# Mathematics and Philosophy

## Introduction to tutors and students

## Why study philosophy?

The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it.

Bertrand Russell, *The Philosophy of Logical Atomism*

If mathematics is the language of science and fundamental truths about the world we live in, then the philosophy of maths is the study of that language. We can prove all sorts of things in mathematics, from Pythagoras' theorem to Fermat's last theorem, but is there a limit to what we can prove? Is it possible for some things to be true, but unprovable? What we'd like is a mathematical language which is **complete** (we can prove all possible true statements), **consistent** (we cannot prove false things), and **decidable** (we have a decision procedure for deciding which statements are true and which are not). Kurt Gödel showed in 1931 that you cannot have a system which is both complete and consistent. This seems like a fundamental problem for mathematics.

## What's in the course?

In the first three years of the course, your time is split equally between maths and philosophy. The maths that you do is predominantly pure - there is no statistics in the course, and very little applied maths. The philosophy courses you take will be a combination of general philosophy (taken with the other courses that include philosophy like PPE and Physics & Philosophy) and specific philosophy of mathematics.

As part of your fourth year, you can choose to take a mathematics dissertation, a philosophy thesis, or both. In the fourth year you can also choose the combination of maths and philosophy courses you take - up to purely mathematics courses, with no philosophy, or vice versa.

If you'd like to look at the complete list of courses, you can look through the course handbook.

## Who is this course good for?

- If you enjoy logic and thinking outside the box.
- If you want a mathematics course which is primarily focused on pure mathematics.
- If you're interested in thinking about broad philosophical questions, as well as learning mathematics.

## Problems to think about

**Logic**

The following problems were published by Charles Dodgson (Lewis Carroll), a professor of Mathematics at Christ Church college, Oxford and the author of Alice in Wonderland. The problems were published in his book "Symbolic Logic" in 1897 (which is free to read via Project Gutenburg), and the aim is to find an inescapable conclusion using all the prepositions.

- Babies are illogical;
- Nobody is despised who can manage a crocodile;
- Illogical persons are despised.

To solve this, we need to consider the universe - that is, what are the subjects of the puzzle? We can see from the above that the universe is the set of all persons. Let us use the following notation $a$ = able to manage a crocodile; $b$ = babies; $c$ = despised; $d$ = logical; $\to$ for implies; and ~ for not. Using the above prepositions we can say:

- $b \to $~$d$
- $a \to $~$c$
- ~$d \to c$

Using all of these, we can say $b \to $~$d \to c \to $~$a$. That is, babies cannot manage crocodiles.

Now try these:

- No ducks waltz;
- No officers ever decline to waltz;
- All my poultry are ducks.

- All humming-birds are richly coloured;
- No large birds live on honey;
- Birds that do not live on honey are dull in colour.

- No birds, except ostriches, are 9 feet high;
- There are no birds in this aviary that belong to any one but me;
- No ostrich lives on mince-pies;
- I have no birds less than 9 feet high.

- The only animals in this house are cats;
- Every animal is suitable for a pet, that loves to gaze at the moon;
- When I detest an animal, I avoid it;
- No animals are carnivorous, unless they prowl at night;
- No cats fails to kill mice;
- No animals ever take to me, except what are in this house;
- Kangaroos are not suitable for pets;
- None but carnivora kill mice;
- I detest animals that do not take to me;
- Animals, that prowl at night, always love to gaze at the moon.

## Employment prospects

The joint courses in Mathematics and Philosophy provide you with the opportunity to attain high levels of two quite different kinds of widely applicable skills. Mathematical knowledge and the ability to use it is a key element in tackling quantifiable problems and is the most highly developed means of obtaining knowledge through purely abstract thinking. Philosophical training teaches you to analyse issues, often by questioning received assumptions, and to articulate that analysis clearly.

Our Maths and Philosophy graduates go on to a wide range of careers, including journalism, the civil and diplomatic services, and software development. The degree is very well respected amongst employers, who value the wide ranging skill set graduates learn.

For more information about our graduates and their experiences, see the Careers section.

## For more information

For more information about the Maths and Philosophy course, you can look at the departmental prospectus.

Try the Oxford Podcasts series on General Philosophy (a series of lectures given to first year undergraduates).

If you're interested in learning more about how maths and philosophy intertwine, some recommended books are *Gödel, Escher, Bach: An Eternal Golden Braid* by Douglas Hofstadter, *The Problems of Philosophy* by Bertrand Russell and, for a more general introduction to philosophy, *What Does It All Mean?* by Thomas Nagel.

On the web, there's an excellent introduction to the concept of different sizes of infinity by Jeremy Kun.