1. Philosophy and History of Mathematics
A Brief Introduction: Week 1

2. Course Overview
Welcome to City Lit’s “Mathematical Explorations” project!
This course
HP092: Philosophy of Mathematics & Logic (Autumn term)
HP093: Philosophy of Infinity (Spring term)
HP096: History & Philosophy of Calculus (Summer term)
One-day summer schools

3. What is Philosophy?
Philosophy is a part of Western culture and is bound up intimately with its history, especially between about 1600 and 1900.
“The Department of Lost Causes” – we don’t answer questions very often!
Maths is also part of culture, and claims to be a “science” in the broad sense. Until quite recently, mathematics was a common skill among intellectuals and was not highly specialised.

4. Branches of Philosophy
Epistemology
Metaphysics and Ontology
Language and Logic
Mind and consciousness
Value (including ethics, politics and aesthetics)
We will concentrate on the first two, and a little of the third. You can do City Lit courses in all these areas.

5. Why Philosophy of Maths?
Some philosophical questions about maths to keep in mind:
What is mathematics?
Is mathematics “knowledge”? Does it offer certainty?
What kinds of things are the objects maths deals with?
Why is maths so useful (especially in the natural sciences)?
What methods are legitimate in mathematical research?
Are there things we can’t know about maths even in principle?

6. Plato’s Allegory of the Cave
Appears near the end of his long dialogue The Republic (c380 BCE).
The main concern of this text is political organisation, but this passage is by far the most famous part.
Key terms: idealism, materialism, form, philosopher, enlightenment.

7. Euclid’s Elements
This is a compendium of geometrical knowledge that dates from c300 BCE.
It was universally taught in schools the West until the 19th century, and is still the basis of some maths education today, especially in the US.
Key terms: axiom, definition, postulate, common notion, proof.