New book: Philosophy makes better mathematicians

Does mathematics consist of absolute truths, and are mathematical results always indisputable? Most people would probably respond yes without thinking twice, but the answer is actually also in part no. Mathematics can also be approached from a philosophical angle - and it is important to do so. Otherwise, we cannot ask the big, important questions in life, writes University of Southern Denmark-scientist in a new book.

The author is Jessica Carter, Associate Professor at the Department of Mathematics and Computer Science at University of Southern Denmark. She teaches philosophy, and together with Tinne Hoff Kjeldsen from Roskilde University Center in Denmark she has written about the importance of learning some philosophy and history in order to become a competent and reflective mathematician.

The book is called International Handbook of Research in History, Philosophy and Science Teaching and it will be published by Springer Verlag in June 2013. The two authors have contributed with the chapter: The Role of History and Philosophy in University Mathematics Education.

Many believe that mathematics cannot be discussed. But it can - and it is important to do so. When discussing mathematics from a philosophical point of view, we stop learning equations and formulas by heart and start talking about all the different ways we can work with mathematics and the places it can take us.

According to Jessica Carter it is important to stamp out the long-lived myth that mathematics is indisputable.

There are many examples of matters within mathematics that can be discussed. I do not believe that the rigor of mathematics is up for discussion, but I think we should deal with the fact that it is changing and evolving as all other sciences.

A philosophical question, that can evoke many thoughts from a mathematician, is: Has mathematics always existed, and do we know about it, because we have discovered it? Or has man created mathematics?

The Greek philosopher Plato believed that mathematics is eternal, because concepts and ideas have their own existence. Opposite to this idea later philosophers believed that concepts and ideas cannot exist independently of man, and therefore mathematics is something that has been invented.

When I teach philosophy to our mathematics students and ask them to think about this question, my main point is not to give them an answer to the question. The aim is rather to teach the mathematics students to think philosophically about mathematics, to think critically, to ask questions, to analyze arguments and to assess assumptions. Mathematics is a subject which, like all other subjects has undergone development, and mathematicians, too, get smarter by thinking philosophically, reflecting and putting the subject into perspective.

Several centuries before Christ the Greek thinker Pythagoras believed that numbers were the basis of everything in the world. He explained the world from natural numbers and operated only with these and fractions (the ratio of natural numbers).

Pythagorean numbers were only positive integers, and this limited toolbox worked fine until one of hisstudents discovered that the diagonal of a square with sides being integers could not be described only using integers.The problem was that the square root of 2 was not a rational number, but an irrational number. Immediately the problem was removed by throwing the presumptuous student in the sea, where he drowned - a radical, though not long-lasting solution. There was no getting around the fact that integers and fractions were not enough to describe the world. The mathematical way to solve the problem was to expand the concept of numbers and introduce irrational numbers. Negative numbers were also needed, but they came only later.

This story illustrates that mathematics is not fixed. Even mathematics has undergone an evolution, andit will continue to do so. Today's mathematicians are in the process of developing new mathematics, andmathematics students should be made aware of this, says Jessica Carter.