This question is rarely asked, but it seems students are uneasy not to have some kind of answer! They know that mathematics has many applications, and that this is a good reason for studying it. But that leaves open the question of:
what is it about mathematics that allows so many applications?
Here is a quick answer which I believe gives a flavour of the subject and a lead into further development of this description. Mathematics gives a language for rigorous expression of intuitions so as to allow
Often we want to know not only that something is true but why it is true. This search for understanding is basic to the progress of mathematics. The search for understanding often leads to the development of new worlds of mathematics, whose applications then ring down the centuries.
Note that engineering and physics depend on the mathematical basis of the calculations in those subjects.
The capacity of mathematics for description is shown by one example: the mathematics of quantum mechanics describes the real world in a way inaccessible to common sense and common language.
The use of abstraction and symbolic representation to show appropriate information ansd structure is well known, as in the famous map of the London Underground, which is held up as an example of good design. Of course the use of appropriate notation is a key to the success of mathematics. An early example is the introduction of the equals sign, =, by Robert Recorde in 1557.
The use of mathematics for analogy is suprisingly neglected in the teaching of mathematics. See our exposition on knots for the analogy between prime knots and prime numbers. To obtain such analogies, the use of abstraction is an essential part. It is abstraction by which we compare structures. See an article on category theory as a basis for analogy and comparison (pdf).
I gave a presentation on knots to schools in Leicester in the 1980s, and a teacher afterwards came up to me and said that was the first time he had heard the word `analogy' used in relation to mathematics! Of course analogies are not between things but between the relations between things, and it is the power of abstraction to allow far reaching analogies.
Our knot exhibition also uses the theme of knots to illustrate some basic methods of mathematics:
Thus the applications come after the previous necessary developments.
Mathematics has developed a range of new structures and of new worlds, suggesting the relevance of a quotation on the role of the poet.
The great geneticist Dobzhansky has written: `Nothing in biology is comprehensible except in the context of evolution'. How then can we explain mathematics in this context? Perhaps the idea of structure gives a clue. Organisms must make some model of the environment to survive, and these models or maps reveal, and simplify considerably, the structure of the environment, to be efficient. The model, or map, has to be smaller than the original! So the organism has some idea of the landscape in which it lives. This model also has to record how the landscape behaves, it has to give the landscape structure.
So in mathematics we get the developing notion of `mathematical structure'. There was an emphasis on this in the great advances in exposition by Bourbaki, but which still tried to reduce mathematics to logic and set theory. We now know, through Godel and others, that this project was doomed, but that also that the attempt was valuable. (See Yuri Manin's `Georg Cantor and His Heritage': ) There seem to great advantages in following the categorical approach: certainly the literature on category theory shows a strong interest in discussing the aims of the subject, and of mathematics!
For further discussion, see the article The methodology of mathematics, R. Brown and T. Porter. In that article we quote some views of the famous physicist E. Wigner, of which a sample is:
"Mathematics is the science of skilful operations with concepts and rules
invented just for this purpose. [this purpose being the skilful
"The principal emphasis is on the invention of concepts.
"The depth of thought which goes into the formation of mathematical
concepts is later justified by the skill with which these concepts are used.
The paper on Making a Mathematical Exhibition also discusses these issues, since we needed to decide what we were trying to say about mathematics in order to structure the exhibition. The exhibition illustrates how applications come after various basic procedures which are esentially moving towards understanding are carried out. This is the basic case for abstract mathematics.
See also my small article on the links of Bangor with the great master of the invention of new mathematical concepts and language, Alexander Grothendieck.
In 1964 I met Sam Ulam at a conference in Syracuse, Sicily ,(my first international conference). He remarked that young people may think that the most ambitious aim would be to solve a famous conjecture. He urged that pursuing that aim might distract them from developing the mathematics most appropriate to them. I found it interesting that someone as good as Ulam should urge this view! See also a discussion on famous problems.
Finally, Jan Abas has urged that I should state strongly that a part of the attraction of mathematics is the notion of beauty. This is a quality he conveys with his web site on Islamic Art. This link with beauty and imagination is also the motivation behind the web site of the Center for the Popularisation of Mathematics.
Here is a link in French developing some of these ideas.
See also the site of the juggler Colin Wright for views on the nature of mathematics.
Download Yu L. Manin's articles on Mathematical knowledge: internal, social and cultural aspects, Truth as value and duty: lessons of mathematics.
Here is a link to a set of my articles on popularisation and teaching, including one on quality assurance, and another on category theory as a basis for analogy and comparison (pdf), and on Promoting Mathematics . See also the mathoetic mode.
February 14, 2005.
revised slightly May 28, 2008; December 29, 30, 2008; September 10, 2009; January 23, 2010.