What is the course about?
This course assumes you are familiar with the material covered in HP103 Philosophy of Mathematics 1: geometry.
What kinds of things are numbers? How do arithmetic and geometry relate? 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? Can infinity be a suitable topic for mathematics?
These are the kinds of questions animating the philosophy of mathematics. They were given new impetus by the radical developments of the nineteenth century, including: non-Euclidean geometry; the creation of new number systems; set theory; and the innovations in logic. We will provide an overview of what was at stake as mathematics was transformed 200 years ago and examine how it led to new questions about its nature and its foundations. We will then look at the lively debates that ensued. We will conclude by considering contemporary debates about the nature of mathematical objects.
What will we cover?
We begin with an introduction to what happened in nineteenth century mathematics. We will study the main ideas and a few techniques. Familiarity with this material provides us with historical understanding and something to philosophise about!
Then we look at the controversies over the philosophy of mathematics in the early twentieth century, in particular the attempts to found mathematics on a logical, axiomatic base with set theory. We will study the reactions against this project associated with 'intuitionist' and 'constructivist' accounts of mathematical objects and proof.
We conclude with a survey of the current state of play in the philosophy of mathematics and the return to Platonism as the framing reference for discussion of 'ideal' objects.
• Mathematical truths: are they timeless, universal, necessary and objective?
• What is mathematical reasoning and its form of proof?
• What kinds of things are numbers?
• How is mathematics applicable to natural and human sciences?
• Is mathematics just the formal manipulation of symbols? Is it therefore like a game?
• Proof by contradiction and its discontents: is there something wrong with classical logic?
• Contemporary platonism, anti-platonism and phenomenology.
What will I achieve?
By the end of this course you should be able to...
• Interpret the philosophical significance of modern developments in mathematics.
• Sketch arguments about proof and meaning in mathematics.
• Reflect on the relationship between contemporary philosophy of mathematics and its ancient roots.
• Assess arguments over the nature of mathematical objects and foundations.
• Examine the motivations for different approaches to logic and axiomatics.
• Describe the positions of key modern figures such as Frege, Russell, Hilbert, Wittgenstein and Brouwer.
• Pursue your own investigations in contemporary thinking about mathematics.
What level is the course and do I need any particular skills?
This is an intermediate course. It also assumes you are familiar with the content of its companion course, HP 101 Philosophy of Mathematics 1: geometry which starts in September 2017. This course asks philosophical questions about mathematics, but no specific mathematical knowledge is required. We will remain at the level of concepts and ideas and aim to keep the technical mathematical material to a minimum. For both philosophy and mathematics a willingness to engage with new ideas is much more important than prior knowledge. If you have not studied any philosophy before you should consider a beginner's course. We suggest HP013 Philosophy & History of Mathematics: A Brief Introduction (runs in November).
How will I be taught, and will there be any work outside the class?
These classes use a mixture of lecture, structured discussion-based activities and problem-solving in small groups. Optional online resources and exercises will be suggested if you want to consolidate or extend the material outside class.
Are there any other costs? Is there anything I need to bring?
No equipment will be required besides pen and paper or other means for taking notes. Access to the internet is advantageous but not required.
When I've finished, what course can I do next?
This course is ideal preparation for HP 101 Advanced Philosophy of Mathematics: Gödel which is offered in Summer term in the same time-slot.
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