a logical calculus ABSTRACT: It is an interesting fact about the history of modern logic, that both Boole and Frege (and others) gave solutions to the same logical problem in order to show the power of their respective notations. In this talk I will present the problem and discuss Boole and Frege's solutions. This will highlight an underappreciated aspect of Boole's work and of its difference with Frege's betterknown approach, which sheds light on the concepts of 'calculus' and 'mechanization' and on their history. While Frege's outlook has dominated philosophical thinking about logical symbolism, Boole's idea of an intrinsic link between a 'calculus' and a 'directive method' to solve problems presents a new perspective on the role of notations in logic and mathematics. (This talk is based on joint work with David Waszek.) 
Friday April 30 10:00 am 
Ellen Abrams Cornell University 
'Which shall be regarded as the best?': Axiom Systems and American Mathematics ABSTRACT: In the early twentieth century, researchers in the United States engaged with foundational studies in mathematics by building and evaluating axiom, otherwise known as postulate, systems. At the same time, their contemporaries were evaluating the meaning and politics of knowledge more broadly. In this talk, I explore the ways in which the study of postulates in the United States was tied to important Progressive Era questions about the nature of knowledge, the status of the knower, and the development of American Pragmatism. While most investigations of postulate studies have considered their implications within mathematical research and education, I look instead to the role of postulate studies in the professionalization of mathematics in the United States and to its cultural status more broadly. 
Friday May 14 10:00 am 
Jessica Carter Aarhus University 
The Philosophy of Mathematics as Practiced ABSTRACT: For some time now part of the philosophy (and related studies) of mathematics has been referred to as 'the philosophy of mathematical practice'. Although a number of contributions have offered characterisations of this new area, it is still not entirely clearly defined. Moreover, a number of similar terms, such as 'mathematical philosophy' and the 'philosophy of mathematics as practiced', have been used to refer to related areas within the philosophy of mathematics. The talk will present and discuss some of these characterisations. I shall start by giving a brief account of mathematical philosophy as coined by Russell and as part of scientific philosophy at the beginning of the 20th century. The second part will discuss two quite different accounts of the philosophy of mathematical practice, offered by Paolo Mancosu (2008) and JeanPaul Van Bendegem (2014). In the final part I propose a characterisation of the philosophy of mathematics as practiced as the part of philosophy that actively engages with the actual practice of mathematics  past and present. This means, for example, that philosophical questions are based on mathematics itself, or the activities of mathematicians. Furthermore, answers to philosophical questions may draw on mathematical tools or (historical) case studies. The 'activities of mathematicians' could, for example, include what mathematicians say or write about their work or how they go about finding new results. This part will propose some questions that could be relevant to contemporary mathematical practice and presents some illustrative examples. References. P. Mancosu 2008. Introduction. In Paolo Mancosu (ed.), The Philosophy of Mathematical Practice, pp. 121. Oxford: Oxford University Press.  J. P. Van Bendegem 2014. The impact of the philosophy of Mathematical Practice on the Philosophy of Mathematics. In Léna Soler, Sjoerd Zwart, Michael Lynch & Vincent IsraelJost (eds.), Science After the Practice Turn in the Philosophy, History, and Social Studies of Science pp. 215226. New York, USA: Routledge. 
Date  Speaker  Title and Abstract 
Friday September 18 9:00 am 
Professor Edray Goins Pomona College 
The Black Mathematician Chronicles: Our Quest to Update the MAD Pages ABSTRACT: In 1997, Scott Williams (SUNY Buffalo) founded the website "Mathematicians of the African Diaspora,'' which has since become widely known as the MAD Pages. Williams built the site over the course of 11 years, creating over 1,000 pages by himself as a personal labor of love. The site features more than 700 African Americans in mathematics, computer science, and physics as a way to showcase the intellectual prowess of those from the Diaspora. Soon after Williams retired in 2008, Edray Goins (Pomona College), Donald King (Northeastern University), Asamoah Nkwanta (Morgan State University), and Weaver (Varsity Software) have been working since 2015 to update the Pages. Edray Goins led an REU of eight undergraduates during the summer of 2020 to write more biographies for the new MAD Pages. In this talk, we discuss the results from Pomona Research in Mathematics Experience (PRiME), recalling some stories of the various biographies of previously unknown African American mathematical scientists, and reflecting on some of the challenges of running a math history REU. This project is funded by the National Science Foundation (DMS1560394). 
Friday October 2 9:00 am 
Lucas Bang Harvey Mudd College 
Abstraction and Computation : a history of the future ? ABSTRACT: Computer science is a direct descendent of logic and mathematics. The crucial notion that computer science inherits from math is that of abstraction. In this talk, we will analyze the tension created by abstraction as both a discourse of progress and a discourse of forgetting. While doing so, we will draw a thread through the history of logic and mathematics, phenomenological aspects of human experience, intractability results, and social injustices created by computational thinkers. Taken together, these observations compel us to take a long view of the future of computation as it is increasingly woven into the fabric of our everyday lives. 
Friday October 16 9:00 am 
Ekaterina Babintseva Harvey Mudd College 
Of Minds and Computers: Harnessing Mathematical Creativity ABSTRACT: In the mid20th century, "creative thinking" became a prominent category in American psychology and pedagogy. Advanced by cognitive psychologists as both a descriptive and a normative characteristic of the human self, the notion of creative thinking soon came to shape many midcentury debates in mathematics pedagogy. This paper traces the work of the educational psychologists and mathematicians at the University of Illinois who attempted to create special computer software that would teach creative thinking in mathematics. Developed for the University of Illinois' PLATO (Programmed Logic for Automated Teaching Operations) teaching computer, this software sought to introduce students to the intuitive aspect of mathematical thinking. Following this research through the 1960s1970s, this paper discusses how scientists used PLATO as a laboratory for testing midcentury theories of learning and approaches to math education. 
Friday October 30 9:00 am 
Nicolas Michel Utrecht University 
"Tictac geometry": The principle of correspondence from Chasles to Segre ABSTRACT: When, in 1864, the French geometer Michel Chasles (17931880) enunciated and proved what he called the 'principle of correspondence', he envisioned a grand design for this geometrical proposition: to "an infinity of questions", the principle would immediately provide the most general of answers without any computation whatsoever. In this principle, and especially in a textual apparatus thereto associated, Chasles thought he had obtained a device to guide the virtuous practice of geometrical research. In the following decades, this principle would become a staple in the teaching of several Italian mathematicians, most notably that of Corrado Segre (18631924). A mere glance at the notebooks in which Segre prepared his lectures, which are for the most part still extant, shows it occupied a fundamental role in his exposition of algebraic geometry. In a wellknown address given to his students at the university of Torino and published in 1891, Segre returned to Chasles's historical and epistemological writings on geometrical practice, embracing in the surface his promotion of the methods of modern geometry, but also rejecting the idea that their simplicity and immediateness was what made them so desirable. The principle of correspondence, then, featured prominently in Segre's address as one such geometrical method of great value, but whose infinite number of applications did not constitute something worthy of further research. In this talk, I shall compare the images of mathematical life given by Chasles and Segre and, using the principle of correspondence as a guiding thread, explore how these contrasting images resulted in different evaluations of what made the worth of this mathematical result, to what ends it should be used, and how it should be expressed on the page. 
Friday October 30 9:00 am 
Brigitte Stenhouse The Open University 
Translating Laplace's Mécanique Céleste in early 19thcentury Great Britain ABSTRACT: One of the key texts held up as an example of the inferiority of British mathematics in the early nineteenth century was PierreSimon Laplace's Traité de Mécanique Céleste. The work, published in five volumes between 1799 and 1825, was said to reduce the "whole theory of astronomy into one work" and to be incomprehensible to all but a handful of British readers (Playfair, 1808). By 1825 three partial English translations of Mécanique Céleste had been published, each with unique additions and amendments aiming to make the work accessible to a reader with a 'British' mathematical education. Nevertheless, in the late 1820s it was still felt that a good English translation was lacking, and two authors, the Scottish Mary Somerville and the American Nathaniel Bowditch, produced translations which differed widely in style both from each other and from their predecessors. By considering these five translations side by side, we will investigate how different perceived causes of the inferiority of British mathematics led to different methodologies of translation. 
Date  Speaker  Title and Abstract 
Monday September 16 3:00 pm 
Reading discussion

Charlotte Angas Scott (1893) "Paper Read Before the Mathematical Club at Girton College" The Girton Review 
Monday September 30 3:00 pm 
Della Dumbaugh University of Richmond 
Does one man make a team? Solomon Lefschetz as Editor of the Annals of Mathematics ABSTRACT: In late 1907, at the age of 23, the engineer Solomon Lefschetz lost his hands and forearms in a transformer accident. This unfortunate situation prompted Lefschetz to reorient his life towards a career in mathematics where he ultimately played a critical role in the American mathematical community in the twentieth century. He contributed significantly to algebraic topology, its applications to algebraic geometry, and the theory of nonlinear ordinary differential equations. He demonstrated his leadership as a faculty member at the University of Princeton and as President of the American Mathematical Society, among other influential positions. Lefschetz also served as the main editor for the Annals of Mathematics from 1928 to 1958, an important period for the journal. During this time, the Annals became an increasingly wellknown and respected journal. Its rise, in turn, stimulated American mathematics. This talk introduces Lefschetz and explores his role as editor of the Annals, including the papers that were published in the journal, the editorial boards, and the authors of the more than 2000 articles that appeared during his editorship. 
Monday October 14 3:00 pm 
Gabriel Greenberg University of California Los Angeles 
The Semantics of Iconic and Symbolic Representation ABSTRACT: C.S. Peirce distinguished two basic forms of representation: Symbolic representation is exemplified by words, mathematical and logical symbols, and sentences. Iconic representation is exemplified by 3D models, diagrams, maps, and pictures. Ferdinand de Saussure later suggested that symbols bear a merely "arbitrary" relation to their content, while pictures, at least, bear an "inner connection" to theirs. But what the iconic/symbolic distinction really comes to, or if such a distinction is warranted at all, remains hotly contested. In this talk, I will outline a theory of the distinction that is rooted in the distinctive style of semantics embodied by each form of representation. While much recent discussion has focused on the characteristic sign structure exploited in each type of representation, I will argue that we must look instead at the underlying semantic rules. I'll begin by spelling out formal semantics for extremely simple iconic and symbolic systems, which enlist only atomic signs. From there we'll look at schematic semantics for language, diagrams, and pictures. Generalizing from these cases, we see that symbolic representations involve what I will call direct assignments of meaning, while iconic representations involve mediated assignments of meaning. What mediates the relationship between sign and content in iconic representation is a general and natural rule. This analysis provides a formal and philosophical interpretation of Saussure's early intuition, and, I will argue, it explains a rich set of facts about iconicity and symbolism. 
Monday November 11 3:00 pm 
Jeffrey Oaks University of Indianapolis 
Arithmetical proofs in Arabic algebra ABSTRACT: Many people know about geometrical proofs for the solutions to quadratic equations in Arabic algebra, but few are aware that many Arabic authors, starting in the early 11th c., proved them instead in the context of arithmetic. The foundations and approaches of these proofs vary, and there are different motives at play for the switch. I review the arithmetical proofs in five Arabic authors: alKaraji (ca. 1010), Ibn alYasamin (d. 1204), Ibn alBanna' (late 13th c.), alFarisi (late 13th c.), and Ibn alHa'im (1387). 
Monday December 2 3:00 pm 
Michael Friedman Humboldt University Berlin 
On branch points and branch curves at the turn on the 19th century. ABSTRACT: How can one imagine the "bending" of algebraic, complex, projective curves or surfaces? The talk will aim to describe the different ways branch points and branch curves were researched and visualised at the turn of the 19th century. On the one hand, for branch points of complex curves one can find an abundance of visualisation techniques employed: Riemann, Neumann, Klein and von Dyck all promoted numerous forms of visualisation, either in form of twodimensional illustrations or threedimensional material models. On the other hand, For branch (and ramification) curves of complex surfaces, there were hardly any visual representations: When the Italian school of algebraic geometry studied branch curves systematically, at the beginning of the century, only partial illustrations can be seen (if at all), and branch curves were generally made a tool to research surfaces, rather than an object to be researched on its own. 
Date  Speaker  Title and Abstract 
Monday February 11 3:00 pm 
Vincenzo De Risi University of Paris 
Johann Lambert and Modern Axiomatics ABSTRACT: We will discuss the epistemological views of Johann Heinrich Lambert (17281777), as they are presented in his major philosophical works (Neues Organon, 1764, and Anlage zur Architectonik, 1771) and his groundbreaking essay on parallel lines (1766). Lambert had developed a very innovative conception of the nature of definitions, axioms, and their relations. He also developed at some length the idea of a mathematical model stemming from axioms, and had a structural approach to geometry. Lambert's ideas made possible the discovery of nonEuclidean geometry some 50 years later, and influenced the views on mathematics of Bernard Bolzano and others. We will explore these important developments in the history of epistemology that may well represent a watershed between ancient and modern axiomatics. 
Monday February 25 3:00 pm 
Deborah Kent Drake University 
"Glorious beyond description": The U.S. Naval Corps of Professors of Mathematics experience a total solar eclipse in 1869 Abstract: A largescale eclipse expedition in 1869 involved delegations from a curious mix of U.S. governmental offices. In particular, the U.S. Naval Corps of Professors of Mathematics left Washington, D.C., starting in late June to travel newlycompleted railroad to Iowa. The main scientific objectives of the parties were shaped by discoveries from a British eclipse expedition to India in 1868, existing technology, and an American scientific agenda. Although this eclipse event didn't generate the massive public migration of 1878 (or of 2017), the crowds in 1869 nonetheless required police intervention to avoid disrupting the scientific work. This work capitalized on the technologies of westward expansion  the telegraph and the railroad  utilized new developments in astrophotography, and deployed legions of government scientists in a coordinated effort. This great success and attendant publicity boosted American science and laid a foundation for the 1878 eclipse mania. 
Monday March 11 3:00 pm 
Reading discussion

Judy Grabiner (1986) "Computers and the Nature of Man: a historian's perspective on controversies about artificial intelligence" Bulletin of the American Mathematical Society 
Monday April 1 3:00 pm 
Jim Smith San Francisco State University 
Mario Pieri, Overloading, and Information Hiding in 1907 Abstract: Around 1900, constructing real from rational arithmetic required logic beyond the comfort level of many mathematicians. Dedekind's method, streamlined by Pasch, was rather simple: certain settheoretic operations on classes of rationals behaved like familiar operations on real numbers. Russell claimed they really were real numbers. Peano objected: writing x [proper subset of] y when one meant x < y could lead to logical errors. The geometer Mario Pieri worked alongside Peano to formulate the modern axiomatic method. In 1907 Pieri suggested a simple remedy for Peano's problem. It disappeared into history, but the problem reappeared when languages were devised for manipulating coded data of different but analogous types using different algorithms for analogous operations. Pieri's suggestion foreshadowed identifier overloading and information hiding in objectoriented programming. Those techniques enhance reliability by making programming more intuitive, fostering portability, and preventing disruption of lowlevel computations by higherlevel software errors. This interplay of disciplines would be interesting for historians of logic and computer science to explore. 
Monday April 22 3:00 pm 
Sean Walsh UCLA 
Infinitesimals, valued fields, and the orders of infinite smallness Abstract: In the 1960s, Abraham Robinson famously used model theory to defend the coherence of the calculus as based on infinitesimals. In Appendix 2 to his 1974 paper "Differentials, HigherOrder Differentials and the Derivative in the Leibnizian Calculus," Bos argued that Robinson's nonstandard analysis did not take into account the distinct orders of infinite smallness present in the infinitesimals in the historical calculus. In this talk, we describe how incorporating a valuation  in the sense of valued fields can help nonstandard analysis to overcome this deficit. After describing the proposal, we test it out on the historical cases from Euler and Bernoulli to which Bos drew attention. This is based on joint work with Tim Button, and in particular Sections 4.54.6 of the book Philosophy and Model Theory (Oxford University Press, 2018). 
Monday May 6 3:00 pm 
Reading discussion

TBD 
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 
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.