What is the course about?
Mathematics underpins the empirical sciences that investigate the real world, yet its own subject matter is led by abstract proof and deals with 'ideal' objects, such as the number 5. This paradox presents us with a series of questions:
What kind of things are numbers? How does geometry relate to space? What is mathematical proof? Are the truths of mathematics necessary truths? Are they discovered or invented? How do we gain mathematical knowledge and what does it tell us about our cognitive capacities?
This course is a stand-alone introduction to the history and philosophy of mathematics but it also provides the necessary background and understanding for our other courses: Philosophy of mathematics and logic; Philosophy of infinity; Philosophy and history of calculus.
What will we cover?
We begin by looking at the place of mathematics, geometry and logic in Greek thought with a focus on the philosophers Plato and Aristotle and the geometer Euclid. Euclid’s book, The Elements, defined mathematical truth for two millennia. Aristotle and Plato disagreed fundamentally about the nature and purpose of mathematics and its relation to scientific knowledge, establishing concepts and attitudes that continue to frame contemporary thinking.
Aristotle is particularly important to subsequent science because his novel concept of ‘the continuum’ was seen to have resolved the paradoxes attributed to Zeno regarding motion and change in the physical world. We will examine in detail these classic riddles including whether Achilles can ever catch a Tortoise.
From Aristotle and Euclid we will also learn about the classical conceptions of valid proof, axiomatic method and logical deductions, all of which underpin mathematics’ claim to be the science of truth.
We conclude our course with modern responses to these ideas in light of the 16th and 17th century revolutions in natural science and mathematics and their new claims to knowledge. Here we will discuss the contemporary debates and disputes over the relationships between mathematics, reason and truth.
What will I achieve?
By the end of this course you should be able to...
• Distinguish a valid proof from a true statement
• Reproduce some classic proofs in number theory and geometry
• Describe the different positions about mathematics taken by empiricists and rationalists
• Outline Zeno’s paradoxes of motion and how Aristotle attempts to resolve them
• Discuss the nature of numbers and space and their relation to the idea of infinity
• Explain how axiomatic methods in mathematics are meant to establish true knowledge
• Account for the differences in reasoning between inductive and deductive methods and assess their relative merits
• Distinguish various everyday number systems and their axiomatic foundations.
What level is the course and do I need any particular skills?
This is an introductory course. Some simple arithmetic will be used in class, but this is not a mathematics course.
How will I be taught, and will there be any work outside the class?
Each session will be led by a tutor who will introduce the material and facilitate discussions.
Some additional reading will be suggested but there is no required homework.
Are there any other costs? Is there anything I need to bring?
No, just bring whatever you use to make notes.
When I've finished, what course can I do next?
General information and advice on courses at City Lit is available from the Student Centre and Library on Monday to Friday from 12:00 – 19:00.
See the course guide for term dates and further details