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 Mathematics and Philosophy BA Hons (GV15)
Mathematics and Philosophy BA Hons  2018 Entry
UCAS Code
GV15
Entry Year
2018
A Level Requirements
AAB
see all requirements
see all requirements
Duration
Full time 3 Year(s)
Course Overview
Uncover the fundamental workings of the universe and develop a highlevel of reasoning through our exciting and challenging programme. While studying mathematics and philosophy, you will gain a wealth of skills, knowledge and experience, preparing you for your chosen career.
Maths and philosophy are both difficult to concisely define, but at their core, they are concerned with the underlying workings and meaning of the universe. Maths is the study of change, patterns, quantities, structures and space, while philosophy is concerned with fundamental problems in topics such as knowledge and reason.
Over the three years, you will be able to draw on expertise from two specialist departments: Mathematics and Statistics; and Politics, Philosophy and Religion. This is an engaging programme of study and our reputation for excellence in research means that we are able to offer highquality teaching delivered by academics who are leaders in their field.
In first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, problemsolving and quantitative reasoning skills.
During this year, you will also receive an introduction to philosophy, which will provide you with insight into some of the central problems of the discipline and the theories produced in response to them, as well as technical concepts, vocabulary, and some of the techniques of reasoning and analysis associated with the discipline.
In the second year, you will further develop your knowledge in complex and real analysis, and abstract and linear algebra. These highly analytical topics will complement your study of philosophy, which will really begin to expand and develop this year. You will be able to choose from a range of philosophy modules, such as Epistemology, Metaphysics, Modern Political Thought, and Philosophy of Science. This will allow you to build a solid repertoire of philosophy knowledge and analytical skills, while gearing your studies towards your own interests and aspirations.
Our final year offers you the chance to choose from a wide range of specialist modules, allowing you to develop the programme further to suit your interests and guide you to a specific career pathway. These topics include: Combinatorics; Lebesgue Integration; and Representation Theory of Finite Groups, as well as exciting philosophy modules such as Aesthetics, Continental Philosophy, and Moral, Legal and Political Philosophy.
Entry Requirements
Grade Requirements
A Level AAB including A level Mathematics or Further Mathematics OR ABB including A level Mathematics and Further Mathematics
IELTS 6.5 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 35 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics grade A and Further Mathematics grade A
Access to HE Diploma May occasionally be accepted
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
This module provides the student with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. We introduce examples of functions and their graphs, and techniques for building new functions from old. We then consider the notion of a limit and introduce the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and be introduced to rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
Students are introduced to the basic ideas and notations involved in describing sets and their functions. The module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, we can say that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? We develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
Rather than counting objects, we might be interested in connections between them, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. We see how partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes. Students will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

Geometry and Calculus
The main focus of this course is vectors in two and threedimensional space. We start off with the definition of vectors and we see some applications to finding equations of lines and planes. We then consider some different ways of describing curves and surfaces via equations or parameters, and we use partial differentiation to determine tangent lines and planes, as well as using integration to calculate the length of a curve.
In the second half of the course, we study functions of several variables. When attempting to calculate an integral over one variable, we often substitute one variable for another more convenient one; here we will see the equivalent technique for a double integral, where we have to substitute two variables simultaneously. We also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, we discuss how to calculate the areas of various surfaces and the volumes of various solids.

Integration and Differentiation
Building on MATH113, this module explores the familiar topics of integration, series and differentiation, and develops them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: “Can you sum up infinitely many numbers and get a finite number?”. Students will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Introduction to Philosophy
This module introduces students to some of the central problems of philosophy and the theories produced in response to them. It also introduces some of the subject's technical concepts and vocabulary, and some of its techniques of reasoning and analysis. Reading includes both classical and contemporary material.
Philosophy has a significant role to play, both in acquainting students with some of the ideas which have helped shape Western culture, and in the critical understanding of ideas and methods in many other disciplines. The level of the module does not presuppose previous knowledge of philosophy. If students have studied philosophy before, the module will enable them to deepen and broaden their understanding of the subject and to improve their philosophical skills. The module aims not only to inform students with what philosophers have said but also to encourage them to engage with the issues. Topics will be drawn from the range of philosophical problems, approaches, and canonical figures.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic equation and eigenvectors and eigenvalues.

Numbers and Relations
This module introduces the student to logic and mathematical proofs, with emphasis placed on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
We take a look at the language and structure of mathematical proofs in general, emphasising how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas are then applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

Statistics
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society and then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, and this underpins the skills needed for all subsequent statistical modules of the degree.
Core
Year 2

Abstract Algebra
This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here you’ll select a small number of properties which these and other examples have in common, and use them to define a group.
You’ll also consider the elementary properties of groups. It turns out that several surprisingly elegant results can be proved fairly simply! By looking at maps between groups which 'preserve structure' you’ll discover a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same'.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives us a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials, but you’ll meet several less familiar examples too.

Complex Analysis
Complex Analysis has its origins in differential calculus and the study of polynomial equations.
In this module you’ll consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. You’ll use integral calculus of complex functions to find elegant and important results, including the fundamental theorem of algebra, and you’ll also use classical theorems to evaluate real integrals.
The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Linear Algebra II
This module will give you the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
You’ll consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of your study will also involve looking at the concepts of length and angle with regard to vector spaces.

Real Analysis
In this module you’ll take a thorough look at the limits of sequences and convergence of series. You’ll learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
You’ll spend time studying the Intermediate Value Theorem and the Mean Value Theorem, and will discover that they have many applications of widely differing kinds. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next we put the notion of integration under the microscope. Once it’s properly defined (via limits), you’ll learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. You’ll also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.
Core

Constructing Ethics: Christianity and Islam
This module explores the emergence and construction of ethics within the context of two world religions: Christianity and Islam. It examines the ways in which religious attitudes to ethical concern and practice are influenced by traditional, textual and cultural factors.
Some of the ethical concerns to be covered throughout the module are: politics and economics; justice and war; sex and sexual practice; and rights and law. Finally, the module will encourage students to explore some of these areas crossculturally through the consideration of questions of difference and otherness.

Epistemology
The aim of this module is to provide students with a good, broad introduction to some of the key themes in epistemology the theory of knowledge.
It begins with a core question; What is knowledge? This leads on to questions about how knowledge relates to other things, like belief, and truth. Throughout the term students will see that it is much harder to answer the core question than one might initially think, raising a question of why it is so hard to give a clear and general, account of what knowledge is. Students will also look at sources of knowledge  especially, perception, selfknowledge and testimony. The module also explores some of the relationships between epistemology and ethics, ending with the question of whether we ever ought to refrain from seeking knowledge.
By the end of this module, students will be able to understand and discuss critically the central problems and theories of epistemology, and explain how epistemology relates to other areas of philosophy.

Ethics: Theory and Practice
This module aims to provide students with an understanding of some historical and contemporary approaches to the subject of ethics. It addresses central issues by engaging with classical texts in the history of the subject, such as Aristotle’s Nicomachean Ethics, David Hume’s Enquiry Concerning the Principles of Morals, Immanuel Kant’s Groundwork for the Metaphysics of Morals and John Stuart Mill’s Utilitarianism.
The module will also explore selected topics in moral philosophy, such as the nature, strength and weakness of consequentialism, deontology, and virtue theory. In addition to this, students will study topics in metaethics, such as the ‘moral problem’, noncognitivist realism, and quasirealism.
Other topics covered include topics in applied and practical ethics, such as issues of life and death in biomedical practice, the ethics of war, and the ethics of personal life; as well as the nature of moral motivation and moral psychology.

History of Philosophy
Western philosophy has a long and rich history, and many of the questions occupying presentday philosophers have been around for hundreds or even thousands of years.
The exact structure of this module may vary from year to year, but core themes will normally include:
 What is the nature of the mind? How does it relate to the body?
 What is the nature of human knowledge? Can we come to have any reliable knowledge of the world outside our minds?
 Is there a God? What arguments have been made in philosophy historically for the existence of God? How successful are these arguments?
 What is fundamentally real? Minds? Matter? Ideas?
Students will study these problems, amongst others, by close consideration of a selection of texts from the history of Western philosophy. This may include selections from the ancient (classical), medieval, early modern (17th/18th centuries) period, and the 19th century. Thinkers who may be considered include Plato, Aristotle, Augustine, Scotus, Descartes, Locke, Berkeley, Hume, Kant, Hegel, and Nietzsche.

Metaphysics
This module is designed to improve students' knowledge and understanding of some key issues in metaphysics as determined by the syllabus. The module will focus primarily on some issues concerning space and time, the nature of physical objects and persons, and some key philosophical distinctions. Topics will include:
 A priori and empirical; analytic and synthetic; necessary and contingent. Some basic distinctions are examined and compared.
 The status of geometry and the metaphysics of space
 The nature of physical objects; 3D versus 4D treatments
 Personal identity
 The reality or unreality of time.
Studying this module should enable students to see connections between various philosophical issues that should be of value to them with regard to other philosophy modules that they are studying.

Modern Political Thought
The aim of this module is to provide a broad grounding in some important aspects of the discipline of politics that are conceived of as both an attempt to understand the nature of politics and to assess the worth of various political arrangements. It involves consideration of notions such as politics, citizenship, democracy, government, state, welfare, individualism, utilitarianism, conservatism, socialism and, social democracy, together with an examination of the various ways in which political studies have been understood as a disciplined investigation of things political. The module covers four broad topics: freedom, markets and the state; citizenship, nationalism and democracy; equality and welfare; and politics and political science.
The module is divided into two sections over two terms. In the first term students will read, examine and discuss thinkers who make a contribution to the understanding of the notions of liberty and the individual (Hobbes, Locke, J S Mill, and Hayek). In the second term students will explore the thought of thinkers who are associated with the ideas of equality and community (Rousseau, Marx, the Fabians, and Rawls).

Philosophical Questions in the Study of Politics and Economics
This module considers some of the difficulties involved in gaining knowledge about human societies. It focuses especially on economics and politics, disciplines which raise some of the largest questions about society – for example: Who gets what? Who rules whom? Can individual choices generate social change?
In this module students will not address such questions empirically, but instead step back to ask what sort of methods have been used to answer them, what sorts of modes of explanation or understanding are appropriate, and what assumptions are built into the ways economists and political scientists frame their enquiries. The aim of the module, then, is to critically examine methods and assumptions in both disciplines, in order to appreciate the scope and limits of their claims to knowledge.

Philosophy of Science
This module considers philosophical issues that arise in both the natural and social sciences. With regard to the natural sciences, students will consider traditional accounts of scientific method and theorytesting, then examining philosophical challenges to the status of science as a rational form of enquiry. Particular consideration is given to four of the most important twentiethcentury philosophers of science: Popper, Kuhn, Lakatos and Feyerabend.
With regard to the social sciences, the module will ask whether endeavours such as sociology, economics, anthropology and history should really be counted as sciences, and then consider some of the special issues that arise in the study of human society. For example, how are we to understand other societies (for instance, in anthropology)? What is the place for individualism versus collectivism in social explanation (for example, in sociology and history)? What is the scientific status of social models based on postulates of rational choice (for example, in economics and politics)?
No scientific background is assumed on this module.

Philosophy of the Mind
This module aims to introduce students to a wide range of connected topics in the theory of knowledge, philosophy of mind and philosophy of language, drawing on both classical and contemporary writings. It examines issues such as: the nature and justification of our knowledge of the external world, and the relations between knowledge and belief; the mindbody (or mindbrain) problem; the place of mental life and bodily continuity in the identity of individuals; and the different theories of truth, meaning and the languageworld relationship, including logical positivism.
This module begins by examining issues in the metaphysics of mind, before moving on to epistemological issues: How can we gain knowledge of our own mental states, or of other people’s? How should psychologists seek to investigate the mind?
For the most part, this module will be structured around contemporary texts.

Western Philosophy and Religious Thought
This module aims to encourage students to think philosophically about religious issues. Using the work of both classical and contemporary philosophers and religious thinkers, it addresses some of the central philosophical questions raised by religious belief. In addition, students will be encouraged to think historically ad contextually, in order to understand the ways in which the role of philosophy in relation to religion in the west has changed over time.
The module introduces students to the work of some of the most important philosophers from Plato to Wittgenstein and the implications of their thought for religion. It will also address themes and issues which may vary from year to year but will be drawn from the following: the nature of theism; immortality; the problem of evil; religious experience; and the implications of postmodern thought for religious belief.
Optional
Year 3

Aesthetics
This module introduces central issues, problems and theories in philosophical aesthetics by critically examining specific topics in the philosophy of art, and by examining the theories of major figures who have contributed to the tradition of philosophical aesthetics. The module uses concrete examples from most of the arts, including painting, literature, film, and music, to illuminate theoretical debates and issues.
Topics and major aesthetic theorists covered may include the following (note that this list is not exhaustive and indicative only, not all topics will be covered) :
 Aesthetics in the analytic and continental traditions of philosophy
 Aesthetic theories of Plato, Hume, Kant, Schopenhauer, Nietzsche, and others
 Beauty and its definition
 The relations between art, religion and philosophy
 Art and morality
 The culture industry and its impact on our responses to art
 Disinterestedness
 The relations between aesthetics and politics: Should art be politically committed? If so, in what ways?

Applied Philosophy
This module focuses on selected topics in Applied Philosophy. It involves the application of philosophical methods and knowledge to a range of problems that face institutions, professions, policymakers and regulatory bodies. Further questions arise about the nature and limits of Applied Philosophy.
Examples of topics that may be studied include:
 Philosophy of privacy and data protection
 Philosophical bioethics
 Philosophy of the media
 Philosophy and psychiatric classification
 Applied epistemology
 Selected topics in medical ethics
 Philosophy of education
 The metaphilosophy of applied philosophy
 Applied philosophy of language
 Applied philosophy of science and technology

Combinatorics
Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous (for example graphs, lattices, designs and codes). While combinatorics is a huge subject  with many important connections to other areas of modern mathematics  it is a very accessible one.
In this module you’ll be introduced to the fundamental topics of combinatorial enumeration (sophisticated counting methods), graph theory (graphs, networks and algorithms), and combinatorial design theory (Latin squares and block designs). You’ll also explore important practical applications of the results and methods.

Continental Philosophy
This module aims to introduce the work of some key figures in 19th and 20th century continental philosophy, such as Hegel, Marx, Kierkegaard, Nietzsche, Heidegger, Foucault, Hannah Arendt and Habermas. The approach taken is predominantly philosophical rather than historical, and will involve critically examining claims and arguments about such matters as the existence and nature of human freedom, the relationships between knowledge, truth, power and morality, alienation and human labour, and the possibility of mutual recognition and community. It is expected that students will engage with the original texts, formulate the central arguments to be found in them and assess their cogency.
The module begins by looking at Nietzsche’s Toward a Genealogy of Morality, before turning to Foucault, who adapts Nietzsche’s method of historical analysis in order to challenge assumptions about progress toward freedom and welfare in modern societies. Finally students will study Arendt and her political thought on totalitarian politics using a parallel method of historical analysis.

Darwinism and Philosophy
This module will examine philosophical issues that arise in connection with specific sciences, in particular biology and medicine, as opposed to the general philosophy of science.
The following topics will be covered:
 Methodological issues in biology and medicine
 Issues of classification and kinds in biology and medicine – e.g. what is a species, what is a gene, what is a disease?
 Issues to do with the relationship between these sciences and others, in particular the issue of intertheoretic reduction, i.e. can biology be reduced to physics and chemistry?
 Radical critiques of mainstream biology and medicine – e.g. political critiques of sociobiology, the antipsychiatry movement.
 Historical issues the development of these sciences.

Differential Equations
This module considers questions relating to linear ordinary differential equations. While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations as well as allowing us to study certain properties of these solutions.

Dissertation
This module provides an opportunity for students to choose a topic related to some aspect of Politics and International Relations, Philosophy and Religious Studies which particularly interests them, and to pursue it in depth. The topic may be related to work that is being done on a formally taught course, or it may be less directly linked to course work. The intention is that students will develop their research skills and their ability to work at length under their own direction.
Students are expected to start thinking seriously about the 9,00010,000 word dissertation towards the end of the Lent term of their second year, and to submit a provisional topic by the end of that term. Work should be well advanced by Christmas in the third year. The completed dissertation must be submitted by the end of the Lent term in the third year.

Dissertation with external collaboration
This module aims to allow students to pursue independent indepth studies of a topic of their choice, within the scope of their scheme of study. The topic may be related to work that is being done on a formally taught course, or it may be less directly linked to course work.
Students will develop their employability and research skills, and their ability to work independently at length under their own direction with input from an academic supervisor.
The external collaboration will enhance students’ ability to reflect on the impact of academic work. One option is to incorporate work done through the Richardson Institute Internship Programme, but students may also discuss other forms of collaboration with their supervisor.

Elliptic Curves
This module gives you a solid foundation in the basics of algebraic geometry. You’ll explore how curves can be described by algebraic equations, and learn how to understand and use abstract groups in dealing with geometrical objects (curves).
You’ll also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally you’ll learn to perform algebraic computations with elliptic curves.

Financial Mathematics
This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.
Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and BlackScholes models.
Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:
 Conditional expectation
 Filtrations
 Martingales
 Stopping times
 Brownian motion
 BlackScholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated riskneutral probability; performing calculations with the BlackScholes formula; and proving various steps in the derivation of the BlackScholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.
In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the realworld; improve their written and oral communication skills; and develop their critical thinking.

Future generations
What moral obligations do we have towards future generations to those yet to be born, and to people whose very existence (or nonexistence) depends on how we act now?
This module explores this question by examining both a series of practical case studies and some of the main concepts and theories philosophers use when thinking about these issues.
Questions considered include, among a range of others:
 How should we weigh quality against quantity of life?
 Ought we to try significantly to extend the human lifespan (to 150 years or beyond)?
 Is there a moral obligation to refrain from having children, or one to have (more) children?
 Should we use selection techniques to minimise the incidence of genetic disorders and disabilities in future populations? Should parents be allowed to determine the genetic characteristics of their future children?
 How should the interests of nonhuman creatures be weighed against those of humans? How strong are our moral obligations to prevent extinctions, and to preserve wildernesses?

Geometry of Curves and Surfaces
This module is an introduction to smooth curves and surfaces in threedimensional space. You’ll encounter various geometrical properties of these objects, such as length, area, torsion and curvature, and will have the opportunity to explore the meaning of these quantities. You’ll use a variety of examples to calculate their values, and will use them to apply techniques from calculus and linear algebra.

Graph Theory
The study of graphs  mathematical objects used to model pairwise relations between objects  is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.
Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.
While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation.

Groups and Symmetry
In this module you’ll develop the knowledge of groups that you’ve gained in second year. You’ll study ‘direct products’ which are used to construct new groups, while any finite group is shown to ‘factor’ into ‘simple’ pieces. You’ll also consider situations in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.

Hilbert Space
In this module you’ll examine the notion of a norm, which introduces a generalized notion of ‘distance’ to the purely algebraic setting of vector spaces. You’ll learn several quite natural ways to do this, both for vectors of any dimension and for functions. You’ll then focus on the more special notion of an inner product which generalizes angles at the same time as distances.
Once we’ve established these concepts, you’ll have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how you can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, you’ll look at some of the main results of linear algebra, which generalize very nicely to linear operators between Hilbert spaces.

History of Twentieth Century Philosophy
This module focuses upon some key aspects of the history of 20th Century Philosophy.
The module begins by examining a revolution in philosophy at the very start of the 20th century with the origins of analytic philosophy. It then focuses on Wittgenstein’s radical philosophy (or antiphilosophy). Wittgenstein’s own philosophical development brings to the fore a deep schism, or tension, that has existed throughout the century’s philosophy, one which lays between those who hold that philosophy should align itself with natural science and mathematics, and those who reject this view. Students will examine whether philosophy should seek to emulate the natural sciences and illustrate the tension between scientistic and humanistic philosophy via mid20th century debate about the nature of historical explanation.
The final lectures look at the distinction between analytic and continental philosophy in the 20th century, and upon the emergence of applied philosophy later in the century, asking whether philosophy can ever really be applied to reallife problems.

Indian Religious and Philosophical Thought
This module will introduce major themes and issues in Indian philosophy, focusing on the Hindu and Buddhist philosophical traditions. Beginning with philosophical sections in the Upanishads and the dialogues of the Buddha, the module will trace the development of Indian philosophy from the early to the classical periods. Various ethical, metaphysical, and epistemological concepts will be covered, such as: order and virtue (dharma), consequential action (karma), ultimate reality (Brahman), the nature of the self (atman), the highest good (moksha), and the means for attaining knowledge (pramana).
Throughout the module, students will look at the dialogical relationship between the Hindu and Buddhist philosophical traditions, particularly the shared practice of debate.

Issues in the Philosophy of Mind
This module will introduce students to some advanced topics in the philosophy of mind. Through the debates examined students will be exposed to a number of methodological approaches in the philosophy of mind  including the use of empirical evidence in philosophy, conceptual analysis, ordinary language philosophy and thought experimentation.
Topics examined will vary from year to year but may include:
 Consciousness
 Understanding other minds
 Selfknowledge
 Emotions
 Understanding abnormal mental states
 The self
 Perception
 Evolutionary psychology
 Animal, alien, and computer minds
 Mental causation

Lebesgue Integration
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.

Liberals and Communitarians
This module examines the central debates about politics and justice between liberals and communitarians in contemporary AngloAmerican analytic philosophy. Whereas liberals stress the importance of the individual and the need for them to pursue their own good in their own way, communitarians stress the embedded, interconnected, and social nature of the persons and politics.
The module asks three major questions. Firstly what it means to be engaged in political theory. Secondly, how the idea of justice should be understood, and finally, what implications does our view of justice have for our political arrangements?
The module is divided into two main sections. Concentrating first on the central figure of this debate: John Rawls and his seminal work A Theory of Justice. Then looking at how the debate has widened, initially looking at the libertarian criticisms raised by Nozick before moving on to consider the communitarian positions advanced by Sandel, Walzer, Okin, and Pateman; finally considering alternative forms of liberalism offered by Raz, Rorty, and Gray.

Linear Systems
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics courses and provide opportunities for further study. The theory of Linear systems is engineering mathematics.
In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.
Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The course shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables us to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.
The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilize a (A,B,C,D) system.

Logic and Language
This module provides an introduction to formal logic together with an examination of various philosophical issues that arise out of it. The syllabus includes a study of the languages of propositional and quantificational logic, how to formalize key logical concepts within them, and how to prove elementary results using formal techniques.
Additional topics include identity, definite descriptions, modal logic and its philosophical significance, and some criticisms of classical logic.

Metric Spaces
This module gives an introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications. Studying this module will give you a deeper understanding of continuity as well as a basic grounding in abstract topology.
You’ll also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply your knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

Modern Christian thought
This module, for the most part, concentrates on (Protestant) Christian thinkers from the Germanspeaking world. These thinkers have dominated the development of Christian thought in Europe and America until very recent times, when various 'political theologies' (Black, feminist and liberationist) started to erode their influence.
The point of departure on this module must be the Enlightenment and its definitive philosopher  Immanuel Kant. The module begins, therefore, by looking at the challenges facing early nineteenth century theologians, consider the responses of five major Christian thinkers of the nineteenth and twentieth centuries and shall end by exploring the challenges facing Christian thought today.

Modern Religious and Atheistic Thought
This module will examine some of the major debates in religious and atheistic thought, looking in particular at the way in which these debates are framed by a specifically modern epistemological framework, and the ways in which religious thought and atheistic thought might be though to be mutually constitutive and mutually implicated rather than simply oppositional.
The aim of this module is to examine and evaluate some of the most central issues in Enlightenment and postEnlightenment Western religious and atheistic philosophical debates. The module will begin by looking the philosophy of G W F Hegel and its implications for subsequent religious and atheistic thought. It will then proceed to consider the thought of the postHegelian masters of suspicion: Feuerbach, Marx, Freud and Nietzsche. After this, it will look at ways in which religious and atheistic thought have been brought together, as manifested in various forms of Christian atheism. Finally, it will consider postmodern critiques of modern atheism and the nature of the associated return of religion.

Moral, legal and political philosophy
This module will address central issues in contemporary ethical (including metaethical), legal and political philosophy, and will allow a systematic critical exploration of the connections between ideas and arguments in each of the three areas of the subject.
Topics covered will include some of the following: modern theory of moral motivation, value theory, contractualism, the 'moral problem'; responsibility and criminal liability, the justification of punishment, the proper scope of the law; democratic theory, egalitarianism, justice, nationalism, multiculturalism, liberty and human rights.

Multivariate Statistics in Machine Learning
Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.
An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise highdimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.

Number Theory
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?
Questions like these are easy to ask, and to describe to the nonspecialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

Philosophy of Medicine
Are psychopaths evil or sick? Should the NHS pay for the treatment of nicotine addiction? Is it right for shy people to take characteraltering drugs?
Whether a condition is considered a disease often has social, economic and ethical implications. It tends to be taken for granted that what it is to be healthy can be identified and is desirable. Similarly, it is assumed that those who are diseased or disabled can be diagnosed and require help. In this module we question these assumptions via examining the key concepts of normality, disease, illness, mental illness, and disability.

Philosophy of the Human Sciences
This module considers key philosophical issues in the sciences of human societies and social structures, such as sociology, economics or history.
As well as considering whether these subjects should be considered as sciences the module looks at a number of philosophical issues, such as those arising in the understanding of other societies (for instance, in anthropology), individualism versus collectivism in social explanation (for example, in sociology and history), and the scientific status of social models based on postulates of rational choice (for example, in economics and politics).

Political Ideas
This module examines central themes in the liberal branch of contemporary AngloAmerican analytic political philosophy. The liberal positions on justice, liberty, equality, the state, power, rights and utility are all explored. The approach is philosophical rather than applied; focusing on the ideas of liberal politics: how individual liberty can be maximised while not harming others; how an individual philosophical position can guide political determinants of a society and places the developments of liberal ideas in their appropriate historical contexts.
The module also examines the connection between the ideas of liberalism and the idea of democracy to explore the philosophical tensions between the two and how these might be resolved.
The module will include among other topics: questions about justice: analytic philosophy and liberalism; visions of the state: liberalism, republicanism, socialism; liberty and individuality; liberalism and democracy; negative and positive liberty; equality; utility and rights; and toleration and multiculturalism: responses to diversity.

Politics and Ethics in Indian Philosophy
This module will look at Indian source texts on politics and ethics. In particular, it will be looking at sources that explore the concept of dharma, a term that incorporates issues of justice, religion, ethics, duty, and law. The module will examine the sources of dharma both in their own historical and cultural contexts, as well as in the context of contemporary debates in political theory and ethics. The texts examined will include: the inscriptions of Ashoka, the Buddhist Nikayas, the Arthashastra, the Law Codes of Manu, the Mahabharata, and the Kamasutra. These sources are examined in connection with modern political figures, such as Gandhi and Savarkar, as well as in connection with recent debates in India about secularism, democracy and pluralism.

PPR in Education
This module is designed to allow students to gain experience of educational environments, to develop transferable skills, and to reflect on the role and communication of their own discipline. The module is organised and delivered collaboratively between Lancaster University Students’ Union LUSU Involve, the school/college where the placement is based, and the department.
The module will give students experience of classroom observation and experience, teacher assistance, as well as teaching small groups (under supervision). In particular, the module will not only give students the opportunity to observe and experience teaching and learners for themselves, it will also require them to reflect on how their own subject area (Religion, Politics and International Relations, or Philosophy) is experienced by learners, delivered in other parts of the educational sector, and applied in a classroom setting. Students will also be asked to reflect on how teaching and learning at this earlier level combines with what is taught and promoted at the level of Higher education (as experienced in the University).

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First you’ll examine the notion of a probability space through simple examples featuring both discrete and continuous sample spaces. You’ll then use random variables and the expectation to develop a probability calculus, which you can apply to achieve laws of large numbers for sums of independent random variables.
You’ll also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Reading Political Theory
Students will study the thought of two seminal thinkers in political theory. This module provides an opportunity to explore texts slowly, methodically and in depth, allowing students to link that thought to wider literature that has developed as a response to the thinkers' ideas, and see how those ideas linkup into a wider systematic and philosophic whole.
Topics include among many others:
 Plato: forms and the gooditself, the structure of the kallipolis, the lies of the rulers and censorship, women, children, and slaves, freedom and totalitarianism.
 Aristotle: links between politics and ethics, community and the political animal, and natural law
 Kant: freedom and autonomy, roles of the individual, the role of reason, cosmopolitanism, government by agreement, censorship and the state, enlightenment.
 Nietzsche: early and later political philosophy, the role of state, the death of God, the slave revolt in morals, the Ubermensch, sovereign individuals, radical aristocracy.
 Marx: the early Marx, Hegel, speciesbeing, alienation, historical materialism, bourgeois philosophy, critique of the modern state, the communist alternative.

Representation Theory of Finite Groups
This module covers the basics of ordinary representation theory. You’ll learn the concepts of Rmodule and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups. You’ll also learn to perform computations with representations and morphisms in a selection of finite groups.

Rings Fields and Polynomials
This module furthers your knowledge of commutative rings from your second year study.
You’ll be introduced to two new classes of integral domains called Euclidean domains, where you have a counterpart of the division algorithm, and unique factorization domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
You’ll also explore how wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorization of polynomials, carry over to this new setting.

Special Subject: The Imagination
This module will examine philosophical accounts of the imagination. It will look at theories of the nature of the imagination and its connections to other mental states, such as attention, emotion, memory, beliefs, intentions, and desires.
In addition, a range of topics focusing on the role of imagining in a number of different domains will also be explored, including moral judgement, practical reasoning, perception, pictorial experience, and modal thought.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.
Careers
Careers
Maths and philosophy graduates are highly employable, having indepth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including analysis and manipulation, interpretation, logical thinking, problemsolving, and reasoning, as well as adept knowledge of the disciplines. As a result, combining these two subjects opens up a range of opportunities and graduates are highly sought after.
The starting salary for many of these roles is highly competitive, and popular career options include:
 Actuary
 Data Analyst
 Investment Analyst
 Research Scientist
 System Developer
 Teacher
A degree in this discipline can also be useful for roles such as Barrister, Local Government Office, Psychotherapist, Stockbroker, and many more.
Alternatively, you may wish to undertake postgraduate study at Lancaster and pursue a career in research and teaching.
Fees and Funding
Fees
We set our fees on an annual basis and the 2018/19 entry fees have not yet been set.
As a guide, our fees in 2017 were:
UK/EU  Overseas 

£9,250  £16,620 
Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.

Course Overview
Course Overview
Uncover the fundamental workings of the universe and develop a highlevel of reasoning through our exciting and challenging programme. While studying mathematics and philosophy, you will gain a wealth of skills, knowledge and experience, preparing you for your chosen career.
Maths and philosophy are both difficult to concisely define, but at their core, they are concerned with the underlying workings and meaning of the universe. Maths is the study of change, patterns, quantities, structures and space, while philosophy is concerned with fundamental problems in topics such as knowledge and reason.
Over the three years, you will be able to draw on expertise from two specialist departments: Mathematics and Statistics; and Politics, Philosophy and Religion. This is an engaging programme of study and our reputation for excellence in research means that we are able to offer highquality teaching delivered by academics who are leaders in their field.
In first year, you will build on your previous knowledge and understanding of mathematical methods and concepts. Modules cover a wide range of topics from calculus, probability and statistics to logic, proofs and theorems. As well as developing your technical knowledge and mathematical skills, you will also enhance your data analysis, problemsolving and quantitative reasoning skills.
During this year, you will also receive an introduction to philosophy, which will provide you with insight into some of the central problems of the discipline and the theories produced in response to them, as well as technical concepts, vocabulary, and some of the techniques of reasoning and analysis associated with the discipline.
In the second year, you will further develop your knowledge in complex and real analysis, and abstract and linear algebra. These highly analytical topics will complement your study of philosophy, which will really begin to expand and develop this year. You will be able to choose from a range of philosophy modules, such as Epistemology, Metaphysics, Modern Political Thought, and Philosophy of Science. This will allow you to build a solid repertoire of philosophy knowledge and analytical skills, while gearing your studies towards your own interests and aspirations.
Our final year offers you the chance to choose from a wide range of specialist modules, allowing you to develop the programme further to suit your interests and guide you to a specific career pathway. These topics include: Combinatorics; Lebesgue Integration; and Representation Theory of Finite Groups, as well as exciting philosophy modules such as Aesthetics, Continental Philosophy, and Moral, Legal and Political Philosophy.

Entry Requirements
Entry Requirements
Grade Requirements
A Level AAB including A level Mathematics or Further Mathematics OR ABB including A level Mathematics and Further Mathematics
IELTS 6.5 overall with at least 5.5 in each component. For other English language qualifications we accept, please see our English language requirements webpages.
Other Qualifications
International Baccalaureate 35 points overall with 16 points from the best 3 Higher Level subjects including 6 in Mathematics HL
BTEC May be accepted alongside A level Mathematics grade A and Further Mathematics grade A
Access to HE Diploma May occasionally be accepted
STEP Paper or the Test of Mathematics for University Admission Please note it is not a compulsory entry requirement to take these tests, but for applicants who are taking any of the papers alongside Mathematics or Further Mathematics we may be able to make a more favourable offer. Full details can be found on the Mathematics and Statistics webpage.
We welcome applications from students with a range of alternative UK and international qualifications, including combinations of qualification. Further guidance on admission to the University, including other qualifications that we accept, frequently asked questions and information on applying, can be found on our general admissions webpages.
Contact Admissions Team + 44 (0) 1524 592028 or via ugadmissions@lancaster.ac.uk

Course Structure
Course Structure
Many of Lancaster's degree programmes are flexible, offering students the opportunity to cover a wide selection of subject areas to complement their main specialism. You will be able to study a range of modules, some examples of which are listed below.
Year 1

Calculus
This module provides the student with an understanding of functions, limits, and series, and knowledge of the basic techniques of differentiation and integration. We introduce examples of functions and their graphs, and techniques for building new functions from old. We then consider the notion of a limit and introduce the main tools of calculus and Taylor Series. Students will also learn how to add, multiply and divide polynomials, and be introduced to rational functions and their partial fractions.
The exponential function is defined by means of a power series which is subsequently extended to the complex exponential function of an imaginary variable, so that students understand the connection between analysis, trigonometry and geometry. The trigonometric and hyperbolic functions are introduced in parallel with analogous power series so that students understand the role of functional identities. Such functional identities are later used to simplify integrals and to parametrise geometrical curves.

Convergence and Continuity
This module provides a rigorous overview of real numbers, sequences and continuity. Covering bounds, monotonicity, subsequences, invertibility, and the intermediate value theorem, among other topics, students will become familiar with definitions, theorems and proofs.
Examining a range of examples, students will become accustomed to mathematical writing and will develop an understanding of mathematical notation. Through this module, students will also gain an appreciation of the importance of proof, generalisation and abstraction in the logical development of formal theories, and develop an ability to imagine and ‘see’ complicated mathematical objects.
In addition to learning and developing subject specific knowledge, students will enhance their ability to assimilate information from different presentations of material; learn to apply previously acquired knowledge to new situations; and develop their communication skills.

Discrete Mathematics
Students are introduced to the basic ideas and notations involved in describing sets and their functions. The module helps students to formalise the idea of the size of a set and what it means to be finite, countably infinite or uncountably finite. For finite sets, we can say that one is bigger than another if it contains more elements. What about infinite sets? Are some infinite sets bigger than others? We develop the tools to answer these questions and other counting problems, such as those involving recurrence relations, e.g. the Fibonacci numbers.
Rather than counting objects, we might be interested in connections between them, leading to the study of graphs and networks – collections of nodes joined by edges. There are many applications of this theory in designing or understanding properties of systems, such as the infrastructure powering the internet, social networks, the London Underground and the global ecosystem.

Further Calculus
This module extends the theory of calculus from functions of a single real variable to functions of two real variables. Students will learn more about the notions of differentiation and integration and how they extend from functions defined on a line to functions defined on the plane. We see how partial derivatives help us to understand surfaces, while repeated integrals enable us to calculate volumes. Students will also investigate complex polynomials and use De Moivre’s theorem to calculate complex roots.
In mathematical models, it is common to use functions of several variables. For example, the speed of an airliner can depend upon the air pressure, temperature and wind direction. To study functions of several variables, we introduce rates of change with respect to several quantities. We learn how to find maxima and minima. Applications include the method of least squares. Finally, we investigate various methods for solving differential equations of one variable.

Geometry and Calculus
The main focus of this course is vectors in two and threedimensional space. We start off with the definition of vectors and we see some applications to finding equations of lines and planes. We then consider some different ways of describing curves and surfaces via equations or parameters, and we use partial differentiation to determine tangent lines and planes, as well as using integration to calculate the length of a curve.
In the second half of the course, we study functions of several variables. When attempting to calculate an integral over one variable, we often substitute one variable for another more convenient one; here we will see the equivalent technique for a double integral, where we have to substitute two variables simultaneously. We also investigate some methods for finding maxima and minima of a function subject to certain conditions.
Finally, we discuss how to calculate the areas of various surfaces and the volumes of various solids.

Integration and Differentiation
Building on MATH113, this module explores the familiar topics of integration, series and differentiation, and develops them further. Taking a different approach, students will learn about the concept of integrability of continuous functions; improper integrals of continuous functions; the definition of differentiability for functions; and the algebra of differentiation.
Applying the skills and knowledge gained from this module, students will tackle questions such as: “Can you sum up infinitely many numbers and get a finite number?”. Students will also enhance their knowledge and understanding of the fundamental theorem of calculus.

Introduction to Philosophy
This module introduces students to some of the central problems of philosophy and the theories produced in response to them. It also introduces some of the subject's technical concepts and vocabulary, and some of its techniques of reasoning and analysis. Reading includes both classical and contemporary material.
Philosophy has a significant role to play, both in acquainting students with some of the ideas which have helped shape Western culture, and in the critical understanding of ideas and methods in many other disciplines. The level of the module does not presuppose previous knowledge of philosophy. If students have studied philosophy before, the module will enable them to deepen and broaden their understanding of the subject and to improve their philosophical skills. The module aims not only to inform students with what philosophers have said but also to encourage them to engage with the issues. Topics will be drawn from the range of philosophical problems, approaches, and canonical figures.

Linear Algebra
Introducing the theory of matrices together with some basic applications, students will learn essential techniques such as arithmetic rules, row operations and computation of determinants by expansion about a row or a column.
The second part of the module covers a notable range of applications of matrices, such as solving systems of simultaneous linear equations, linear transformations, characteristic equation and eigenvectors and eigenvalues.

Numbers and Relations
This module introduces the student to logic and mathematical proofs, with emphasis placed on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.
We take a look at the language and structure of mathematical proofs in general, emphasising how logic can be used to express mathematical arguments in a concise and rigorous manner. These ideas are then applied to the study of number theory, establishing several fundamental results such as Bezout’s Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorisations.
The concept of congruence of integers is introduced to students and they study the idea that a highest common factor can be generalised from the integers to polynomials.

Probability
Probability theory is the study of chance phenomena, the concepts of which are fundamental to the study of statistics. This module will introduce students to some simple combinatorics, set theory and the axioms of probability.
Students will become aware of the different probability models used to characterise the outcomes of experiments that involve a chance or random component. The module covers ideas associated with the axioms of probability, conditional probability, independence, discrete random variables and their distributions, expectation and probability models.

Statistics
To enable students to achieve a solid understanding of the broad role that statistical thinking plays in addressing scientific problems, the module begins with a brief overview of statistics in science and society and then moves on to the selection of appropriate probability models to describe systematic and random variations of discrete and continuous real data sets. Students will learn to implement statistical techniques and to draw clear and informative conclusions.
The module will be supported by the statistical software package ‘R’, which forms the basis of weekly lab sessions. Students will develop a strategic understanding of statistics and the use of associated software, and this underpins the skills needed for all subsequent statistical modules of the degree.
Core
Year 2

Abstract Algebra
This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here you’ll select a small number of properties which these and other examples have in common, and use them to define a group.
You’ll also consider the elementary properties of groups. It turns out that several surprisingly elegant results can be proved fairly simply! By looking at maps between groups which 'preserve structure' you’ll discover a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same'.
Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives us a way to abstractly model various number systems, proving results that can be applied in many different situations, such as number theory and geometry. Familiar examples of rings include the integers, the integers modulation, the rational numbers, matrices and polynomials, but you’ll meet several less familiar examples too.

Complex Analysis
Complex Analysis has its origins in differential calculus and the study of polynomial equations.
In this module you’ll consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. You’ll use integral calculus of complex functions to find elegant and important results, including the fundamental theorem of algebra, and you’ll also use classical theorems to evaluate real integrals.
The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Linear Algebra II
This module will give you the opportunity to study vector spaces, together with their structurepreserving maps and their relationship to matrices.
You’ll consider the effect of changing bases on the matrix representing one of these maps, and will examine how to choose bases so that this matrix is as simple as possible. Part of your study will also involve looking at the concepts of length and angle with regard to vector spaces.

Real Analysis
In this module you’ll take a thorough look at the limits of sequences and convergence of series. You’ll learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at Alevel.
You’ll spend time studying the Intermediate Value Theorem and the Mean Value Theorem, and will discover that they have many applications of widely differing kinds. The next topic is new: sequences and series of functions (rather than just numbers), which again has many applications and is central to more advanced analysis.
Next we put the notion of integration under the microscope. Once it’s properly defined (via limits), you’ll learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. You’ll also explore some applications of integration that are quite different from the ones in Alevel, such as estimations of discrete sums of series.
Further possible topics include Stirling's Formula, infinite products and Fourier series.
Core

Constructing Ethics: Christianity and Islam
This module explores the emergence and construction of ethics within the context of two world religions: Christianity and Islam. It examines the ways in which religious attitudes to ethical concern and practice are influenced by traditional, textual and cultural factors.
Some of the ethical concerns to be covered throughout the module are: politics and economics; justice and war; sex and sexual practice; and rights and law. Finally, the module will encourage students to explore some of these areas crossculturally through the consideration of questions of difference and otherness.

Epistemology
The aim of this module is to provide students with a good, broad introduction to some of the key themes in epistemology the theory of knowledge.
It begins with a core question; What is knowledge? This leads on to questions about how knowledge relates to other things, like belief, and truth. Throughout the term students will see that it is much harder to answer the core question than one might initially think, raising a question of why it is so hard to give a clear and general, account of what knowledge is. Students will also look at sources of knowledge  especially, perception, selfknowledge and testimony. The module also explores some of the relationships between epistemology and ethics, ending with the question of whether we ever ought to refrain from seeking knowledge.
By the end of this module, students will be able to understand and discuss critically the central problems and theories of epistemology, and explain how epistemology relates to other areas of philosophy.

Ethics: Theory and Practice
This module aims to provide students with an understanding of some historical and contemporary approaches to the subject of ethics. It addresses central issues by engaging with classical texts in the history of the subject, such as Aristotle’s Nicomachean Ethics, David Hume’s Enquiry Concerning the Principles of Morals, Immanuel Kant’s Groundwork for the Metaphysics of Morals and John Stuart Mill’s Utilitarianism.
The module will also explore selected topics in moral philosophy, such as the nature, strength and weakness of consequentialism, deontology, and virtue theory. In addition to this, students will study topics in metaethics, such as the ‘moral problem’, noncognitivist realism, and quasirealism.
Other topics covered include topics in applied and practical ethics, such as issues of life and death in biomedical practice, the ethics of war, and the ethics of personal life; as well as the nature of moral motivation and moral psychology.

History of Philosophy
Western philosophy has a long and rich history, and many of the questions occupying presentday philosophers have been around for hundreds or even thousands of years.
The exact structure of this module may vary from year to year, but core themes will normally include:
 What is the nature of the mind? How does it relate to the body?
 What is the nature of human knowledge? Can we come to have any reliable knowledge of the world outside our minds?
 Is there a God? What arguments have been made in philosophy historically for the existence of God? How successful are these arguments?
 What is fundamentally real? Minds? Matter? Ideas?
Students will study these problems, amongst others, by close consideration of a selection of texts from the history of Western philosophy. This may include selections from the ancient (classical), medieval, early modern (17th/18th centuries) period, and the 19th century. Thinkers who may be considered include Plato, Aristotle, Augustine, Scotus, Descartes, Locke, Berkeley, Hume, Kant, Hegel, and Nietzsche.

Metaphysics
This module is designed to improve students' knowledge and understanding of some key issues in metaphysics as determined by the syllabus. The module will focus primarily on some issues concerning space and time, the nature of physical objects and persons, and some key philosophical distinctions. Topics will include:
 A priori and empirical; analytic and synthetic; necessary and contingent. Some basic distinctions are examined and compared.
 The status of geometry and the metaphysics of space
 The nature of physical objects; 3D versus 4D treatments
 Personal identity
 The reality or unreality of time.
Studying this module should enable students to see connections between various philosophical issues that should be of value to them with regard to other philosophy modules that they are studying.

Modern Political Thought
The aim of this module is to provide a broad grounding in some important aspects of the discipline of politics that are conceived of as both an attempt to understand the nature of politics and to assess the worth of various political arrangements. It involves consideration of notions such as politics, citizenship, democracy, government, state, welfare, individualism, utilitarianism, conservatism, socialism and, social democracy, together with an examination of the various ways in which political studies have been understood as a disciplined investigation of things political. The module covers four broad topics: freedom, markets and the state; citizenship, nationalism and democracy; equality and welfare; and politics and political science.
The module is divided into two sections over two terms. In the first term students will read, examine and discuss thinkers who make a contribution to the understanding of the notions of liberty and the individual (Hobbes, Locke, J S Mill, and Hayek). In the second term students will explore the thought of thinkers who are associated with the ideas of equality and community (Rousseau, Marx, the Fabians, and Rawls).

Philosophical Questions in the Study of Politics and Economics
This module considers some of the difficulties involved in gaining knowledge about human societies. It focuses especially on economics and politics, disciplines which raise some of the largest questions about society – for example: Who gets what? Who rules whom? Can individual choices generate social change?
In this module students will not address such questions empirically, but instead step back to ask what sort of methods have been used to answer them, what sorts of modes of explanation or understanding are appropriate, and what assumptions are built into the ways economists and political scientists frame their enquiries. The aim of the module, then, is to critically examine methods and assumptions in both disciplines, in order to appreciate the scope and limits of their claims to knowledge.

Philosophy of Science
This module considers philosophical issues that arise in both the natural and social sciences. With regard to the natural sciences, students will consider traditional accounts of scientific method and theorytesting, then examining philosophical challenges to the status of science as a rational form of enquiry. Particular consideration is given to four of the most important twentiethcentury philosophers of science: Popper, Kuhn, Lakatos and Feyerabend.
With regard to the social sciences, the module will ask whether endeavours such as sociology, economics, anthropology and history should really be counted as sciences, and then consider some of the special issues that arise in the study of human society. For example, how are we to understand other societies (for instance, in anthropology)? What is the place for individualism versus collectivism in social explanation (for example, in sociology and history)? What is the scientific status of social models based on postulates of rational choice (for example, in economics and politics)?
No scientific background is assumed on this module.

Philosophy of the Mind
This module aims to introduce students to a wide range of connected topics in the theory of knowledge, philosophy of mind and philosophy of language, drawing on both classical and contemporary writings. It examines issues such as: the nature and justification of our knowledge of the external world, and the relations between knowledge and belief; the mindbody (or mindbrain) problem; the place of mental life and bodily continuity in the identity of individuals; and the different theories of truth, meaning and the languageworld relationship, including logical positivism.
This module begins by examining issues in the metaphysics of mind, before moving on to epistemological issues: How can we gain knowledge of our own mental states, or of other people’s? How should psychologists seek to investigate the mind?
For the most part, this module will be structured around contemporary texts.

Western Philosophy and Religious Thought
This module aims to encourage students to think philosophically about religious issues. Using the work of both classical and contemporary philosophers and religious thinkers, it addresses some of the central philosophical questions raised by religious belief. In addition, students will be encouraged to think historically ad contextually, in order to understand the ways in which the role of philosophy in relation to religion in the west has changed over time.
The module introduces students to the work of some of the most important philosophers from Plato to Wittgenstein and the implications of their thought for religion. It will also address themes and issues which may vary from year to year but will be drawn from the following: the nature of theism; immortality; the problem of evil; religious experience; and the implications of postmodern thought for religious belief.
Optional
Year 3

Aesthetics
This module introduces central issues, problems and theories in philosophical aesthetics by critically examining specific topics in the philosophy of art, and by examining the theories of major figures who have contributed to the tradition of philosophical aesthetics. The module uses concrete examples from most of the arts, including painting, literature, film, and music, to illuminate theoretical debates and issues.
Topics and major aesthetic theorists covered may include the following (note that this list is not exhaustive and indicative only, not all topics will be covered) :
 Aesthetics in the analytic and continental traditions of philosophy
 Aesthetic theories of Plato, Hume, Kant, Schopenhauer, Nietzsche, and others
 Beauty and its definition
 The relations between art, religion and philosophy
 Art and morality
 The culture industry and its impact on our responses to art
 Disinterestedness
 The relations between aesthetics and politics: Should art be politically committed? If so, in what ways?

Applied Philosophy
This module focuses on selected topics in Applied Philosophy. It involves the application of philosophical methods and knowledge to a range of problems that face institutions, professions, policymakers and regulatory bodies. Further questions arise about the nature and limits of Applied Philosophy.
Examples of topics that may be studied include:
 Philosophy of privacy and data protection
 Philosophical bioethics
 Philosophy of the media
 Philosophy and psychiatric classification
 Applied epistemology
 Selected topics in medical ethics
 Philosophy of education
 The metaphilosophy of applied philosophy
 Applied philosophy of language
 Applied philosophy of science and technology

Combinatorics
Combinatorics is the core subject of discrete mathematics which refers to the study of mathematical structures that are discrete in nature rather than continuous (for example graphs, lattices, designs and codes). While combinatorics is a huge subject  with many important connections to other areas of modern mathematics  it is a very accessible one.
In this module you’ll be introduced to the fundamental topics of combinatorial enumeration (sophisticated counting methods), graph theory (graphs, networks and algorithms), and combinatorial design theory (Latin squares and block designs). You’ll also explore important practical applications of the results and methods.

Continental Philosophy
This module aims to introduce the work of some key figures in 19th and 20th century continental philosophy, such as Hegel, Marx, Kierkegaard, Nietzsche, Heidegger, Foucault, Hannah Arendt and Habermas. The approach taken is predominantly philosophical rather than historical, and will involve critically examining claims and arguments about such matters as the existence and nature of human freedom, the relationships between knowledge, truth, power and morality, alienation and human labour, and the possibility of mutual recognition and community. It is expected that students will engage with the original texts, formulate the central arguments to be found in them and assess their cogency.
The module begins by looking at Nietzsche’s Toward a Genealogy of Morality, before turning to Foucault, who adapts Nietzsche’s method of historical analysis in order to challenge assumptions about progress toward freedom and welfare in modern societies. Finally students will study Arendt and her political thought on totalitarian politics using a parallel method of historical analysis.

Darwinism and Philosophy
This module will examine philosophical issues that arise in connection with specific sciences, in particular biology and medicine, as opposed to the general philosophy of science.
The following topics will be covered:
 Methodological issues in biology and medicine
 Issues of classification and kinds in biology and medicine – e.g. what is a species, what is a gene, what is a disease?
 Issues to do with the relationship between these sciences and others, in particular the issue of intertheoretic reduction, i.e. can biology be reduced to physics and chemistry?
 Radical critiques of mainstream biology and medicine – e.g. political critiques of sociobiology, the antipsychiatry movement.
 Historical issues the development of these sciences.

Differential Equations
This module considers questions relating to linear ordinary differential equations. While explicit solutions can only be found for special types of equations, some of the ideas of real analysis allow us to answer questions about the existence and uniqueness of solutions to more general equations as well as allowing us to study certain properties of these solutions.

Dissertation
This module provides an opportunity for students to choose a topic related to some aspect of Politics and International Relations, Philosophy and Religious Studies which particularly interests them, and to pursue it in depth. The topic may be related to work that is being done on a formally taught course, or it may be less directly linked to course work. The intention is that students will develop their research skills and their ability to work at length under their own direction.
Students are expected to start thinking seriously about the 9,00010,000 word dissertation towards the end of the Lent term of their second year, and to submit a provisional topic by the end of that term. Work should be well advanced by Christmas in the third year. The completed dissertation must be submitted by the end of the Lent term in the third year.

Dissertation with external collaboration
This module aims to allow students to pursue independent indepth studies of a topic of their choice, within the scope of their scheme of study. The topic may be related to work that is being done on a formally taught course, or it may be less directly linked to course work.
Students will develop their employability and research skills, and their ability to work independently at length under their own direction with input from an academic supervisor.
The external collaboration will enhance students’ ability to reflect on the impact of academic work. One option is to incorporate work done through the Richardson Institute Internship Programme, but students may also discuss other forms of collaboration with their supervisor.

Elliptic Curves
This module gives you a solid foundation in the basics of algebraic geometry. You’ll explore how curves can be described by algebraic equations, and learn how to understand and use abstract groups in dealing with geometrical objects (curves).
You’ll also gain an understanding of the notions and the main results pertaining to elliptic curves, and the way that algebra and geometry are linked via polynomial equations. Finally you’ll learn to perform algebraic computations with elliptic curves.

Financial Mathematics
This module formally introduces students to the discipline of financial mathematics, providing them with an understanding of some of the maths that is used in the financial and business sectors.
Students will begin to encounter financial terminology and will study both European and American option pricing. The module will cover these in relation to discrete and continuous financial models, which include binomial, finite market and BlackScholes models.
Students will also explore mathematical topics, some of which may be familiar, specifically in relation to finance. These include:
 Conditional expectation
 Filtrations
 Martingales
 Stopping times
 Brownian motion
 BlackScholes formula
Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated riskneutral probability; performing calculations with the BlackScholes formula; and proving various steps in the derivation of the BlackScholes formula. They will also be able to describe basic concepts of investment strategy analysis, and perform price calculations for stocks with and without dividend payments.
In addition, to these subject specific skills and knowledge, students will gain an appreciation for how mathematics can be used to model the realworld; improve their written and oral communication skills; and develop their critical thinking.

Future generations
What moral obligations do we have towards future generations to those yet to be born, and to people whose very existence (or nonexistence) depends on how we act now?
This module explores this question by examining both a series of practical case studies and some of the main concepts and theories philosophers use when thinking about these issues.
Questions considered include, among a range of others:
 How should we weigh quality against quantity of life?
 Ought we to try significantly to extend the human lifespan (to 150 years or beyond)?
 Is there a moral obligation to refrain from having children, or one to have (more) children?
 Should we use selection techniques to minimise the incidence of genetic disorders and disabilities in future populations? Should parents be allowed to determine the genetic characteristics of their future children?
 How should the interests of nonhuman creatures be weighed against those of humans? How strong are our moral obligations to prevent extinctions, and to preserve wildernesses?

Geometry of Curves and Surfaces
This module is an introduction to smooth curves and surfaces in threedimensional space. You’ll encounter various geometrical properties of these objects, such as length, area, torsion and curvature, and will have the opportunity to explore the meaning of these quantities. You’ll use a variety of examples to calculate their values, and will use them to apply techniques from calculus and linear algebra.

Graph Theory
The study of graphs  mathematical objects used to model pairwise relations between objects  is a cornerstone of discrete mathematics. As a result, students will develop an appreciation for a range of discrete mathematical techniques while undertaking this module.
Throughout the module, students will also learn about structural notions, such as connectivity, and will explore trees, minor closed families of graphs, matrices related to graphs, the Tutte polynomial of small graphs, and planar graphs and analogues.
While studying these areas, students will gain experience of following and constructing mathematical proofs, and correctly and coherently using mathematical notation.

Groups and Symmetry
In this module you’ll develop the knowledge of groups that you’ve gained in second year. You’ll study ‘direct products’ which are used to construct new groups, while any finite group is shown to ‘factor’ into ‘simple’ pieces. You’ll also consider situations in which a group ‘acts’ on a set by permuting its elements; this powerful idea is used to identify the symmetries of the Platonic solids, and to help study the structure of groups themselves.

Hilbert Space
In this module you’ll examine the notion of a norm, which introduces a generalized notion of ‘distance’ to the purely algebraic setting of vector spaces. You’ll learn several quite natural ways to do this, both for vectors of any dimension and for functions. You’ll then focus on the more special notion of an inner product which generalizes angles at the same time as distances.
Once we’ve established these concepts, you’ll have the opportunity to study geometrical ideas like orthogonality, which can be seen to apply to much more general spaces than Euclidean spaces of three (or even n) dimensions, notably to infinite dimensional spaces of functions. For example, Hilbert space theory shows how Fourier series are really another case of expressing an element in terms of a basis, and how you can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, you’ll look at some of the main results of linear algebra, which generalize very nicely to linear operators between Hilbert spaces.

History of Twentieth Century Philosophy
This module focuses upon some key aspects of the history of 20th Century Philosophy.
The module begins by examining a revolution in philosophy at the very start of the 20th century with the origins of analytic philosophy. It then focuses on Wittgenstein’s radical philosophy (or antiphilosophy). Wittgenstein’s own philosophical development brings to the fore a deep schism, or tension, that has existed throughout the century’s philosophy, one which lays between those who hold that philosophy should align itself with natural science and mathematics, and those who reject this view. Students will examine whether philosophy should seek to emulate the natural sciences and illustrate the tension between scientistic and humanistic philosophy via mid20th century debate about the nature of historical explanation.
The final lectures look at the distinction between analytic and continental philosophy in the 20th century, and upon the emergence of applied philosophy later in the century, asking whether philosophy can ever really be applied to reallife problems.

Indian Religious and Philosophical Thought
This module will introduce major themes and issues in Indian philosophy, focusing on the Hindu and Buddhist philosophical traditions. Beginning with philosophical sections in the Upanishads and the dialogues of the Buddha, the module will trace the development of Indian philosophy from the early to the classical periods. Various ethical, metaphysical, and epistemological concepts will be covered, such as: order and virtue (dharma), consequential action (karma), ultimate reality (Brahman), the nature of the self (atman), the highest good (moksha), and the means for attaining knowledge (pramana).
Throughout the module, students will look at the dialogical relationship between the Hindu and Buddhist philosophical traditions, particularly the shared practice of debate.

Issues in the Philosophy of Mind
This module will introduce students to some advanced topics in the philosophy of mind. Through the debates examined students will be exposed to a number of methodological approaches in the philosophy of mind  including the use of empirical evidence in philosophy, conceptual analysis, ordinary language philosophy and thought experimentation.
Topics examined will vary from year to year but may include:
 Consciousness
 Understanding other minds
 Selfknowledge
 Emotions
 Understanding abnormal mental states
 The self
 Perception
 Evolutionary psychology
 Animal, alien, and computer minds
 Mental causation

Lebesgue Integration
Introducing the Lebesgue integral for functions on the real line, this module features a classical approach to the construction of Lebesgue measure on the line and to the definition of the integral. The bounded convergence theorem is used to prove the monotone and dominated convergence theorems, and the results are illustrated in classical convergence problems including Fourier integrals.
Among the range of topics addressed on this module, students will become familiar with Lebesgue's definition of the integral, and the integral of a step function. There will be an introduction to subsets of the real line, including open sets and countable sets. Students will measure of an open set, and will discover measurable sets and null sets. Additionally, the module will focus on integral functions, along with Lebesgue's integral of a bounded measurable function, his bounded convergence theorem and the integral of an unbounded function. Dominated convergence theorem; monotone convergence theorem.
Other topics on the module will include applications of the convergence theorems and Wallis's product for P. Gaussian integral, along with some classical limit inversion results and the Fourier cosine integral. Students will develop an understanding of Dirichlet's comb function, Archimedes' axiom and Cantor's uncountability theorem, and will learn to prove the structure theorem for open sets. In addition, students will be able to prove covering lemmas for open sets, as well as understanding the statement of Heine—Bore theorem, as well as understanding the concept and proving basic properties of outer measure. As well as understanding inner measure. Finally, students will be expected to prove Lebesgue's theorem on countable additivity of measure.

Liberals and Communitarians
This module examines the central debates about politics and justice between liberals and communitarians in contemporary AngloAmerican analytic philosophy. Whereas liberals stress the importance of the individual and the need for them to pursue their own good in their own way, communitarians stress the embedded, interconnected, and social nature of the persons and politics.
The module asks three major questions. Firstly what it means to be engaged in political theory. Secondly, how the idea of justice should be understood, and finally, what implications does our view of justice have for our political arrangements?
The module is divided into two main sections. Concentrating first on the central figure of this debate: John Rawls and his seminal work A Theory of Justice. Then looking at how the debate has widened, initially looking at the libertarian criticisms raised by Nozick before moving on to consider the communitarian positions advanced by Sandel, Walzer, Okin, and Pateman; finally considering alternative forms of liberalism offered by Raz, Rorty, and Gray.

Linear Systems
The aim of this module is to provide third year students with more options of applicable topics which draw upon second year pure mathematics courses and provide opportunities for further study. The theory of Linear systems is engineering mathematics.
In the mid nineteenth century, the engineer Watt used a governor to control the amount of steam going into an engine, so that the input of steam reduced when the engine was going too quickly, and the input increased when the engine was going too slowly. Maxwell then developed a theory of controllers for various mechanical devices, and identified properties such as stability. The crucial idea of a controller is that the output can be fed back into the system to adjust the input.
Many devices can be described by linear systems of differential and integral equations which can be reduced to a standard (A,B,C,D) model. These include electrical appliances, heating systems and economic processes. The course shows how to reduce certain linear systems of differential equations to systems of matrix equations and thus solve them. Linear algebra enables us to classify (A,B,C,D) models and describe their properties in terms of quantities which are relatively easy to compute.
The module then describes feedback control for linear systems. The main result describes all the linear controllers that stabilize a (A,B,C,D) system.

Logic and Language
This module provides an introduction to formal logic together with an examination of various philosophical issues that arise out of it. The syllabus includes a study of the languages of propositional and quantificational logic, how to formalize key logical concepts within them, and how to prove elementary results using formal techniques.
Additional topics include identity, definite descriptions, modal logic and its philosophical significance, and some criticisms of classical logic.

Metric Spaces
This module gives an introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications. Studying this module will give you a deeper understanding of continuity as well as a basic grounding in abstract topology.
You’ll also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply your knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

Modern Christian thought
This module, for the most part, concentrates on (Protestant) Christian thinkers from the Germanspeaking world. These thinkers have dominated the development of Christian thought in Europe and America until very recent times, when various 'political theologies' (Black, feminist and liberationist) started to erode their influence.
The point of departure on this module must be the Enlightenment and its definitive philosopher  Immanuel Kant. The module begins, therefore, by looking at the challenges facing early nineteenth century theologians, consider the responses of five major Christian thinkers of the nineteenth and twentieth centuries and shall end by exploring the challenges facing Christian thought today.

Modern Religious and Atheistic Thought
This module will examine some of the major debates in religious and atheistic thought, looking in particular at the way in which these debates are framed by a specifically modern epistemological framework, and the ways in which religious thought and atheistic thought might be though to be mutually constitutive and mutually implicated rather than simply oppositional.
The aim of this module is to examine and evaluate some of the most central issues in Enlightenment and postEnlightenment Western religious and atheistic philosophical debates. The module will begin by looking the philosophy of G W F Hegel and its implications for subsequent religious and atheistic thought. It will then proceed to consider the thought of the postHegelian masters of suspicion: Feuerbach, Marx, Freud and Nietzsche. After this, it will look at ways in which religious and atheistic thought have been brought together, as manifested in various forms of Christian atheism. Finally, it will consider postmodern critiques of modern atheism and the nature of the associated return of religion.

Moral, legal and political philosophy
This module will address central issues in contemporary ethical (including metaethical), legal and political philosophy, and will allow a systematic critical exploration of the connections between ideas and arguments in each of the three areas of the subject.
Topics covered will include some of the following: modern theory of moral motivation, value theory, contractualism, the 'moral problem'; responsibility and criminal liability, the justification of punishment, the proper scope of the law; democratic theory, egalitarianism, justice, nationalism, multiculturalism, liberty and human rights.

Multivariate Statistics in Machine Learning
Using the classical problem of data classification as a running example, this module covers mathematical representation and visualisation of multivariate data; dimensionality reduction; linear discriminant analysis; and Support Vector Machines. While studying these theoretical aspects, students will also gain experience of applying them using R.
An appreciation for multivariate statistical analysis will be developed during the module, as will an ability to represent and visualise highdimensional data. Students will also gain the ability to evaluate larger statistical models, apply statistical computer packages to analyse large data sets, and extract and evaluate meaning from data.

Number Theory
Number theory is the study of the fascinating properties of the natural number system.
Many numbers are special in some sense, eg. primes or squares. Which numbers can be expressed as the sum of two squares? What is special about the number 561? Are there short cuts to factorizing large numbers or determining whether they are prime (this is important in cryptography)? The number of divisors of an integer fluctuates wildly, but is there a good estimation of the ‘average’ number of divisors in some sense?
Questions like these are easy to ask, and to describe to the nonspecialist, but vary hugely in the amount of work needed to answer them. An extreme example is Fermat’s last theorem, which is very simple to state, but was proved by Taylor and Wiles 300 years after it was first stated. To answer questions about the natural numbers, we sometimes use rational, real and complex numbers, as well as any ideas from algebra and analysis that help, including groups, integration, infinite series and even infinite products.
This module introduces some of the central ideas and problems of the subject, and some of the methods used to solve them, while constantly illustrating the results with exercises and examples involving actual numbers.

Philosophy of Medicine
Are psychopaths evil or sick? Should the NHS pay for the treatment of nicotine addiction? Is it right for shy people to take characteraltering drugs?
Whether a condition is considered a disease often has social, economic and ethical implications. It tends to be taken for granted that what it is to be healthy can be identified and is desirable. Similarly, it is assumed that those who are diseased or disabled can be diagnosed and require help. In this module we question these assumptions via examining the key concepts of normality, disease, illness, mental illness, and disability.

Philosophy of the Human Sciences
This module considers key philosophical issues in the sciences of human societies and social structures, such as sociology, economics or history.
As well as considering whether these subjects should be considered as sciences the module looks at a number of philosophical issues, such as those arising in the understanding of other societies (for instance, in anthropology), individualism versus collectivism in social explanation (for example, in sociology and history), and the scientific status of social models based on postulates of rational choice (for example, in economics and politics).

Political Ideas
This module examines central themes in the liberal branch of contemporary AngloAmerican analytic political philosophy. The liberal positions on justice, liberty, equality, the state, power, rights and utility are all explored. The approach is philosophical rather than applied; focusing on the ideas of liberal politics: how individual liberty can be maximised while not harming others; how an individual philosophical position can guide political determinants of a society and places the developments of liberal ideas in their appropriate historical contexts.
The module also examines the connection between the ideas of liberalism and the idea of democracy to explore the philosophical tensions between the two and how these might be resolved.
The module will include among other topics: questions about justice: analytic philosophy and liberalism; visions of the state: liberalism, republicanism, socialism; liberty and individuality; liberalism and democracy; negative and positive liberty; equality; utility and rights; and toleration and multiculturalism: responses to diversity.

Politics and Ethics in Indian Philosophy
This module will look at Indian source texts on politics and ethics. In particular, it will be looking at sources that explore the concept of dharma, a term that incorporates issues of justice, religion, ethics, duty, and law. The module will examine the sources of dharma both in their own historical and cultural contexts, as well as in the context of contemporary debates in political theory and ethics. The texts examined will include: the inscriptions of Ashoka, the Buddhist Nikayas, the Arthashastra, the Law Codes of Manu, the Mahabharata, and the Kamasutra. These sources are examined in connection with modern political figures, such as Gandhi and Savarkar, as well as in connection with recent debates in India about secularism, democracy and pluralism.

PPR in Education
This module is designed to allow students to gain experience of educational environments, to develop transferable skills, and to reflect on the role and communication of their own discipline. The module is organised and delivered collaboratively between Lancaster University Students’ Union LUSU Involve, the school/college where the placement is based, and the department.
The module will give students experience of classroom observation and experience, teacher assistance, as well as teaching small groups (under supervision). In particular, the module will not only give students the opportunity to observe and experience teaching and learners for themselves, it will also require them to reflect on how their own subject area (Religion, Politics and International Relations, or Philosophy) is experienced by learners, delivered in other parts of the educational sector, and applied in a classroom setting. Students will also be asked to reflect on how teaching and learning at this earlier level combines with what is taught and promoted at the level of Higher education (as experienced in the University).

Probability Theory
This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.
First you’ll examine the notion of a probability space through simple examples featuring both discrete and continuous sample spaces. You’ll then use random variables and the expectation to develop a probability calculus, which you can apply to achieve laws of large numbers for sums of independent random variables.
You’ll also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

Reading Political Theory
Students will study the thought of two seminal thinkers in political theory. This module provides an opportunity to explore texts slowly, methodically and in depth, allowing students to link that thought to wider literature that has developed as a response to the thinkers' ideas, and see how those ideas linkup into a wider systematic and philosophic whole.
Topics include among many others:
 Plato: forms and the gooditself, the structure of the kallipolis, the lies of the rulers and censorship, women, children, and slaves, freedom and totalitarianism.
 Aristotle: links between politics and ethics, community and the political animal, and natural law
 Kant: freedom and autonomy, roles of the individual, the role of reason, cosmopolitanism, government by agreement, censorship and the state, enlightenment.
 Nietzsche: early and later political philosophy, the role of state, the death of God, the slave revolt in morals, the Ubermensch, sovereign individuals, radical aristocracy.
 Marx: the early Marx, Hegel, speciesbeing, alienation, historical materialism, bourgeois philosophy, critique of the modern state, the communist alternative.

Representation Theory of Finite Groups
This module covers the basics of ordinary representation theory. You’ll learn the concepts of Rmodule and of group representations, the main results pertaining to group representations, and will handle basic applications in the study of finite groups. You’ll also learn to perform computations with representations and morphisms in a selection of finite groups.

Rings Fields and Polynomials
This module furthers your knowledge of commutative rings from your second year study.
You’ll be introduced to two new classes of integral domains called Euclidean domains, where you have a counterpart of the division algorithm, and unique factorization domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.
You’ll also explore how wellknown concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorization of polynomials, carry over to this new setting.

Special Subject: The Imagination
This module will examine philosophical accounts of the imagination. It will look at theories of the nature of the imagination and its connections to other mental states, such as attention, emotion, memory, beliefs, intentions, and desires.
In addition, a range of topics focusing on the role of imagining in a number of different domains will also be explored, including moral judgement, practical reasoning, perception, pictorial experience, and modal thought.
Optional
Lancaster University offers a range of programmes, some of which follow a structured study programme, and others which offer the chance for you to devise a more flexible programme. We divide academic study into two sections  Part 1 (Year 1) and Part 2 (Year 2, 3 and sometimes 4). For most programmes Part 1 requires you to study 120 credits spread over at least three modules which, depending upon your programme, will be drawn from one, two or three different academic subjects. A higher degree of specialisation then develops in subsequent years. For more information about our teaching methods at Lancaster visit our Teaching and Learning section.
Information contained on the website with respect to modules is correct at the time of publication, but changes may be necessary, for example as a result of student feedback, Professional Statutory and Regulatory Bodies' (PSRB) requirements, staff changes, and new research.

Calculus

Careers
Careers
Maths and philosophy graduates are highly employable, having indepth specialist knowledge and a wealth of skills. Through this degree, you will graduate with a comprehensive skill set, including analysis and manipulation, interpretation, logical thinking, problemsolving, and reasoning, as well as adept knowledge of the disciplines. As a result, combining these two subjects opens up a range of opportunities and graduates are highly sought after.
The starting salary for many of these roles is highly competitive, and popular career options include:
 Actuary
 Data Analyst
 Investment Analyst
 Research Scientist
 System Developer
 Teacher
A degree in this discipline can also be useful for roles such as Barrister, Local Government Office, Psychotherapist, Stockbroker, and many more.
Alternatively, you may wish to undertake postgraduate study at Lancaster and pursue a career in research and teaching.

Fees and Funding
Fees and Funding
Fees
We set our fees on an annual basis and the 2018/19 entry fees have not yet been set.
As a guide, our fees in 2017 were:
UK/EU Overseas £9,250 £16,620 Channel Islands and the Isle of Man
Some science and medicine courses have higher fees for students from the Channel Islands and the Isle of Man. You can find more details here: Island Students.
Funding
For full details of the University's financial support packages including eligibility criteria, please visit our fees and funding page
Students also need to consider further costs which may include books, stationery, printing, photocopying, binding and general subsistence on trips and visits. Following graduation it may be necessary to take out subscriptions to professional bodies and to buy business attire for job interviews.
The Department
29%
Average time in lectures, seminars and similar
39%
Average assessment by coursework