Theories of Mind and Mathematics

The course website is this syllabus: http://math.northwestern.edu/~theojf/FreshmanSeminar2014/.

Any student with a documented disability needing accommodations is requested to speak directly to the Office of Services for Students with Disabilities (SSD; 847-467-5530) and to the instructor as early as possible in the quarter (preferably within the first two weeks of class). All discussions will remain confidential.

Table of Contents

• Guiding questions
• Methods and course requirements
  • A comment about grades and feedback
• Copyright notice and academic integrity
• Required readings
  • Active reading
  • Calendar, with links to readings and handouts
• Recommended readings
  • Writing manuals

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Guiding questions

What is it that mathematicians accomplish? Does mathematics accurately describe the world? Is mathematics consistent? Is science consistent? How can mere humans comprehend and communicate mathematics? How can mere humans comprehend human thought?

One feature of human consciousness is our ability to make choices and to create art; are these abilities consistent with a materialist scientific understanding of the universe? Another feature of human consciousness is our ability to think about ourselves; is a computer conscious if it can report on its own status? Mathematics can be used to study the structure of mathematics, and humans can think about the structure of thought; how are these forms of self-reference related, and does this relation shed light on the aforementioned questions?

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Methods and course requirements

We will spend the majority of class time discussing the assigned readings. Students should arrive having thought about the readings and prepared to share those thoughts; see Active reading. We will also devote some class time to peer review and other activities in small groups. Because in-class discussion is a core component of the class, daily attendence and daily participation are required. Except in rare instances, two or more absences will negatively impact your final grade.

This class will also demand a fair amount of work outside of class time, including reading and writing assignments. You should expect to devote six hours per week outside of class. If you find that you are working significantly more than this, let me know.

The requirements for the class fall into the following four categories:

1. Reading and discussion.
2. Four short (two to three pages) essays.
3. One long (eight to ten pages) final paper.
4. One short (five to seven minutes) presentation at the end of class symposium.

A comment about grades and feedback

As with, I hope, all of your classes, the primary purpose of this class is to learn. More specific purposes include: improvement as a writer; a deeper understanding of "understanding." Notably absent from this list is anything about grades. I think of grades as a necessary evil, and I find they can be a distraction from learning. As such, especially on the first few essays, I will provide detailed qualitative feedback, but I won't provide a "grade." As the course progresses, I will start to provide more "quantitative" feedback so that you have a sense of how I think you're doing. I realize that for some of you, not getting grades on the first few assignments will be a bit distressing; I hope that instead you can change any distress into relief that you can focus on improving as a writer and thinker rather than on improving some assigned letter.

Feedback, of course, is a two-way relationship. This course will be most successful if you provide me feedback throughout the quarter. Some topics I would like to solicit feedback on include the course's content, style, level, and workload. So feel free at any time to send me an email, stop by my office, or leave an anonymous note in my mailbox.

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Copyright notice and academic integrity

Articles linked below are the intellectual property of their authors, and in many cases the legal property of their publishers. They are made available here compliant with academic fair use guidelines. The copies posted here should not be downloaded or distributed for any reason except nonprofit academic scholarship; any other use may constitute a violation of federal copyright law.

Federal copyright law is only one of many principals guiding academic intellectual property. More damaging to academic communities is the misappropriation of others' ideas, otherwise known as plagiarism. (Such misappropriation is not necessarily illegal. For example, mathematical theorems and scientific results can be neither copyrighted nor patented, but claiming others' results as your own is dishonest, and is grounds for ejection from the academy. Similarly, it is immoral but not illegal to plagiarize the 1912 Encyclopedia Britannica.) For detailed discussion of academic integrity, see http://www.weinberg.northwestern.edu/handbook/integrity/.

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Required readings

This class will involve quite a lot of reading — in many weeks, over 100 pages. We will alternate between

• Douglas R. Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, Vintage Books Edition, Random House, 1980. (Henceforth abbreviated GEB.)

Active reading

Scholarly reading is more than words passing in front of eyes. It requires pauses to think, to organize, and to look things up. Always have paper and pen with you as you read. Moreover, please get into the habit of looking up everything mentioned in a reading: find every in-text citation in the bibliography, and encode the author and title; find every work of visual art on Google Images, and listen to a bit of every work of music on either Youtube or Naxos Music Library.

For each reading, keep track of those passages that:

1. Are the core of the reading.
2. You don't understand.
3. You do understand, but disagree with.
4. Are intriguing and easily overlooked.

Calendar, with links to readings and handouts

I will occasionally update this calendar with links to handouts and other in-class material. All readings should be completed by the start of class on the day listed. All essays are due by email by 9am on the day listed.

Tuesday, April 1 Introduction.
Handouts: Syllabus. The MU puzzle. Thinking about the MU puzzle.

Thursday, April 3 Read by the start of class:

• GEB pp. 3–74 (Introduction: A Music-Logical Offering – Chapter III: Figure and Ground).

Tuesday, April 8 First short essay due by email by 9am.
Read by the start of class:

• William P. Thurston, On proof and progress in mathematics, Bulletin of the American Mathematical Society, vol. 30, no. 2 (April 1994), pp. 161–177.

Thursday, April 10 Read by the start of class:

• GEB pp. 75–176 (Contracrostipunctus – Chapter VI: The Location of Meaning).

Tuesday, April 15 Second short essay due by email by 9am.
Read by the start of class:

• Roger Penrose, The Road to Reality: A complete guide to the laws of the universe, Chapter 1: The roots of science, pp. 7‐24, Knopf, 2006. The linked PDF also contains Chapters 2 and 3, which describe in more detail than we will have time for some of the mathematics related to the course.

Thursday, April 17 Read by the start of class:

• GEB pp. 177–272 (Chromatic Fantasy, And Feud – end of Part I).

Tuesday, April 22 Read by the start of class:

• Eugene Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Communications in Pure and Applied Mathematics, Vol. 13, No. 1 (February 1960), pp. 1–14.

Thursday, April 24 Read by the start of class:

• GEB pp. 275–336 (Prelude... – ... Ant Fugue).

Tuesday, April 29 Third short essay due by email by 9am.
Read by the start of class:

• Thomas Nagel, What is it like to be a bat?, The Philosophical Review , Vol. 83, No. 4 (October, 1974), pp. 435–450. Reprinted in Mortal Questions, pp. 165–180, Cambridge University Press, 1979.

Thursday, May 1 Read by the start of class:

• GEB pp. 337–390 (Chapter XI: Brains and Thoughts – Chapter XII: Minds and Thoughts).

Tuesday, May 6 Read by the start of class:

• Kathleen Akins, What is it like to be boring and myopic?, Dennet and His Critics: Demystifying Mind, pp. 124–160, Blackwell, 1993.

Thursday, May 8 Read by the start of class:

• GEB pp. 391–479 (Aria with Diverse Variations – Chapter XV: Jumping out of the System).

Tuesday, May 13 Fourth short essay due by email by 9am.
Read by the start of class:

• Roger Penrose, Mathematical intelligence, What is Intelligence, pp. 107–136, Press Syndicate of the University of Cambridge, 1994.

Thursday, May 15 Read by the start of class:

• GEB pp. 480–548 (Edifying Thoughts of a Tobacco Smoker – Chapter XVI: Self-Ref and Self-Rep).

Tuesday, May 20 Read by the start of class:

• GEB pp. 549–585 (The Magnificrab, Indeed – Chapter XVII: Church, Turing, Tarski, and Others).

• GEB pp. 586–632 (SHRDLU, Toy of Man's Designing – Chapter XVIII: Artificial Intelligence: Retrospects).

Thursday, May 22 Flip through, but don't feel obligated to read carefully:

• GEB pp. 633–680 (Contrafactus – Chapter XIX: Artificial Intelligence: Prospects).

• GEB pp. 681–742 (Sloth Canon – Six-Part Ricercar).

Tuesday, May 27 and Thursday, May 29 In-class symposium.

Tuesday, June 10 Final paper due via email by 9am.

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Recommended readings

The following list consists primarily of texts I considered, perhaps only briefly, for this class. (Some are too technical, some not quite on topic, etc.) The list is organized roughly with the order of material in the class, although no such organization can be exact. The list illustrates two good principles of scholarship: when studying an article, look at other pieces by the same author; when studying an article, look at other pieces in the same collection. Entries beginning with an asterisk (*) are on reserve at the main library.

• Douglas Hofstadter, I Am a Strange Loop, Basic Books, 2007. Hofstadter has written prolifically on topics related to this class, so look at some of his other writings as well.

• Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Knopf, 2006. We will read Chapter 1. The book is technical in places, but excellent for those interested in math and physics.
• Bertrand Russell, The study of mathematics, The New Quarterly, Vol. 1, pp. 31–44, November 1907.
• Jordan Ellenberg, The last great problem, The Believer, November 2003.
• Paul Lockhart, A mathematician's lament, 2002.
• Arthur Jaffe and Frank Quinn, "Theoretical mathematics": Toward a cultural synthesis of mathematics and theoretical physics, Bulletin of the American Mathematical Society, vol. 29, no. 1 (July 1993), pp. 1–13.
• Terence Tao, There’s more to mathematics than rigour and proofs, 2009.
• Eugenia Cheng, Mathematics, Morally, 2004.
• *Tom Stoppard, Arcadia, 1993.
• Richard P. Feynman, Six Easy Pieces, 1995.
• G. H. Hardy, Mathematical Proof, Mind, New Series, Vol. 38, No. 149 (Jan., 1929), pp. 1-25/
• G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1967.
• Philip Ball, Beauty ≠ Truth, aeon Magazine, May 19, 2014.

• R. P. Boas, Can we make mathematics intelligible?, The American Mathematical Monthly, Vol. 88, No. 10 (Dec., 1981), pp. 727-731.
• Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts, Mathematical Writing, 1987.
• Jean-Pierre Serre, How to write mathematics badly, 2012.
• *Leonard Gillman, Writing Mathematics Well: A Manual for Authors, Mathematical Association of America, 1987.
• *Mary-Claire van Leunen, A Handbook for Scholars, Knopf, 1978.

• *Jean Khalfa, ed., What is Intelligence, Cambridge University Press, 1994. In particular: Daniel Dennet, Language and intelligence, pp. 161–178. We will read the chapter by Penrose therein.
• *Bo Dahlbom, ed., Dennett and His Critics: Demystifying Mind, Blackwell, 1993. We will read the chapter by Akins therein.
• *Thomas Nagel, Mortal Questions, Cambridge University Press, 1979. We will read chapter 12 therein.
• Jennifer Freyd, Shareability: the social psychology of epistemology, Cognitive Science, 7, pp. 191-210, 1983.

• *Roger Penrose, The Emperor’s New Mind: Concerning computers, minds, and the laws of physics, Oxford University Press, 1989. In his words, "a semi-popular but lengthy account".
• *Roger Penrose, Shadows of the Mind, Oxford University Press, 1994. His most complete account.
• *Roger Penrose, et al. The Large, the Small, and the Human Mind, Cambridge University Press, 1995.
• *Roger Penrose, On the physics and mathematics of thought, The Universal Turing Machine: A Half-Century Survey, pp. 491–522, Oxford University Press, 1988. It's also worth looking at other articles in this volume.
• *Roger Penrose, Gödel, the mind, and the laws of Physics, Kurt Gödel and the Foundations of Mathematics, pp. 339–358, Cambridge University Press, 2011. The article by Hilary Putnam in the same volume (The Gödel theorem and human nature, pp. 325–338) is also worth looking at: Putnam argues against the positions held by Lucas and Penrose, and in favor of those held by Chomsky and Hofstadter.

• *Ernest Nagel and James R. Newman, Gödel's Proof, edited and with a new foreword by Douglas R. Hofstadter, New York University Press, 2001.
• Apostolos Doxiadis and Christos H. Papadimitriou, Logicomix: an epic search for truth, Bloomsbury, 2009. A graphic novel about the foundations of mathematics.
• *Tom Stoppard, Arcadia, 1993. A play about poetry and mathematics and chaos and academe.
• Ralph P. Boas, Rose Acacia, Lion Hunting and Other Mathematical Pursuits, Mathematical Association of America, 1995.
• John Berger, Ways of Seeing, Viking, 1973. A foundational piece about the relation between viewer and art.
• Scott Kim, Inversions: a catalog of calligraphic cartwheels, McGraw-Hill, 1981.
• Anything by Greg Egan, a science fiction writer who focuses on strong AI. For example, Dark Integers, Asimov's Science Fiction, 2007.
• Anything by Jorge Luis Borges.

Writing manuals

If you don't already own one, please procure a writing manual. The most standard writing manual, especially within the humanities, is:

• MLA Handbook for Writers of Research Papers, 7th edition, Modern Language Association, 2009.

• The Chicago Manual of Style, 16th edition, University of Chicago Press, 2010.

Many professional organizations publish their own writing manuals. Within the social sciences, the most popular is:

• Publication Manual of the American Psychological Association, 6th Edition, American Psychological Association, 2009.

• Leonard Gillman, Writing Mathematics Well: A Manual for Authors, Mathematical Association of America, 1987.

• Mary-Claire van Leunen, A Handbook for Scholars, Knopf, 1978.

There are also many good online resources, which might suffice (but I recommend owning a physical manual nevertheless). One particularly good one is:

• The Purdue OWL, Purdue U Writing Lab, 2010, https://owl.english.purdue.edu/owl/.

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