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Types of Mathematics

What is mathematics? Someone doing their Leaving Certificate would probably have numbers, maybe functions in their answer. Here are some descriptions from famous mathematicians - in most cases their descriptions would appear to have been coloured by their specialist area of mathematics.

``Mathematics in its widest significance is the development of all types of formal, necessary, deductive reasoning.'' Whitehead, Logician
``It is not the essence of mathematics to be conversant with the ideas of number and quantity.'' Boole. Boole who lived in Cork is famous for his Algebra of Sets.

Most mathematicians would regard Set Theory as an essential part of the language of the subject, but

``Later generations will regard set theory as a disease from which one has recovered.'' Poincaré

Nothing about numbers so far. Even when they enter, it is as

``Mathematics is not the art of computation but of minimal computation.'' Anonymous

In Computer Science, complexity theory is exactly about minimal computation. But most mathematicians will be driven by a desire not just to solve a given problem but to learn from their solution how they can most easily solve any similar problems. This is part of the intertwining of Pure and Applied Mathematics which is also well expressed in:

``The Science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain with the aid of experience.'' Kant
``Most of the best mathematical inspiration comes from experience.'' Von Neumann

Here we have the reasoning and inspiration of Pure Mathematics being driven by the experience of Applied Mathematics. On the same sort of point

``The paradox is now fully established that utmost abstractions are the true weapons with which to control our thoughts of concrete facts.'' Whitehead.

But it can be very difficult to see this in first year.

While learned mathematicians wax eloquently on the beauty of their chosen field, others have expressed the frustration, enjoyment and humour of a life dedicated to reasonable thought.

``I have had the results for a long time, but I do not know yet know how to arrive at them'' Gauss
``If I only had the theorems, then I could find the proofs easily enough.'' Riemann

But who gets the most enjoyment?

``The most interesting moments are not where something is proved but where a new concept is involved.'' Kaplansky

On the usefulness of mathematics we might jokingly quote

``God exists since mathematics is consistent and the devil exists since we can't prove this consistency.'' Weyl

Now here is a bit more information on what we mean when we talk about Pure Mathematics, Applied Mathematics, and so on. As we said earlier, mathematics in the broad sense develops as a continuous interplay between application and theory. The different sorts of mathematics arise as different stages of this interplay. At the risk of boring the reader we emphasise,

``Mathematical Science is an indivisible whole, an organisation whose vitality depends on the connections between its parts. Advancement in mathematics is made by simplification of methods, the disappearance of old procedures which have lost their usefulness and the unification of fields until then foreign.'' Hilbert

Applied Mathematics is using mathematics to solve real world problems. In this field it is essential to be able to apply many different mathematical techniques, and be able to handle problems involving data where a knowledge of statistics becomes important. It is also necessary to be able to take a practical problem, from engineering for example, and turn it into a mathematical problem; this is referred to as mathematical modelling.

``The number and rate of applications of mathematics is increasing and the equipment the students need to enable unforeseen applications, is not specialised mathematics but that core of the most general kind which will enable them to investigate new applications.'' Henkin

Pure Mathematics usually enters when the application has become abstracted. Now the basic concepts are the focus of attention. The subject is studied for its own interest, seeking out the inherent beauty, in the knowledge that the more deeply the subject is developed the better the applications will be later. In the pure mathematics courses a student will have an opportunity to see how centuries of widely varying applications have led to a number of abstract theories. These could include Differential Geometry, Algebraic Topology, Abstract Algebra, Functional Analysis and more.

Theoretical Physics is interested in physical systems. The objective is to understand nature at its most basic level and thus be in a position to predict the future behaviour of any system in which one is interested. To do this a physicist must have a sound grasp of basic physical laws and be able to unravel the implications of the laws using mathematics. Some of the basic tools for a theoretical physicist are: classical mechanics, electromagnetic theory, fluid mechanics, statistical mechanics, quantum mechanics and relativity.

Computing - at a theoretical level - tries to identify which tasks can be automated on a computer and which tasks can not. This involves looking for good solutions to, or theoretical difficulties in, computational problems; typically it involves establishing the correctness and efficiency of computer programs. Although the area deals sometimes with abstractions like formal logic, it requires diverse methods from discrete mathematics and difference equations for example, and therefore some of it is applied rather than pure mathematics. At a practical level, computer science deals with the design and implementation of modern computer software. The design and implementation of computer hardware is more a branch of electronic engineering than a mathematical topic and does not play any significant part in the curriculum of the computing courses on offer to mathematics students.

Numerical Analysis is concerned with numerical calculations of all kinds. The growth of numerical analysis as a branch of mathematics has been especially rapid in the last fifty years because of the development of electronic computers. It is concerned with the mathematical and experimental study of robust and efficient algorithms for obtaining approximate numerical solutions to problems in all areas of science and engineering. Since the early 1980's it has become a very exciting field to work in because of the availability of powerful computers at reasonable cost, and recent developments are based on a range of new parallel architectures. The efficient exploitation of these new computers will be a major concern of numerical analysis for the foreseeable future.

Statistics involves the use of modern techniques in mathematics and computing to analyse the data that arise in many areas of business and science. Recent trends involve a resurgence of interest in methods for informal exploratory analysis of data. This relies increasingly on friendly computing environments supported by efficient algorithms. The first year course discusses such procedures. The second year course lays down a more formal modelling probability framework. Later courses involve more advanced applications to time series, management science methods, and linear statistical models.

No matter which course option is chosen, care is taken to make sure that all students are computer literate and are exposed to important ideas from the different areas.

If you would like more information you could usefully consult the Encyclopedia Brittanica which has excellent essays on most mathematical subjects.