Cambridge Mathematical Tripos
Encyclopedia
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
. It is the oldest Tripos
Tripos
The University of Cambridge, England, divides the different kinds of honours bachelor's degree by Tripos , plural Triposes. The word has an obscure etymology, but may be traced to the three-legged stool candidates once used to sit on when taking oral examinations...
(subject) that is examined in Cambridge.
Origin
In its classical nineteenth-century form, the tripos was a distinctive written examination of undergraduate students of the University of CambridgeUniversity of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
. Prior to 1824, the Mathematical Tripos was formally known as the "Senate House Examination". From about 1780 to 1909, the "Old Tripos" was distinguished by a number of features, including the publication of an order of merit of successful candidates, and the difficulty of the mathematical problem
Mathematical problem
A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's...
s set for solution. By way of example, in 1854, the Tripos consisted of 16 papers spread over 8 days, totaling 44.5 hours. The total number of questions was 211. The actual marks for the exams were never published, but there is reference to an exam in the 1860s where, out of a total possible mark of 17,000, the senior wrangler achieved 7634, the second wrangler 4123, the lowest wrangler around 1500 and the lowest scoring candidate obtaining honours (the wooden spoon
Wooden spoon (award)
A wooden spoon is a mock or real award, usually given to an individual or team which has come last in a competition, but sometimes also to runners-up. Examples range from the academic to sporting and more frivolous events...
) 237; about 100 candidates were awarded honours and the 300-odd below that level were known as poll men. The questions for the 1841 examination may be found within the Cambridge University Magazine.
Influence
According to the study Masters of Theory: Cambridge and the Rise of Mathematical Physics by Andrew Warwick, during this period the style of teaching and study required for the successful preparation of students had a wide influence: on the development of 'mixed mathematics' (a precursor of later applied mathematicsApplied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
and mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, with emphasis on algebraic manipulative mastery); on mathematical education; as vocational training for fields such as astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
; and in the reception of new physical theories, particularly in electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
as expounded by James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
.
Early history
The early history is of the gradual replacement during the middle of the eighteenth century of a traditional method of oral examination by written papers, with a simultaneous switch in emphasis from LatinLatin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
disputation to mathematical questions. That is, all degree candidates were expected to show at least competence in mathematics. A long process of development of coaching – tuition usually outside the official University and college courses – went hand-in-hand with a gradual increase in the difficulty of the most testing questions asked. The standard examination pattern of bookwork (mostly memorised theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s) plus rider (problems to solve, testing comprehension of the bookwork) was introduced.
Wranglers and their coaches
The list of wranglers, that is, the candidates awarded a first-class degree, became in time the subject of a great deal of public attention. The coaches, of whom Edward RouthEdward Routh
Edward John Routh FRS , was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the nineteenth century...
was the most outstanding, assumed a para-academic status. The level of technique required of the candidates was high, and the time pressure in the examinations acute. It became common for those with a first degree in mathematics elsewhere to come to Cambridge to take part in the Tripos, as a second degree.
1909 Tripos reforms
The reforms implemented in 1909 did much to dismantle the old Mathematical Tripos system. It continued as an examination.The influence persists. In Cambridge terms, it has done much to support the particular kind of mathematical approach of the University's Faculty of Mathematics
Faculty of Mathematics, University of Cambridge
The Faculty of Mathematics at the University of Cambridge comprises the Department of Pure Mathematics and Mathematical Statistics and the Department of Applied Mathematics and Theoretical Physics . It is housed in the Centre for Mathematical Sciences site in West Cambridge, alongside the Isaac...
. G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
, one of those most responsible for the changes, was concerned in particular to assert the importance of pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
. The undergraduate course of mathematics at Cambridge still reflects a historically-broad approach; and problem-solving skills are tested in examinations, though the setting of excessively taxing questions has been discouraged for many years.
Today's Mathematical Tripos
Today, the Mathematical Tripos course comprises three undergraduate years (Parts IA, IB and II) which qualify a student for a BA degree, and an optional one year graduate course (Part III) which qualifies a student for an MMath (with BA) if theyare a Cambridge fourth year student or a MASt (Master of Advanced Studies) degree
if they come from outside just to do Part III.
Assessment is mostly by written examination at the end of each academic year, with some coursework elements in the second, third and fourth years.
During the undergraduate part of the course, students are expected to attend around 12 one-hour lectures per week on average, together with two supervisions. Supervisions are informal sessions in which a small group of students - normally a pair - goes through previously completed example sheets under the guidance of a faculty member, college fellow or graduate student. Each example sheet takes about 10 hours to complete, but this is highly dependent on the individual student and the difficulty of the example sheet in question.
During the first two years (Parts IA and IB) the schedule of courses is quite rigid, and students have relatively little choice. Courses in these years cover pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
(algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
and analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
); applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
(electromagnetism
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
); and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
. Until recently, some students took options within the Tripos that allowed them to give up some Mathematics courses in exchange for courses in Physics or Computer Science, with the possibility of changing to those subjects at the end of the first year - however, the Computer Science option was discontinued from the 2008-2009 academic year.
During the third year (Part II), a wider choice of courses is available and a student will typically begin to specialise in either pure mathematics or applied mathematics.
CATAM (Computer-Aided Teaching of All Mathematics)
One component of the mathematical Tripos is CATAM, a series of computational projects in parts IB and II. CATAM projects are completed in the first two terms of the academic year, with the first set of projects due in soon after the first term.Part IA
There are two options; Mathematics and Mathematics with Physics. Mathematics with Computer Science was recently dropped, although those taking Computer Science can opt to study Papers 1 and 2 during the first year of their course. In the second option, the Numbers and Sets and Dynamics and Relativity courses are exchanged for the Physics courses from Part 1A of the Natural Sciences Tripos.Michaelmas term
There are four lecture courses, each consisting of twenty-four one-hour lectures each and all lectured in the Cockcroft Lecture Theatre. These are 'GroupsGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
', 'Vectors and Matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
', 'Number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s and Sets', and 'Differential Equations'.
Lent term
Again, there are four lecture courses, each consisting of twenty-four one-hour lectures each and all lectured in the Cockcroft Lecture Theatre: 'AnalysisMathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
I', 'Probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
', 'Vector Calculus' and 'Dynamics
Dynamics (mechanics)
In the field of physics, the study of the causes of motion and changes in motion is dynamics. In other words the study of forces and why objects are in motion. Dynamics includes the study of the effect of torques on motion...
and Relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
'.
Easter term
There is a series of lecture courses examined in the second year - attendance is optional, since they will be repeated next year, but it is generally recommended that students attend at least one or two. They are held in the Mill Lane lecture rooms, and consist of 'MetricMetric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
and Topological Spaces
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
', 'Variational Principles
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
', 'Optimisation
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
', 'CATAM', and the non-examinable 'Concepts in Theoretical Physics'.
The year is examined over four 3-hour papers in late May or early June - these are the Mathematical Tripos
Tripos
The University of Cambridge, England, divides the different kinds of honours bachelor's degree by Tripos , plural Triposes. The word has an obscure etymology, but may be traced to the three-legged stool candidates once used to sit on when taking oral examinations...
Examination Papers.
Part IB
Courses in Part IB include options from pure maths (such as 'Linear Algebra', 'Groups, Rings and Modules', 'Analysis II', 'Metric and Topological Spaces' and 'Geometry') and applied maths (such as 'Quantum Mechanics', 'Electromagnetism', 'Fluid Dynamics').Part II
The final year of the bachelors degree contains over 30 courses, and one can study areas of maths such as Differential Geometry, Galois TheoryGalois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, General Relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, Algebraic Topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, Electrodynamics, Partial Differential Equations, Stochastic Process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
, Graph Theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
and Fluid Dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
.
Part III
The famous one year Part III course offers over 80 different courses, and one can study areas of maths such as Lie Groups, SupersymmetrySupersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
, Functional Analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, Ramsey Theory
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
, Black Holes, Astrophysical Fluid Dynamics, Elliptic Curves, Representation Theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, Quantum Field Theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, Stochastic Calculus
Stochastic calculus
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes...
, Spectral Geometry
Spectral geometry
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other...
, Algebraic Number Theory
Algebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
, Biostatistics
Biostatistics
Biostatistics is the application of statistics to a wide range of topics in biology...
and Additive Combinatorics.
External links
On the importance of the Tripos in the history of mathematics in Britain: search on "tripos" in
For statistics on the number of graduates (men and women) between 1882 and 1940 see:
For the present-day Tripos see:
- Cambridge University: Guide to the Mathematical Tripos (pdf)
- Actual examination papers from 2001-2009
- The Cambridge Maths faculty's site explaining the CASM
- Nelson, GrahamGraham NelsonGraham A. Nelson is a British mathematician and poet and the creator of the Inform design system for creating interactive fiction games. He has also authored several IF games, including the acclaimed Curses and Jigsaw , using the experience of writing Curses in particular to expand the range of...
. "Miss Warren’s Profession" Eureka 51, 1992. Critique of Part III.