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.

This module offers a critical introduction to Chinese philosophy by focusing on its diversity and contemporary relevance. The module introduces the main schools of thought that emerged during the classical era (one of the most vibrant periods of Chinese philosophy), including Daoism, Confucianism, Legalism, Mohism, and the school of names.

Students will be introduced to the main concepts used in classical Chinese philosophy as well as the central issues debated by classical Chinese philosophers, such as whether human nature is good or bad, whether one should engage in society or retrieve from it in order to live a fulfilled life, whether humans are at the centre of the world or simply a part of it, and whether language is enhancing human potential or limiting it, to name but a few examples.

Can we know what it is like to be a fish? Why do we spontaneously try to save a child who is in danger? Is a white horse a horse? Should we rejoice in non-action, engage in politics, or do both at the same time? Is there such a thing as feminist philosophy in ancient China? These are some of the issues that will be discussed during the term.

This module will offer an introduction to feminist philosophy by addressing the question of what feminist philosophy is and providing an overview of some important debates in feminist philosophy including the debate on the concept “woman” and the distinction between sex and gender, the literature on intersectionality and the relationship between gender-based and other forms of oppression, and feminist thinking on care, marriage and the family. These issues are among the most important ones in contemporary feminist philosophy as well as being issues that occupy a prominent place in public debate. Further, discussing them will allow us to explore different traditions and approaches to feminist philosophy, analysing contributions of analytic and continental feminist philosophers, of liberal feminist philosophers and those critical of the liberal tradition, and centring feminist perspectives that are often marginalised.

This module considers a range of issues currently being debated by political philosophers and political theorists. Specific topics may change slightly, but the current plan is cover the following, with attention to questions of freedom and justice throughout:

• Business corporations and employment
• Racism and sexism
• Democracy
• Climate politics
• Structural injustice and sweatshop labour
• Public health and state interventions

Studying this module should improve students’ knowledge and understanding of some key issues in metaphysics as determined by the syllabus. This focuses primarily on some issues concerning space and time, the nature of physical objects and persons, and some key philosophical distinctions. Studying this module should also enable them to see connections between various philosophical issues that should be of value to them with regard to other philosophy modules that they are studying.

This course explores ideas central to any understanding of politics. It focuses on two related themes: Equality, and Community. In the course we will explore the thought of thinkers who are associated with these ideas of equality and community (Rousseau, Marx, the Fabians, and Rawls). By the end of the course, you will have an understanding of the key ideas of the thinkers under review and be able to assess the contribution that these thinkers have made to our wider understanding of politics. You will also be able to recognise the relevance of these thinkers to our current political debates, and be able to employ their ideas within those debates. Additionally, you will be able to evaluate the key features of an argument, be confident to express your own views, and evaluate the responses of others.

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.

This course covers nineteenth-century philosophy, a crucial period in several ways: there was a new attention to history and the relation between philosophy and history; there was the rise of socialism and its impact on philosophy; and there were philosophical criticisms of Christianity, which were met by explicit defences of Christianity by some philosophers. We explore these issues through the work of six figures in nineteenth-century German and British philosophy: Hegel, Feuerbach, and Marx; Nietzsche, Cobbe, and Besant.

This module examines some theoretical issues involved in gaining knowledge about human societies. We will look at the role of theories and models in economics and political science, the special nature of "social institutions," and whether economic and political knowledge can be separated from value-judgments:

• Rational choice theory and models based on it
• Social norms and cooperation
• John Searle’s theory of “institutional facts”
• The nature of money and different accounts of power
• Whether values can or should be kept out of economics and political science
• Some ways in which states and markets are related

This course considers philosophical issues that arise in connection with the sciences. It 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.

It is not assumed that students 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.

Knowledge is an essential aspect of our social lives. This module focuses on a range of real world social, ethical and political problems involving knowledge. Topics include: problems of epistemic injustice (where people are not believed because of identity prejudice); whether virtues of open-mindedness might provide a solution to epistemic injustice. A proper understanding of the ethics and politics of knowledge requires us to examine both doubt and ignorance. We consider whether systemic racism is sustained by an active kind of “wilful” ignorance. We explore how powerful corporations seek to deliberately engineer doubt to further their interests. We examine political deception and the idea of “Post-truth” politics. In the final section we turn to the limits of seeking knowledge and how to balance the interests that states and corporations have in knowing personal information, against our interests in keeping such information private.

This module aims (a) to provide students an introduction to issues within the philosophy and politics of higher education, and (b) to help students to reflect about their own position in, and aims while at the university. During the modules, students will consider key questions regarding the aims of university education and its political context and history, as well as dedicate time to thinking about how their own studies fit into those aims and context, and what they wish to achieve during their undergraduate degrees. Twice weekly lectures will be supplemented by seminars, but also by regular ‘keynotes’ in which those working within and alongside higher education will present their own views and approach to higher education. Invited figures may include senior figures from within Lancaster University, representatives of UCU, the Student Union, and local stakeholders.

This course covers core theoretical questions surrounding the nature and status of normative claims: those involving moral, political, or other values. We explore the whether normative claims admit of truth or falsity, or whether they are merely expressions of preference: whether such claims can be objectively warranted, or are ultimately ‘subjective’. Themes to be treated include: the meaning of words such as ‘ought’ and ‘good’; the relationship between values and facts; the Frege-Geach problem; the place of motivation in value judgments. Key ‘metaethical’ theories, such as naturalistic and non-naturalistic realism, emotivism, and prescriptivism, will be outlined and explored.

This module will introduce students to some of the most well-known women philosophers from ancient India, China, and Greece. The content of the module will be philosophical sources that were either: authored by women, include views, voices, and/or characters that claim to represent a woman’s perspective, or that are explicitly about women. In addition to reading this source material, the module will also develop the awareness of and skills to address some of the unique challenges of studying women philosophers, particularly in contexts where it is not clear if women composed the sources attributed to them.

This module focuses on selected topics in Applied Philosophy. Applied Philosophy 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.

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.

This module will trace the development of Buddhist thought, from its emergence in India in the fifth century BCE, through its development across Asia, particularly China, but also Tibet and Japan. It will also look at how Buddhist philosophy has been received by Western philosophers from the 19th century. In addition to tracking and analysing key concepts, such as not-self, dependent origination, emptiness, and Buddha nature, this module will examine themes that pervade Buddhist philosophy in its various contexts, such as the relationships between teaching and practice, philosophy and literature, and religion and politics. Although the regional emphasis will change depending on the lecturer, this module will offer students an in-depth exploration of one of the most enduring, voluminous, and influential philosophical traditions of the world.

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.

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 course 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 ow 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.

PPR.399 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 write a dissertation of 9,000-10,000 words. They are expected to start thinking seriously about the 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 at the start of Summer Term in the third year. To help students prepare for work on the dissertation, there will be an introductory talk on topics relating to doing one's own research and planning and writing a dissertation. A course handout will be available setting out in more detail the requirements for the dissertation and giving full details of lectures, supervision arrangements and assessment.

The aim of this module is to allow students to pursue independent in-depth studies of a topic of their choice, within the scope of their 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. Students will develop their employability and research skills, and their ability to work independently at length under their own direction with input from external collaborators and 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.

Students are expected to start thinking seriously about the dissertation towards the end of the Lent term of the second year, and to submit a provisional topic by the end of that term. Work should begin during the Summer term of the second year and a draft plan must be approved by the end of the Summer term. Work should be well advanced by Christmas in the third year. The completed dissertation must be submitted at the start of Summer Term in the third year. To help students prepare for work on the dissertation, there will be an introductory talk on topics relating to doing one’s own research and planning and writing a dissertation. A course handout will be available setting out in more detail the requirements for the dissertation and giving full details of lectures, supervision arrangements and assessment.

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.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?
• 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?

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.

This course considers conceptual questions around 'health' and 'disease' (and related concepts of 'disability', 'normality', 'medicine', 'treatment') and explores how these relate to issues of health policy. We start by considering concepts of health and disease:

• Does whether a condition is a disease depend purely on matters of biological fact?
• Does a condition also have to be harmful to count as pathological?
• Is there any distinction that can be drawn between mental and physical disorders?
• Is it justified to treat people with mental disorders differently, e.g. in involuntary treatment?
• Should psychopaths who commit horrible crimes be considered to suffer from a disorder, or are they evil?
• What does it mean to say that someone is ‘normal’?

Many critics worry about medicalisation, and think that ever more conditions are coming to be considered diseases. Is this true, and does it matter?

We’ll also consider conceptual issues connected to treatment. ‘Evidence Based Medicine’ aims to employ treatments that have been shown to work. But, how can it be determined whether a treatment works? What should the aims of therapy be? What is the distinction between medicines and other drugs?

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 special subject focuses on feminist philosophy and in particular the study of women and feminists in the history of philosophy, using nineteenth-century British philosophy as a case study. The course provides an in-depth understanding of debates around women in the history of philosophy, the relation between feminism and women, and how to research and study historical women philosophers who until recently have been omitted from the canon. This will provide important transferable skills in doing research in the digital world, including working with digital archives and historical journals. The course will allow students to undertake a sustained piece of independent research on a historical essay of their choice by a woman philosopher from nineteenth-century Britain. Students taking this course will not merely be learning about philosophy as done by others; they will be doing cutting-edge philosophical research themselves.

This module will examine key sources in the history of Indian political philosophy from ancient times to the present. We will begin by looking at the most influential political sources from ancient India, including the inscriptions of King Ashoka and the Arthashastra. Some of the questions we will be asking are how the ideas in these texts speak to modern debates about secularism, pluralism, and civil religion. We will then turn our attention to the modern period, reading the political thought of figures such as Gandhi, Ambedkar, and Ashis Nandy. We will look at how these and other modern political thinkers draw from premodern Indian traditions, as well as how they engage with and critique Western political ideas from Indian perspectives.

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.

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.

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.

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.

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 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.

This course will examine some of the core philosophical questions raised by warfare and conflict. We will look at the ethics of war and killing, but also at more neglected philosophical issues in this area, and non-Western approaches as well as classic texts in the Western tradition.

We will do so by examining some of the central dilemmas faced by soldiers, policy makers and non-combatants, in the form of a weekly question for discussion. These questions include: Can war be beautiful? When, if ever, should we go to war? What counts as legitimate action in war? What, if anything, do we owe to our enemies? Is soldiering a good life? What does technological development mean for warfare? What should a responsible citizen do when their country is, or looks about to be, at war? Who has the epistemic authority to speak about war? Is war always tragic?

This module will introduce students to the thought of Sigmund Freud, the founder of psychoanalysis. As well as providing a firm grounding in Freud's own thought, it will also raise questions about the implications of Freud's thought for philosophy. To this end, it will examine Paul Ricoeur's pioneering work on Freud and philosophy and will also look at the work of subsequent thinkers who have explored the ramifications of Freud's work for philosophy, especially Jacques Lacan, Michel de Certeau and Slavoj Žižek. The module will raise questions about the extent to which philosophy should respond to some of the insights of psychoanalysis.

This module introduces central issues, problems and theories in philosophical aesthetics by critically examining a number of central topics including: the nature of aesthetic experience; the objectivity of aesthetic judgement; emotional responses to fiction; the moral and cognitive value of art; the aesthetic value of nature. In addition to central philosophical discussions, various findings from empirical psychology and neuroscience will also be used. Although examples from all of the arts will be employed throughout the course, the emphasis will be on the wider issues just listed, and not exclusively focussed on art. That is, aesthetics will be explored as an important area of the philosophy of value in general.

This module considers key philosophical issues in the sciences of human mind, behaviour and social structures, such as psychology, psychiatry, sociology, economics and history. Topics to be considered may include the status of reason-based explanations of human behaviour, the legitimacy of psychoanalytic explanations, the understanding of other societies, individualism versus collectivism in social explanation, and the scientific status of social models based on postulates of rational choice.

This course 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; its focus is 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 course 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 course is a survey of major topics and concepts in Anglo-American liberal political ideas. The syllabus will include the following topics: questions about justice; visions of the state; negative and positive liberty; equality, utility and rights; toleration and multiculturalism; neutrality and the market.

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 critically explores a range of key topics in the ethics and politics of communication. In the first half of the course, we begin by an introduction to some basic concepts in linguistics and philosophy of language – especially to do with the practical side of communication. We then focus on (a) how certain kinds of communication can bring about ethical change (e.g. making something permissible); (b) upon whether lying and other kinds of deception are permissible, and if so, when. In the second half we turn to some broadly political issues: whether political lying is justified in a way that everyday lying is not. We consider three domains where freedom of communication is both important and contentious: freedom of speech, freedom of the press, and freedom on social media, including the challenges posed by “content moderation”.

In the Twentieth Century, Western philosophy underwent a number of fundamental “turns” — the linguistic turn, the phenomenological turn, the postmodern turn. Some of these changes were viewed as “revolutions” in philosophy. At the extreme end, there were even arguments that Western philosophy, as conceived since Plato, was finished. In this module we explore some of these key transformations. We consider the “linguistic” turn, and the formation of “analytic philosophy” at the turn of the C20. One central figure of this linguistic turn is Ludwig Wittgenstein. But Wittgenstein shifts from being at the centre of analytic philosophy to arguing that philosophy is finished. At the same time, philosophy also undergoes a phenomenological turn. We focus on how this leads, via Sartre, to a revival of existentialism. The contrasts between French philosophy and English-speaking philosophy become even more pronounced in the final third of the C20, with post-structuralism and post-modernist philosophy viewed by the “analytic” philosophy community as not even being a kind of philosophy. We assess the roots of, and justification of, this “analytic/continental” divide.