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What is mathematics for Ehrhard Behrends
This is a very difficult question, and there is probably no one answer to which all mathematicians would subscribe. We have collected a number of partial answers. There is great variation in the ambitions of the authors and the demands placed by them on the reader
For Ehrhard Behrends
For those of you who do not yet know much about mathematics, here you will find a few important facts as we present a brief introduction to the subject. Of course, we shall have to be very superficial, since mathematics has been pursued by some of the world’s best minds for over 2500 years. For those wanting more, opportunities to learn about many aspects of mathematics can be found at this website.
As an illustration, let us consider a simple example. You would like to improve the appearance of your living room with new wall-to-wall carpeting. The room measures four by five meters, and you wish to know how many square meters of carpeting to order. It is unlikely that any readers will find themselves mentally overtaxed by this little problem: You simply multiply four times five to get twenty, and the problem is solved. You need to order twenty square meters of carpeting. But what, in fact, has just occurred? A problem from everyday life was translated into another world, the world of mathematics, and then solved. The solution is then translated back into the terms of the original real-world problem. In this case, not much knowledge was necessary, just a bit of plane geometry and some simple arithmetic.
The same idea is behind all applications of mathematics: Objects from the “real” world are translated into mathematical objects; the “real” problems become mathematical problems; and those problems must be solved. In conjunction with the “World Mathematical Year 2000,” this translation principle was incorporated in a poster design by the Danish mathematician Vagn Lundsgaard Hansen: In the picture you can see the 1624-meter Great Belt Bridge (Storebæltsbroen), the longest suspension bridge in Europe. It links the Danish islands Funen and Zealand.
In this translation process—the technical term is mathematical modelling—a number of mathematical subdisciplines play a role. Sometimes—as in our example—numbers alone suffice, while on other occasions the need will arise for functions or vectors or probability spaces or sets or some other notion. The solution of such mathematical problems requires some specialized knowledge, and sometimes one has to be content with only a good approximation to an exact solution. And finally, today one would generally be quite lost without the aid of a computer.
We Have Three More Important Observations:
1. This approach actually works! The technique of making use of the “somehow mathematical structure” of the universe has been applied since the beginning of the modern era with overwhelming success. As a vision of the world, this idea was formulated as early as Pythagoras, who maintained that “all is number,” while the victory lap, so to speak, was taken with Galileo’s statement, “The book of nature is written in the language of mathematics.”
In sum: Mathematics is useful.
In many areas of technology, economics, and science it is impossible to do without mathematics. For example, a mathematical application that in everyday life is invisible but is frequently used—perhaps even by you, for example in automated banking by phone or over the Internet—is the encryption of electronic communications, for example using the RSA encryption algorithm. New areas of application are being constantly discovered, and the requisite mathematics frequently has to be developed from scratch. It is clear, then, that employment opportunities for mathematicians will be bright for a good many years to come.
2. Mathematics also creates models for describing reality. The properties of such models are derived from the fundamental assumptions of the given theory—the so-called axioms—using rigorous methods of proof. In this way, mathematicians obtain truths that have a timeless and objective significance. For example, if we interest ourselves in the natural numbers 1, 2, 3, 4, …, then we quickly discover that some numbers have a special property with respect to multiplication: We call those numbers that are divisible by only themselves and 1 prime numbers (examples are 17, 41, 101). The first significant properties of prime numbers were discovered long ago, by the ancient Greeks. One such fact is that every whole number can be written uniquely as the product of prime numbers. Another is that there are infinitely many prime numbers. Many mathematicians consider the discovery of interesting general truths about mathematical objects to be the most important aspect of their subject, and many have spent most of their lives in single-minded and exhaustive pursuit of some question that seems to them to be of great importance.
In Sum: Mathematics is fascinating.
3. The fact that we can attain a better understanding of the universe with mathematics than with merely a qualitative description raises a number of fundamental questions. Why is this so? In precisely what sense do we obtain a better knowledge of the universe? Is mathematics provably reliable, that is, do we have a guarantee that the results established by mathematical methods are correct?
In sum: Some careful thought leads very quickly to a number of interesting philosophical problems.
On the website www.mathematics-in-europe.eu you will find information primarily on the first two of the above points. Some aspects of such philosophical problems are dealt with on the site philosophy of mathematics, and one can also find a discussion of existence statements. Of particular importance to us is to make clear the significance of mathematics as a part of present-day technological and scientific culture.
Have fun surfing!
E. Behrends, FU Berlin, Germany