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.

After the crisis in mathematics ushered in by Russell's paradox, developments in formal logic offered renewed hope for foundational security. This course covers its major success stories—namely, the development of a formal notion of syntax and semantics for which the classical rules of inference are sound (everything they prove is true) and complete (everything true is provable). We also face new problems, like Skolem's paradox, which notes that many logical theories cannot single out their intended models. We'll conclude by turning to intuitionistic logic, exploring how these sorts of problems afford opportunities to characterize hidden structure in our reasoning.

What is philosophy, anyway? In this introductory course, we aim to figure that out by getting our hands dirty and doing it. We will explore traditional and contemporary responses to questions that have kept folks across generations up at night, including: Should you believe what you hear? What do we know (if anything)? What does it mean to have a mind? Are our best scientific theories true descriptions of the world? We conclude by reflecting on what philosophy has been in the past and what we want it to be going forward. 

Scientific theories are powerful! They make stunningly accurate predictions and offer satisfying explanations of phenomena around us. But what makes these theories so successful? How much do they really tell us about the world? And just what makes an explanation "scientific," anyhow? We will review historical and contemporary approaches to answering these questions, covering topics such as the reality of unobservable entities, how we choose between competing theories, how old theories reduce to new ones (if they do at all), and the nature of scientific explanation. This course is self-contained; no previous scientific or mathematical training 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.