Mathematics and Philosophy BA/BSc

at University of Warwick

Core modules

There are two routes through the degree: the three year BA/BSc in Mathematics and Philosophy and the four-year BSc with Specialism in Logic and Foundations. You will be eligible for transfer to the Specialism in Logic and Foundations degree based on your first year exam results.

If you remain on the Mathematics and Philosophy route, you may choose to apply for an intercalated year, spent either studying abroad or on a work placement. This extends the duration of your degree to four years, with your third year spent abroad or on placement, and will be reflected in your degree qualification (i.e. BSc Mathematics and Philosophy with Intercalated Year).
 

Year One
Mind and Reality
Look around. What if all your experiences were the products of dreams, or neuroscientific experiments? Can you prove they aren’t? If not, how can you know anything about the world around you? How can you even think about such a world? Perhaps you can at least learn about your own experience, what it’s like to be you. But doesn’t your experience depend on your brain, an element of the external world? This course will deepen your understanding of the relationship between your mind and the rest of the world.

Logic 1: Introduction to Symbolic Logic
This module teaches you formal logic, covering both propositional and first-order logic. You will learn about a system of natural deduction and understand how to demonstrate that it is both sound and complete. You will learn how to express and understand claims using formal techniques, including multiple quantifiers. Key concepts you will consider are logical validity, truth functionality and formal proof quantification.

Linear Algebra
Linear algebra addresses simultaneous linear equations. You will learn about the properties of vector spaces, linear mappings and their representation by matrices. Applications include solving simultaneous linear equations, properties of vectors and matrices, properties of determinants and ways of calculating them. You will learn to define and calculate eigenvalues and eigenvectors of a linear map or matrix. You will have an understanding of matrices and vector spaces for later modules to build on.

Mathematical Analysis I/II
Analysis is the rigorous study of calculus. In this module, there will be a considerable emphasis throughout on the need to argue with much greater precision and care than you had to at school. With the support of your fellow students, lecturers, and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to solve each kind of problem, to the point where you can develop your own methods for solving problems. The module will allow you to deal carefully with limits and infinite summations, approximations to pi and e, and the Taylor series. The module also covers construction of the integral and the Fundamental Theorem of Calculus.

Foundations
It is in its proofs that the strength and richness of mathematics are to be found. University mathematics introduces progressively more abstract ideas and structures and demands more in the way of proof until much of your time is occupied with understanding proofs and creating your own. Learning to deal with abstraction and with proofs takes time. This module will bridge the gap between school and university mathematics, taking you from concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.

This module also looks at algorithms and operational complexity, including cryptographic keys and RSA.

Differential Equations
Can you predict the trajectory of a tennis ball? In this module you cover the basic theory of ordinary differential equations (ODEs), the cornerstone of all applied mathematics. ODE theory proves invaluable in branches of pure mathematics, such as geometry and topology. You will be introduced to simple differential and difference equations and methods for their solution. You will cover first-order equations, linear second-order equations and coupled first-order linear systems with constant coefficients, and solutions to differential equations with one-and two-dimensional systems. We will discuss why in three dimensions we see new phenomena, and have a first glimpse of chaotic solutions.

Geometry and Motion
Geometry and motion are connected as a particle curves through space, and in the relation between curvature and acceleration. In this course you will discover how to integrate vector-valued functions and functions of two and three real variables. You will encounter concepts in particle mechanics, deriving Kepler’s Laws of planetary motion from Newton’s second law of motion and the law of gravitation. You will see how intuitive geometric and physical concepts such as length, area, volume, curvature, mass, circulation and flux can be translated into mathematical formulas, and appreciate the importance of conserved quantities in mechanics.

Introduction to Abstract Algebra
This course will introduce you to abstract algebra, covering group theory and ring theory, making you familiar with symmetry groups and groups of permutations and matrices, subgroups and Lagrange’s theorem. You will understand the abstract definition of a group, and become familiar with the basic types of examples, including number systems, polynomials, and matrices. You will be able to calculate the unit groups of the integers modulo n.

Introduction to Probability
This module takes you further in your exploration of probability and random outcomes. Starting with examples of discrete and continuous probability spaces, you will learn methods of counting (inclusion-exclusion formula and multinomial coefficients), and examine theoretical topics including independence of events and conditional probabilities. You will study random variables and their probability distribution functions. Finally, you will study variance and co-variance, including Chebyshev’s and Cauchy-Schwarz inequalities. The module ends with a discussion of the celebrated Central Limit Theorem.

Year Two
Logic II: Metatheory
In this module, you will learn about the metatheory of propositional and first-order logic; to understand the concept of a sound and complete proof system similar to that of Logic I. You will study elementary set theory and inductive definitions and then consider Tarski's definitions of satisfaction and truth, proceeding to develop the Henkin completeness proof for first-order logic. You will learn to appreciate the significance of these concepts for logic and mathematics, with the ability to define them precisely.

Algebra I: Advanced Linear Algebra
On this course, you will develop and continue your study of linear algebra. You will develop methods for testing whether two general matrices are similar, and study quadratic forms. Finally, you will investigate matrices over the integers, and investigate what happens when we restrict methods of linear algebra to operations over the integers. This leads, perhaps unexpectedly, to a complete classification of finitely generated abelian groups. You will be familiarised with the Jordan canonical form of matrices and linear maps, bilinear forms, quadratic forms, and choosing canonical bases for these, and the theory and computation of the Smith normal form for matrices over the integers.
 

Mathematical Analysis III
In the first half of this module, you will investigate some applications of year one analysis: integrals of limits and series; differentiation under an integral sign; a first look at Fourier series. In the second half you will study analysis of complex functions of a complex variable: contour integration and Cauchy’s theorem, and its application to Taylor and Laurent series and the evaluation of real integrals.

Year Three
Set Theory
Set theoretical concepts and formulations are pervasive in modern mathematics. They provide a highly useful tool for defining and constructing mathematical objects as well as casting a theoretical light on reducibility of knowledge to agreed first principles. You will review naive set theory, including paradoxes such as Russell and Cantor, and then encounter the Zermelo-Fraenkel system and the cumulative hierarchy picture of the set theoretical universe. Your understanding of transfinite induction and recursion, cardinal and ordinal numbers, and the real number system will all be developed within this framework.

Year Four

Dissertation

Or

Third Year Maths Essay
(BSc with Specialism in Logic and Foundations only)

Optional modules
Optional modules can vary from year to year. Example optional modules may include:

• Commutative Algebra
• Knot Theory
• Logic III: Incompleteness and Undecidability
• Philosophy of Mathematics
• Metaphysics
• Computability Theory
   

All applicants, regardless of their visa requirements, are required to show that their ability to understand and express themselves in both written and spoken English is sufficiently high for them to derive full benefit from their degree course. The level of competence required depends on what type of degree you wish to study. Many applicants can demonstrate this through a satisfactory level of achievement in, for example, GCSE English Language or the English courses within the International Baccalaureate.

Applicants whose academic qualifications meet the University’s admissions requirements, but whose English language qualifications are not accepted as equivalent, may be offered a place at Warwick conditional on an acceptable English language qualification being achieved before they join the University. A number of English language tests are accepted; some of the most frequently offered ones are detailed below.

Please note that certain English tests (including IELTS and PTE) must be taken within two years and one month prior to the start of your course at Warwick. Applicants whose tests fall outside this timescale will need to retake a suitable English language test.

Applicants who require a visa to study in the UK may, depending on the circumstances, be asked to provide evidence of a secure English language test (e.g. IELTS for UKVI (Academic)), rather than an examination (such as International Baccalaureate English) due to the visa requirements set by the UK government.

ENGLISH LANGUAGE REQUIREMENTS FOR UNDERGRADUATE STUDENTS (2022 Entry)

If you do not meet the English Language Requirements for your course in full, you may still be eligible for the course on condition of:

• Meeting the English Language Requirements for our Pre-sessional English course, and
• Successfully completing the Pre-sessional English course

Please use the table below to find the relevant band for the course(s) in which you are interested. You can sort the table by clicking on the heading of the column you wish to sort by.

International Tuition Fees:

£19750 per year

Graduates from our Philosophy single and joint honours degrees have gone on to pursue careers as:

• Authors, writers and translators
• Legal professionals
• Marketing professionals
• Management consultants and business analysts
• Chartered and certified accountants
• Teaching and educational professionals