What is the course about?
This course assumes you are familiar with the material covered in HP103 Philosophy of Mathematics 1: geometry & HP091 Philosophy of mathematics 2: arithmetic. These courses are offered in Autumn 2017 and Spring 2018.
The 1931 publication of Kurt Gödel's "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" upended contemporary understandings of mathematical foundations and philosophy of mathematics. The two "incompleteness theorems" proved that the notions of truth and provability had to be considered separately and that finite arithmetic could not be set on a foundation that could demonstrate its own consistency.
Popular accounts of these results tend to skimp on the technical underpinnings of the theorems and their proofs, leaving the keen amateur more than a little mystified. Our six-week course will aim to show a bit more of their workings and explain how the theorems led to related findings about computability and algorithms associated with Alan Turing, Alonzo Church and Alfred Tarski.
What will we cover?
We will begin by looking at the concern for the foundations and formalisation of mathematics at the beginning of the last century: the context for Gödel's work. We will then look at each completeness theorem in turn as they focus on truth and provability and the related problem of demonstrating the consistency of a formal axiomatic system. We will consider the consequences for philosophy of mathematics and logic along with a look at how these results informed the early days of computer science. We will also consider the relation of the incompleteness theorems to popular claims about the relativism of knowledge and the 'post-truth' condition.
What will I achieve?
By the end of this course you should be able to...
• Explain why mathematics was concerned with axiomatic approaches.
• Outline why consistency and completeness are thought to be essential properties of axiomatic systems.
• Discuss what kind of algorithms can produce lists effectively.
• State Gödel’s Incompleteness Theorems and outline the proof of the first.
• Explain what the difference between truth and provability is.
• Explain why you might conclude with Tarski that truth is not a syntactic property of a system.
• Outline how issues in computation such as 'decidability' and the 'halting' problem are related to Gödel's theorems.
What level is the course and do I need any particular skills?
This is an advanced philosophy course. This material is normally only taught to advanced undergraduates or postgraduates studying Philosophy & Mathematics, Mathematics or Computer Science.
This should not be your first experience of philosophy and we will assume that you are familiar with the material on the intermediate courses in Philosophy of Mathematics (HP 103 and HP 091). This course is designed to follow on from those two courses and is taught in the same time slot.
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?
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