In 2009, the Department of Philosophy at Carnegie Mellon University will hold a three-week summer school in logic and formal epistemology for promising undergraduates in philosophy, mathematics, computer science, linguistics, and other sciences. The goals are to

• introduce promising students to cross-disciplinary fields of research at an early stage in their career; and

• forge lasting links between the various disciplines.

The summer school will be held from Monday, June 8 to Friday, June 26, 2009. There will be morning and afternoon lectures and daily problem sessions, as well as planned outings and social events.

 

The summer school is free. That is, we will provide:

Categories and Structures

Monday, June 8 to Friday, June 12

Instructor: Steve Awodey

 

If you had to choose one word to characterize and summarize the modern approach common to mathematics, logic, theoretical computer science, linguistics, cognitive psychology, and related sciences, likely that word would be "structure". The emphasis on structural aspects in these and related disciplines has proven amazingly fruitful in recent decades, leading to impressive theoretical and practical advances. Much of this work has gone hand in hand with the development of the notion of abstract structure as an object of study in itself, notably in category theory. The theory of categories and functors, originally a branch of abstract algebra, is the mathematical explication and investigation of the notion of abstract structure; it has turned out to be a simple and powerful tool with far-reaching applications. We will introduce students to the basic concepts of category theory and show through numerous examples how these concepts can be used to capture and investigate structural features of the various sciences.

 

 

 

 

 

Decisions and Games

Monday, June 15 to Friday, June 19

Instructor: Teddy Seidenfeld

 

How to choose rationally? One perspective on a choice problem is to frame it with a single decision maker, you, and to consider what follows from the simple, prudential consideration that you should not pass up sure-gains. That is, you should not choose option 1 if you are sure to do better under option 2 regardless which state of nature obtains. Another perspective on choice is to frame a problem as involving multiple decision makers (who may have competing goals with your own), and to consider what follows from the equally simple prudential consideration that you should not permit the opponents to take advantage of you. That is, you should not play the game in a way that allows the opponents to make you worse off than you might be by playing differently, particularly when they benefit at your expense by doing so. These two perspectives on how to choose are not exclusive. A decision problem where you are uncertain about which state of nature obtains can be converted into a 2-person (zero-sum) game where you play against Nature. In this component of the summer school, we will examine decision theory and game theory in order to understand what each has to teach us about choosing rationally.

 

 

 

 

 

Logic and Formal Verification

Monday, June 22 to Friday, June 26

Instructor: Jeremy Avigad

 

Since the early twentieth century, it has been understood that mathematical arguments can be represented in formal axiomatic theories, at least in principle. Until recently, however, spelling out the basic logical inferences underlying even elementary mathematical arguments was too difficult to carry out in practice. With the advent of computational "proof assistants" the situation has changed, and such systems are now being used to verify the both the correctness of mathematical proofs and the correctness of hardware and software design. In this component of the summer school, we will consider some of the logical and computational mechanisms that have been developed to support these efforts, experiment with the Isabelle proof assistant, and explore surrounding epistemological issues.