This course weaves a number of threads together to examine the nature of logic and computation, as well as how these relate to human reasoning and understanding. Essentially, we will be examining what symbols are, and from where they get meaning. In many ways, this is a highly philosophical and theoretical enterprise, but it is relevant to exploring the ultimate limitations and possibilities of computer systems, which are essentially formal symbol-processing machines. Toward the end of the course, we will discuss some computer systems that perform various types of "human-like reasoning".
A formal system can be considered as an unambiguous and "content-independent" way of manipulating and interpreting symbols. Some of the most sophisticated formal systems are various types of logic,which represent some of the major efforts to come up with a scientific ways of investigating how humans (and possibly machines) can reason. From this perspective, reasoning involves the manipulation of symbols that actually "mean something useful". That is, they correspond to objects, relations, and concepts that prove useful in the real world.
One of the most interesting and paradoxical aspects of reasoning systems (whether formal or not) arises when they start reasoning about themselves. It turns out that the question of self-reference poses some interesting puzzles, but also provides some insights into the kinds of things we might hope to accomplish by studying formal reasoning systems.
The text for this class is Hofstadter's "Goedel, Escher, Bach: An Eternal Golden Braid". In this book, Hostadter investigates the questions of symbols and meaning, and also has a lot of fun with word-play, paradoxes, and relating different kinds of concepts (such as logic, art, and music) to each other. We will have some some fun with these concepts, too. But ultimately you will be learning about how think about and how to think with formal systems such as logic and computers. You will learn about how symbols produce meanings constrained both by their structure and the structure of whoever or whatever is interpreting the symbols. You will learn that there are more things you can do with formal systems than you may have previously thought. And you will learn there are some important things that can provably not be accomplished by formal systems. The ultimate focus of the course will be on logic, computation, and reasoning, but there will be many diversions into how these topics relate to music and art.
Following is a more specific listing of some of the topics we will cover over the course of the semester:
Because of this structure, it is essential that you come to class each day having completed the reading and prepared to ask questions and engage in discussion. Some of the discussions, particularly of the details of logic and formal reasoning, will get quite tricky, and you need to stay on top of these. I will also be providing you with occasional assignments to give you a more concrete understanding of some of the concepts.
In addition to doing the reading for every class, you will have the assignment each class session of bringing at least three written questions. This is a way for me to keep tabs on your progress with the reading, but also provides an important way to stimulate discussion. Please use your list of questions to bring up topics in class that you would like to discuss, then turn them in to me at the end of each class.
I will generally do my grading on a 100 point scale. Points do not map directly to letter grades until everything is tallied at the end of the semester. However, as a rough guide, I generally consider 90 points or more to be ``A'' work, 80 points or more to be ``B'' work, 70 points or more to be ``C'' work, and 60 points or more to be ``D'' work.
I also hold the philosophy that A's are reserved for those who demonstrate a real mastery of the subject matter, in addition to performing superior work on all of the assignments. Doing an adequate job on all of the assignments alone will likely earn you an A-minus. To receive an exceptional grade, you should do exceptional work, finding ways to go beyond my expectations for the basic project requirements. For every assignment, I will be inclined to reward extra effort, innovation, industriousness, etc.