What is the Philosophy of Maths?

Mathematics is a specialised, unique and extremely successful area of human activity, and a lot of philosophical questions arise from it. What exactly is mathematics? Do mathematical objects exist? How do we acquire mathematical knowledge? Why is maths so useful in science? What do we mean by 'proof'?

Here is a very brief glossary of the terms used in the subject, with references to places where you can find out more.

Category Theory

An alternative to Set Theory. Category theory takes the view that mappings between objects are the important thing in maths (rather than the objects themselves).

Epistemology

Questions of knowledge, in particular how we aquire mathematical knowledge.

Intutionism

The belief that mathematical objects must be mentally constructed before they can be said to exist, it is not sufficient to prove their existence indirectly.

Formalism

The belief that mathematical statements are just strings of symbols that we manipulate according to rules we make up, with only limited meaning.

Formal System

A set of rules for manipulating strings of symbols. Almost all notation in maths forms a formal system, they are the basic tools for doing maths.

Logicism

The belief that maths reduces to logic, that the truths of mathematics are analytic (logically necessary).

Ontology

Questions of existence. Do mathematical objects exist? If so, in what sense? If not, why do we talk about them?

Platonism

The belief that mathematical objects have an objective existence as abstract objects, independently of us and of the physical world.

Set Theory

The most usual foundational theory for maths - an extensive and very rigorous theory of 'sets' (collections of objects) was developed during the 20th century, and almost all modern maths can be embedded in this theory.

Structuralism

The view that maths deals with structures, i.e. collections of objects with relations defined between them, rather than individual objects.

Back to Homepage