Mathematics and Philosophy

Core Modules

Year 1

You will take the following modules in Mathematics:

• Calculus

  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.

• Calculus II

  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.

• Introduction to Pure Mathematics

  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.

• Linear Algebra I

  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.

You will take the following modules in Philosophy:

• Introduction to Modern Philosophy

  The ‘new philosophy’ of the seventeenth century set the modern philosophical agenda by asking fundamental questions concerning knowledge and understanding and the relation between science and other human endeavours, which subsequently became central to the European Enlightenment. This module aims to familiarise you with the origins of empiricist and rationalist/idealist thought, focussing on the work of Descartes and Locke. The module will enable you to develop your close reading skills, and will give you the opportunity to see how arguments are developed across the length of philosophical texts.

• Problems of Knowledge

  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?

   

• Philosophical Methods

  In every aspect of our lives we are inundated by information and misinformation, claims and counter-claims: some people tell us we should believe this; others that we should believe that. Decisions have to be made; possible evidence has to be sifted; reasons have to be given; arguments have to be propounded; risks evaluated. All this requires the ability to reason critically: to distinguish between bad arguments and good ones, supporting evidence from mere distraction. Everybody has the basic ability to do this, but it is not always as developed we need it to be: and in this complex world being able to present your point forcefully and rationally is vitally important. The aim of this module is to help you develop the skills required to get the most out of their degree and beyond. 

• Introduction to Ancient Philosophy

  In this module you will develop an understanding of ancient philosophical ideas and the ways in which philosophical arguments are presented and analysed. You will look at the thought and significance of the principal ancient philosophers, from the Presocratics to Aristotle, and examine sample texts such as Plato's 'Laches' and the treatment of the virtue of courage in Aristotle, 'Nicomachean Ethics' 3.6-9.

Year 2

You will take the following modules in Mathematics:

• Vector Calculus

  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.

• Probability Theory

  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.

• Linear Algebra II

  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.

You will choose one of the following:

• Kant

  The module looks at key texts by Immanuel Kant which are the foundation of Modern European Philosophy. These texts raise questions concerning the status of human knowledge and the nature and justification of human action that have concerned philosophers ever since. The module considers Kant's Critique of Pure Reason. The core theme of the module is how philosophy responds to the situation in which it can no longer rely on theological support for its claims about truth and morality. This raises questions about the nature of the human subject that are evident in the conjunction of the massive success of the modern natural sciences with an abiding worry as to whether sceptical objections to establishing true knowledge can be overcome. Kant sees these issues in terms of 'transcendental philosophy' establishing the limits of knowledge by seeing what the necessary conditions of knowledge are.

• Empiricism and Rationalism

  This module will explore the central developments in modern philosophy occurring between the foundation of modern empiricism and rationalism by Locke and Descartes in the 17th century, and the emergence of Kant’s philosophical system in the late 18th century. The module will look at three of the key figures from the two traditions, exploring the key theories they expound, and the arguments used to support these theories. The figures covered will depend on the research specialisms of the module convenor, but a typical syllabus would involve reading works by Spinoza, Leibniz, and Hume. Looking at these philosophers over a number of weeks will allow you to develop your close reading skills, and to see how the arguments put forward by these philosophers work together to produce a systematic metaphysical worldview.

You will take the following module in Philosophy:

• Mind and World

  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.

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

• All modules are core

Year 2

• Vector Calculus

  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.

• Probability Theory

  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.

• Contemporary Epistemology

  There has been a sharp revival of interest in fundamental questions relating to knowledge in recent years. These include the status of testimonial knowledge; the extent to which possession of knowledge requires one or more virtues; the suggestion that knowledge can be a group rather than an individual achievement; the idea that it is unjust to place people in positions where they cannot acquire knowledge that might empower them; the relationship between knowing how to do something and that something is the case; the role of bias, discrimination and presupposition. Building on the first year module on epistemology, this module focusses on one or more of these and investigates them in depth.

• Ancient Ethics

  This module will introduce ancient Greek ethics, primarily focusing on the ideas of Socrates (as presented in Plato’s early dialogues) and Aristotle. The first part will look at key themes in Socratic-Platonic ethics, examining material from a range of Platonic dialogues, including (but not limited to) the Protagoras, Gorgias, and Euthydemus. It will consider topics such as virtue, knowledge, ignorance, and weakness of will. The second part will focus on Aristotle’s ethics, as presented in his Nicomachean Ethics, and will look at topics such as happiness, character, virtue, pleasure, and the ideal life. Subsequent developments in ancient Greek ethics (i.e. Epicureanism, Stoicism) will be covered in the companion module ‘PY2218/PY3218 Hellenistic Philosophy’, although each module is designed to stand without the other. Where relevant, aspects of ancient ethics from other traditions (Indian, Chinese) may also be incorporated in order to broaden and diversify the curriculum. Although focused on historical texts, the module will be primarily concerned with the philosophical problems that they raise.

• Philosophy of Psychiatry

  This module covers various issues in the philosophy of psychiatry. Addressing these issues requires the application of insights from a range of philosophical fields, including philosophy of science, philosophy of mind, philosophy of medicine, practical ethics, and metaphysics. Studying philosophy of psychiatry can be a great way to think about some difficult, highly theoretical philosophical issues including free will, mental causation, and explanation, all of which find natural application in the field of psychiatry. Philosophy of Psychiatry is also one of the few areas of philosophy that routinely combines both ‘analytic’ and ‘continental’ philosophical perspectives.

• Philosophy of Medicine and Bioethics

  We will draw on issues in philosophy of science and ethics to understand and attempt to solve conceptual problems arising in medicine and the biomedical sciences. Among other things, we will consider what a disease is, whether we own our bodies (and body parts), what is involved in informed consent, and what is properly involved in decision-making in medicine.

• Medieval Philosophy

  This module will examine a range of key thinkers and themes in medieval philosophy, from the fourth to the fourteenth century, telling the story of the development and transmission of philosophical ideas along the way. It will begin in late antiquity, showing the ways in which medieval thought was built on the ancient Greek philosophical tradition. It will outline the transmission of Greek thought to the Arabic-speaking world, examine a number of Arabic philosophers, and consider the impact of Arabic thought on medieval philosophy in Paris. It will conclude with Duns Scotus, active in fourteenth century Paris and Oxford. Topics discussed will focus on problems in metaphysics, such as the nature of existence, universals, the mind, and time. The relationship between philosophy and theology (or reason and faith) will be a continuing theme. The primarily metaphysical content will make this module a companion to ‘PY2217/PY3217 Ancient Metaphysics’, although each module is designed to stand without the other. It will examine (in translation) texts originally written in Greek, Arabic, and Latin.

Year 3

• Quantum Theory 1

  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.

• Mathematics Project

  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.

• Number Theory

  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.

• Ancient Metaphysics

  This module will focus on the metaphysics of Plato and Aristotle, concentrating on Plato’s theory of Ideas and Aristotle’s response to it. The first part will look at Plato’s theory of Ideas as developed in his middle period dialogues and his own criticisms of it in his later works, drawing on dialogues such as the Republic, Sophist, and Parmenides. The second part will look at Aristotle’s criticisms of Plato and his own theory of hylomorphism, developed in response (in the Metaphysics, Physics, and elsewhere). Central philosophical themes will include the ontological status of ideas, the role of universals, causation, the nature of matter, and problems relating to knowledge. The metaphysics of Plato and Aristotle are foundational for the subsequent history of philosophy, some of which will be explored in the companion module ‘PY2219/PY3219 Medieval Philosophy’, although each module is designed to stand without the other. Although focused on historical texts, the module will be primarily concerned with the philosophical problems that they raise. 

• Philosophy and Literature
• Aesthetics

  The aim of this module is to consider the main directions of eighteenth-century and post-Kantian aesthetics, in particular the issues that have arisen about what it means to consider objects—whether art or nature—aesthetically, and an analysis of concepts bound up with this “aesthetic attitude”, such as disinterestedness, beauty and the sublime. Each week will focus on one issue surrounding the question of taste, of judgements of beauty and the sublime and of the aesthetic experience, from Hume, through Kant, to the present day. Particularly attention will be paid to non-artistic aesthetic experiences, such as those of the natural world, and, as much as feasible, attention will also be paid to art and aesthetics produced outside the modern European tradition, such as aesthetics in the Islamic world. 

• Art and Philosophy

  The module will provide the opportunity for you to apply theoretical skills developed in relation to philosophy of art and aesthetics to practical problems in some of the following domains: curating, gallery education, artistic practice, art criticism and the management of cultural institutions. After an initial five weeks of theoretical discussions around historical and contemporary art, you will work closely with the curators of Royal Holloway's art collection to gain familiarity with and apply knowledge and skills to the exhibition of and reflection on situated artworks.

• Hegel

  German idealism sets itself the task of satisfying three main aims: systematizing Kant’s philosophy by finding necessary premises for its conclusions; providing a rigorous demonstration of the laws of thought; and ensuring that satisfying these aims satisfies the third aim of proving that reason is not the product of a purposeless, mechanistic world, but is itself an absolutely free purposive activity. This module investigates Hegel’s Phenomenology of Spirit as an attempt to satisfy these aims. We will explore Hegel’s distinctive and influential criticisms of Kant, his development of dialectic as a method of deriving the laws of thought, and his argument that reason is absolutely free. We will pay special attention to his successive, unfolding theses for the essentially self-conscious character of consciousness, the essentially recognitive character of self-consciousness, and the essentially historical character of recognition.

• Philosophy Dissertation

  You will demonstrate your skills as an independent learner by embarking upon a substantial piece of written work of between 8,000 and 10,000 words in length. You will be guided by a dissertation supervisor, but will choose your own topic, approach, and philosophical sources.