The Abstract Mathematical Universe

What Are Numbers?

In this course, Professor Kevin Buzzard (Imperial College London) explores numbers, in particular the counting numbers and their abstract nature. In the first mini-lecture, we consider what ‘2 + 2 = 4’ really means and introduce the idea of an abstract ‘Mathematical Universe’ that can be used to describe real-world events. In the second mini-lecture, we explore the concept of infinity, looking specifically at Fermat’s Last Theorem and the Collatz Conjecture. In the third mini-lecture, we discuss finite number systems, looking particularly at the system of numbers modulo 10. In the fourth mini-lecture, we look at functions and how they are related to sequences. In the fifth mini-lecture, we see that while computers can master games with finite dimensions and a finite set of rules, such as chess, to an extent far beyond that of a human, humans are much better at working in the abstract and infinite Mathematical Universe than computers are. The sixth mini-lecture explores the principle of mathematical induction through the lens of Giuseppe Peano’s three rules that can be used to define the counting numbers. In the seventh mini-lecture, we use Peano’s rules to define addition and multiplication. In the eighth mini-lecture, we apply our understanding of Peano’s rules to our original question from the first mini-lecture, ‘What is ‘2 + 2 = 4?,’ and consider what advancements the development of computers might bring to the Mathematical Universe.

The Abstract Mathematical Universe

In this mini-lecture, we explore the motivations behind studying the abstract nature of mathematics. In particular, we consider: (i) what ‘2 + 2 = 4’ really means by looking at adding together familiar objects such as pens and elephants; (ii) the abstract noun ‘four’ and how mathematicians seek to explain the abstract nature of something being ‘four’; (iii) how adding one to four, and so on, produces an infinite collection of numbers; (iv) the Universe, which physicists can show to be finite, and how this is different from the abstract ‘Mathematical Universe’; (v) the number line, which is infinite and lives in the abstract Mathematical Universe of pure mathematics; and (vi) how working in the abstract Mathematical Universe can be used to model the real Universe, for example, in the work done by applied mathematicians studying sound waves, statisticians modelling viruses, and computer scientists creating unbreakable codes.