What is Modern Mathematics? (For The Non-Mathematician)
Despite being responsible for much of the modern world, nearly no one understands what modern mathematics is, and why it is so successful. There’s also some advice for how to learn and get into modern mathematics, and how to study the Cambridge Maths Tripos from the comfort of your own home!
(disclaimer: blackboards are now rarely used). You’ll also have to excuse my quaint English idioms, such as calling mathematics maths, rather than math, and for me drinking tea while writing the article.
What Mathematics used to be
Before we begin with modern mathematics, we need to understand traditional mathematics. This was primarily geometrical (i.e. visual reasoning) and algebra of the real numbers. This is the sort of thing you encountered as a young child: circles, triangles, addition, subtraction, equations and their roots.
You can do a lot of maths with this sort of thing. Newton originally built calculus with geometry; most competition mathematics, such as the International Mathematics Olympiad, predominantly uses this.
Now sometimes this maths was axiomatic. That means there are a certain number of assumed things, from which you build the rest. Maybe you’ve heard of Euclid’s postulates? These were the axioms for geometry. Sometimes…
