Math as Philosophy

3 Credits · Level: Undergraduate

Here is a stunning fact: the world can be understood mathematically. This fact underlies our success in science, computers, and even our private every-day reasoning processes. But how is this so? Why is mathematical thinking so astonishingly useful to help us understand the world around us? In this course, we examine the conceptual foundations of mathematics. No prior mathematical knowledge is required for this course. This is not a course about doing calculations. It is about abstract structures, and how we use such structure in our thinking. Throughout the course, we will ask the following questions. How do we organize things into collections, and networks? Does "+" (adding) mean what you think it means? Are there numbers that can't be enumerated (even by God), and if so, how do we even know about such spooky numbers? How do computers work, and how could it possibly all boil down to just ones and zeros? Are there math problems that can't be solved (even with an infinitely powerful computer)? And how do we even know how to figure out the answer to that? Finally, how do computer simulated neural networks "learn," and how much is it like human learning?