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After completing this course students are able to:
- Discuss critically, and situate in scholarly and historical context, reflections on the nature of mathematics.
- Draw on a range of examples to discuss the role of mathematics in society, culture, and human thought.
- Read simple mathematical proofs and explain the ideas and methodologies involved.
- Reason rigorously and conceptually about fundamental notions in a simple geometric setting.
- Relate fundamental aspects of mathematical reasoning to its broader meaning and purpose.
- Discuss the nature of mathematical knowledge and axiomatic-deductive systems, especially on the basis of key foundational concepts of Euclidean and non-Euclidean geometry.
Relationship between Assignments and Course Goals
class participation: 1, 2, 3, 5, 6
assignments: 3, 4, 5, 6
midterm exam: 1, 2, 5, 6
Final exam: 1, 2, 5, 6
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In this course we take a big-picture look at the nature of mathematics and its role in human thought, emphasizing its interactions with society, history, philosophy, science, and culture. The course involves studying key mathematical arguments in some detail, but our goal is not to develop a repertoire of technical skills. Instead, we study accessible mathematical topics specifically selected for their rich interconnections with cultural context, and integrate our mathematical work with reflections on its broader meaning and implications.
Instead of the drill and practice problems of a traditional mathematics class, we approach mathematics through seminar discussions, hands-on activities, and readings connecting it to broader issues. We thus analyze a selection of emblematic and important mathematical proofs and use them as a platform for reflecting on the nature of mathematics. In parallel, we read excerpts from seminal historical texts across the ages as well as modern scholarship from a wide range of academic disciplines that shed light on the interplay between mathematics and its societal and intellectual context. We focus especially on geometry, from the origins of mathematical reasoning in early civilizations, to Euclid’s Elements that was the gold standard of exact reasoning for millennia and the model for countless philosophical systems, to the projective geometry of Renaissance art, to the more modern non-Euclidean geometry that overturned conventional wisdom about the nature of human spatial perception and the shape of space.
Format
We interleave mathematical topics with seminar discussions of a rich array of short readings that connect the material to a broader cultural, philosophical, and historical context. We study a selection of mathematical proofs and topics drawn from geometry with an emphasis on conceptual understanding. To this end we supplement textual mathematical sources with physical models and hands-on activities that allow us to experience and explore geometry in a concrete way.
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