Mathematics and Philosophy

This module builds on the binary operations studies in previous modules, such as addition or multiplication of numbers and composition of functions. Here, students will select a small number of properties which these and other examples have in common, and use them to define a group.

They will also consider the elementary properties of groups. By looking at maps between groups which 'preserve structure', a way of formalizing (and extending) the natural concept of what it means for two groups to be 'the same' will be discovered.

Ring theory provides a framework for studying sets with two binary operations: addition and multiplication. This gives students 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; several less familiar examples will also be explored.

Complex Analysis has its origins in differential calculus and the study of polynomial equations. In this module, students will consider the differential calculus of functions of a single complex variable and study power series and mappings by complex functions. They will use integral calculus of complex functions to find elegant and important results and will also use classical theorems to evaluate real integrals.

The first part of the module reviews complex numbers, and presents complex series and the complex derivative in a style similar to calculus. The module then introduces integrals along curves and develops complex function theory from Cauchy's Theorem for a triangle, which is proved by way of a bisection argument. These analytic ideas are used to prove the fundamental theorem of algebra, that every non-constant complex polynomial has a root. Finally, the theory is employed to evaluate some definite integrals.The module ends with basic discussion of harmonic functions, which play a significant role in physics.

Students will be provided with the foundational results and language of linear algebra, which they will be able to build upon in the second half of Year Two, and the more specialised Year Three modules. This module will give students the opportunity to study vector spaces, together with their structure-preserving maps and their relationship to matrices.

They will 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 their study will also involve looking at the concepts of length and angle with regard to vector spaces.

A thorough look will be taken at the limits of sequences and convergence of series during this module. Students will learn to extend the notion of a limit to functions, leading to the analysis of differentiation, including proper proofs of techniques learned at A-level.

Time will be spent studying the Intermediate Value Theorem and the Mean Value Theorem, and their many applications of widely differing kinds will be explored. 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, the notion of integration will be put under the microscope. Once it is properly defined (via limits) students will learn how to get from this definition to the familiar technique of evaluating integrals by reverse differentiation. They will also explore some applications of integration that are quite different from the ones in A-level, such as estimations of discrete sums of series.

Further possible topics include Stirling's Formula, infinite products and Fourier series.

The aim of this module is to give you a good, broad introduction to some of the key themes in epistemology (the theory of knowledge). We begin with the question what is knowledge? This then leads us on to questions about how knowledge relates to other things, like belief, and truth. Our answers to these questions have implications for how we think about the structure of knowledge (e.g., must all of our knowledge rest upon a “firm foundation”?). Throughout the term we will see that it is much harder to answer our core question than you might think and this raises the question of why it is so hard to give a clear, general, account of what knowledge is. We also look at different sources of knowledge - especially, perception, self-knowledge and “testimony” (other people’s say-so) and, towards the end of term explore some of the relationships between epistemology and ethics, ending the term with the question whether we ever ought to refrain from seeking knowledge.

This module will consider some of the major issues currently being debated by political philosophers and political theorists. Specific topics may change from year to year, but issues usually covered include some of the following:

• - State power and citizens’ obligations
• - Equality between social groups
• - Material equality
• - Environmental politics
• - Public goods and state action
• - Politics and regulation of business activity
• - Global justice

In this module we give you the opportunity to develop your knowledge and understanding of some of the key issues in metaphysics. We focus primarily on issues concerning space and time, the nature of physical objects and persons, and some key philosophical distinctions. Studying this module can also enable you to see connections between various philosophical issues, which can be of value for other philosophy modules.

In this module we will be looking at a variety of views about the nature of mind and mental phenomena and how they fit into the natural world. We begin with the classic Cartesian account of mind: substance dualism. We then turn to current behaviourist, materialist, and functionalist theories of mind. Some of the larger questions we will be considering are: How are behaviour and mental states related to each other? Are minds really just brains? Or are minds more like computers? Next we consider some of the most perplexing problems about the nature of mind, currently occupying philosophers. How do our thoughts manage to reach out to reality and be about anything, especially when many of the things we think about don’t exist? Do mental states have causal powers of their own or do they somehow inherit them from the causal powers of brains? And finally, can we explain the mystery of consciousness?

Moral philosophy is the systematic theoretical study of morality or ethical life: what we ought to do, what we ought to be, what has value or is good. This module engages in this practice by critical investigation of some of the following topics, debates, and figures: value and valuing; personhood/selfhood; practical reason; moral psychology; freedom, agency, and responsibility; utilitarianism and its critics; virtue ethics and its critics; deontology and its critics; contractarianism and its critics; the nature of the good life; the source and nature of rights; the nature of justice; major recent and contemporary figures, such as Bernard Williams, Martha Nussbaum, Peter Railton, Christine Korsgaard, Philippa Foot, Allan Gibbard, Simon Blackburn; major historical figures such as Aristotle, David Hume, Immanuel Kant, John Stuart Mill, G. E. Moore.

Our aim in this module is to consider some of the big philosophical questions underlying social sciences. Economics and politics raise both deep philosophical questions about society and subjectivity; for example: Who gets what? Who rules whom? Who, or what, decides? In this module we will investigate a variety of methods that attempt to address these questions, and what answers might be possible. In sum, the aim is to examine methods and assumptions across central movements in the social sciences, politics and economics, from a philosophical perspective to see the troubles and possibilities in each.

This module considers philosophical issues that arise in connection with the sciences. We will consider what scientific method is, how science relates to the rest of knowledge, whether it provides an ideal model for rational inquiry in general, and whether we should think of science as describing reality. In the first few weeks we will consider traditional accounts of scientific method and theory-testing, and then examine philosophical challenges to the status of science as a rational form of enquiry. We give particular consideration to three of the most important twentieth-century philosophers of science: Popper, Kuhn, and Feyerabend. Next we will consider whether and in what sense we should be confident that our best current scientific theories are accurate descriptions of reality. We do not assume that you have an extensive knowledge of science: the relevant scientific concepts will be presented in a simple and accessible way, and there will be no maths.

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 meta-philosophy of applied philosophy
• Applied philosophy of language
• Applied philosophy of science and technology

Bayesian statistics provides a mechanism for making decisions in the presence of uncertainty. Using Bayes’ theorem, knowledge or rational beliefs are updated as fresh observations are collected. The purpose of the data collection exercise is expressed through a utility function, which is specific to the client or user. It defines what is to be gained or lost through taking particular actions in the current environment. Actions are continually made or not made depending on the expectation of this utility function at any point in time.

Bayesians admit probability as the sole measure of uncertainty. Thus Bayesian reasoning is based on a firm axiomatic system. In addition, since most people have an intuitive notion about probability, Bayesian analysis is readily communicated.

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, students will 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). They will also explore important practical applications of the results and methods.

Students’ knowledge of commutative rings as gained from their second year of study in Rings and Linear Algebra will be built upon, and an introduction to the fourth year Galois Theory module will be provided.

They will be introduced to two new classes of integral domains called Euclidean domains, where they have a counterpart of the division algorithm, and unique factorisation domains, in which an analogue of the Fundamental Theorem of Arithmetic holds.

How well-known concepts from the integers such as the highest common factor, the Euclidean algorithm, and factorisation of polynomials, carry over to this new setting, will also be explored.

This module introduces three of the most important thinkers from the ‘continental’ tradition of philosophy, with a focus on moral and political questions. The aim is to give you an understanding of their main ideas and help you develop your own critical perspective on them. We begin by looking at Friedrich Nietzsche’s provocative account of the origins and development of morality. We then turn to Michel Foucault, who adapts Nietzsche’s method of historical analysis in order to challenge our assumptions about progress, freedom and welfare in modern societies. Finally, we turn to Hannah Arendt. Using a parallel method of historical analysis, Arendt examines the social and political elements that came together in the disaster of Nazi and Stalinist totalitarianism.

The module will look at philosophical issues that arise out of Darwin’s theory of evolution. These include questions about how best to understand the theory of evolution, and questions about what evolution implies for our view of the world, and in particular of ourselves. The module breaks down into three broad areas:

• Different ways to understand the theory of evolution, e.g. Is evolution, as some would have us believe, all about genes? Is natural selection the only important factor in evolution?
• Conceptual issues relating to biology, e.g. How do we define ‘function’? Is there one right way to classify living things?
• Implications of Darwinism for understanding human nature, e.g. Does the fact that we have evolved affect how we should see human nature? Why are evolutionary theories of human nature so controversial? Does Darwinism have any implications for moral questions?

Questions relating to linear ordinary differential equations will be considered during this module. Differential equations arise throughout the applications of mathematics, and consequently the study of them has always been recognised as a fundamental branch of the subject. The module aims to give a systematic introduction to the topic, striking a balance between methods for finding solutions of particular types of equations, and theoretical results about the nature of solutions.

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.

This module provides you with an opportunity to choose a topic related to some aspect of Politics and International Relations, Philosophy and Religious Studies which particularly interests you, 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. We encourage you to develop your research skills, and your ability to work at length under your own direction. You submit a 9,000 - 10,000 word dissertation by the end of the Lent term in your third year. To help you prepare for work on the dissertation, typically there is an introductory talk in second year on topics relating to doing one's own research and planning and writing a dissertation

The aim of this module is to allow you to pursue independent in-depth studies of a topic of your choice, within the scope of your scheme of study. The topic will be formulated in dialogue with one or more external collaborator(s) and may be related to work that is being done on a formally taught course, or it may be less directly linked to course work. You will have the opportunity to develop your employability and research skills, and your ability to work independently at length under your own direction with input from external and an academic supervisor. The external collaboration will give you the chance to enhance your ability to reflect on the impact of academic work. One option is to incorporate work done through the Richardson Institute Internship Programme, but you may also discuss other forms of collaboration with their supervisor. The completed dissertation is usually submitted at the start of Summer Term in the third year. To help you prepare for work on the dissertation, typically there is an introductory talk in second year on topics relating to doing one’s own research and planning and writing a dissertation.

What moral obligations do we have towards future generations – to people who are yet to be born, and to merely possible people whose existence (or non-existence) depends on how we decide to act now? In this module, we explore this question in detail by examining both a series of case studies and some of the main concepts and theories that philosophers use when thinking about these issues

The questions considered normally include:

• Is there a moral obligation to refrain from having children (e.g. for environmental reasons) and what measures may governments take to encourage or enforce population control? Conversely, might there be a moral obligation 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 use these techniques to determine the characteristics of their future children?
• How should we weigh quality against quantity of life? Would a world with a relatively small number of ‘happier’ people be preferable to one with many more ‘less happy’ ones?
• When considering long-term environmental issues (e.g. climate change, nuclear power) and long-term financial issues (e.g. national debt and pensions) how should we balance the interests and rights of people who exist now against those of future people?
• When considering the future, how should the interests of non-human creatures be weighed against those of humans? How strong are our moral obligations to prevent extinctions, and to preserve wildernesses? Would considerably extending the human life span (to 150 years or beyond) be defensible if this meant that fewer ‘new’ people could be born?

The topic of smooth curves and surfaces in three-dimensional space is introduced. The various geometrical properties of these objects, such as length, area, torsion and curvature, will be explored and students will have the opportunity to discover the meaning of these quantities. They will then use a variety of examples to calculate these values, and will use those values to apply techniques from calculus and linear algebra.

A number of well-known concepts will be encountered, such as length and area, and some new ideas will be introduced, including the curvature and torsion of a curve, and the first and second fundamental forms of a surface. Students will learn how to compute these quantities for a variety of examples, and in doing so will develop their geometric intuition and understanding.

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.

Students will develop the knowledge of groups that they gained in second year during the Groups and Rings module. ‘Direct products’, which are used to construct new groups, will be studied, while any finite group will be shown to ‘factor’ into ‘simple’ pieces.

Situations will be considered 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.

Finally, students will prove some interesting and important results, known as 'Sylow’s theorems', relating to subgroups of certain orders.

Students will examine the notion of a norm, which introduces a generalised notion of ‘distance’ to the purely algebraic setting of vector spaces. They will learn several quite natural ways to do this, both for vectors of any dimension and for functions. Focus will then be on the more special notion of an inner product which generalises angles at the same time as distances.

Once these concepts have been established, students will 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 people can use orthogonality to find best approximations to a given function by functions of a prescribed type. Finally, students will look at some of the main results of linear algebra, which generalise very nicely to linear operators between Hilbert spaces.

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.

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
• Self-knowledge
• Emotions
• Understanding abnormal mental states
• The self
• Perception
• Evolutionary psychology
• Animal, alien, and computer minds
• Mental causation

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.

This module examines the central debates about politics and justice between liberals and communitarians in contemporary Anglo-American 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.

Statistical inference is the theory of the extraction of information about the unknown parameters of an underlying probability distribution from observed data. Consequently, statistical inference underpins all practical statistical applications.

This module reinforces the likelihood approach taken in second year Statistics for single parameter statistical models, and extends this to problems where the probability for the data depends on more than one unknown parameter.

Students will also consider the issue of model choice: in situations where there are multiple models under consideration for the same data, how do we make a justified choice of which model is the 'best'?

The approach taken in this course is just one approach to statistical inference: a contrasting approach is covered in the Bayesian Inference module.

The 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.

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 high-dimensional 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.

Students will be given an opportunity to consider key issues in the teaching and learning of mathematics during this module. Whilst it is an academic study of mathematics education and not a training course for teachers, it does provide an excellent foundation for a PGCE especially in preparing students to write academically.

Having studied mathematics for many years, students are well-placed to reflect upon that experience and attempt to make sense of it in the light of theoretical frameworks developed by researchers in the field. This module will help them with this process so that as mathematics graduates they will be able to contribute knowledgeably to future debate about the ways in which this subject is treated within the education system.

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 Black-Scholes 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
• Black-Scholes formula

Throughout the module, students will learn key financial maths skills, such as constructing binomial tree models; determining associated risk-neutral probability; performing calculations with the Black-Scholes formula; and proving various steps in the derivation of the Black-Scholes 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 real-world; improve their written and oral communication skills; and develop their critical thinking.

An introduction to the key concepts and methods of metric space theory, a core topic for pure mathematics and its applications, is given during this module. Studying this module will give students a deeper understanding of continuity as well as a basic grounding in abstract topology. With this grounding, they will be able to solve problems involving topological ideas, such as continuity and compactness.

They will also gain a firm foundation for further study of many topics including geometry, Lie groups and Hilbert space, and learn to apply their knowledge to areas including probability theory, differential equations, mathematical quantum theory and the theory of fractals.

This module, for the most part, concentrates on (Protestant) Christian thinkers from the German-speaking 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.

The aim of this module is to examine and evaluate some of the most central issues in Enlightenment and post-Enlightenment Western religious and atheistic philosophical debates. The module will begin by looking at the philosophy of G W F Hegel and its implications for subsequent religious and atheistic thought. We will then proceed to consider the thought of the post-Hegelian ‘masters of suspicion’: Feuerbach, Marx, Freud and Nietzsche. After this, we will look at ways in which religious and atheistic thought have been brought together, as manifested in various forms of ‘Christian atheism.’ Finally, we will consider postmodern critiques of modern atheism and the nature of the associated ‘return of religion.’

This module will address central issues in contemporary ethical (including meta-ethical), 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.

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 non-specialist, 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.

The aim of this module is to provide you with a through grounding in some of the central issues in philosophical aesthetics within the continental European tradition. The module introduces these issues by looking at the work of some of the most important philosophers who have written in this tradition. These philosophers are not only important in their own right and because of the influence that they have had and continue to have, but also because their work provides a way in to key debates and issues in aesthetics, as well as to enrich experience of and critical engagement with contemporary art in all its forms.

Are psychopaths evil or sick? Should the NHS pay for the treatment of nicotine addiction? Is it right for shy people to take character-altering 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.

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).

This module examines central themes in the liberal branch of contemporary Anglo-American 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.

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.

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).

This module is ideal for students who want to develop an analytical and axiomatic approach to the theory of probabilities.

First the notion of a probability space will be examined through simple examples featuring both discrete and continuous sample spaces. Random variables and the expectation will be used to develop a probability calculus, which can be applied to achieve laws of large numbers for sums of independent random variables.

Students will also use the characteristic function to study the distributions of sums of independent variables, which have applications to random walks and to statistical physics.

This module will examine the philosophical accounts of the imagination. We 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, as well as to other phenomena such as dreams. In addition, a range of topics focussing 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

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 anti-philosophy). 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 mid-20th 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 real-life problems.