Theories of Mind and Mathematics

Tuesdays and Thursdays, 2:00–3:30, University Library 3322
Instructor: Theo Johnson-Freyd. Office Hours: MW 1–3 and by appointment, in Lunt 308.
Email: Website:

The course website is this syllabus:

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

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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.
Details will be discussed in class.

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

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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

and individual essays (listed on the calendar below). GEB is a long book, and touches on many topics. A non-exhaustive list includes: number theory; logic; computer science; artificial intelligence; cognitive psychology; modern art; Baroque music; Zen Buddhism; quantum physics; molecular biology. In places it is profound, in places it is pedantic, and in places it is quite dated.

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.
Bring your notes with you to class, and be prepared to discuss your selections.

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:

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

Handout: Thinking about thinking about the MU puzzle.

Thursday, April 10 Read by the start of class:

Handout: First essay introductions.

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

Handout: Kunst der Lesethemen.

Thursday, April 17 Read by the start of class:

Tuesday, April 22 Read by the start of class:

Handouts: Grading Rubric. Four example MathReviews: MR1249357, MR1202292, MR530196, MR2360307.

Thursday, April 24 Read by the start of class:

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

Handouts: Peer Review Worksheet, Summary and Review.
Strongly recommended: Master’s Recital: Lucien Werner, conductor, 6:00pm–7:30pm, Lutkin Hall. Three Bach concertos for trumpet, violin, and cello. Admission is free.

Thursday, May 1 Read by the start of class:

Handout: Counterpoint.

Tuesday, May 6 Read by the start of class:

Thursday, May 8 Read by the start of class:

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

Handout: Eliezer Yudkowski, "(The cartoon guide to) Löb's theorem".

Thursday, May 15 Read by the start of class:

Tuesday, May 20 Read by the start of class:

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

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

Read by the start of class:

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

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

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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.

Texts on the nature of physics and mathematics: On writing mathematics: The theory of mind and cognitive psychology: Roger Penrose is one of the 20th century's great mathematicians and theoretical physicists, and he has written extensively on the relations between Gödel's theorem, artificial intelligence, and cutting-edge physics. He comes to a markedly different conclusion than Hofstadter does. Most of the items below are quite technical, but worth a look: Other:

Writing manuals

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

Another very popular option is: These manuals are regularly updated.

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

Within mathematics there are two professional organizations, the Mathematical Association of America, and the American Mathematical Society. The former has published a writing manual, which is available on reserve at the main library: The other main writing manual for mathematics, which is more detailed than Gillman's pamphlet and is also on reserve, is: Note that whereas the APA regularly updates their manual, the AMS does not.

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

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