BA (Hons) Mathematics with Philosophy (Placement Year) Bachelor's degree at Lancaster University

From questions about truth, justice, and knowledge to debates over freedom, power, and human purpose - how has philosophical inquiry shaped cultural, political, and scientific life across centuries? In this module you will explore the major ideas and traditions that have guided the development of Western thought. Specific thinkers examined will vary from year to year, but they will include philosophers whose ideas have helped shape philosophical viewpoints, categories and boundaries in the western philosophical tradition. You will be encouraged to think about the problems and limitations of different thinkers’ approaches, and their impact on the way we practice and understand the boundaries

What is real, how can we know, and how can we check our reasoning? In this module you will study philosophical tools for reasoning and arguing (critical thinking) and discover fundamental philosophical questions about knowledge (epistemology) and the nature of reality (metaphysics). In studying critical thinking, you will learn methods of constructing and analysing arguments and acquire basic logical terminology. In exploring epistemology, you’ll discuss questions such as: what exactly is it to ‘know’ something? Can we know anything at all? Are there alternative knowledges? In metaphysics, you will consider questions such as: what is the fundamental nature of reality? How are we to understan

At university, emphasis is placed on understanding general mathematical theorems. They apply in many different cases, and understanding why a result is true enables us to creatively use the underlying ideas to tackle new problems. Study the language and structure of mathematical proofs, illustrated by results from number theory. You will see the concept of congruence of integers, which is a simplified form of arithmetic where seemingly impossible problems become solvable. In relation, you’ll encounter the abstract idea of an equivalence relation. Sets and functions form the basic language of mathematics. You will study functions of a real variable and abstract functions between arbitrary set

Interested in how mathematicians build theories from basic concepts to complex ideas, like eigenvalues and integration? Journey from polynomial operations to matrices and calculus through this module. Starting with polynomials and mathematical induction, you will learn fundamental proof techniques. You will explore matrices, arrays of numbers encoding simultaneous linear equations, and their geometric transformations, which are essential in linear algebra. Eigenvalues and eigenvectors, which characterise these transformations, will be introduced, highlighting their role in applications including population growth and Google's page rankings. Next, we will reintroduce you to calculus, from its

An introduction to the mathematical and computational toolsets for modelling the randomness of the world. You will learn about probability, the language used to describe random fluctuations, statistics and the mathematical techniques used to extract meaning from data. You will explore how computing tools can be used to solve challenges in scientific research, artificial intelligence, machine learning and data science. You will develop the axiomatic theory of probability, discover the theory and uses of random variables and investigate how theory matches intuitions about the real-world. You will then dive into statistical inference, learning to select appropriate probability models to describ

Symmetry is central to our understanding of a range of subjects, from the structure of molecules to the roots of polynomials. In this module, you will see how group theory naturally appears whenever we look at symmetry. Using familiar examples, including symmetries of regular polygons, rotations and reflection matrices, roots of 1 in the complex plane, and permutations, you will define what makes a group and how this can provide a unifying language, highlighting connections between seemingly different subjects. You will then transition into mathematical analysis, developing an approach to sequences, limits, and continuity that provides the foundation for calculus. Examining a range of exampl

Ever wondered about the hidden structures that govern mathematics? Algebra is more than just equations, it's the language of symmetry and structure, underpinning subjects ranging from geometry and quantum mechanics to number theory and cryptography. The main frameworks for modern algebra are group theory and ring theory. Group theory topics include classifying symmetries, the symmetric group, Lagrange's theorem and the first isomorphism theorem. Similarly, ring theory explores the notions of subrings, ideals, and homomorphisms in an example-driven methodology, using abstract number systems, polynomial structures, and matrices. This module introduces the essential theory and techniques for al

The success of Newton/Leibniz’s calculus raises the question: what happens if we replace the real numbers with the complex numbers? After all, their arithmetic structure is similar, and we can measure distances between points in both. You will learn how to define the derivative of a complex function as usual and explore the behaviour of functions that are complex differentiable. Everything resembles the real case, ultimately leading to the astonishing result that if a complex function can be differentiated once, it can be differentiated infinitely often and is expressed by its Taylor series. Integral calculus for complex functions opens a route towards evaluating definite integrals that cann

Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning. You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem. Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving

Continuing with your study into real numbers, you will explore their completeness (the idea that there are no ‘gaps’, unlike in the rationals). This completeness will be used to understand the limits of sequences, convergence of series, and power series. This framework will allow for precision when exploring continuity, differentiability, and integrability of functions of a real variable, providing an improved foundation for calculus. That will enable you to understand when it is appropriate to use calculus; for instance, in proving theorems in other areas of mathematics, such as mathematical physics, probability and number theory. The cornerstone of mathematical analysis is the construction

In the nineteenth and twentieth centuries, a new approach to philosophy began to emerge that questioned and interrogated the inherited western philosophical tradition from Plato to Kant. This new approach was later described as ‘continental’ philosophy. In this module, you will discover some of the key thinkers from this continental tradition of philosophy. The particular philosophers will vary from year to year, but will include pioneering thinkers who have been particularly influential on later continental philosophers (for example, Hegel, Kierkegaard, Nietzsche and Wittgenstein) as well as more recent continental thinkers themselves (for example, Lyotard, Derrida, Levinas, Badiou, Žižek,

Critically engage with questions and debates about our socially connected lives and the ways in which we interact and act on the world through language and communication, individually and as a society, to shape knowledge and reality. In this module you will gain the skills and insight to ask questions which change each year but may include: How does communication work in our individual and collective lives? How might certain kinds of communication bring about ethical and political change (for example, by making something permissible or changing the boundaries of acceptable political discourse)? Are lying and other kinds of deception permissible, and if so, when and for whom? What does freedo

What does it take to have a mind? How does science work? Does human reason equip us to understand the external world? In this module we explore the nature of consciousness and reality, and the methods by which we understand them, focussing on key debates in the philosophy of mind and philosophy of science. In the first part of the module, you will explore what it takes to have a mind, examining and critiquing some of the wide range of answers philosophers have offered to this intractable problem. Questions you will investigate include: What is the relationship between the mind and the brain? How do animal minds or artificial intelligence fit into our understanding of thought and consciousnes

Create a portfolio of investigative and critical writing which explores a particular philosophical topic in depth. In this module you will be guided with expert support from Lancaster philosophers to develop your philosophical and independent study skills. Through deep engagement with a specific topic you will develop your ability to assess philosophical arguments and make independent judgements, informed by reasoning and evidence. You will engage with a text, problem, figure or body of work chosen by an academic within the philosophy team at Lancaster who is a specialist on the topic and work with their expert support, in groups and independently. Project topics offered each year will be dr

Explore how philosophical thinking can tackle real-world problems, from personal dilemmas to global challenges, by linking abstract ideas to concrete decisions. In this module you will engage with an applied philosophical challenge, using your philosophical skills to provide compelling reasons in favour of your solution. Through production of a short podcast or in-person presentation, you will also develop your ability to engage in philosophical argumentation beyond the written word. And along the way, you’ll gain the tools to think clearly, act responsibly, and engage thoughtfully with the complex world around you. Specific topics studied each year draw on Lancaster’s wide range of applied

You will spend this year working in a graduate-level placement role. This is an ideal opportunity to gain experience in an industry or sector that you might be considering working in once you graduate. Although it's up to you to find your placement we'll support you all the way. Our Careers Service will provide guidance on CVs, applications, interview techniques and creating a digital profile.

Commutative rings generalise both integers and polynomials and they play a very important role in a wide area of mathematics. As well as being important in algebra, they sit at the heart of algebraic approaches including geometry and number theory, in part because rings of functions occur so naturally there, as they do in analysis. At this stage, you will already know how to factor and divide integers and polynomials. Therefore, a crucial question is to understand the factorisability and divisibility properties in more general commutative rings. For example, what is the analogue of the set of prime integers, or which are the invertible elements? You will seek to answer these questions, begin

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance. You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proo

An inner product space is a real or complex vector space, equipped with certain extra structure that formalises the geometrical notion of orthogonality. It turns out that each inner product space has an intrinsic notion of distance, allowing us to discuss convergence and completeness. Complete inner product spaces are known as Hilbert spaces. The theory of Hilbert spaces blends linear algebra and (real) analysis. It is a natural and powerful tool for studying problems of quantitative approximation. Furthermore, it provides an abstract framework that can be applied to diverse areas of maths, from differential equations and spectral theory to quantum mechanics and stochastic processes. This mo

Knots play a fundamental role in many areas of mathematics, from pure topology and algebra through to quantum field theory and protein-folding. Develop tools to measure knottedness, including geometrical ideas like curvature, knot invariants like the Jones polynomial, and the crucial concept of the fundamental group, which has applications in topology far beyond detecting knots.

Linear systems of differential and integral equations provide a mathematical model for a wide range of real-world devices, including communication systems, 5G networks, electrical circuits, heating systems and economic processes. Mathematical analysis of these models gives insight into the behaviour of these devices, with applications in automatic control, signal processing, wireless communications and numerous other areas. Linear systems are considered in continuous time that reduce to a standard (A,B,C,D) state space representation. Via the Laplace transform, these are reduced further to rational transfer functions. Linear algebra enables us to classify and solve (A,B,C,D) models, while we

The module commences by looking at classical methods of encryption, discussing their advantages, disadvantages and efficiency. You will also investigate statistical attacks on these methods of encryption and the need for better methods. After this, you will explore modern methods of encryption that are used in the real-world and rely on the robustness of modular arithmetic. While most encryption methods are still considered secure, you will review potential attacks on these systems (e.g. factorisation algorithms) and situations where bad key generation or implementation has occurred. Production of a big enough quantum computer renders the above schemes useless. Therefore, you will dive into

A metric space consists of a set, whose elements are called points, and a notion of distance between points governed by three simple rules, abstracted from basic properties of Pythagorean distance in the Euclidean plane. In examples, ‘points’ may be functions where uniformity of convergence can be captured, or binary sequences with applications in computer science, or even subsets of a Euclidean space delivering fractal sets as limits. Topology goes further, abstracting the notions of continuity and convergence, rendering a teacup and doughnut indistinguishable. A topological space equips each of its ‘points’ with its so-called ‘neighbourhoods’. The few simple principles governing these unlo

From music, film, and television to sports, fashion, and digital media, explore how cultural texts raise philosophical questions about identity, morality, power, and meaning, and examine how popular culture both reflects and shapes the way we understand ourselves and the world around us. In this module you will look at how philosophers have understood the production, circulation and reception of popular culture, and how it bears on our own autonomy, agency, power, and identity. You will study themes which may include: Philosophical approaches to mass culture, cultural value, art, and aesthetic judgement Authorship, mass production, genre, kitsch, remix, and the ontology of reproducible artwo

Interconnected global crises and states of ‘polycrises’ or ‘wicked problems’ impact upon the daily lives of millions of people across the globe. Environmental, financial, security, diplomatic, political and military concerns all pose acute problems of knowledge and understanding, require individual and collective action, and raise questions around duties and rights for addressing multi-faceted complex problems. Philosophical reasoning can play a key role in helping individuals, politicians, states and societies navigate these challenges, and in shaping and critiquing the principles for taking action. In this module you will actively work on developing your own philosophical contributions to

In this module you will develop either a single extended piece or a portfolio of independent philosophical work. This may take the form of either (1) a philosophical dissertation or (2) a communication and engagement portfolio of outward-facing philosophical work targeting a diverse range of audiences. 1) Dissertation Independent research and sustained long-form writing making up the core of professional academic philosophical study. With the Dissertation option, you have the opportunity to demonstrate your research and writing ability via completion of an independent dissertation project. You will identify a specific philosophical topic from the wide range of research specialisms within Phi

Engage with cutting-edge philosophical research, working with an academic philosopher on the topic of their live philosophical project and expertise. In this module you may be: Reviewing and critically commenting on the chapters of a manuscript Reading an academic’s recent publications and coming up with further questions and challenges to build on their ideas Discussing your module supervisor's new research and it's relevancy to contemporary philosophical debates In student-led and discussion-based workshops you will present your own summaries and arguments, take part in guided debates, and work on the challenge of your final written piece: a paper which is not only informed by, but may its

Study the structure of intricate mathematical objects, such as groups and rings, by looking at linear approximations of them. Linear approximation is such a fundamental idea that it extends throughout mathematical sciences, cropping up in quantum physics and topological data analysis. Explore representations of finite groups before passing to algebras and modules, which are ‘vector spaces’ over rings. You will look at the atomic theory of representations: the simple and indecomposable representations that are their building blocks. Can we describe all the building blocks? Attempting to answer this leads us to complete reducibility for representations of finite groups (Maschke's theorem) and

Engage with a significant philosophical text or collection of texts, working with an academic philosopher on the topic of their live philosophical project and expertise. In this module you will: Read deeply Develop interpretations Make reasoned assessments Find and engage with secondary literature Contribute to contemporary understanding and critique of your text(s) In student-led and discussion-based workshops, you will present your own philosophical interpretations and arguments, take part in guided debates, and work on a portfolio of critical readings. In doing so you will be joining the practice, shared by all professional philosophers, of contributing to the understanding and developmen