Claremont History and Philosophy of Mathematics Seminar

Founded 2016


The Claremont History and Philosophy of Mathematics Seminar meets on Mondays from 3:00-4:00 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, 


Fall 2017 Schedule


Date Speaker Title and Abstract

Monday

October 2

3:00 pm

Erich Reck

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

John Stillwell

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. 308--329

Monday

November 20

3:00 pm

Abram Kaplan

Columbia University 

 

The geometry of situation in seventeenth-century 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

Theodora Dryer

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 twentieth-century 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 model-based 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 computer-based algorithmic cultures that remain the dominant order of today.





Spring 2017 Schedule


Spring 2017 


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. 62--89 

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 (1560-1621) 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 "Holmes-Moriarty Paradox." In "The Final Problem," Holmes and Moriarty allegedly arrive at their untimely deaths, with no witnesses, at Reichenbach Falls. The "Holmes-Moriarty 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

Understanding complex physical systems through the use of simulations often takes on a narrative character. That is, scientists using simulations seek an understanding of processes occurring in time by generating them from a dynamic model, thereby producing something like a historical narrative. This talk focuses on simulations of the Diels-Alder reaction, which is widely used in organic chemistry. It calls on several well-known works on historical narrative to draw out the ways in which use of these simulations mirrors aspects of narrative understanding: Gallie for "followability" and "contingency"; Mink for "synoptic judgment"; Ricoeur for "temporal dialectic"; and Hawthorn for a related dialectic of the "actual and the possible". Through these reflections on narrative, the talk aims for a better grasp of the role that temporal development sometimes plays in understanding physical processes and of how considerations of possibility enhance that understanding.





Fall 2016 Schedule


Fall 2016 


Date Speaker Title and Abstract

Monday

September 19

3:00 pm

Jed Buchwald

CalTech 

The Newtonian Origins of Experimental Error

In the mid-1750s the English mathematician Thomas Simpson tried to convince astronomers that it was a good idea to average multiple measurements. He had much work to do, because neither astronomers nor physicists were in the habit of combining multiple measurements to produce a best final value. How then did experimenters and observers work with discrepant data before statistical methods became common at the beginning of the 19th century? We will first tour the worlds of Tycho Brahe, Robert Hooke, René Descartes and Johannes Hevelius to see how they worked with data. And then we turn to the young Isaac Newton, who developed an altogether novel way with measurements, the very way that became ever after the foundation of experimental method, for it was Newton who first decided that bad numbers could be put together to generate good ones.

Monday

October 10

3:00 pm

Jim Hoste

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 non-alternating 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?

The consensus for the last century or so has been that diagrammatic proofs are not genuine proofs. Recent philosophical work, however, has shown that (at least in some circumstances) diagrams can be perfectly rigorous. The implication of this work is that, if diagrammatic reasoning in a particular field is illegitimate, it must be so for local reasons, not because of some in-principle illegitimacy of diagrammatic reasoning in general. In this talk, I try to identify some of the reasons why geometers in particular began to reject diagrammatic proofs. I argue that the reasons often cited nowadays -- that diagrams illicitly infer from a particular to all cases, or can't handle analytic notions like continuity -- played little role in this development. I highlight one very significant (but rarely discussed) flaw in diagrammatic reasoning: diagrammatic methods don't allow for fully general proofs of theorems. I explain this objection (which goes back to Descartes), and how Poncelet and his school developed around 1820 new diagrammatic methods to meet this objection. As I explain, these new methods required a kind of diagrammatic reasoning that is fundamentally different from the now well-known diagrammatic method from Euclid's Elements. And, as I show (using the case of synthetic treatments of the duals of curves of degrees higher than 2), it eventually became clear that this method does not work. Truly general results in "modern" geometry could not be proven diagrammatically.

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

I examine Euclid's fifth postulate---known also as the parallel postulate---in the light of E, a recent formalization of Euclid's diagrammatic method of proof in the Elements. In doing so, I consider how Euclid's approach to geometric lines as unbounded objects can be understood as constructive in a modern foundational sense.





Spring 2016 Schedule


Spring 2016 


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. 153--193 

Friday

April 29

Tom Archibald

Simon Fraser University 

Abstract mathematics and axiomatization in mid-20th c. America: Halmos, Dieudonné, and Measure.

"It is always rash to make predictions, but the reviewer cannot help thinking that, despite its intrinsic merits, this book, as well as its brethren of the same tendency, will in a few years have joined many another obsolete theory on the shelves of the Old Curiosity Shop of mathematics." Jean Dieudonné, review of K. Mayhofer's "Inhalt und Mass'', 1953.
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

In the mid-1750s the English mathematician Thomas Simpson tried to convince astronomers that it was a good idea to average multiple measurements. He had much work to do, because neither astronomers nor physicists were in the habit of combining multiple measurements to produce a best final value. How then did experimenters and observers work with discrepant data before statistical methods became common at the beginning of the 19th century? We will first tour the worlds of Tycho Brahe, Robert Hooke, René Descartes and Johannes Hevelius to see how they worked with data. And then we turn to the young Isaac Newton, who developed an altogether novel way with measurements, the very way that became ever after the foundation of experimental method, for it was Newton who first decided that bad numbers could be put together to generate good ones.