by Eberhard Zeidler

Part I

Napoleon went to Egypt in 1798 with an expedition corps of thirty-eight thousand soldiers. Confronted with the silent eternity of the Pyramids of Gizeh, he was said to have cried out: "Soldiers! From the height of these pyramids, forty centuries look down upon you!" The cave paintings found in France and Spain bear witness to an amazing sense of shape and forms. They are separated from the present-day omnipresent computer by one-hundred and fifty centuries. Mathematics originated in numbers and simple geometric figures. Since the classical period of ancient Greek mathematics that is associated with names like Plato, Euclid, Archimedes and Diophantos, mathematics has always been like a Sphinx as an allegory for both very esoteric and highly practical science.

Mathematics is both a challenge to the human intellect and an indispensable tool of technology.

In other words, mathematics is both abstract and practical. Mathematics has applications ranging from solving complex problems in engineering such as designing a Boeing 777 at the computer all the way to logic and epistemology in philosophy. The mathematician Godefrey Harold Hardy (1877-1947) who taught at Oxford and Cambridge wrote: "A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." The English mathematician and philosopher Bertrand Russel (1872-1970), who was a friend of Hardy, reminds us: "Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world must conform."

What is the significance of mathematics? Probably the most profound answer to this question is:

Mathematics is an organ for knowledge, man's mental eye that allows him to venture into areas of knowledge extraordinarily remote from his everyday world of experience.

This does not only concern the depths of the cosmos, but also the processes on an atomic and subatomic scale that completely different laws hold for than we are used to in everyday life. The only way they can be comprehended is with mathematics. The more we remove ourselves from our world of everyday experience in high technology, the more we need to apply mathematical methods. A case in point is miniaturising circuits in computers. This has reached a point that as much as ten per cent of the heat developed there is caused by the complex properties of the ground state of a quantum field based upon the Casimir effect. In turn, we can only mathematically comprehend the Casimir effect using the abstract methods of quantum field theory. One of the most amazing things people notice about mathematics time and again is the fact that one and the same mathematical method can be applied to a wide variety of different problems.

This underscores the character of mathematics as a science that transcends all boundaries.

A case in point is using the same mathematical method for analysing chaotic processes and predicting whether a human heart, a car motor or a star is threatened by an infarction. This is a method that was developed in astrophysics. Another example is the young Einstein (1879-1955), who developed a theory of Brownian random motion of tiny particles in liquids in 1905. Brownian motion is a phenomenon that the English botanist Robert Brown first observed under the microscope in 1827 and it developed into a mathematical theory of random processes. Today, it is used in a number of areas, including determining the values of derivations on financial markets. Robert Merton and Myron Scholes received the Nobel Prize in Economy for it in 1997. We could list other examples. For instance, Johann Radon delved into the innermathematical problem of reconstructing the shape of a geometric figure from its layer sections in 1918. This Radon transformation is frequently used in computer tomography. It was developed by Alan Cormack and Godefrey Hounsfield and they received the Nobel Prize in Medicine for it in 1975. Computer tomography is completely pain-free, showing that mathematics is a blessing for humankind since introducing radiopaque medium to visualise the human brain was very painful. Furthermore, Francis Crick and James Watson would not have been able to do their ingenious work decoding the double helix structure of DNA based upon X-ray structure analysis without the meshing of biology, chemistry, mathematics and physics. And they also received the Nobel Prize in Medicine in 1962 for their work. Our last example is the explosion in computer efficiency about thirty years ago, which was made possible by developing new types of mathematical methods for calculating huge electric circuits. They lead to systems of ordinary differential equations which have the unpleasant mathematical property of stiffness.

Figure 1 (see right side) shows a picture from the Hubble Space Telescope that people call Einstein's Cross. This is a single very distant quasar whose light comes from the early era of the universe. It passes through a galaxy and is refracted by the gravitation of this galaxy's stars according to Einstein's General Relativity Theory, producing multiple images of the quasar in the telescope. The mathematics used for describing these gravitational lenses is the same as is used in geometric optics for lens systems on earth that was founded by Fermat (1601-1665) and Huygens (1629-1695), which was also developed later on. The French mathematician Henri Poincaré (1854-1912) is one of the great men of mathematics who developed the theory of dynamic systems, and he defined mathematics as follows: "La mathmatique est l'art de donner le meme nom des choses diffrentes"

There was an anthology that the Springer Publishing House brought out recently with the title of ,,Mathematics Unlimited - 2001 and Beyond". Eighty leading authors in the world draft a fascinating image of mathematics for a broad public. In the spirit of Carl Friedrich Gauss (1777-1855), one of the greatest mathematicians of all time, this book makes no distinction between pure and applied mathematics. After all, nature, technology and medicine pose questions that cannot be answered by thinking in a limited frame, rather they require the force of the full range of mathematics. You may find the following lines under the portrait of Gauss in the German Museum of masterpieces of the natural sciences and technology in Munich: "His spirit lifted the deepest secrets of numbers, space, and nature; he measured the orbits of the planets, the form and the forces of the earth; in his mind he carried the mathematical science of a coming century." The topics covered by this anthology above not only include a myriad of profound innermathematical questions having a significant effect upon modern theoretical physics, but also have a series of relations to the real world we are living in:

• algorithms on computers and scientific computing,
• how the human brain functions,
• how the human heart and circulatory system function,
• astrophysics and cosmology,
• the optimal control of technical regulating systems,
• building and programming high-performance computers,
• calculating the parameters of new materials,
• performing calculation for huge circuits in computers,
• computer tomography and image processing in medicine,
• elastic media,
• elementary particles,
• financial markets,
• liquids and turbulence,
• gases and shock waves,
• life insurance and other risk insurance,
• mathematics in the entertainment industry,
• modelling internet,
• molecular genetics,
• planning liver operations,
• protein folding,
• quantum computers,
• computer simulations,
• statistical processing of large data sets,
• the structure of DNA,
• encoding data,
• calculating weather and climate.

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• Part I
• Part II
• Part III
• Part IV
• References

About the Autor

Prof. Dr. Eberhard Zeidler wurde 1940 in Leipzig geboren. Dort studierte er Mathematik und Physik. 1974 wurde er zum ordentlichen Professor für Analysis an die Universität Leipzig berufen. Zusammen mit Prof. Dr. Jürgen Jost und Prof. Dr. Stefan Müller gründete er 1996 das Max-Planck-Institut für Mathematik in den Naturwissenschaften in Leipzig und war von 1996 bis 2003 dessen geschäftsführender Direktor. Er ist Mitglied der Deutschen Akademie der Naturforscher Leopoldina. Für sein Lebenswerk erhielt er den Alfried Krupp Wissenschaftspreis 2006 der Alfried Krupp von Bohlen und Halbach-Stiftung.
Prof. Zeidler starb im November 2016.

Figure 1: Einstein's Cross