What's Mathematics?

by Jörg Zuther

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What's Mathematics?

If you calculate something you don't do mathematics, you apply mathematics. Overall, calculating is a boring activity that just needs a high degree of concentration (it may be exciting if you know some tricks to win a calculation contest) and should be left to pocket calculators and computers. That doesn't mean that people shouldn't learn to calculate, since it is difficult to see how they could otherwise get a feeling for numbers enabling them to detect absurd results provided by computers and pocket calculators that are defect, incorrect programmed, or feeded with false data.

Mathematics can be seen as the bases-for-all-kinds-of-calculations creating science. You feel uneasy about several aspects of a complicated and seemingly chaotic reality and think that you might feel better if you could control these aspects. If you are a mathematician, you begin to create an abstract model of the reality containing parameters corresponding to the aspects mentioned above. You hope that you have included all relevant related factors into your model and that the relations between these factors as implemented in your model are correct. Then your model serves as a basis for calculations whose results may be reinterpreted to the real world. (An example for such a model is offered by our ordinary numbers (together with operations like their addition and their multiplication) we use every day to calculate prices, quantities etc. This model can only be applied to situations where the real entities behave sufficiently constant. E.g. you can add a litre of water to two litres of water and get three litres of water. This is not necessarily true if you add a litre of one chemical substance to two litres of another chemical substance: If there's a chemical reaction you might get only one or two litres instead of three. Don't tell me that you can't add apples and pineapples - one apple plus two pineapples are three pieces of fruits.)

This is the 'applied' point of view of mathematics. But mathematics can also be regarded as the science of the abstract structures (such as numbers, circles, triangles, functions, networks etc.). Many mathematicians are not interested in the appliability of their models or of the abstract structures they investigate - they're interested in the abstract structures by themselves. If you forget all complications of the reality then you often can build up beautiful and elegant theories due to the simplicity of the basics (which is the reason why mathematics can also be seen as an art). These theories allow a deeper understanding of the abstract structures (whether appearing in the above mentioned models or not), and sometimes a pure theoretical structure is recognized as useful for some application or inspires an application no one would have discovered without the knowledge of the structure.

If you have problems to understand what a mathematician is doing, then you should read The spirit of mathematical research by Frank Wikström. Further, you could have a view on Mathematics - What can I do with this degree?, a .pdf-file provided by the Career Services at The University of Tennessee, Knoxville, USA.

Approaches of other people to the question: What's mathematics?

• Mathematics by Somos (by Michael Somos)
  

• What Kind Of Thing Is A Number?
  a talk with Reuben Hersh

• Interactive Mathematics Miscellany and Puzzles: Manifesto (by Alexander Bogomolny)
  mathematics seen in a sociological and educational way

• Real World Mathematics (by the Schools of California Online Resources for Education (SCORE))
  a growing collection of materials showing where mathematics is used in the real world

• Mathematics Department (of David Levin's Learning@Web.Sites)
  a huge list of excellent and/or cool web resources concerning the many facets of mathematics