Lecture 1

Introduction

Practical arrangements

• There are two lectures a week:

  Monday 11am

  and

  Tuesday 10am

• Notes will be placed online on the course webpage before each lecture.

• Each week you will have a tutorial (on Thursday or Friday).

• You should find the homework online. Starting next week, you should do these exercises (on the webpage) in your own time and hand them in at that week’s problem class. We will mark them, and return them to you before the problem class the week after that. If you’d like more feedback (on any of the solutions), please ask at the problem class or by email.

• The challenge problem is usually hard: you’re not expected to attempt it, but might enjoy doing so.

  Do it on a separate sheet of paper.

More practicalities

• I offer online surgery hours each week. During that time you can book a 15 minute slot in my Google Calendar if you need extra help with the lecture material or exercises. These are at:

  Thursday 12-1pm.

  If you want to talk at some other time, email me to ask!

• There will (very soon) be a discussion board on Blackboard. Please do use it!

  If you email me certain sort of questions, I might ask you to put them on the discussion board for the benefit of others.

• At the end of this year there will be an exam, covering both semesters’ material. This will count for

  80% of the module.

• There will be an online test each week, released each Tuesday at 11am (immediately after the lecture) and due in at 11am on Monday. These will count for

  20% of the module.

• For this course you have three hours of contact time per week (two hours of lectures, one hour of problem classes). You’re supposed to spend approximately as much time again (three more hours each week) in private study for this course, reading the notes and working on problems.

  If you do not do this, you will not be able to catch up in the run-up to the exams.

Things to do if you get stuck:

• Use the notes/slides/videos online.

• Ask your colleagues.

• Ask me.

• Search the web.

Things not to do:

• Hope it will sort itself out.

• Leave it until the time of the exam.

What is mathematics?

It’s hard to say what maths is. It is rather easier to say a few things about what mathematics is not.

Contrary to popular belief, mathematics is not the study of numbers.

Of course, the study of numbers is:

• part of mathematics,

• very useful in other parts of mathematics, and

• particularly useful in applications outside mathematics.

So we’ll see lots of stuff about numbers in this course, and in other courses. But what else is there, if it’s not all about numbers?

Here are a few pointers. These are just supposed to be a handful of examples rather than a big list of everything!

Ideas of space

Here are two knotted loops of string:

Are they the same? That is, if I had one, could I manipulate it so as to look like the other?

Knot theory — the study of problems just like this — is nowadays a thriving corner of mathematics. But knots are not numbers, and this question is not a question about numbers. Numbers might be useful in solving it, though!

This is just one of many examples of ideas of space in modern mathematics.

Ideas to do with space are nowadays of core importance in physics, just as numbers have. The world is made of space with interesting things in, after all.

Ideas of configuration

Is it possible to have a party of ten people, where everyone is a friend or a friend-of-a-friend of everyone else, and where everyone has exactly three friends present?

It’s true that the numbers three and ten appear in this problem. But it’s not really a problem about numbers: it’s a problem about social networks and how they can be configured.

The study of networks (social and otherwise) has become known as graph theory. The subject of combinatorics encompasses this and many other kinds of configuration problem.

This has great application in computer science: after all, computer networks are examples of networks.

So what is mathematics?

It’s hard to say! Perhaps you’ll form an opinion yourselves over the next three or four years.

My working definition will be:

Mathematics is the rigorous study of abstract systems.

Let’s look at what that means.

Mathematics deals with simplifications, which are sometimes absurdly unrealistic. Mathematicians talk about a line of length $1$, even if there’s no ruler able to tell the difference between $0.999$ and $1.001$. They talk about perfect circles and lines with zero width.

Perhaps paradoxically, it’s because of the unrealistic simplification that mathematics is able to describe the real world so well.

When mathematics models the behaviour of a spacerocket, treating the rocket as perfectly round and ignoring the dust and the small lumps of bird mess is the the right way to get an answer that’s good enough.

One has to be very careful, but the abstraction of mathematics has been an amazing tool. For example, it may be true that nothing is perfectly round, but many things are so nearly round as to make their real shape irrelevant.

The purpose of choosing an abstraction is to give us something we can be completely certain about.

Every move must be fully justifiable (and fully justified, if you’re trying to persuade people). There must be no risk of confusion or mistakes.

If we want to take liberties in our arguments then there’s not much point in making an abstraction in the first place.

Rigour

Everyone knows that there are $64$ squares on a chessboard. After all, chessboards are $8\times 8$ grids, and $8\times 8 = 64$.

That said, here’s a square on a chessboard:

Obviously I’ve tricked you: I’m using the phrase “square on a chessboard” to mean two different things. But it’s a good example of where mathematicians might be careful to be precise, to ensure that no mistakes are made. Things will get more complicated, and the need for care will be greater in future!

Here is a cautionary tale:

• In 1852, Francis Guthrie asked:

  Can every map be coloured with only four colours so that any two neighbouring regions have different colours?

  Here’s a vintage map of England and Wales (and, bizarrely, the Isle of Man) coloured in this way:

That’s not a proof, but it is some evidence that it might be possible.

• In 1879, Alfred Kempe offered a proof that the answer is yes.

• In 1880, Peter Tait offered another, different, proof that the answer was yes. Mathematicians were satisfied, and stopped trying to prove it!

• In 1890, Percy Heawood pointed out that Kempe’s proof contained a big mistake.

• In 1891, Julius Petersen pointed out that Tait’s proof also contained a big mistake. Now, after twelve years spent believing the problem had been solved, and the answer was yes, mathematicians realised that in fact, they still had no idea.

• In 1976, Kenneth Appel and Wolfgang Haken offered a new proof that the answer was indeed yes!

• As of 2020, the argument of Appel and Haken has been checked many times, and is accepted as a complete solution.

The mistakes of Kempe and Tait are not particularly complicated. What caused this 12-year period of confusion was a lack of sufficient rigour.

In this course we will learn the basics of correct, logical argument. If you get the right final answer but your justification is incorrect or incomprehensible, you will deserve (and you will probably get) very few marks. Kempe and Tait had the right final answer too, but they had no way of knowing that.

At times this may seem like an unnecessary burden: especially when you feel that the right answer is “obvious”. However, if you don’t spend time in shallow water learning how to swim, you’ll never be comfortably able to swim in deep water.

Sets and functions

Having spent the best part of an hour trying to persuade you that mathematics is not about numbers, most of this semester will now be about numbers.

But in order to study them properly, we’ll need to start at the beginning, by talking about sets.

Sets of numbers

One problem with numbers is that there are several different sorts of them.

We’re probably used to several sorts already:

natural numbers

integers

rational numbers

real numbers

complex numbers

We’ll go into more detail later in the course.

We often need to say which we mean, in order to avoid confusion and error. For example, it’s certainly possible that I might invite $3$ friends over for dinner, but it’s hard to invite $-5$ friends or $3/4$ friends or $\sqrt{2}$ friends over.