# Mathematics and Philosophy BA (Joint Hons)

• Offers study abroad opportunities
• Offers a Year in China

## Key information

• Course length: 3 years
• UCAS code: GV15
• Year of entry: 2018
• Typical offer: A-level : ABB / IB : 33 including 6 in HL Mathematics with no score less than 4 / BTEC : Applications considered

### Programme Year One

Students take four modules from the Philosophy Year One programme.

Philosophy modules:

• Reading and Writing Philosophy 1
• Reading and Writing Philosophy 2 Critical, Analytical, and Creative Thinking
• Symbolic Logic 1

Students take the core foundation modules from the Mathematics Year One programme:

• Foundation Module I: Calculus
• Foundation Module II: Complex Numbers and Linear Algebra
• Foundation Module III: Multivariable Calculus

And one of the following:

• Numbers, Groups and Codes
• Mathematical Reasoning and Problem Solving

#### Year One Compulsory Modules

• ##### Calculus I (MATH101)
Level 1 15 First Semester 80:20 1.       To introduce the basic ideas of differential and integral calculus, to develop the basic  skills required to work with them and to  apply these skills to a range of problems. 2.       To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity. 3.       To introduce the notions of sequences and series and of their convergence. differentiate and integrate a wide range of functions;​sketch graphs and solve problems involving optimisation and mensuration​understand the notions of sequence and series and apply a range of tests to determine if a series is convergent
• ##### Calculus II (MATH102)
Level 1 15 Second Semester 80:20 ·      To discuss local behaviour of functions using Taylor’s theorem. ·      To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals. use Taylor series to obtain local approximations to functions; ​obtain partial derivaties and use them in several applications such as, error analysis, stationary points change of variables​evaluate double integrals using Cartesian and Polar Co-ordinates​
• ##### Introduction to Linear Algebra (MATH103)
Level 1 15 First Semester 80:20 To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.      To develop geometrical intuition in 2 and 3 dimensions.      To introduce students to the concept of subspace in a concrete situation.    To provide a foundation for the study of linear problems both within mathematics and in other subjects. manipulate complex numbers and solve simple equations involving them    ​solve arbitrary systems of linear equations​understand and use matrix arithmetic, including the computation of matrix inverses​compute and use determinants​understand and use vector methods in the geometry of 2 and 3 dimensions​calculate eigenvalues and eigenvectors and, if time permits, apply these calculations to the geometry of conics and quadrics
• ##### Reading and Writing Philosophy 1 (PHIL107)
Level 1 15 First Semester 0:100 To introduce the academic skills and knowledge necessary for the critical reading and writing of philosophy.​To foster in students an appreciation of the value of philosophy.​To enable students to read effectively and to takes notes efficiently.To develop students'' skill in presenting complex ideas to an audience and in practicing the intellectual virtues associated with philosophical discussion.  To promote students'' skill in writing rigorously argued, well-written and well-presented philosophical essays. ​​To promote students'' research skills.​ ​Students will be able to explain and evaluate some work relevant to a selected specialist topic in ethics. (This topic may vary from year to year. Examples include: human treatment of animals; ethics and the environment.)​Students will be able to explain and evaluate some central work about political liberty.​Students will be able to give structured seminar presentations and to conduct discussion in a manner that displays the intellectual virtues associated with philosophy. ​Students will be able to write essays that embody a philosophically-informed approach to argumentation.Students will be able to use the Harvard referencing system. ​Students will be able to conduct independent research in support of their work, using appropriate print and online resources (including the Routledge Encyclopedia of Philosophy and the Philosopher''s Index). ​
• ##### Reading and Writing Philosophy 2 (PHIL108)
Level 1 15 Second Semester 0:100 To consolidate the academic skills and knowledge necessary for the critical reading and writing of philosophy. To consolidate students'' appreciation of the value of philosophy.​To consolidate students'' ability to read and take notes effectively.​To consolidate students'' skill in presenting complex ideas to an audience and in practising the intellectual virtues associated with philosophical discussion.​​To consolidate students'' skill in writing rigorously argued, well-written and well-presented essays.To consolidate students'' research skills. ​To consolidate students'' appreciation of, and ability to use, forward-facing feedback and review. Students will be able to explain and evaluate some central work from the early modern period, covering the following topics: (i) perception; (ii) personal identity; (iii) freedom and determinism.Students will develop greater skill and confidence in giving structured seminar presentations and in conducting discussion in a manner that displays the intellectual virtues associated with philosophy. Students will develop greater skill and confindence in writing essays that embody a philosophically-informed approach to argumentation.​Students will be able to use the Harvard system of referencing. ​Students will be able to conduct independent research in support of their work, using appropriate print and online resources (including the Routledge Encyclopedia of Philosophy and the Philosopher''s Index).

#### Year One Optional Modules

• ##### Numbers, Groups and Codes (MATH142)
Level 1 15 Second Semester 80:20 ·         To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory. ·         To give an appreciation of the utility and power of group theory as the study of symmetries. ·         To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques. ​​ Use the division algorithm to construct the greatest common divisor of a pair of positive integers;                                                          ​Solve linear congruences and find the inverse of an integer modulo a given integer; ​Code and decode messages using the public-key method​Manipulate permutations with confidence​Decide when a given set is a group under a specified operation and give formal axiomatic proofs; ​Understand the concept of a subgroup and use Lagrange''s theorem;​Understand the concept of a group action, an orbit and a stabiliser subgroup​ Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;
• ##### Critical, Analytical and Creative Thinking (PHIL112)
Level 1 15 First Semester 60:40 To introduce students to the concepts and methods of informal logic and to enable students to use these concepts and methods in assessing arguments both within and outside philosophy. ​To help students to think more logically themselves, and to locate and remove inconsistencies in their own thoughts. ​To introduce students to methods of causal, statistical and probabilistic reasoning and to enable students to identify and avoid causal, statistical and probabilistic fallacies. ​To enable students to think creatively about problems and to come up with rational solutions to them, and to make logical decisions in the light of available evidence. ​Students will able to explain and apply the basic concepts of logic.​Students will be able to identify conclusions and premises in arguments, including hidden premises.Students will be able to reconstruct and evaluate arguments. ​Students will be able to distinguish between reasoning and rhetoric and to identify fallacies and rhetorical ploys in arguments. ​Students will be able to distinguish between deductive and inductive infererence, including distinguishing between different types of inductive inference (enumerative, statistical, causal, analogical).​Students will be able to tell when a given set of statements is logically consistent and when it is not.​Students will be able to explain some of the problems with relativism about truth.Students will be able to explain and apply some of the basic principles of statistics and of probablity theory. ​​Students will be able to demonstrate creative thinking by spotting possibilities missed by less creative thinkers.
• ##### Introduction to Logic (PHIL127)
Level 1 15 Second Semester 60:40 To introduce students to the concepts, a language of, and the methods of, classical sentential logic.To introduce students to a language of classical quantificational logic. Students will be able to explain and apply the basic concepts of classical sentence logic. Students will be able to translate from English into sentence logic and vice versa.​Students will be able to construct and use truth tables.​Students will be able to construct proofs in natural deduction for sentence  logic. Students will be able to translate from English into quantificational logic and vice versa.​
• ##### Ethics (PHIL101)
Level 1 15 First Semester 85:15 ​Students will become familiar with key concepts in ethics – bothmeta-ethics and normative ethics.​  ​​Students will gain an acquaintance with the main approaches to moral theory (such as virtue ethics, deontology, consequentialism), as well as key debates in meta-ethics (subjectivism vs objectivism, naturalism vs non-naturalism).​​Students will tackle central questions in ethics, such as ‘is a good action more about good intentions than beneficial outcomes?’, ‘does lying possess an objective property of badness?’, ‘ought different people to follow different moral codes?’ and ‘what activities lead to a good life?’.​ ​Students will be able to distinguish between some main concepts in ethical debates, past and present.​Students will be able to explain recent developments in ​meta-ethics and normative ethics.​Students will be able to evaluate some of the main theories in the history of moral philosophy and contemporary ethics.Students will be able to analyse concepts and arguments relating to ethical issues.​Students will be able to identify philosophical assumptions underlying ethical claims and judgments.Students will be able to structure a discussion of issues in ethics.​​Students will be able to speak with confidence and clarity on issues of moral philosophy.Students will be able to explain details of canonical texts in moral philosophy.​Students will be able to articulate and defend basic positions in classic and contemporary moral philosophy.Students will be able to write coherently and rigorously about abstract philosophical issues raised by ethical debates.
• ##### Introduction to Statistics (MATH162)
Level 1 15 Second Semester 80:20 To introduce topics in Statistics and to describe and discuss basic statistical methods. To describe the scope of  the application of these methods. to describe statistical data;​ to use the Binomial, Poisson, Exponential and Normal distributions; ​to perform simple goodness-of-fit tests​to use the package Minitab to present data, and to make statistical analysis
• ##### Mind, Knowledge and Reality (PHIL103)
Level 1 15 First Semester 60:40 To introduce students to some of the main topics in metaphysics: God, the mind/body problem, personal identity, time and free will. To introduce students to the philosophical system of Rene Descartes.​ Students should be able to distinguish between sound and unsound arguments.Students should be able to build a case for a specific metaphysical position, by weighing theoretical virtues, such as Occam''s razor, and metaphysical principles, such as the conceivability principle and the principle of sufficient reason.​​Students should be able to extract an argument from text, render put it into standard form, and critically evaluate its premises.​Students should be able to explain Descartes'' philosophical system.Students should be able explain the basic issue, and the standard views, pertaining to five topics in contemporary metaphysics: God, personal identity, consciousness, free will and time.​​Students should be able to able to argue for a specific view pertaining to five issues in contemporary metaphysics: God, personal identity, consciousness, free will and time.​Students should be able to discuss reality in the partially abstract manner distinctive of metaphysical thought.
• ##### Political Philosophy (PHIL102)
Level 1 15 Second Semester 60:40 Students will be introduced to the theories and arguments of some of the most important philosophers and of the western tradition of political thought, such as Plato, Aristotle, Hobbes, Locke, Marx and Mill. Students will be introduced to some of the main concepts in political philosophy, including political obligation, democracy, community, rights, liberty, justice and property.​ Students will be able to distinguish some main concepts in political philosophical debates.​Students will be able to distinguish between different ways of understanding  concepts in political philosophical debates.Students will be able to explain and evaluate some of the main theories in the history of political philosophy.​Students will be able to analyse concepts and arguments relating to political issues.​Students will be able to identify philosophical assumptions underlying political claims.​Students will be able to structure a discussion of issues in political philosophy.​Students will be able to speak with confidence and clarity on issues of political philosophy​.Students will be able to explain details of canonical texts in political philosophy.Students will be able to articulate and defend basic positions in political philosophy.​Students will be able to write coherently and rigorously about abstract philosophical issues raised by political debates.
• ##### Philosophy and the Arts (PHIL110)
Level 1 15 Second Semester 60:40 ​​​​​To consider philosophically relevant questions and concepts pertaining to the scope of art and the evaluation of artworks.To enable students to reflect philosophically about their intuitions regarding the arts and about their appreciation of particular artistic media.​ Students will be able to examine whether the concept of art may apply to objects and activities from different historical periods and cultural contexts.​Students will be able to consider critically the impact that cultural institutions and their practices may have on philosophical theorising concerning the arts.​Students will be able to assess the view that artistic value is a matter of subjective response to it.Students will be able to analyse the character of self-expression through art, and assess its significance in evaluating artworks.​Students will be able to evaluate the argument that artistic intentions must inform our appreciation of works of art.​Students will be able to define and expound the conception of beauty in a narrow and in a wide sense.​Students will be able to outline and discuss the significance of the distinction between artistic and aesthetic properties.​Students will be able to argue for or against the view that artworks are unrepeatable.​Students will be able to interpret the ways in which content and meaning is attributed to art that does not seem to represent anything.​Students will be able to provide a critical account of the possible links between seeking truth and creating good art.​Students will be able to discuss whether art can function as a vehicle for demonstrating what is morally good.

### Programme Year Two

Students choose modules to the value of four units from the Philosophy Year Two programme and four from Mathematics.

#### Year Two Compulsory Modules

• ##### Logic (PHIL207)
Level 2 15 First Semester 60:40 To introduce students to the language and methods of classical quantificational logic.To enable students to use trees for both sentence logic and quantificational logic.​​To relate quantificational logic to the philosophy of language. ​ Students will be able to explain and apply the basic concepts of classical quantificational logic.Students will consolidate their skill in ​translating from English into quantificational logic and vice versa.Students will be able to construct proofs in natural deducation for valid sequents of quantificational logic. Students will be able to test sets of formulas for consistency using trees and to assess sequents of sentence logic and sequents of quantificational logic for validity using trees.​Students will be able to explain Russell''s theory of definite descriptions and formally to represent sentences that use definite descriptions in a Russellain manner using the notation of quantificational logic.Students will be able to define, both formally and informally, some formal properties of relations (i.e., reflexivity, symmetry, transitivity and related properties)​ and to represent these properties using diagrams.

#### Year Two Optional Modules

• ##### Theory of Knowledge (PHIL212)
Level 2 15 Second Semester 60:40 To introduce students to the main philosophical questions concerning the concept of knowledge.  To introduce students to the main views taken by historical and contemporary philosophers concerning the concept of knowledge. ​To enable students to form and defend their own views concerning the concept of knowledge. ​Students will be able to discuss the main philosophical questions concerning perception.​​Students will be able to name and discuss direct realism, indirect realism, and anti-realism concerning perception.Students will be able to discuss the main philosophical questions concerning memory.​Students will be able to name and discuss realism and anti-realism about the memory.​​Students will be able to discuss the main philosophical questions concerning introspection.​Students will be able to name and discuss realism and anti-realism about the self.​​Students will be able to discuss the main philosophical questions concerning reason.​​​Students will be able to name and discuss Kripke''s views and the traditional view concerning a priority, necessity, and analyticity.​Students will be able to discuss the main philosophical questions concerning testimony.​​​Students will be able to name and discuss both the direct-source and the indirect-source view of testimony.Students will be able to discuss the main philosophical questions concerning the structure of knowledge.​​​​​​Students will be able to name and discuss foundationalism and coherentism, and various views on which beliefs are suitable to be in the foundations (strong classical foundationalism, weak classical foundationalism, and theistic foundationalism).​Students will be able to discuss the main philosophical questions concerning the definition of ‘knowledge''.Students will be able to deploy Gettier-style examples, and name and discuss the tripartite definition of ''knowledge'', internalism, conclusive justificationism, externalism, reliabilism, and proper functionalism.​​​Students will be able to discuss the main philosophical questions concerning scepticism.​​Students will be able to name and discuss global scepticism, 1st-order scepticism, 2nd-order scepticism, the verity principle, the necessity principle, and the infallibility principle, and various responses such as contextualism.​​Students will be familiar with, and able to discuss, scepticism concerning induction.​​​Students will be able to discuss Hume''s doubts concerning induction.
• ##### Philosophy of Religion (PHIL215)
Level 2 15 First Semester 60:40 ​To introduce the current state of discussion concerning the concept of God. ​​To introduce the major arguments for, and the major arguments against, the existence of God. ​To enable the student to clarify and develop his or her own views on whether God exists and what God is like. ​Students will be able to list the four main approaches to the concept of God (universal revelational theology, purely biblical theology, creation theology, and perfect-being theology), and discuss their strengths and weaknesses.​Students will be able to name the main attempts at defining ‘omnipotence'', and their strengths and weaknesses, with particular reference to the paradox of the stone.​Students will be able to discuss the main attempts at defining ''omniscience'', and discuss their strengths and weaknesses, with particular reference to the problem of freedom and foreknowledge.​​​Students will be able to discuss the main attempts at defining divine goodness, and discuss their strengths and weaknesses, with particular reference to the problem of whether God can be good if he is unable to sin.​​Students will be able to discuss and evaluate the ontological argument in its versions by Anselm, Descartes, and Plantinga.​Students will be able to discuss and evaluate the cosmological argument in its versions by Aquinas, Leibniz, and van Inwagen.​Students will be able to discuss and evaluate the design argument in its versions by Aquinas, Paley, and Swinburne.​​Students will be able to discuss and evaluate Plantinga''s view that we need no argument or evidence to believe in God rationally, James''s view that faith is a gamble like the leap of a mountaineer, and Pascal''s view that faith is pragmatically justified.​Students will be able to discuss and evaluate the logical and evidential arguments for atheism ​from the existence of evil, and the various defences and theodicies in response to them, in particular the free-will defence and the greater-good defence.
• ##### Liberty, Justice and the Good Society (PHIL219)
Level 2 15 Second Semester 60:40 ​Students will be invited to consider the theories and arguments of thinkers who have shaped contemporary political philosophy, such as John Rawls, Robert Nozick and Michael Walzer. Students will be asked to consider some of the main concepts in political philosophy, including freedom, equality and justice.​Students will be invited to appreciate the variety of philosophical issues raised by contemporary political debates around controversial topics, such as feminism and multiculturalism.​ Students will be able to distinguish some of the main concepts in debates within contemporary political philosophy.​Students will be able to explain different ways of understanding concepts employed in debates in contemporary political philosophy.Students will be able to explain and evaluate some of the main theories in contemporary political philosophy.​Students​ will be able to analyse concepts and arguments relating to contemporary political issues.Students will be able to identify philosophical assumptions underlying political questions and claims.Students will be able to speak with confidence and clarity on issues of contemporary political philosophy.Students will be able to explain details of influential texts in recent political philosophy.​Students will be able to articulate and defend positions on issues in contemporary political philosophy.​Students​ will be able to write coherently and rigorously about philosophical issues raised by current political controversies.​Students will be able to structure a discussion of issues in contemporary political philosophy.
• ##### Metaphysics (PHIL228)
Level 2 15 Second Semester 60:40 To provide an introduction to some of the most significant debates in contemporary metaphysics;  topics include:  change and persistence, objects and properties, space and time. ​​Students will be able to identify the main issues and positions in contemporary metaphysical discussions of space, time, persistence, properties, substance, persons, modality and existence.Students will be able to explain the main strengths and weaknesses of these positions.​​Students will be able to identify the historical contexts of some of these positions.​​Students will be able to construct a positive case for a specific metaphysical position, by appealing to theoretical virtues (e.g. simplicity), metaphysical principles (e.g. the principle of sufficient reason) and thought experiments which evoke powerful intuitions.​​Students will further develop their abilities to extract arguments from texts, render them in standard form, and assess the soundness of their premises and the validity of their structures.​​Students will be able to think more creatively about metaphysical issues.​​Students will be able to explain the competing positions in contemporary meta-metaphysics.​
• ##### Ancient Greek Philosophy (PHIL237)
Level 2 15 First Semester 60:40 To consider the theories and arguments of some of the most important ancient philosophers, in particular Plato and Aristotle. To consider key ethical, epistemological and metaphysical concepts relevant to ancient philosophy, and their interconnections.To analyse and practise the dialectical skills portrayed in the ancient texts. Students will be able to explain and evaluate some of the main theories in ancient Greek philosophy.​Students will be able to analyse concepts and arguments relating to classic ethical, epistemological and/or metaphysical issues.​Students will be able to identify points of agreement and disagreement between different philosophies.​Students will be able to structure a discussion of central issues in ancient Greek philosophy​.​Students will be able to engage dialectically with positions in ancient Greek philosophy and to articulate the implications of these positions.​Students will be able to present their ideas with clarity and confidence.​Students will be able to write coherent, structured and informative accounts of abstract philosophical issues.​​Students will be able to highlight the social and political dimensions of ancient Greek philosophy, in terms of justice and the ''good life''.​Students will be able to compare and contrast philosophy as an intellectual activity and philosophy as a ''way of life'', with reference to Socratic enquiry.​​Students will be able to reflect on the significance of ancient Greek philosophy for the emergence of Western ideals about democracy and rational discourse.
• ##### Moral Philosophy (PHIL239)
Level 2 15 First Semester 60:40 Students will be invited to consider the theories and arguments of some of the most important contemporary moral philosophers focused on normative and applied ethics, including James Rachels, Peter Singer and Bernard Williams.Students will be asked to consider some of the main concepts in moral philosophy, including consequentialism, deontology, virtue, impartiality, agent-relativity/neutrality and speciesism.​Students will be invited to appreciate the variety of philosophical issues raised by morality and a range of controversial practices such as punishment, abortion, euthanasia and the treatment of nonhuman animals.​ Students will be able to distinguish some of the main concepts in moral philosophical debates.Students will be able to distinguish between different ways of understanding  concepts in moral philosophical debates.​Students will be able to explain and evaluate some of the main theories in contemporary moral philosophy​Students will be able to analyse concepts and arguments relating to ethical issues.​Students will be able to identify philosophical assumptions underlying ethical claims.​Students will be able to structure a discussion of issues in moral philosophy​Students will be able to speak with confidence and clarity on issues of moral philosophy​.​Students will be able to explain details of influential texts in recent moral philosophy.​Students will be able to articulate and defend positions in moral philosophy.​Students will be able to write coherently and rigorously about abstract philosophical issues raised by ethical controversies.​
• ##### Business Ethics S1 (PHIL271)
Level 2 15 First Semester 0:100 ​​To introduce and explain major contemporary perspectives on corporate behaviours.​To introduce moral perspectives as they relate to managerial decision making and corporate structures.To make students familiar with a range of recurrent ethical problems arising in business. ​To improve students'' skills in identifying and analyzing ethical issues that managers and employees face.​To give students practice in formulating, defending, and planning the implementation of action plans managing ethical dilemmas.​ Students will be able to discuss the main theories concerning the place of ethics in business. ​​Students will be able to state the broad principles of , and discuss the strengths and weaknesses of,basic moral theories, such as consequentialism​​​​Students will  be able to state and discuss the broad ethical principles concerning  costs and benefits, the challenge posed by  uncertainty, professional roles, profits and the right of shareholder interests, and affirmative action. .​​Students will be able to state and discuss the broad ethical principles concerning the obligations of complex organizations with respect to loyalty and whistle-b lowing, insider trading, customer responsibility, and corporate responsibility. Students will be able to state and discuss the broad ethical principles concerning social justice and executive compensation.​Students will be able to consider an ethical approach as a basis for sustainable marketing.
• ##### Chinese Philosophy (PHIL220)
Level 2 15 First Semester 60:40 ​To examine the ways in which philosophy in Classical Chinese civilisation develops in the Hundred Schools period after the dialogue between Kongzi and Mozi, and to relate fundamental Chinese concepts to Western counterparts. ​To investigate what is distinctive about classical Chinese approaches to questions of ontology, social harmony, personal morality and soteriology. ​Students will be able to engage in informed discussions about the concepts and categories in which  philosophical discussions were conducted in ancient China.​Students will develop skills in developing and contextualising new information about other worldviews.​Sudents will be enabled to assimilate alternative cultural perspectives from which to view their own traditions.​Students will be able to explain and evaluate some of the main theories propounded in the classical period of Chinese thought.​Students will be able to discuss the problem of​ cultural relativism informed by an understanding of a particular alien pattern of thinking.​Students will be able to relate classical Chinese thought to European philosophical interests.
• ##### Classical Mechanics (MATH228)
Level 2 15 Second Semester 90:10 To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems. ​the motions of bodies under simple force systems, including calculations of the orbits of satellites, comets and planetary motions ​ motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravitry over the Earth''s surface  ​rigid body dynamics using centre of mass, angular momentum and moments of inertia
• ##### Commutative Algebra (MATH247)
Level 2 15 Second Semester 90:10 To give an introduction to abstract commutative algebra and show how it both arises naturally, and is a useful tool, in number theory. After completing the module students should be able to: • Work confidently with the basic tools of algebra (sets, maps, binary operations and equivalence relations). • Recognise abelian groups, different kinds of rings (integral, Euclidean, principal ideal and unique factorisation domains) and fields. • Find greatest common divisors using the Euclidean algorithm in Euclidean domains. • Apply commutative algebra to solve simple number-theoretic problems.
• ##### Complex Functions (MATH243)
Level 2 15 First Semester 80:20 To introduce the student to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory. After completing this module students should:  -  appreciate the central role of complex numbers in mathematics; -  be familiar with all the classical holomorphic functions; -  be able to compute Taylor and Laurent series of such functions; -  understand the content and relevance of the various Cauchy formulae and theorems; -  be familiar with the reduction of real definite integrals to contour integrals; -  be competent at computing contour integrals.
• ##### Engineering Mathematics II (MATH299)
Level 2 7.5 First Semester 90:10 To introduce some advanced Mathematics required by Engineers, Aerospace Engineers, Civil Engineers and Mechanical Engineers. To develop the students ability to use the mathematics presented in the module in solving problems. A good knowledge of matrices and their use to solve systems of linear equations.​ An understanding of how to find eigenvalues and eigenvectors.A good knowledge of multi-variable calculus​.
• ##### Field Theory and Partial Differential Equations (MATH283)
Level 2 7.5 First Semester 80:20 To introduce students to the concepts of scalar and vector fields. To develop techniques for evaluating line, surface and volume integrals. To introduce students to some of the basic methods for solving partial differential equations After completing the module, students should be able to: Evaluate Grad, Div, Curl and Laplacian operators in Cartesian and polar coordinates Evaluate line, double and volume integrals Have a good understanding of the physical meaning of flux and circulation  Be able to solve simple boundary value problems for the wave equation, diffusion equation and Laplace''s equation
• ##### Financial Mathematics I (MATH267)
Level 2 15 First Semester 90:10 This module is to provide an understanding of the fundamental concepts of financial mathematics, and how these concepts are applied in calculating present and accumulated values for various streams of cash flows. Students will also be given an introduction to financial instruments, such as derivatives, the concept of no-arbitrage. 1. Understand and calculate all kind of rates of interest, find the future value and present value of a cash flow, and write the equation of value given a set of cash flows and an interest rate.  ​2. Derive formulae for all kinds of annuities​3. Given an annuity with level payments, immediate (or due) , payable m-thly, (or payable continuously), and any three of present value, future value, interest rate, payment, and term of annuity, calculate the remaining two items. ​4. Given an annuity with non-level payments, immediate (or due) , payable m-thly, (or payable continuously), the pattern of payment amounts, and any three of present value, future value, interest rate, payment, and term of annuity, calculate the remaining two items. ​5. Calculate the outstanding balance at any point in time. ​6. Calculate a schedule of repayments under a loan and identify the interest and capital components in a given payment. ​7. Given the quantities, except one, in a sinking fund arrangement calculate the missing quantity. ​8. Calculate the present value of payments from a fixed interest security, bounds for the present value of a redeemable fixed interest security. ​9. Given the price, calculate the running yield and redemption yield from a fixed interest security. ​10. Calculate the present value or real yield from an index-linked bond. ​11. Calculate the price of, or yield from, a fixed interest security where the income tax and capital gains tax are implemented. ​12. Calculate yield rate, the dollar-weighted and time weighted rate of return, the duration and convexity of a set of cash flows. ​13. Describe the concept of a stochastic interest rate model and the fundamental distinction between this and a deterministic model.
• ##### Financial Mathematics II (MATH262)
Level 2 15 Second Semester 90:10  to provide an understanding of the basic financial mathematics theory used in the study process of actuarial/financial interest,  to provide an introduction to financial methods and derivative pricing financial instruments ,  to understand some financial models with applications to financial/insurance industry,  to prepare the students adequately and to develop their skills in order to be ready to sit for the CT1 & CT8 subject of the Institute of Actuaries (covers 20% of CT1 and material of CT8). After completing the module students should: (a) Understand the assumptions of CAMP, explain the no riskless lending or borrowing and other lending and borrowing assumptions, be able to use the formulas of CAMP, be able to derive the capital market line and security market line, (b) Describe the Arbitrage Theory Model (APT) and explain its assumptions, perform estimating and testing in APT, (c) Be able to explain the terms long/short position, spot/delivery/forward price, understand the use of future contracts, describe what a call/put option (European/American) is and be able to makes graphs and explain their payouts, describe the hedging for reducing the exposure to risk, be able to explain arbitrage, understand the mechanism of short sales, (d) Explain/describe what arbitrage is, and also the risk neutral probability measure, explain/describe and be able to use (perform calculation) the binomial tree for European and American style options, (e) Understand the probabilistic interpretation and the basic concept of the random walk of asset pricing, (f) Understand the concepts of replication, hedging, and delta hedging in continuous time, (g) Be able to use Ito''s formula, derive/use the the Black‐Scholes formula, price contingent claims (in particular European/American style options and forward contracts), be able to explain the properties of the Black‐Scholes formula, be able to use the Normal distribution function in numerical examples of pricing, (h) Understand the role of Greeks , describe intuitively what Delta, Theta, Gamma is, and be able to calculate them in numerical examples.
• ##### Geometry of Curves (MATH248)
Level 2 15 Second Semester 90:10 To introduce geometric ideas and develop the basic skills in handling them. To study the line, circle, ellipse, hyperbola, parabola,  cubics and many other curves. To study theoretical aspects of parametric, algebraic and projective curves. To study and sketch curves using an appropriate computer package. After completing this module students should be able to: - use a computer package to study curves and their evolution in both parametric and algebraic forms. -determine and work with tangents, inflexions, curvature, cusps, nodes, length and other features. -calculate envelopes and evolutes. - solve the position and shape of some algebraic curves including conics.   The first learning outcome is assessed by coursework, the others by both coursework and examination.
• ##### Group Project Module (MATH206)
Level 2 15 Second Semester 0:100 ·         To give students experience of working effectively in small groups. ·         To train students to write about mathematics. ·         To give students practice in delivering presentations. ·         To develop students’ ability to study independently. ·         To prepare students for later individual project work. ·         To enhance students’ appreciation of the connections between different areas of mathematics. ·         To encourage students to discuss mathematics with each other. Work effectively in groups, and delegate common tasks.​Write substantial mathematical documents in an accessible form. ​Give coherent verbal presentations of more advanced mathematical topics.  ​Appreciate how mathematical techniques can be applied in a variety of different contexts
• ##### Hellenistic, Neoplatonic, Byzantine and Medieval Philosophy (PHIL221)
Level 2 15 Second Semester 60:40 To consider the theories and arguments of some of the most important philosophers of the Hellenistic, late Antique and Medieval periods.To study key ethical, epistemological and metaphysical concepts and their interconnections.To enable students to analyse and practise the dialectical skills portrayed in the texts examined. Students will be able to explain and evaluate some of the main theories in ancient and/or medieval philosophyStudents will be able to analyse concepts and arguments relating to classic ethical, epistemological and/or metaphysical issuesStudents will be able to structure a discussion of central issues in ancient and/or medieval philosophy​Students will be able to identify points of agreement and disagreement between different philosophiesStudents will be able to dialectically engage with positions in ancient and/or medieval philosophy and articulate their implicationsStudents will be able to present their ideas with clarity and confidenceStudents will be able to develop in writing coherent, structured and informative accounts on abstract philosophical issues
• ##### Introduction to Methods of Operational Research (MATH261)
Level 2 15 First Semester 90:10 ​After completing the module students should:Appreciate the operational research approach.Be able to apply standard methods to a wide range of real-world problems as well as applications in other areas of mathematics.Appreciate the advantages and disadvantages of particular methods.Be able to derive methods and modify them to model real-world problems.Understand and be able to derive and apply the methods of sensitivity analysis.  ​ ​​Appreciate the operational research approach.​Be able to apply standard methods to a wide range of real-world problems as well asapplications in other areas of mathematics.​Appreciate the advantages and disadvantages of particular methods.​Be able to derive methods and modify them to model real-world problems.​​Understand and be able to derive and apply the methods of sensitivity analysis.  Appreciate the importance of sensitivity analysis. ​
• ##### Introduction to the Methods of Applied Mathematics (MATH224)
Level 2 15 Second Semester 90:10 To provide a grounding in elementary approaches to solution of some of the standard partial differential equations encountered in the applications of mathematics. To introduce some of the basic tools (Fourier Series) used in the solution of differential equations and other applications of mathematics. After completing the module students should: -               be fluent in the solution of basic ordinary differential equations, including systems of first order equations; -               be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations; -               be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations; -               be able to solve simple first order partial differential equations; -               be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.
• ##### Introduction to Methods of Operational Research (MATH261)
Level 2 15 First Semester 90:10 ​After completing the module students should:Appreciate the operational research approach.Be able to apply standard methods to a wide range of real-world problems as well as applications in other areas of mathematics.Appreciate the advantages and disadvantages of particular methods.Be able to derive methods and modify them to model real-world problems.Understand and be able to derive and apply the methods of sensitivity analysis.  ​ ​​Appreciate the operational research approach.​Be able to apply standard methods to a wide range of real-world problems as well asapplications in other areas of mathematics.​Appreciate the advantages and disadvantages of particular methods.​Be able to derive methods and modify them to model real-world problems.​​Understand and be able to derive and apply the methods of sensitivity analysis.  Appreciate the importance of sensitivity analysis. ​
• ##### Life Insurance Mathematics I (MATH273)
Level 2 15 First Semester 100:0 Provide a solid grounding in the subject of life contingencies for single life, and in the subject of the analysis of life assurance and life annuities, including pension contracts.Provide an introduction to mathematical methods for managing the risk in life insurance,?Develop skills of calculating the premium for a certain life insurance contract, including allowance for expenses and profits?Prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT5 subject of the Institute and Faculty of Actuaries. Be able to explain and analyze the factors that affect mortality, simple life assurance and life annuity contracts. ​Understand the concept (and the mathematical assumptions) of the future life time random variables in continuous and discrete time, ​Be able to define the survivals probabilities and the force of mortality of the (c) section of the Syllabus, explain these types of probabilities and the force of mortality intuitively, be able to calculate the different types of the survival probabilities in theoretical and numerical examples. ​Understand the concept of the De Moivre, Makeham, Gompertz, Weibull and the exponential law (constant force of mortality) for modelling fractional ages, explain the basic difference between the laws above, be able to use these laws to calculate the survival probabilities of (c) of the Syllabus in numerical examples. ​Understand, define/calculate and derive the expected present values of all types of the life assurances of (d) of the Syllabus. Derive relations between life assurances both in continuous and discrete time, be able to use recursive equations for the calculation of the expected present value of different types of life assurances, calculate the variance of the present values for basic forms of life assurances.​Be able to derive the distributions and the moment/variance of the aforementioned future lifetimes, be able to make graphs of these future life times.
• ##### Linear Algebra and Geometry (MATH244)
Level 2 15 First Semester 90:10 To introduce general concepts of linear algebra and its applications in geometry and other areas of mathematics. After completing the module students should be able to: • appreciate the geometric meaning of linear algebraic ideas, • appreciate the concept of an abstract vector space and how it is used in different mathematical situations, • apply a change of coordinates to simplify a linear map, • manipulate matrix groups (in particular Gln, On and Son), • understand bilinear forms from a geometric point of view.
• ##### Mathematical Models: Microeconomics and Population Dynamics (MATH227)
Level 2 15 First Semester 90:10 1.             To provide an understanding of the techniques used in constructing, analysing, evaluating and interpreting mathematical models. 2.             To do this in the context of two non-physical applications, namely microeconomics and population dynamics. 3.             To use and develop mathematical skills introduced in Year 1 - particularly the calculus of functions of several variables and elementary differential equations. After completing the module students should be able to: -               Use techniques from several variable calculus in tackling problems in microeconomics. -               Use techniques from elementary differential equations in tackling problems in population dynamics. -               Apply mathematical modelling methodology in these subject areas. All learning outcomes are assessed by both examination and course work.
• ##### Mathematics Education and Communication (MATH291)
Level 2 15 Whole Session 0:100 ​Improving communication skills.Exposing students to current pedagogical practice​ and issues related to child protectionEncouraging students to reflect on mathematics with which they are familiar in a teaching context. Confidence in planning and presenting mathematics to school-age children.​Knowledge of current best pedagogical practice and child protection issues.Ability to work in a team.Understanding the role of outreach in mathematics education.
• ##### Mathematics International Exchange Module I (MATH271)
Level 2 60 First Semester 0:100 This is an exchange module. Compatible modules are selected by the outgoing students with approval by the programme director and should mirror as closely as possible the modules normally taking at Liverpool during the same period. The aim of the exchange is to provide the student with experience of living in another country and sampling different academic environments. By the end of this module the student should have: -achieved learning outcomes equivalent to 60 credits contributing to their current programme as agreed by their programme director -enhanced their knowledge and skills in specific areas of their current programme
• ##### Mathematics International Exchange Module II (MATH272)
Level 2 60 Second Semester 0:100 This is an exchange module. Compatible modules are selected by the outgoing students with approval by the programme director and should mirror as closely as possible the modules normally taking at Liverpool during the same period. The aim of the exchange is to provide the student with experience of living in another country and sampling different academic environments. By the end of this module the student should have: -achieved learning outcomes equivalent to 60 credits contributing to their current programme as agreed by their programme director. -enhanced their knowledge and skills in specific areas of their current programme
• ##### Metric Spaces and Calculus (MATH241)
Level 2 15 First Semester 90:10 To introduce the basic elements of the theories of metric spaces and calculus of several variables. After completing the module students should: Be familiar with a range of examples of metric spaces. Have developed their understanding of the notions of convergence and continuity. Understand the contraction mapping theorem and appreciate some of its applications. Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map. Understand the inverse function and implicit function theorems and appreciate their importance. Have developed their appreciation of the role of proof and rigour in mathematics.
• ##### Numerical Methods (MATH266)
Level 2 15 Second Semester 90:10 To provide an introduction to the main topics in Numerical Analysis and their relation to other branches of Mathematics After completing the module students should be able to: • write simple mathematical computer programs in Maple, • understand the consequences of using fixed-precision arithmetic, • analyse the efficiency and convergence rate of simple numerical methods, • develop and implement algorithms for solving nonlinear equations, • develop quadrature methods for numerical integration, • apply numerical methods to solve systems of linear equations and to calculate eigenvalues and eigenvectors, • solve boundary and initial value problems using finite difference methods.
• ##### Operational Research: Probabilistic Models (MATH268)
Level 2 15 Second Semester 90:10 To introduce a range of models and techniques for solving under uncertainty in Business, Industry, and Finance. The ability to understand and describe mathematically real-life optimization problems.​Understanding the basic methods of dynamical decision making.​Understanding the basics of forecasting and simulation.​The ability to analyse elementary queueing systems.
• ##### Ordinary Differential Equations (MATH201)
Level 2 15 First Semester 90:10 To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations. To illustrate the breadth of applications of ODEs and fundamental importance of related concepts. After completing the module students should be:  - familiar with elementary techniques for the solution of ODE''s, and the idea of reducing a complex ODE to a simpler one; - familiar with basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions; - well versed in the solution of linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix; - aware of a range of applications of ODE.
• ##### Statistical Theory and Methods I (MATH263)
Level 2 15 Second Semester 85:15 To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation.  The emphasis is on applications. After completing the module students should have a conceptual and practical understanding of a range of commonly applied statistical procedures.  They should have also developed some familiarity with the statistical package MINITAB.
• ##### Statistical Theory and Methods II (MATH264)
Level 2 15 Second Semester 90:10 To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory. After completing the module students should understand basic probability calculus. They should be familiar with a range of techniques for solving real life problems of the probabilistic nature.
• ##### Vector Calculus With Applications in Fluid Mechanics (MATH225)
Level 2 15 First Semester 85:15 To provide an understanding of the various vector integrals, the operators div, grad and curl and the relations between them. To give an appreciation of the many applications of vector calculus to physical situations. To provide an introduction to the subjects of fluid mechanics and electromagnetism. After completing the module students should be able to: -     Work confidently with different coordinate systems. -     Evaluate line, surface and volume integrals. -     Appreciate the need for the operators div, grad and curl together with the associated theorems of Gauss and Stokes. -     Recognise the many physical situations that involve the use of vector calculus. -     Apply mathematical modelling methodology to formulate and solve simple problems in electromagnetism and inviscid fluid flow. All learning outcomes are assessed by both examination and course work.

### Programme Year Three

Students choose modules to the value of four units from the Philosophy Year Three programme and four Mathematics modules. Students may choose to undertake a project as one of the Mathematics options – this may be on a topic related to Philosophy.

#### Year Three Optional Modules

• ##### Philosophy Dissertation (PHIL306)
Level 3 30 Whole Session 0:100 The aim is for the student to choose a topic of special interest in philosophy and conduct research into this area of interest via reading and private study under the supervison of the supervisor to whom they have been allocated. The student will produce a systematic piece of written work, organised in chapters/sections in the manner of professional and published work in philosophy, so as to show that the research referred to in the Aims has been mastered in a way appropriate to someone with a grasp of the practice of professional philosophy.
• ##### Mind, Brain and Consciousness (PHIL309)
Level 3 15 First Semester 60:40 To give students an understanding of the main developments in twentieth century analytic philosophy of mind: dualism, behaviourism, identity theory and functionalism.​To give students a grasp of cutting-edge debates in philosophy of mind concerning (i) the place of consciousness in nature, (ii) the relationship between consciousness and thought, (iii) artificial intelligence. Students should be able to explain the history of twentieth century analytic philosophy of mind.​Students should be able to explain cutting edge contemporary debates on (i) the place of consciousness in nature, (ii) the relationship between thought and consciousness, (iii) artificial intelligence.Students should be able to build a case for a specific view concerning (i) the place of consciousness in nature, (ii) the relationship between thought and consciousenss, (iii) aritificial intelligence. ​Students should be able to explain the main strengths and weaknesses of dominent theories on these three things in the philosophical literature.​Students should further develop their abilities to extract arguments from texts, render them in schematic form, and assess the soundness of their premises and the validity of their structures.​Students should be able to think more creatively about the relationship between thought, consciousness and the physical world.
• ##### Philosophy of Language (PHIL310)
Level 3 15 Second Semester 60:40 To study some of the main issues in the contemporary philosophy of language. ​Students will be able to explain the point of compositional theories of meaning. ​Students will be able to explain the nature and purpose of Frege''s sense-reference distinction.Students will be able to explain Russell''s Theory of Descriptions.​Students will be able to explain the difference between extensionality and intensionality and be able to evaluate some of the problems connected with these notions.​Students will be able to explain and evaluate sceptical approaches to meaning, such as Quine''s and Kripke''s.​Students will be able to explain and evaluate Davidson''s programme of radical interpretation.​Students will have be able to explain rival theories of truth and evaluate their relative merits.​Students will have be able to explain the connections between the notions of truth and meaning, and be able to evaluate the debate between realists and anti-realists.​
• ##### Aesthetics (PHIL316)
Level 3 15 First Semester 60:40 Students will be introduced to arguments of some of the most important philosophers on art, aesthetics and cultural theory, including Kant, Hegel, Danto and Tolstoy.Students will consider key concepts and theories in aesthetics, including the aesthetic judgement, disinterestedness, the institutional theory of art, the nature of representation and expression and feminist and post-modern critiques.Students will be encouraged to make connections between works of art and artistic practices of the past and present. Students will be able to analyse key concepts and arguments relating to aesthetics and art.​Students will be able to structure discussion of issues in aesthetics.​Students will be able to identify links between influential philosophical theories and artistic practices.​Students will be able to articulate and defend positions in aesthetics and philosophy of art.​Students will be able to present their ideas with clarity and confidence.​Students will be able to develop in writing coherent, structured and informative accounts on abstract philosophical issues.​
• ##### Indian Philosophy (PHIL326)
Level 3 15 Second Semester 60:40 To examine the ways in which philosophy in Classical India develops as a dialogue between thinkers of Buddhist and Brahminical persuasions and to relate fundamental Indian metaphysical concepts to Western counterparts.​To investigate what is distinctive about Indian approaches to questions of ontology, soteriology, social harmony, and morality. Students will be able to engage in informed discussions identifying and evaluating the concepts and categories in which philosophical discussions were conducted in India.​Students will able to be enabled to assimilate a differentview Western philosophical traditions from the perspective of Indian philosohical traditions.​Students will be able to contextualise information about the Indian worldviews under discussion.  Students will be able to think more imaginatively by empathising with unfamiliar outlooks on life. ​Students will be able to engage in debate informed by an awareness of the particularity and peculiarities of Western philosophical positions.
• ##### Existentialism (PHIL332)
Level 3 15 First Semester 60:40 To consider the theories and arguments of some of the most important existentialist philosophers, such as Kierkegaard, Nietzsche, Heidegger and Sartre.​To analyse some of the main themes of existentialist philosophy, such as the nature of subjectivity, the possibility of freedom, the death of God, time, nothingness.​To relate the philosophical issues raised by existentialism to lived practice and to concrete examples.​ Students will be able to explain and evaluate some of the main theories in existentialism.​Students will be able to analyse key concepts and arguments relating to the existentialist movement.​Students will be able to structure discussion of issues around existentialist metaphysics and ethics.​Students will be able to identify the relevance of existentialist philosophy to their own lives.​Students will be able to articulate and defend positions relating to existentialist themes.​​Students will be able to present their ideas with clarity and confidence.​​Students will be able to writing coherent, structured and informative accounts on abstract philosophical issues.​​
• ##### Actuarial Models (MATH376)
Level3
Credit level15
SemesterSecond Semester
Exam:Coursework weighting100:0
Aims 1 Be able to understand the differences between stochastic and deterministic modelling 2 Explain the need of stochastic processes to model the actuarial data 3 Be able to perform model selection depending on the outcome from a model. 4 Prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT4 subject of the Institute of Actuaries.
Learning Outcomes 1 Understand Use Markov processes to describe simple survival, sickness and marriage models, and describe other simple applications, Derive an appropriate Markov multi-state model for a system with multiple transfers, derive the likelihood function in a Markov multi-state model with data and use the likelihood function to estimate the parameters (with standard errors). 2 The Kaplan-Meier (or product limit) estimate, the Nelson-Aalen estimate , Describe the Cox model for proportional hazards Apply the chi-square test, the standardised deviations test, the cumulative deviation test, the sign test, the grouping of signs test, the serial correlation test to testing the adherence of graduation data, 3 Understand the connection between estimation of transition intensities and exposed to risk (central and initial exposed to risk) , Apply exact calculation of the central exposed to risk,
• ##### Analysis and Number Theory (MATH351)
Level 3 15 First Semester 100:0 To provide an introduction to uniform distribution modulo one as a meeting ground for other branches of pure mathematics such as topology, analysis and number theory. To give an idea of how these subjects work together in this context to shape modern analysis by focusing on one particular rich context. After completing the module students should be able to: understand completions and irrationality; understand diophantine approximation and its relation to uniform distribution; appreciate that analysis has a complex unity and to have a feel for basic computations in analysis. calculate rational approximations to real and p adic numbers and to put this to use in number theoretic situations. calculate approximations to functions from families of simpler functions e.g. continuous functions by polynomials and step functions by trigonometric sums. work with basic tools from analysis, like Fourier series and continuous functions to prove distributional properties of sequences of numbers.
• ##### Applied Probability (MATH362)
Level 3 15 First Semester 100:0 To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for ‘‘dynamic" events occuring over time. To familiarise students with an important area of probability modelling. 1. Knowledge and Understanding After the module, students should have a basic understanding of: (a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes (b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain (c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.2. Intellectual Abilities After the module, students should be able to: (a) formulate appropriate situations as probability models: random processes (b) demonstrate knowledge of standard models (c) demonstrate understanding of the theory underpinning simple dynamical systems3. General Transferable Skills (a) numeracy through manipulation and interpretation of datasets (b) communication through presentation of written work and preparation of diagrams (c) problem solving through tasks set in tutorials (d) time management in the completion of practicals and the submission of assessed work (e) choosing, applying and interpreting results of probability techniques for a range of different problems.
• ##### Applied Stochastic Models (MATH360)
Level 3 15 Second Semester 100:0 To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of stochastic model building for ''dynamic'' events occurring over time or space. To enable further study of the theory of stochastic processes by using this course as a base. After completing the module students should have a grounding in the theory of continuous-time Markov chains and diffusion processes. They should be able to solve corresponding problems arising in epidemiology, mathematical biology, financial mathematics, etc.
• ##### Cartesian Tensors and Mathematical Models of Solids and VIscous Fluids (MATH324)
Level 3 15 First Semester 100:0 To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials. Cartesian tensors are first introduced. This is followed by modelling of the mechanics of continuous media. The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity. After completing the module, students should be able to understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, and apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.
• ##### Chaos and Dynamical Systems (MATH322)
Level 3 15 Second Semester 100:0 To develop expertise in dynamical systems in general and study particular systems in detail. After completing the module students should be able to:understand the possible behaviour of dynamical systems with particular attention to chaotic motion;  ​be familiar with techniques for extracting fixed points and exploring the behaviour near such fixed points; ​understand how fractal sets arise and how to characterise them.
• ##### Combinatorics (MATH344)
Level 3 15 First Semester 90:10 To provide an introduction to the problems and methods of Combinatorics, particularly to those areas of the subject with the widest applications such as pairings problems, the inclusion-exclusion principle, recurrence relations, partitions and the elementary theory of symmetric functions. After completing the module students should be able to:     -           understand of the type of problem to which the methods of Combinatorics apply, and model these problems; -           solve counting and arrangement problems; -           solve general recurrence relations using the generating function method; -           appreciate the elementary theory of partitions and its application to the study of symmetric functions.
• ##### Differential Geometry (MATH349)
Level 3 15 Second Semester 85:15 This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space.  While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering. 1. Knowledge and understanding After the module, students should have a basic understanding of a) invariants used to describe the shape of explicitly given curves and surfaces, b) special curves on surfaces, c) the difference between extrinsically defined properties and those which depend only on the surface metric, d) understanding the passage from local to global properties exemplified by the Gauss-Bonnet Theorem. 2. Intellectual abilities After the module, students should be able to a) use differential calculus to discover geometric properties of explicitly given curves and surface, b) understand the role played by special curves on surfaces. 3. Subject-based practical skills Students should learn to a) compute invariants of curves and surfaces, b) interpret the invariants of curves and surfaces as indicators of their geometrical properties. 4. General transferable skills Students will improve their ability to a) think logically about abstract concepts, b) combine theory with examples in a meaningful way.
• ##### Frontiers of Ethics (PHIL302)
Level 3 15 First Semester 60:40 To consider conceptual and ethical issues arising from matters of global concern, such as international justice, war, and the environmental crisis.​ To consider arguments and assumptions underlying a range of claims concerning such issues as disability, global citizenship, climate change and the ethical status of nature. To examine difficulties for traditional philosophical approaches raised by such issues and recent theoretical developments relevant to them. Students will be able to distinguish between some of the main concepts involved in philosophical debates arising from matters of current global concern.​Students will be able to distinguish between different ways of understanding  concepts in philosophical debates arising from from matters of global concern.​Students will be able to explain and evaluate some of the main theories in debates about matters of disability, global justice, just war, environmental justice and environmental ethics.​Students will be able to analyse concepts and arguments relating to current ethical issues.​Students will be able to identify philosophical assumptions underlying ethical claims.​Students will be able to structure a philosophical discussion of current ethical issues.​​Students will be able to speak with confidence and clarity on current ethical issues.​Students will be able to explain details of texts shaping current philosophical debates about matters of global concern.Students will be able to articulate and defend positions in current philosophical debates about matters of global concern.​Students will be able to write coherently and rigorously about abstract philosophical issues raised by current ethical controversies.​
• ##### Further Methods of Applied Mathematics (MATH323)
Level 3 15 First Semester 100:0 To give an insight into some specific methods for solving important types of ordinary differential equations. To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics. To build on the students'' existing knowledge of partial differential equations of first and second order. After completing the module students should be able to: -     use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations. -     solve simple integral extremal problems including cases with constraints; -     classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases; -     classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions.   [This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]
• ##### Group Theory (MATH343)
Level 3 15 First Semester 90:10 To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results. Understanding of abstract algebraic systems (groups) by concrete, explicit realisations (permutations, matrices, Mobius transformations).​The ability to understand and explain classification results to users of group theory.​The understanding of connections of the subject with other areas of Mathematics.​To have a general understanding of the origins and history of the subject.
• ##### Life Insurance Mathematics II (MATH373)
Level 3 15 First Semester 100:0 Provide a solid grounding in the subject of life contingencies for multiple-life, and in the subject of the analysis of life assurance, life annuities, pension contracts, multi-state models and profit testing. ​​Provide an introduction to mathematical methods for managing the risk in life insurance.​Analyze problems of pricing and reserving in relation to contracts involving several lives.​Prepare the students to sit for the exams of CT5 subject of the Institute of Actuaries.Be familiar with R programming language to solve life insurance problems. Be able to explain, define and analyze the joint survival functions.​Understand the concept (and the mathematical assumptions) of the joint future life time random variables in continuous and discrete time and monthly. Be able to derive the distributions and the moment/variance of the joint future lifetimes.Be able to define the survivals probabilities/death probabilities of either or both two lives, explain these types of probabilities and the force of interest intuitively, be able to calculate the different types of the survival/death probabilities in theoretical and numerical examples. ​​Understand, define and derive the expected present values of different types of the life assurances and life annuities for joint lives, be able to calculate the expected present values of the joint life assurances and life annuities in theoretical and numerical examples.​Be familiar with R solfware and uses in actuarial mathematics
• ##### Linear Statistical Models (MATH363)
Level 3 15 First Semester 100:0 ·      to understand how regression methods for continuous data extend to include multiple continuous and categorical predictors, and categorical response variables. ·      to provide an understanding of how this class of models forms the basis for the analysis of experimental and also observational studies. ·      to understand generalized linear models. ·      to develop familiarity with the computer package SPSS. After completing the module students should be able to:         understand the rationale and assumptions of linear regression and analysis of variance. ·      understand the rationale and assumptions of generalized linear models. ·      recognise the correct analysis for a given experiment. ·      carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results. ·      carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results. ·      perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.
• ##### Mathematical Economics (MATH331)
Level 3 15 Second Semester 100:0 ·      To explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur. ·      To see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc.. ·      To treat fully a number of specific games including the famous examples of "The Prisoners'' Dilemma" and "The Battle of the Sexes". ·      To treat in detail two-person zero-sum and non-zero-sum games. ·      To give a brief review of n-person games. ·      In microeconomics, to look at exchanges in the absence of money, i.e. bartering, in which two individuals or two groups are involved.   To see how the Prisoner''s Dilemma arises in the context of public goods. After completing the module students should: ·      Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences. ·      Be able to formulate, in game-theoretic terms, situations of conflict and cooperation. ·      Be able to solve mathematically a variety of standard problems in the theory of games. ·      To understand the relevance of such solutions in real situations.
• ##### Mathematical Physics Project (MATH334)
Level 3 15 Second Semester 0:100 To study in depth an area of theoretical physics and report on it. After completing the project with suitable guidance the student should have  ·      understood an area of advanced theoretical physics ·      had experience in consulting relevant literature ·      gained experience in using appropriate mathematics ·      made a critical appraisal of the current understanding of the area ·      learnt how to construct a written essay and given an  oral presentation
• ##### Mathematical Risk Theory (MATH366)
Level 3 15 Second Semester 100:0  to provide an understanding of the mathematical risk theory used in the study process of actuarial interest,  to provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities),  to develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry,  to prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth). After completing the module students should be able to: (a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria''s, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules. (b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables, (c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables. (d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships , (e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples, (f) Understand and be able to use Panjer''s equation when the number of claims belongs to theR(a, b, 0) class of distributions, use the Panjer''s recursion in order to derive/evaluate the probability function for the total aggregate claims, (g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group are assumed to have deterministic variables, (h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation), (i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process, (j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg''s inequality for exponential and mixtures of exponential claim severities, (k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities, (l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical examples (using convolutions), (m) Derive Lundberg''s equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes, (n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data, (o) Explain the difference and adjust the chain ladder method, when inflation is considered, (p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table), (q) Use loss ratios to estimate the eventual loss and hence outstanding claims, (r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method), use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor).
• ##### Medical Statistics (MATH364)
Level 3 15 Second Semester 100:0 The aims of this module are to: demonstrate the purpose of medical statistics and the role it plays in the control of disease and promotion of health explore different epidemiological concepts and study designs apply statistical methods learnt in other programmes, and some new concepts, to medical problems and practical epidemiological research enable further study of the theory of medical statistics by using this module as a base. identify the types of problems encountered in medical statistics​demonstrate the advantages and disadvantages of different epidemiological study designs ​apply appropriate statistical methods to problems arising in epidemiology and interpret results ​explain and apply statistical techniques used in survival analysis ​critically evaluate statistical issues in the design and analysis of clinical trials ​discuss statistical issues related to systematic review and apply appropriate methods of meta-analysis ​apply Bayesian methods to simple medical problems.
• ##### Networks in Theory and Practice (MATH367)
Level 3 15 First Semester 100:0 To develop an appreciation of network models for real world problems. To describe optimisation methods to solve them. To study a range of classical problems and techniques related to network models. After completing the module students should  .      be able to model problems in terms of networks. ·      be able to apply effectively a range of exact and heuristic optimisation techniques.
• ##### Number Theory (MATH342)
Level 3 15 Second Semester 100:0 To give an account of elementary number theory with use of certain algebraic methods and to apply the concepts to problem solving. After completing this module students should be able to understand and solve a wide range of problems about the integers, and have a better understanding of the properties of prime numbers.
• ##### Numerical Analysis for Financial Mathematics (MATH371)
Level3
Credit level15
SemesterSecond Semester
Exam:Coursework weighting80:20
Aims 1 To provide basic background in solving mathematical problems numerically, including understanding of stability and convergence of approximations to exact solution. 2 To acquaint students with two standard methods of derivative pricing: recombining trees and Monte Carlo algorithms. 3 To familiarize students with implementation of numerical methods in a high level programming language.
Learning Outcomes Awareness of the major issues when solving mathematical problems numerically.

Ability to analyse a simple numerical method for convergence and stability

Ability to formulate approximations to derivative pricing problems numerically.

​Ability to program matlab for pricing options

• ##### Philosophy of Law (PHIL341)
Level 3 15 Second Semester 60:40 ​The module aims ​​to get the student thinking about our (and other) legal system, and the philosophical basis, if any, of the system, its strengths and weakeness, and how, if at all, it might be improved. ​Students will familiarize themselves with the rudiments of the English-and-Welsh legal system.​Students will acquaint themselves with the chief questions in the philosophy of law.​Students ​​​​​will acquaint themselves with the main views on the chief questions in the philosophy of law.​Students will develop their own views on the chief questions in the philosophy of law.
• ##### Philosophy of Play and the VIrtual (PHIL343)
Level 3 15 First Semester 0:100 ​To introduce students to the main contemporary issues around play and games.To develop students'' understanding of the relationships between play, labour and virtuality.To enable students to reflect on their own preconceptions of play and value. ​Students will be able to explain the importance of play as a topic for study.Students will be able to analyse common topics of discourse around play and games, especially digital games: violence, addiction, therapeutic and educational effects, and gamification.​Students will be able to identify philosophical issues ariding from specific games/instances of play.​Students will be able to explain some of the philosophical literature around play, make-believe, choice and responsibility, and virtual worlds.​Students will be able to trace connections between surface controversies and deeper philosophical concerns.​​Students will develop their ability to reflect on their own preconceptions and how these contribute to both philosophical and popular discourse.​
• ##### Philosophy of the Future (PHIL312)
Level 3 15 Second Semester 60:40 ​To provide an introduction to debates concerning the philosophical implications of foreseeable future technological innovations. To examine the relevance of metaphysical and ethical considerations to future technological and scientific developments.​ Students will be able to identify the main issues and positions in contemporary philosophical discussions of issues such as human enhancement, existential risks, teleportation, time travel, the technological singularity, the simulation argument, the feasibility and desirability of uploading into virtual worlds.  ​​​Students will be able to explain the main strengths and weaknesses of these positions.​​​Students will be able to explain the relevance of metaphysical and ethical considerations to debates concerning these issues. ​​Students will be able to think more creatively about philosophical issues​.​Students will be able to structure philosophical arguments relating issues raised by future technological developments.​​Students will be able to articulate and defend specific positions in current philosophical debates concerning likely future developments in science and technology.​​Students will be able to write coherently and rigourously about the philosophical issues raised by future technological developments.
• ##### Population Dynamics (MATH332)
Level 3 15 First Semester 100:0 - To provide a theoretical basis for the understanding of population ecology - To explore the classical models of population dynamics - To learn basic techniques of qualitative analysis of mathematical models ​ ​The ability to relate the predictions of the mathematical models to experimental results obtained in the field.The ability to recognise the limitations of mathematical modelling in understanding the mechanics of complex biological systems.The ability to use analytical and graphical methods to investigate population growth and the stability of equilibrium states for continuous-time and discrete-time models of ecological systems.​
• ##### Projects in Mathematics (MATH399)
Level 3 15 Whole Session 0:100 a) To study in depth an area of pure mathematics and report on it; or b) To construct and study mathematical models of a chosen problem; or c) To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.; or d) To study in depth a particular problem in statistics, probability or operational research. a) (Pure Maths option) - After completing the report with suitable guidance the student should have · gained a greater understanding of the chosen mathematical topic · gained experience in applying his/her mathematical skills · had experience in consulting relevant literature · learned how to construct a written project report · had experience in making an oral presentation b) (Applied Mathematics) - After completing the project with suitable guidance the students should have: - learned strategies for simple model building - gained experience in choosing and using appropriate mathematics - understood the nature of approximations used - made critical appraisal of results - had experience in consulting related relevant literature - learned how to construct a written project report - had experience in making an oral presentation. c) (Applied Maths/Theoretical Physics) - After researching and preparing the mathematical essay the student should have: · gained a greater understanding of the chosen mathematical topic · gained an appreciation of the historical context · learned how to abstract mathematical concepts and explain them · had experience in consulting related relevant literature · learned how to construct a written project report · had experience in making an oral presentation. d) (Statistics, Probability and Operational Research) - After completing the project the student should have: · gained an in-depth understanding of the chosen topic · had experience in consulting relevant literature · learned how to construct a written project report; · had experience in making an oral presentation. e) Mathematics in Society Projects. Only available to G1X3 students Students interested in doing such a project should see Dr A Pratoussevitch and Dr T Eckl initially.
• ##### Quantum Mechanics (MATH325)
Level 3 15 First Semester 100:0 The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time.  The aim of the module is to lead the student to an understanding of the way that relatively simple mathemactics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world. After completing the module students should be able to solve Schrodinger''s equation for simple systems, and have some intuitive understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.
• ##### Relativity (MATH326)
Level 3 15 First Semester 100:0 To impart (i)              a firm grasp of the physical principles behind Special and General Relativity and their main consequences; (ii)           technical competence in the mathematical framework of the subjects - Lorentz transformation, coordinate transformations and geodesics in Riemann space; (iii)          knowledge of some of the classical tests of General Relativity - perihelion shift, gravitational deflection of light; (iv)          basic concepts of black holes and (if time) relativistic cosmology. After  completing this module students should (i)              understand why space-time forms a non-Euclidean four-dimensional manifold; (ii)           be proficient at calculations involving Lorentz transformations, energy-momentum conservation, and the Christoffel symbols. (iii)          understand the arguments leading to the Einstein''s field equations and how Newton''s law of gravity arises as a limiting case. (iv) be able to calculate the trajectories of bodies in a Schwarzschild space-time.
• ##### Riemann Surfaces (MATH340)
Level 3 15 Second Semester 100:0 To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory. Students should be familiar with themost basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.
• ##### Statistical Methods in Actuarial Science (MATL374)
Level 3 15 First Semester 100:0 ​1.    Provide a solid grounding in analysis of general insurance data,Bayesian credibility theory and the loss distribution concept. 2.    ​Provide an introduction to statistical methods for managing risk innon-life insurance and finance. 3.    ​Prepare the students adequately to sit for the exams of CT6 subject ofthe Institute of Actuaries. ​Be able to apply the estimationmethods described in (b) of the Syllabus for the distribution described in (a)of the Syllabus, be able to make hypothesis testing described in (b) of theSyllabus for the distribution described in (a) of the Syllabus.​Be able to estimate the parameters ofthe loss distributions when data complete/incomplete using the method of moments and the method of maximum likelihood, be able tocalculate the loss elimination ratio.​Understand and use the Buhlmann model, the Buhlmann-Straub model, beable to state the assumptions of the GLM models – normal linear model,understand the properties of the exponential family.​Be able to express the values of thelife assurances in (d) of the Syllabus and the life annuities in (f) of theSyllabus in terms of the life table functions. Be able to use approximationsfor the evaluation of the life assurances in (d) of the Syllabus and the lifeannuities in (f) of the Syllabus based on a life table.​Be able to describe the properties ofa time series using basic analytical and graphical tools.​Understand the definitions,properties and applications of well know time seriesmodels, fit time series models to practical data sets and select the suitablemodels, be able to perform simple statistical inference (forecasting) based onthe fitted models, estimate and remove possible trend and seasonality in a timeseries, analyse the residuals of a time series using stationary models.
• ##### Stochastic Modelling in Insurance and Finance (MATL480)
Level M 15 First Semester 100:0 This module aims to introduce students to theadvanced mathematical techniques underlying financial markets and theassociated stochastic analysis. ​To provide an introduction to key financial derivatives and their properties​To provide an understanding of the modelling processes used in financial derivatives e.g. binomial tree, Brownian motion​​To provide a general foundation to the stochastic theory behind financial derivatives.​To provide a general methodology for the pricing and hedging of financial derivatives​A critical awareness of issues in the field of financial derivatives
• ##### Theory of Statistical Inference (MATH361)
Level 3 15 Second Semester 90:10 To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options. After completing the module students should have a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.  The students should also gain an appreciation of the blossoming area of Bayesian approach to inference
• ##### Wittgenstein (PHIL340)
Level 3 15 First Semester 60:40 To study the later Wittgenstein. Topics will include: background in the Tractatus - the limits of language and the nature of ethical and religious discourse; rule-following and the private - the limits of language and the nature of ethical and religious discourse; rule-following and the private language argument; the nature and prospects of philosophy; epistemology and certainty. Students will be able to explain how the Tractatus influenced Wittgenstein''s later philosophy.Students will be able to explain and assess both the Augustinian picture of language and Wittgenstein''s criticism of it​.Students will be able to explain the rule-following considerations and their importance to Wittgenstein and contemporary philosophy of language​.Students will be able to explain and assess the private language argument and its importance to contemporary philosophy of mind​.Students will be able to explain and assess Wittgenstein''s technical notion of ''criterion'' and its philosophical significance​.Students will be able to explain and assess Wittgenstein''s later epistemology​.

The programme detail and modules listed are illustrative only and subject to change.

#### Teaching and Learning

In studying Philosophy you will learn how to defend your views with reasoned arguments, and to assess the arguments of others. Argumentative skills are learned through attending lectures and reading philosophical texts, developed by group seminar discussions, and formally assessed through essays and exams. You complete modules to the value of 120 credits per year, from a wide range of options available. Most modules employ a blend of lectures, seminars and online support materials. You will learn by reading and studying outside class time, by attending and participating in classes, by doing coursework and, for dissertations, via one-to-one meetings with a supervisor. There is also scope, both formally in the placement module and informally, for you to develop practical skills by volunteering.

#### Assessment

Philosophy employs a mixture of modes of assessment: exams and coursework in many different varieties including essays, oral presentations, dissertations, exercises, and supported independent work (eg in the placement module).