This one credit module is designed for students preparing to do advanced work in SCT during Fall 2020. In a series of workshops, students will work on formulating clear lines of inquiry and developing a research plan for their advanced work in SCT. Students will look at various examples of advanced work as presented by current seniors. Various SCT faculty members will present techniques for designing a research project. Students will meet with their section leaders to begin preparing for their senior work. Assessment will be based on the development of an individual research proposal with an initial bibliography.
Advanced mathematics is largely about logical argument, as much as it is computation or calculation. Over time, as each generation extended their ideas into new realms, they looked at the logical arguments of their predecessors and found that there were gaps, elisions, things that were not fully understood. One could imagine that this process might continue forever, but it does not: in the early twentieth century, a number of mathematicians (most notably Hilbert) completed the project of “hitting bedrock”, finding a clear demarcation line between certainty and uncertainty. This required a clarification about what mathematics and reasoning are about (“formalism”). These profound ideas deserve to be better known outside of professional mathematics. (Later, some mathematicians (notably Gödel) used this framework to show the limitations of the framework, in a different and more precise sense than before.) In this class we will go through the development of these ideas. No mathematical knowledge will be assumed.