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How are Maths and Philosophy related?

The relationship between maths and philosophy may not be obvious at first, but on closer inspection, the intersection is rich: the language of logic, set theory and the very concept of number are central to both disciplines. 

The Oxford Maths and Philosophy degree combines two difficult, abstract but fascinating disciplines.

What’s so special about Maths and Philosophy?

Maths and Philosophy is a fairly niche course, only offered at a smattering of universities (24). Notably, it is offered at Oxford, but not Cambridge. It is also offered at Exeter, King’s College London, Leeds, Birmingham, Bristol, Warwick, York and St Andrew’s.

An A-Level in Philosophy is not a prerequisite. It is not necessary, as the course is built upon the premise of no prior knowledge in philosophy. You will do an introductory philosophy course in your first year, where you will learn the foundations of philosophy: the key philosophers, their theories, and how to structure and debate a philosophical argument. 

Which A-Levels Do I Need to Take?

I would definitely recommend having an essay-based A-Level under your belt. I did English A-Level, and the skills I obtained writing essays throughout my sixth form have proved invaluable in my first two years at Oxford.

For the Oxford course, Maths and Further Maths (both with an A* (!!)) are compulsory. I also sat Physics, although this has not directly related to my course in any way, as the maths you study at Oxford is very pure (more algebraic than applied!). Some of my friends studying the Oxford Maths and Philosophy degree also sat History, Religious Studies, Philosophy and Ethics, Chemistry, or Economics as their other A-Levels. 

If you get the opportunity, an EPQ (Extended Project Qualification) is a good litmus test as to whether Maths and Philosophy at Oxford is the course for you. I did an EPQ on the relationship between maths and music, and although not directly related to philosophy, it helped me hone my essay-writing skills, especially for explaining mathematical concepts.

It also showed me I enjoyed the independent reading, research and writing that is crucial to the philosophy side of the course. 

What maths does the course cover?

Maths and Philosophy is the course for you if you love mathematics, but you are also interested in developing your ability to think about mathematics, and what mathematical systems actually mean, faced with the scale of the world.

The course takes the abstract elements of mathematics, such as set theory and logic, and makes them even more abstract, considering, for example:

  • Whether a set is an object
  • What a number actually is
  • Whether we can truly know the basic axioms of mathematics, such as 1+1 does not equal zero
  • Whether mathematical truths are innate within us
  • Whether there is any world at all

Details can be found in our guide to the Oxford Maths course, or on the Oxford Maths and Philosophy course page.

Does the course cover any intersections between Maths and Philosophy?

In your first year you will also study logic, both an introductory course and a more advanced elements of deductive logic course. This will be the biggest crossover between the two disciplines. 

  • Introductory Logic: You will learn about the language of predicate logic, translating English sentences into this formal language, and consider possible ambiguities that could arise. You will also learn about proof systems, including natural deduction and truth tables. 

Predicate logic is a system that converts everything from English to formal language, such as the statement “if John goes to the shops he will buy crisps,” will be formalised as “P → Q” where P is defined to mean “John goes to the shop” and Q as “he will buy crisps” and “ → “ is the logical symbol meaning “if…then…”. 

Truth Tables are central to proofs in logic. For example, if you had the statement “John is tired and Mary is tired,” you could formulate this in the language of predicate language, as detailed above, and this would be P∧Q, where the “∧” symbol is used to mean “…and…”.

This would then be formalised in a truth table as: 

PQP∧Q
TTT
TFF
FTF
FFF

To formulate the truth table, you consider every combination of P and Q being false (F) and true (T) and consider the outcome this would have for the statement. It is clear that “John is tired and Mary is tired” is true only if “John is tired” (P is true) and “Mary is tired (Q is true.) 

  • Elements of Deductive Logic: This course will formalise many of the basic principles you established in Introductory Logic. It will prove the consistency of the proof systems you used, as well as proofs about the natural numbers, and key logical terms such as completeness and compactness. 

What Philosophy will you study?

In your first year you will also sit a philosophy paper which will be essay-based, unlike logic. You will have a section on:

  • General Philosophy: In this you study notable philosophers such as Descartes, Hume and Locke and explore some key philosophical issues such as mind and body, knowledge and scepticism, induction and God and evil. 

The kind of questions you will answer in this General Philosophy paper vary from “Does free will exist?” to “How do we know we have any knowledge?” to “The sun rose today, does that mean it will rise tomorrow?” 

Frege

  • You will also have an essay section focusing on Frege and his Foundations of Arithmetic. Here you will explore Frege’s ideas about numbers, and how numbers are built up and established. This is an introduction to the more in-depth Philosophy of Mathematics paper you will study for your final honours school. 

Conclusion

As you can see, the Oxford Maths and Philosophy degree is an intense and diverse course, exploring not only the two individual disciplines in depth but also the intersection between them. It is rewarding and challenging. If you love maths, but aren’t ready to give up reading and essay-writing, this is the course for you! 

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