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Mathematics with Philosophy

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Mathematics with Philosophy

BSc
  • UCAS code G1V5
  • Option 3 years full time
  • Year of entry 2021

The course

This three-year course combines two of the most fundamental and intellectually stimulating forms of human enquiry. The idea of using mathematics to describe the universe was first introduced by the philosophers of ancient Greece but it is as relevant as ever today. Questions such as "What is a number?" or "Is mathematics discovered or invented?" are deeply philosophical. By studying both subjects you will not only master the skills of handling complex data and finding creative solutions to problems, but you will also be introduced to the beautiful world of abstract ideas, and encouraged to analyse challenging issues, question your assumptions and communicate your thoughts with clarity. You will gain a unique insight into the world of logic that bridges the two disciplines and you will open doors to a diverse range of career opportunities.

Our modular structure gives you the flexibility to tailor your studies to your own interests, and we offer a friendly and motivating learning environment, with a strong focus on small group teaching. Mathematics is one of the oldest academic disciplines and yet it sits at the heart of modern science and technology. Led by experts in the field, our core modules will give you a grounding in the key methods and concepts that underpin the subject, as well as practical skills that are widely transferable in the world of work. Our curriculum covers pure and applied mathematics, statistics and probability, the mathematics of information, financial markets, and more.

You'll also learn about the fundamentals of ancient and modern philosophy, the philosophy of politics, and the art of argument and persuasion. We address some of the most important political, cultural and ethical issues in the world today and tackle fundamental questions about knowledge, reasoning, our views on the universe and the impacts they have on our lives. We also have a vibrant Philosophy Society. We take a uniquely collaborative and interdisciplinary approach to the subject, by looking beyond the confines of the analytic or European tradition to disciplines across the art, humanities and social sciences. You will learn from seasoned philosophers who are published authorities in their field.

Your mathematical studies will make up 75% of your overall degree, while philosophy will contribute the remaining 25%.

Our flexible degree programmes enable you to apply to take a Placement Year, which can be spent studying abroad, working or carrying out voluntary work. You can even do all three if you want to (minimum of three months each)! To recognise the importance of this additional skills development and university experience, your Placement Year will be formally recognised on your degree certificate and will contribute to your overall result. Please note conditions may apply if your degree already includes an integrated year out, please contact the Careers & Employability Service for more information. Find out more

  • Study two of the world’s oldest and most widely applicable academic subjects.
  • Learn from renowned mathematicians and inspirational philosophy teachers, with the flexibility to tailor your studies to your own interests.
  • Our philosophy courses are taught in a lively international community of students in the Department of Politics and International Relations.
  • Benefit from our strong focus on small group teaching.

Core Modules

Year 1
  • In this module, you will develop an understanding of the key concepts in Calculus, including differentiation and integration. You will learn how to factorise polynomials and separate rational functions into partial fractions, differentiate commonly occurring functions, and find definite and indefinite integrals of a variety of functions using substitution or integration by parts. You will also examine how to recognise the standard forms of first-order differential equations, and reduce other equations to these forms and solve them.

  • In this module you will develop an understanding of the fundamental algebraic structures, including familiar integers and polynomial rings. You will learn how to apply Euclid's algorithm to find the greatest common divisor of two integers, and use mathematical induction to prove simple results. You will examine the use of arithmetic operations on complex numbers, extract roots of complex numbers, prove De Morgan's laws, and determine whether a given mapping is bijective.

  • In this module you will develop an understanding of key mathematical concepts such as the construction of real numbers, limits and convergence of sequences, and continuity of functions. You will look at the infinite processes that are essential for the development of areas such as calculus, determining whether a given sequence tends to a limit, and finding the limits of sequences defined recursively.

  • In this module you will develop an understanding of the calculus functions of more than one variable and how it may be used in areas such as geometry and optimisation. You learn how to manipulate partial derivatives, construct and manipulate line integrals, represent curves and surfaces in higher dimensions, calculate areas under a curve and volumes between surfaces, and evaluate double integrals, including the use of change of order of integration and change of coordinates.

  • In this module you will develop an understanding of basic linear algebra, in particular the use of matrices and vectors. You will look at the basic theoretical and computational techniques of matrix theory, examining the power of vector methods and how they may be used to describe three-dimensional space. You will consider the notions of field, vector space and subspace, and learn how to calculate the determinant of an n x n matrix.

  • Knowledge is often thought to be the highest achievement of rational creatures, the thing that distinguishes us from other animals and is the basis of our ability to predict and control our environment. Beginning with the most Platonic of questions—‘what is knowledge?’—this course introduces you to basic topics in contemporary epistemology. Among the questions it goes on to address are: why is knowledge valuable?; how do we acquire knowledge and how do we pass it on to others?; how do we become better knowers?; is there such a thing as collective knowledge?; do animals have knowledge?; is there such a thing as knowledge at all?

     

You will take one of the following:
  • In this module you will develop an understanding of the formal study of arguments through the two basic systems of modern logic - sentential or propositional logic and predicate logic. You will learn how to present and analyse arguments formally, and look at the implications and uses of logical analysis by considering Bertrand Russell’s formalist solution to the problem of definite descriptions. You will also examine the the broader significance of findings in logic to philosophical inquiry.

  • In this module you will develop an understanding of the relationship between the mind and the brain. You will examine the key theories, from Descartes' dualist conception of the relationship between mind and body through to Chalmers's conception of consciousness as 'the hard problem' in the philosophy of mind. You will also consider some of the famous thought experiments in this area, including Descartes's and Laplace's demons, the Chinese Room and the China Brain, Mary and the black-and-white room, and the problem of zombie and bat consciousness.

  • In this module you will develop an understanding of the central problems and debates within moral philosophy and aesthetics. You will look at questions relating to both metaphysical and ethical relativism, including the ways we view our moral commitments within the world, how the individual is related to society, and the value and nature of the work of art. You will also examine approaches from the history of philosophy, including the Anglo-American tradition and recent European philosophy.

Year 2
  • In this module you will develop an understanding of vectors and matrices within the context of vector spaces, with a focus on deriving and using various decompositions of matrices, including eigenvalue decompositions and the so-called normal forms. You will learn how these abstract notions can be used to solve problems encountered in other fields of science and mathematics, such as optimisation theory.

  • In this module you will develop an understanding of the basic complex variable theory. You will look at the definitions of continuity and differentiability of a complex valued function at a point, and how Cauchy-Riemann equations can be applied. You will examine how to use a power series to define the complex expontential function, and how to obtain Taylor series of rational and other functions of standard type, determining zeros and poles of given functions. You will also consider how to use Cauchy's Residue Theorem to evaulate real integrals.

  • In this module you will develop an understanding of the major debates in European and some Anglo-American philosophy. You will look at the key texts by eighteenth and nineteenth-century philosophers Immanuel Kant and Georg Wilhelm Friedrich Hegel, examining the continuing significance of their ideas. You will consider the major epistemological, ethical and aesthetical issues their idea raise, and the problems associated with the notion of modernity. You will also analyse the importance of the role of history in modern philosophy via Hegel's influence.

  • In this module you will develop an understanding of how the rationalist and empiricist traditions in philosophy influence contemporary thought in the philosophy of mind. You will look at the continuing relevance of the mind-body problem to the question of what it is to be a human being and consider the connections between the analytic and European traditions in philosophy with respect to language, subjectivity, and the phenomenology of experience. You will also examine the importance of consciousness to contemporary debates in philosophy, psychology and cognitive science.

  • This course introduces you to the basics of Python programming for building solutions to mathematics-based tasks. This will encourage a deeper understanding of the mathematics that you will learn across your degree, developing general mathematical skills and group working. Additionally, the module will provide guidance through the process of applying for a summer internship or job, as well as reviewing the range of career options available to Mathematicians upon graduation.

Year 3
  • All modules are optional

Optional Modules

There are a number of optional course modules available during your degree studies. The following is a selection of optional course modules that are likely to be available. Please note that although the College will keep changes to a minimum, new modules may be offered or existing modules may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.

Year 1
  • In this module you will develop an understanding of how mathematics has been used to describe space over the last 2,500 years. You will look at ruler and compass constructions from ancient Greece, the influence of algebra on geometry in the renaissance, and the intricate and beautiful fractal patterns developed by Benoît Mandelbrot in the 1970s. You will learn to sketch simple curves using polar coordinates, draw and classify conics, and use simple arguments to distinguish between countable and uncountable sets.

  • In this module, you will develop an understanding of how the techniques for solving differential equations can be applied to describe the real world. You will look at situations from balls flying through the air to planets orbiting the stars, including why the moon continues to orbit the Earth and not the Sun. You will consider the chatotic motion of a pendulum, and examine Einstein's theory of special relativity to describe the propagation of matter and light at high speeds.

  • In this module you will develop an understanding of the notion of probability and the basic theory and methods of statistics. You will look at random variables and their distributions, calculate probabilities of events that arise from standard distributions, estimate means and variances, and carry out t tests for means and differences of means. You will also consider the notions of types of error, power and significance levels, gaining experience in sorting a variety of data sets in a scientific way.

Year 2
  • In this module you will develop an understanding of the concepts of scalar and vector fields. You examine how vector calculus is used to define general coordinate systems and in differential geometry. You will learn how to solve simple partial differential equations by separating variables, and become familiar with how these concepts can be appield in the field of dynamics of inviscid fluids.

  • In this module you will develop an understanding of statistical modelling, becoming familiar with the theory and the application of linear models. You will learn how to use the classic simple linear regression model and its generalisations for modelling dependence between variables. You will examine how to apply non-parametric methods, such as the Wilxocon and Kolmogorov-Smirnov goodness-of-fit tests, and learn to use the R open source software package.

  • In this module you will develop an understanding of ring theory and how this area of algebra can be used to address the problem of factorising integers into primes. You will look at how these ideas can be extended to develop notions of 'prime factorisation' for other mathematical objects, such as polynomials. You will investigate the structure of explicit rings and learn how to recognise and construct ring homomorphisms and quotients. You will examine the Gaussian integers as an example of a Euclidean ring, Kronecker's theorem on field extensions, and the Chinese Remainder Theorem.

  • In this module you will develop an understanding of the convergence of series. You will look at the Weierstrass definition of a limit and use standard tests to investigate the convergence of commonly occuring series. You will consider the power series of standard functions, and analyse the Intermediate Value and Mean Value Theorems. You will also examine the properties of the Riemann integral.

  • In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

  • In this module you will develop an understanding of the basic concepts of graph theory and linear programming. You will consider how railroad networks, electrical networks, social networks, and the web can be modelled by graphs, and look at basic examples of graph classes such as paths, cycles and trees. You will examine the flows in networks and how these are related to linear programming, solving problems using the simplex algorithm and the strong duality theorem.

  • In this module you will develop an understanding of the concepts arising when the boundary conditions of a differential equation involves two points. You will look at eigenvalues and eigenfunctions in trigonometric differential equations, and determine the Fourier series for a periodic function. You will learn how to manipulate the Dirac delta-function and apply the Fourier transform. You will also examine how to solve differential equations where the coefficients are variable.

  • In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

  • In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

  • Introduction to European Philosophy 2: The Critique of Idealism
  • Varieties of Scepticism
  • Philosophy and the Arts
  • The Philosophy of Religion
  • Philosophy and Literature
  • The Good Life in Ancient Philosophy
Year 3
  • You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 7,000 words in length, and give a ten-minute presentation outlining your findings.

  • In this module you will develop an understanding of how prime numbers are the building blocks of the integers 0, ±1, ±2, … You will look at how simple equations using integers can be solved, and examine whether a number like 2017 should be written as a sum of two integer squares. You will also see how Number Theory can be used in other areas such as Cryptography, Computer Science and Field Theory.

  • In this module you will develop an understanding the different classes of computational complexity. You will look at computational hardness, learning how to deduce cryptographic properties of related algorithms and protocols. You will examine the concept of a Turing machine, and consider the millennium problems, including P vs NP, with a $1,000,000 prize on offer from the Clay Mathematics Institute if a correct solution can be found.

  • In this module you will develop an understanding of quantum theory, and the development of the field to explain the behaviour of particles at the atomic level. You will look at the mathematical foundations of the theory, including the Schrodinger equation. You will examine how the theory is applied to one and three dimensional systems, including the hydrogen atom, and see how a probabilistic theory is required to interpret what is measured.

  • In this module, you will develop an understanding of non-linear dynamical systems. You will investigate whether the behaviour of a non-linear system can be predicted from the corresponding linear system, and see how dynamical systems can be used to analyse mechanisms such as the spread of disease, the stability of the universe, and the evolution of economic systems. You will gain an insight into the 'secrets' of the non-linear world and the appearance of chaos, examining the significant developments achieved in this field during the final quarter of the 20th Century.

  • In this module you will develop an understanding of the main priciples and methods of statstics, in particular the theory of parametric estimation and hypotheses testing.You will learn how to formulate statistical problems in mathematical terms, looking at concepts such as Bayes estimators, the Neyman-Pearson framework, likelihood ratio tests, and decision theory.

  • In this module you will develop an understanding of some of the descriptive methods and theoretical techniques that are used to analyse time series. You will look at the standard theory around several prototype classes of time series models and learn how to apply appropriate methods of times series analysis and forecasting to a given set of data using Minitab, a statistical computing package. You will examine inferential and associated algorithmic aspects of time-series modelling and simulate time series based on several prototype classes.

  • In this module you will develop an understanding of the mathematics of communication, focusing on digital communication as used across the internet and by mobile telephones. You looking at compression, considering how small a file, such as a photo or video, can be made, and therefore how the use of data can be minimised. You will examine error correction, seeing how communications may be correctly received even if something goes wrong during the transmission, such as intermittent wifi signal. You will also analyse the noiseless coding theorem, defining and using the concept of channel capacity.

  • In this module you will develop an understanding of how financial markets operate, with a focus on the ideas of risk and return and how they can be measured. You will look at the random behaviour of the stock market, Markowitz portfolio optimisation theory, the Capital Asset Pricing Model, the Binomial model, and the Black-Scholes formula for the pricing of options.

  • In this module you will develop an understanding of some of the standard techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by inclusion and exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

  • In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

  • In this module you will develop an understanding of the elementary ideas of electromagnetism. You will learn how to calculate electric fields and electric potentials from given fixed charge distributions and how to calculate magnetic fields and vector potentials from given steady current distributions. You will examine the magnetic effects of currents, including electromagnetic induction and displacement currents, and analyse the Biot-Savart law and Ampere's law. You will examine Maxwell's equations, and the properties of electromagnetic waves in free space, as well as electric and magnetic dipoles and the electromagnetism of matter.

  • In this module you will develop an understanding of Field Theory. You will learn how to express equations such as X2017=1 in a formal algebraic setting, how to classify finite fields, and how to determine the number of irreducible polynomials over a finitie field. You will also consider some the applications of fields, including ruler and compass constructions and why it is impossible to generically trisect an angle using them.

  • In this module you will develop an understanding of a range of methods for teaching children up to A-level standard. You will act act as a role model for pupils, devising appropriate ways to convey the principles and concepts of mathematics. You will spend one session a week in a local school, taking responsibility for preparing lesson plans, putting together relevant learning aids, and delivering some of the classes. You will work with a specific teacher, who will act as a trainer and mentor, gaining valuable transferable skills.

  • In this module you will develop an understanding of a range methods used for testing and proving primality, and for the factorisation of composite integers. You will look at the theory of binary quadratic forms, elliptic curves, and quadratic number fields, considering the principles behind state-of-the art factorisation methods. You will also look at how to analyse the complexity of fundamental number-theoretic algorithms.

  • In this module you will develop an understanding of efficient algorithm design and its importance for handling large inputs. You will look at how computers have changed the world in the last few decades, and examine the mathematical concepts that have driven these changes. You will consider the theory of algorithm design, including dynamic programming, handling recurrences, worst-case analysis, and basic data structures such as arrays, stacks, balanced search trees, and hashing.

  • In this module you will develop an understanding of how the theory of ideal fluids can be used to explain everyday phenomena in the world around us, such as how sound travels, how waves travel over the surface of a lake, and why golden syrup (or volcanic lava) flows differently from water. You will look at the essential features of compressible flow and consider basic vector analysis techniques.

  • In this module you will develop an understanding of how the Rayleigh-Ritz variational principle and perturbation theory can be used to obtain approximate solutions of the Schrödinger equation. You will look at the mathematical basis of the Period Table of Elements, considering spin and the Pauli exclusion principle. You will also examine the quantum theory of the interaction of electromagnetic radiation with matter.

  • In this module you will develop an understanding of the probabilistic methods used to model systems with uncertain behaviour. You will look at the structure and concepts of discrete and continuous time Markov chains with countable stable space, and consider the methods of conditional expectation. You will learn how to generate functions, and construct a probability model for a variety of problems.

  • In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the no-cloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

  • In this module you will develop an understanding of the role of mathematics and statistics in securities markets. You will investigate the validity of various linear and non-linear time series occurring in finance, and apply stochastic calculus, including partial differential equations, for interest rate and credit analysis. You will also consider how spot rates and prices for Asian and barrier exotic options are modelled.

  • In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

  • In this module you will develop an understanding of public key cryptography and the mathematical ideas that underpin it, including discrete logarithms, lattices and elliptic curves. You will look at several important public key cryptosystems, including RSA, Rabin, ElGamal encryption and Schnorr signatures. You will consider notions of security and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext attack.

  • In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

  • In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

  • In this module you will develop an understanding of geometric objects and their properties. You will look at objects that are preserved under continuous deformation, such as through stretching or twisting, and will examine knots and surfaces. You will see how colouring a knot can be used to determine whether or not it can be transformed into the unknot without any threading. You will also consider why topologists do not distinguish between a cup and a donut.

  • Modern European Philosophy 1: Husserl to Heidegger
  • Modern European Philosophy 2: Critical Theory and Hermeneutics
  • The Varieties of Scepticism
  • The Philosophy of Religion
  • Philosophy and Literature
  • The Good Life in Ancient Philosophy

The course has a flexible, modular structure and you will take a total of 12 course units at a rate of four, 30-credit modules per year. In addition to compulsory modules in years 1 and 2, you will choose from a range of optional courses each year. Some contribute 15 credits to your overall award while others contribute the full 30.

We use a variety of teaching methods and there is a strong focus on small group teaching. You will attend 12 to 15 hours of formal teaching in a typical week. Our mathematics courses are delivered through lectures, seminars, group tutorials, statistics and IT classes, and problem solving workshops. You will also be expected to work on worksheets, revision and project work in your own time. In year 2, much of our mathematics teaching is delivered through lectures, workshops and practical classes, and in year 3, mostly through relatively small group lectures and supervised project work. Philosophy is taught through a combination of lectures, large and small seminars and occasionally through one-to-one tutorials. Outside of class time you will work on group projects and wide-ranging but guided independent study. You will be supported in both subjects by the extensive resources available on Moodle, our e-learning facility.

Assessment is through a mixture of coursework and end-of-year examinations, depending on the courses you choose to take. Statistics and computational courses in mathematics may include project work and tests. All students will work in small groups to prepare a report and an oral presentation on a mathematical topic of their choice, which contributes towards one of the core subject marks in year 2, and two of the optional mathematics units in year 3 are examined solely by a project and presentation. The results of the first year assessments qualify you to progress to the second year but do not contribute to your final degree award. The second and final year results do contribute to the final degree result.

You will be required to take a study skills course during year 1, to equip you with and enhance the writing skills you will need to be successful in your degree. This does not count towards your final degree but you are required to pass it to progress to your second year.

96% Percentage of our graduates in work or undertaking further study within six months of leaving.

Source: DLHE, 2017

98% Percentage of our mathematics students who are satisfied overall.

Source: NSS, 2017

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