Joel David Hamkins, Professor of Logic, Oxford University This lecture is based on chapter 1 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, https://mitpress.mit.edu/books/lectures-philosophy-mathematics. Lecture 1. Numbers Numbers are perhaps the essential mathematical idea, but what are numbers? There are many kinds of numbers—natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more—and these number systems provide a fruitful background for classical arguments on incommensurability and transcendentality, while setting the stage for discussions of platonism, logicism, the nature of abstraction, the significance of categoricity, and structuralism. See lecture course information, including the schedule of topics, at http://jdh.hamkins.org/lectures-on-the-philosophy-of-mathematics-oxford-mt20/.