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What is Research in Mathematics?

You might have your own idea about the meaning of the word "Mathematics". This idea probably developed during your school and college career and will certainly develop if you opt for an MA (or even a Ph.D.) in Mathematics. Instead of giving a complete answer to the above question, some ideas about the nature of mathematics as a science with active research are explained.

The following example might be a useful illustration of the more general comments which follow.
In 1637 the following famous conjecture was formulated by Pierre de Fermat:

Whatever the value of the natural number n > 2 is, there do not exist positive natural numbers a,b and c which satisfy an + bn = cn.

Even though many of the finest mathematicians studied the problem, for more than three centuries, nobody was able to give a correct mathematical proof of this conjecture.

If you are interested in the history of this problem or if you like to know what historians found out about Fermat's life (and many other mathematicians) you could follow these external links. Simon Singh's Fermat Links might also be of interest. An interview with Andrew Wiles is available, the person who finally succeeded in proving this conjecture.

In printed form, more about Fermat's conjecture can be found in the following books which are written for a general audience:

Singh, Simon Fermats Enigma - The Epic Quest to Solve the Worlds Greatest Mathematical Problem
Aczel, Amir D. Fermats Last Theorem - Unlocking the Secret of an Ancient Mathematical Problem
Ribenboim, Paulo Fermats Last Theorem for Amateurs

Despite its apparent simplicity this problem turned out to be unusually difficult. Attempts to solve it lead to the genesis of algebraic number theory, nowadays a fully recognised branch of modern mathematics.

How is it possible that such an elementary question leads to the development of a completely new branch of mathematics?

Basically, the reason lies in the core of the mathematical method which is used to tackle such problems. This method proceeds roughly as follows. First, with precise analytic methods the essence of the problem is isolated. Based on this analysis a system of concepts and notions is developed, adequate to express the main structural features of the problem.
Using the principles of logical consistency, within the newly created abstract building, valid statements are formulated. Among them, those which help to solve the original problem are sought. Proceeding this way, very often new questions arise, which are tackled by the same method. Sometimes, the reasons for certain identities are deeply hidden. They might be revealed only after proceeding through several levels of abstraction. In this way, mathematics proceeds from simple fundamental terms to more complex structures, sometimes spawning a new branch of mathematics.

Therefore, mathematics as a scientific subject is not at all static, and the study of mathematics is very far from pure learning by heart.

The student, who pursues research in mathematics, develops specific qualities, which are described next.
The analysis of the core problem, as explained above, requires a good intuition apart from the ability for abstraction. The reason is that there are no recipes which allow us to create an appropriate idealized form of the actual problem. In the proof phase, there is a demand for imaginativeness a profound knowledge of the basics of the theory and tenacity. However, the most important capacity, which is needed to successfully pursue mathematical research, is the joy in working with abstract terms and the joy of creating knowledge which will still be true in thousand years, because the meaning of "true" and "false" in mathematics is not subject to subjective opinions.

The scientific training gained through research in mathematics for an MA thesis, is based on learning mathematical theories and methods as well as the acquisition of specific ways of thinking and efficient working methods.
This training leads to a way of thinking, which is characterised by abstraction abilities, creativity and tenacity. Qualities, which are nowadays in demand in wide ranges of professional life including education at schools and universities. Flexibility and openness, attributes developed during successful MA studies in mathematics, very often form an advantage against competitors in the job market.

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