AK2021 Foundational Issues in Mathematics 7.5 credits
The subject matter of mathematics is -- or, at any rate, appears to be -- a universe of abstract objects such as numbers, functions, and sets. Such objects can hardly be located in space and time; nevertheless we can study them and get to know their properties. How is this possible, and how reliable is such knowledge? Do numbers even exist independently of us, or are they rather some kind of mental construction? Is mathematical truth the same thing as provability, or may there be mathematical facts beyond the scope of rational inquiry? And can we ever be confident that our mathematical theories are free of contradiction?
In the course, we will study how three schools of thought in the philosophy of mathematics -- logicism, intutionism, and finitism -- have approached these issues, from a conceptual/philosophical point of view as well as from a technical/mathematical one. We will also find reason to acquaint ourselves with the traditional set-theoretical construction of mathematical number systems.