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 it was mostly intuitionistic. For instance, as a child you don’t learn geometry as a set of axioms and subsequent deductions! Socrates argues as much when teaching a boy how to deduce geometrical truths.
The important thing is that these areas of maths are tied to things which are (fairly) concrete: numbers and shapes. Numbers are nicely represented by a line anyway, so this is conceptually very grounded.
Hilbert led the drive in the 20th century to formalise mathematics as a set of axioms and then deductions. He succeeded, against stiff opposition. (You may laugh, but Hilbert viewed his opponents in the same way the Avengers viewed Thanos. Perhaps this is what the MCU was inspired by?)
Hilbert’s opponents thought the new axioms weren’t obviously true , which is a fair point— geometry is intuitive, set theoretic axioms are not! The intuitionists preferred constructionist proofs. This is where to argue that a certain object exists, you show how to construct it. While this is quite enlightening, it is normally easier to prove that if the thing did not exist, this would be a contradiction.
It’s also worth noting that mathematics has always contained the seeds of this. Take the infinite parallel lines of Euclid’s geometry. They’re creations of our mind rather than physical objects. Is an infinite parallel line much more abstract than the constructions of sets?
A technical aside on the Law of the Excluded Middle (optional)
The intuitionists denied the law of the excluded middle. They thought that the statement (not (not P(x))) is a different statement to P(x). A proof by contradiction shows that the statement (not (not P(x))) is true by starting from the assumption of (not P(x)) and then showing this to be false.
This may seem pedantic, but logic actually had to be reformulated to be used in quantum physics, where the law of the excluded middle does not hold! (This is called ‘Quantum Logic’). Moreover, there are statements in mathematics which have been proven to be answerable as both true and false (see Kleene, Introduction to Mathematical Logic for the details), which goes against the law of the excluded middle.
In contrast, while supposing the existence of something which turns out to be impossible and then proving this is very very abstract, actually demonstrating a way to construct something’s existence is much more concrete — but harder.
The underlying issue is that mathematics is based off many preconceived notions of what truth is, which seem to lack the precise nature needed to understand their mathematical implications. While in the world we see and act in things do appear to be true or false — hence Aristotle’s creation of the Law of the Excluded Middle — the applicability of our woolly intuition may not be suitable for the precise language of mathematics. But how can we sensibly question a concept like truth, whose discussion requires understanding the concept?
Abstract, generalise, formalise
Modern mathematics approaches things differently.
It primarily studies structures whose interactions have certain patterns. For instance, it turns out the geometric properties needed to build calculus can be boiled down to: (a) a metric and (b) a space with certain properties. On reflection, this makes sense. While we notion of shortest distance on a ruler makes sense, a different notion might eb needed for a measure of distance in friend networks, or in similarities between lines of code. Abstracting so that many results are common across all these diverse areas kills many birds with one stone.
(It turns out that many properties of calculus in spaces with distances can be abstracted further to the idea of a space matched with sets called the ‘open sets’. This last area is called topology, and the idea is that the key part of lots of analysis of spaces isn’t to do with the distances, but to do with how certain sets behave under operations. For instance, the idea in calculus of small changes can be captured in how functions map these open sets.)
By abstracting, generalising and formalising several things are achieved:
(1) Results are broad.
Thus, calculus designed for the real number line can be applied to the integers under the p-adic metric (yes — you can perform calculus on integers with the right distance function. It’s just that this distance function isn’t very sensible for most applications!). Concepts used in 3D space can be used for analysis of probability, functions , statistics, or in higher dimensional space (e.g. analysing 10 dimensional space is not so much different from 3 dimensional space for a mathematician)
(2) Different areas of mathematics can be linked up.
By classifying things very carefully with the right notation, you can draw on results from across mathematics. Thus proofs about prime numbers often involve arguments from topology, functional analysis and calculus. Clearly making such wide connections is beyond humans without extremely astute classification of properties so we can apply results from elsewhere. (A common tactic for this is to find a function which at certain values expresses a result about primes, but by its larger domain we can use results from elsewhere. The famous Riemann-Zeta function connects a statement about the distribution of primes to a function which takes in complex numbers as its input)
A formalised language means everyone speaks maths the same way.
(4) Avoiding errors.
As mathematical results build off previous ones, if an error creeps in it might infect a whole body of mathematics. The ultra-high levels of rigour are like an extreme quality check to prevent the spread of contagion.
How do I learn it?
The result of this abstraction, generalisation and formalising is that it’s difficult to start. Complicated symbols and abstract statements abound.
When learning mathematics nowadays, you are learning how to think and speak in a new way. So it’s hard and takes time. But the reward in what you can say, express and do is enormous.
Imagine trying to talk about art or politics or football without language?
This is the extent to which you miss out, if you don’t learn the language of the universe, mathematics.
Some good books and resources
When reading maths books, it is important to remember that these are not ordinary books. Maybe it’ll take an hour to do a page — and that’s fine! It’s much more important to make sure you understand each step of the reasoning, and then to spend some time puzzling over the problems. The learning curve is steep at first, but soon you’ll get in the swing of things.
If you are about an A-level (17/18yr) maths standard, and keen then:
Calculus for the Ambitious — Tom Korner
Introduction to Real Analysis, Bartle and Sherbet
Calculus — Spivak
Cambridge Lecture Notes — written up by Dexter Chua, https://dec41.user.srcf.net/notes/
I am currently studying the maths tripos in my own time, alongside my main degree. If you do this, then the exercise sheets are essential. It’s hard.
How To Prove It — Vellemann (this can be a little slow, but is a great introduction to the sort of abstract thinking needed)
If you are rusty on math, I’d recommend making sure your basic knowledge of calculus and algebra is up to speed before attempting these.
Once you’ve got your teeth stuck in, most authors provide some recommendations where to go next. I’m a big fan of Terence Tao’s Analysis I and II, as well as of Kolmogorov and Fomin’s Introduction to Analysis