**Mathematics and the Humanities**

**Instructors**: Michael Harris (Mathematics) and Gayatri Chakravorty Spivak (English and Comparative Literature)

**Location**: Mathematics building, room 312, and by Zoom; information on Courseworks

**Office hours** Harris, Tuesday and Wednesday, 11-12 (starting September 15), or by appointment

**Schedule**: The class meets once weekly. Topics and readings for each session are listed below.

This course is being taught by two senior faculty members who are theorists and practitioners in disciplines as different as mathematics and the humanities at large. The instructors believe that in today's world, the different ways in which theoretical mathematics and the humanities mold the imaginations of students and scholars, should be brought together, so that the robust ethical imagination that is needed to combat the disintegration of our world can be produced. Except for the length of novels, the reading is no more than 100 pages a week.

Our general approach is to keep alive the disciplinary differences between literary/philosophical (humanities) reading and mathematical writing. Some preliminary questions we have considered are: the survival skills of the logicist school over against the Foundational Crisis of the early 20th century; by way of Wittgenstein and others, we ask, Are mathematical objects real? Or are they linguistic conventions? We will consider the literary/philosophical use of mathematics, often by imaginative analogy; and the role of the digital imagination in the humanities: Can so-called creative work as well as mathematics be written by machines? Guest faculty from other departments will teach with us to help students and instructors understand various topics. We will close with how a novel animates “science” in prose, stepping out of the silo of disciplinary mathematics to the arena where mathematics is considered a code-name for science: Christine Brooke-Rose’s novel *Subscript*.

http://www.columbia.edu/cu/bulletin/uwb/#/cu/bulletin/uwb/subj/MATH/GU4200-20203-001

**Structure of the course**

Students will write a one-page response paper to each week’s reading, to be submitted in the evening before class. This will enable us to shape class discussion to the students’ needs. In light of the general difficulty of applying the usual evaluation techniques in the context of the pandemic, we have decided to devote the final week and the final 30 minutes of the previous week to a presentation, by each student, of the most seminal question that they have learned to ask as a result our time together. The students have a choice of turning this into a research paper but this is not a requirement.

This is a restricted course. There will be no incompletes. Class participation 25%, response papers 55%, Presentation of seminal question 20%.

**Weekly syllabus**

The readings have been indicated by authors’ last names. Each week's lead instructor is indicated after the week's topic. Students will be sent detailed reading assignments most weeks. Scanned texts will be available at the Courseworks page under Files. Class details might change in response to pandemic developments.

9/9 Week 1: *Introduction* (Harris and Spivak)

Three texts are proposed as an introduction to the literary representation of the inner lives and goals of mathematicians: a scholarly biography of a 19th century mathematician, revolutionary, and literary intellectual who was also the first woman to hold a professorship in Europe in any subject; an extended essay on the analogies between the mathematical and poetic imaginations by one of the most influential living mathematicians; and an extended essay on the same theme by an author of fiction, in the form of a comparative biography of two siblings. A biography of a 21st century Russian mathematician committed to the ethics of the profession is optional reading.

TEXTS: Koblitz (1993) ; Gessen (2009); Olsson (2019); Mazur (2004), Chapter 1

9/16 *Week 2: Deconstruction* (Spivak)

In his *first* publication, the young Jacques Derrida (Algerian Jews are granted French citizenship only in 1961) reads “The Origin of Geometry,” the last fragment of Edmund Husserl’s *last* (and unfinished) book on *The Crisis of European Knowledge [Wissenschaft]*, claiming Europe as a Jew in Nazi Germany, facing a rejection by his best student Martin Heidegger. The essay investigates how questions of history are being negotiated – in the name of Galileo -- at the same time as the “purity” of geometry. An example of Humanities reading. The responsibility of translation – German to French to English, will be considered where possible.

TEXTS: Derrida/Husserl (1989)

9/23 *Week 3: Logicism* (Harris)

Among philosophers in the English-speaking world, the most influential approaches to mathematics, even today, hold that its aim is to produce sentences in an unambiguous language, or strings of symbols, whose validity depends only on its internal relations (“syntax”) and that leaves no room for empirical verification or interpretation. The classic formulations of this position were the logicism of Frege and Russell and David Hilbert’s formalism. Gödel’s incompleteness theorem showed the limitations of the logicist and formalist programs but they remain the starting point for most reflections about truth in mathematics and are indispensable in understanding the aims of artificial intelligence. The class will read excerpts from Frege and other pioneers of mathematical logic, and the graphic novel *Logicomix* by Doxiadis and Papadimitriou.

We will also study Turing’s highly influential essay on mechanical proof, which examines the implications of a mathematical practice in the absence of human intervention, and explain the basic idea of Gödel's proof.

(discussion with Alma Steingart as guest)

TEXTS: Frege (1991) and van Heijenoort (1967), selections;

Doxiadis and Papadimitriou (2009); Turing (1948); Wolchover (2020)

Engelhardt (2018), pp. 27-33, pp. 70-85;

Wittgenstein (1922), sections 6.1, 6.2. Optional readings TBD.

9/30 *Week 4: Wittgenstein Questions the Discipline * (Spivak)

This week we will consider Wittgenstein’s and Quine’s treatment of mathematical discourse as it relates to the architectonics of their general concerns. The *Tractatus* will have been discussed by Michael Harris in the previous session and we must keep that discussion in mind. In addition to texts by Wittgenstein (*Remarks on Foundations of Mathematics* and *Philosophical Investigations*) and some seminar notes from Quine translated by Derrida, and Nelson Goodman, we will dip into Jean-François Lyotard, *The Differend* (p. 136-8), Saul Kripke’s *Wittgenstein: On Rules and Private Language* (p.141-3). Terry Eagleton, “Wittgenstein’s Friends,” and Henry Staten, *Wittgenstein and Derrida*, p. 64-108 will be optional.

TEXTS: Wittgenstein (1953), p. 1-21; Quine (1962); Goodman (1983), Chapter 3; Lyotard (1983), Kripke (1982), Eagleton (1982), Staten (1984).

Optional readings, Wittgenstein (1922), Chapter 3; Wittgenstein (1983), TBD.

10/7 *Week 5: Eternity vs. Collective Practice* (Harris)

Mathematics is often presented as the only human practice that provides access to timeless and universal truths; it is claimed that mathematical reasoning is valid not only everywhere on earth but can even provide a means of communication with extraterrestrial beings. Imre Lakatos’s classic *Proofs and Refutations* argues instead for a vision of mathematics as the theater of constant negotiation over meaning. Badiou's philosophy does not deny Lakatos's analysis but still points to mathematics as a human activity capable of reaching for eternal validity in the form of "works" (oeuvres). Harris's essay of 2019 uses Andrew Wiles's solution of the 350-year old problem of Fermat to illustrate a more nuanced view of mathematical practice.

TEXTS: harris-2019, highlighted parts;

Lakatos, (2015), available online at https://clio.columbia.edu/catalog/11744050?counter=3 read at least the Preface by Paolo Mancosu, the Author's Introduction, and Chapter 1, to p. 45;

Badiou, *L'immanence des vérités* (text available on Coursework files) pp. 582-6, 588-91; Badiou (2008), chapters 2 and 4; Optional: Badiou, (2016), Chapters 1, 2

10/14 *Week 6: Mathematics in the Humanities* (Plotnitsky/Spivak)

Literary and philosophical figures have assumed the mathematical in various ways. Our examples are Edgar Allan Poe’s “The Purloined Letter,” and the use of Mathematics by the philosophical psychoanalyst Jacques Lacan, including his disagreement with the sentiments expressed in Poe’s story. We have invited Arkady Plotnitsky to lead us in this discussion by way of his essay on Lacan and mathematics. Tom Stoppard's *Arcadia*, tracing modernism back to the Enlightenment and placing mathematics at the center of the intellectual transformations of the 18th century, will be optional reading. When reading *Arcadia*, students are encouraged rather than consider the massive body of criticism of this point of view, the light touch commentary to be found in, for example, W.B. Yeats’s “Fragments I” in *The Tower *and the lines beginning “Measurement began our might” in the poem “Under Ben Bulben.”

TEXTS: Poe (1845); Lacan (1952-1960); Stoppard (1993); Plotnitsky, “On Lacan and Mathematics” (2009); Spivak, “Crimes of Identity” (2014), selection; W.B. Yeats, selections; Engelhardt (2018).

10/21 *Week 7: Mathematical Genres* (Harris)

In contrast to the logicist vision of mathematics as strings of symbols dependent on no external reference, Ording's text explores the multiplicity of mathematical genres by showing how a single banal mathematical observation can be used to illustrate a variety of wildly different discourses, each inscribed in its own social and literary context. Harris's essay focuses on a single genre — the mathematical "trick" — usually considered as marginal, and asks whether the demarcation of the central concerns of mathematics from its margins mirrors the mid 20th century opposition of high and low culture. If time permits, we will also attempt an analysis of the narrative structure of a short proof without any reference to content, following Harris (2012).

TEXTS: Ording (2019), proofs 3, 7, 10, 28, 30, 45, 52, 81, 84, 93; Harris (2015), chapter 8 “The Science of Tricks” online through Columbia libraries, pp. 1-17.

Optional reading: André (2017), Harris (2012), 162-168

10/28 *Week 8: Computational Humanities* (Tenen/Spivak)

Mathematical methods, including computer science and statistics, have become increasingly influential in the humanistic disciplines, including literary studies. This week's readings review some of the arguments for and against what has come to be known as computational humanities. The discussion will focus on the limitations and the scope of this work and its relation to the tradition of close engagement with texts. We have invited Columbia’s own Dennis Tenen, a rising star in Computational Humanities, to help teach the class.

TEXTS: Truitt (2016), selections; Da (2019); Tenen, “Distributed Agency in the Novel” (2018); Piper (2019).

11/4 *Week 9: Can Mathematics Be Done by Machine?* (Harris)

Computer scientists have long pursued the objective of a mechanization of mathematical proof, with some going so far as to claim that human mathematicians will be obsolete by the end (or the middle) of the present century. What mechanization would entail has been analyzed by philosophers, sociologists, and historians, as well as by mathematicians and computer scientists. This is the first of two classes devoted to mechanization and its implications for mathematics. The discussion continues Week 12, when we explore the profound implications that the mechanization of mathematics has for humanistic disciplines as well.

(with the participation of Kevin Buzzard of Imperial College, London)

TEXTS: Harris (2012); selections from MacKenzie (2001), Dick (2015), and conference videos to be announced.

Optional reading: Avigad (2018), Daston (2018).

11/11 *Week 10: The Mathematical Imagination outside of Mathematics* (Spivak)

In addition to an extended discussion of Christine Brooke Rose’s novel *Subscript*, this week will provide an opportunity to explore the influence of mathematical structuralism, as exemplified by the Bourbaki group, both outside mathematics in French structuralism and within mathematics, especially with the work of Alexander Grothendieck.

(possible discussion with Stuart Firestein as guest)

TEXT: Brooke-Rose (1999), Bourbaki (1950)

11/18 *Week 11: Bringing It Back to Method* (Harris)

We examine the foundations of reasoning as conceived by two ancient mathematical traditions: Greek geometry, as recorded in Euclid's Elements, and the Nyaya school of classical Indian logic. A key text for the form is Reviel Netz's analysis of the importance of diagrams as tools of geometric reasoning.

`Nyaya and Mathematics (with Arindam Chakrabarti as guest) `

Link to videos for Week 11)

TEXTS: Netz (1999), Robson (2008), Graham/Kantor (2009), excerpts; Høyrup (1989).

12/2 *Week 12: Can Mathematics Be Done by Machine, Part 2: Implications for Humanities* (Harris and Spivak)

This is the continuation of the discussions of Weeks 8 and 9. Mechanization of mathematics and computational humanities are similar responses to very similar epistemological concerns. In both cases, the aim is to attain optimally objective knowledge by relying on techniques designed to neutralize possible subjective effects. In the European Humanities, the longstanding mechanical intuition of the subject in Descartes and German classical philosophy has been hard to grasp in English translation since the seventeenth century (see Etienne Balibar, *Citizen Subject*). This tradition is there in Freud and Lacan, again not grasped by U.S. psychologism. Spivak will add an article and an excerpt (a Nachlasse), to emphasize that this is a caution against the possibility of the objective as such.

The class will end with 3 students presenting 1 seminal question.

TEXTS: Balibar (2017), p.56-73; Porter (1996); Duncan (1973); Spivak, “Why Study the Past?” (2012); “Outside in the Funding Machine” (2018).

12/9 *Week 13: Colloquium *

Each of 15 students is responsible for one seminal question

Link to page with presentations.