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Course Description

Summary

In the eighteenth and nineteenth centuries, mathematics underwent a vast expansion, into new, exciting, and increasingly counter-intuitive realms. The subject risked mystification and mutual incomprehensibility between experts in different sub-fields. In the first part of the twentieth century, a group of French mathematicians, under the pseudonym Bourbaki, undertook an ultimately successful program to use the foundation of set theory to put all of mathematics onto a common conceptual and logical foundation.

In this framework, every concept in mathematics is ultimately a set with some sort of structure (and structure can itself be defined in terms of sets). This class is an introduction to this framework.

The class has two intended audiences. First, the class will be an essential foundation for students wishing to take advanced mathematics or computer science classes. Every advanced math text is written, more or less explicitly, in this framework, and computer science reasoning is in large part based on these principles. In particular, this class is a prerequisite for Discrete Mathematics, which is recommended for all math and CS students. 

Second, the class is designed to be of interest more broadly, as part of the liberal arts. The Bourbaki framework is one of the great achievements of human thought, and, beyond that, set theory has its own beauty, in particular in making clear sense of the infinite, making it possible to reason precisely about it. Students will learn to make precise arguments, wrestle with concepts of infinity, and see a tour of other concepts within mathematics, such as topology.

Topics include:

• the language of set theory
• the formalization of relations and functions
• like versus equal: equivalence relations and quotient spaces
• examples of structures in mathematics: graphs (networks), modular arithmetic, groups, vector spaces, metric spaces
• the infinite: cardinal and ordinal numbers
• what is a real number? the formalization of the continuum
• proofs
• beyond the Bourbaki framework: categories

There are no prerequisites, not even from high school mathematics. Everything in the class is developed from scratch.

NOTE: This class is a reworking of MAT 2378 Logic and Proof: The Art of Mathematics and the Limits of Knowledge, focusing more deeply on fewer topics. There is substantial overlap between that course and this one. Therefore this class is not generally recommended for those who have taken MAT 2378. Contact the instructor for exceptions.