Folks usually hear about Gödel’s famous incompleteness theorems—slogan-ized as “logic can’t prove its own consistency”—before they ever see them. Some bemoan the results as a fundamental limit on the certainty of our reasoning; others celebrate them as evidence that it’s impossible to reduce human thought to a computer program. We’ll take a close look and decide for ourselves. In addition to building the tools to state and prove Gödel’s theorems, we will look at another curious feature of first-order logic—non-standard models of arithmetic—and assess how second-order logic fares against analogs of Gödel’s arguments.

Scientific theories have enjoyed much success. They afford us tremendous power to predict and explain phenomena in the world around us. In light of this power, you might wonder why it is these theories are so successful. This question invariably leads to others. For instance: how much do our chosen theories tell us about the world—must the unseen entities referenced by scientific explanations exist? And just what counts as a “scientific explanation” anyhow? This course will equip you with the tools necessary to begin answering these questions. We will survey classic and contemporary debates in the philosophy of science, including: the reality of unobservable entities posited by theories; the nature of scientific explanation; how we choose between competing theories; and how we confirm existing theories. We will also consider applications to examples from the physical sciences. However, this course is self-contained. No previous familiarity with any particular physical or mathematical theory is required.

Quantum theory is one of the most successful scientific theories we have. Its predictive success is astonishing. Suppose your friend guesses the distance from New York to Los Angeles and turns out to be correct to within a hand's width; that's how accurately quantum mechanics predicts the maximum wavelength of light that will dislodge electrons from helium atoms. But as successful as quantum theory is, it is also one of the most puzzling theories we have: to try to make sense of it, physicists have talked of many worlds hidden from our own, and of cats in 'superpositions' of being alive and dead. This course focuses on three puzzles of the quantum. First, we will discuss ontology: we will investigate what sorts of descriptions of the world are consistent with quantum theory. Second, we will discuss logic: we will debate whether there can be more than one logic that's correct and whether quantum theory pushes us to revise our logical notions. Third, we will discuss probability: we will ask how we ought to interpret the probabilities that quantum theory assigns to measurement events. What is the world like, how should we reason about it, and what are probabilities? These questions, fascinating in their own right, are also essential to understanding and improving our scientific inquiries.