PLP: An introduction to mathematical proof

The main idea of this text is to teach you how to write correct and clear mathematical proofs. In order to learn to prove things we will study some basic analysis. We will prove many things about the basic properties of numbers sets and functions — like

In order to make sense of this statement we need to understand how to extend the idea of “more” from the context of finite quantities, where we are used to “more” and “less”, to the domain of infinite quantities. We’ll have to define ideas about sets and functions, manipulate and combine them with logic. Along the way we will need to think hard about how to communicate the mathematics that we are doing, so that you, and others, can follow what we are doing.

Hence, a critical part of this subject is to learn to communicate mathematics — not just do mathematics. Mathematics is not simply number crunching or using formulas — this is using mathematics. Mathematics is also about understanding and reasoning and most importantly proving things. Neither of these aspects is more important than the other. Up until now you have mostly done the former, and the aim of this text is to help you get better at the latter.

It is crucial that we are able to explain to others why what we know is true is actually true — this is what proofs are for. Think of the proof as a dialog between you and the reader — you have to make every (reasonable) effort to be clear, precise and accurate. Always think of the reader when you are writing. It is important to argue and write well — it is a useful skill both at university and beyond in the so-called “real world”.

The authors have spend a lot of time reading other people’s work (mostly student work, but also articles written by professional mathematicians with years of experience) and puzzling over the lines written on the page — sometimes legible, sometimes not (especially in exams). And finally after sweating for ages you realise what they were trying to say. In some circumstances one can, of course, contact the writer and ask them “What did you mean?” However, this is frequently not the case — all one has is what is written on the page.

When you do mathematics (and other activities) there is a huge difference between reading and doing. This is especially the case with proofs. So while reading the text is a good way to learn some ideas and get a feeling for some of the stuff, it is really no substitute for doing the exercises. That is where you will really learn.

Behind every proof you read (and you write) lies a good bit of work. You cannot generally look at a problem and write out the proof all fine first go. You need to do some rough work to map out the structure of the proof and the details. Then after this you write out the proof nicely and neatly. Making sure that you present you work well forces you to think about what you are writing down. The investment in your hard work writing, pays off for the people reading your work.

Learning to write proofs takes time and effort. But the rewards are well worth it.