There is no doubt that mathematics occupies a special position among the sciences and in the educational system. Partly, this is due to the fact that mathematics is applied to all the other sciences and therefore an adequate set of mathematical skills is a prerequisite for anybody willing to do science.
But what does a mathematician do, and why do we need to care about the development of abstract mathematics?
Allow me to suggest the following quotes by three distinguished mathematicians: “A mathematician, like a painter or a poet, is a maker of patterns.” (G. H. Hardy, 1877-1947).
“Symmetry is a vast subject, significant in art and nature. Mathematics lies at its root, and it would be hard to find a better one on which to demonstrate the working of the mathematical intellect.” (H. Weyl, 1885-1955).
“The task of mathematics and mathematicians is to articulate the visible regularities in the physical and mental worlds, and to find new structural patterns unperceivable by the direct intuition and common sense.” (M. Gromov, 1943- ).
The notion of pattern is central to understanding. The brain is a recogniser of patterns. There are patterns in the sky and in the sea. But there are also patterns in thought, language, music, social behaviour, the singing of birds, the binomial theorem, and the equations of mathematical physics.
We live in a patterned universe, the cosmos, which means ordered and patterned. What makes mathematics so infinitely adaptable is that it reduces all information, be it to do with spaceships or DNA, into patterns that can be abstractly analysed and interpreted.
The concept of symmetry, like that of pattern, is an extremely wide-ranging one. Its significance in the context of spatial structures and visually striking patterns is easily appreciated. But symmetry has incredibly wide ramifications. The force of gravity, which determines the large-scale structure of the universe, is radially symmetric, and so is the electrical force that holds electrons in orbits around the nucleus. In living organisms, symmetry relates to health in mysterious ways; for example, symmetric flowers produce more nectar than asymmetric ones. Symmetry gives the human brain the ability to predict. If we can see only a small amount of a symmetric structure, then we can predict the rest. The fact that nature prefers symmetry has been exploited by theoretical physicists to make many powerful predictions and develop deeper theories.
The process of mathematics involves recognising patterns and symmetries, making conjectures or predictions, and then proving or disproving these conjectures through rigorous logical analysis. Very often it happens that the process itself is very little related to the outside world, but the essence of mathematics is precisely this process, and independently as to whether it is directly useful, this process includes skills that can be applied to problems and situations in any field of knowledge. That is what, I believe, makes mathematics so uniquely important.
Emanuel Chetcuti is an associate professor at the Department of Mathematics at the Faculty of Science of the University of Malta. In 2006, he was awarded the Birkhoff-von Neumann prize by the International Quantum Structures Association.
Sound Bites
• In 1873, G. Cantor shook mathematics to the core when he discovered that the real numbers outnumber the integers, even though there are infinitely many of both. Moreover, beyond the size of the set of real numbers lies an interminable progression of evermore enormous, indeed all endless, entities. By the 1940 work of K. Gödel, and the 1963 work of P. Cohen, the question of whether the size of the set of real numbers coincides with the second smallest infinite has been shown to be independent of the standard ZFC axioms of set theory; one can have it either way, depending on which axiom one “forces” into the system!
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DID YOU KNOW?
• The German Renaissance artist Albrecht Dürer (1471-1528) is acknowledged worldwide for his paintings, drawings, and copper engravings. His watercolours also mark him as one of the first European landscape artists. Less known is his work on mathematics. He understood that a well-founded knowledge of geometry is a prerequisite to a successful artistic creativity.
• Early enough he studied the Latin translation of Euclid’s Elements, which had appeared in 1505. He also participated in drawing a map of the northern hemisphere of the Earth, together with Johannes Stabius. Although he does not seem to have met with any of the major Italian mathematicians of the time, on his journeys to Italy, he did meet Jacopo de Barbieri, who told him of the mathematical work of Alberti and Pacioli and its importance to the theory of beauty and art. Historically, most important is the first German book on Geometry under the title Four Books on Measurement, which appeared in 1525. In this work, Dürer defines many special curves like the Pascal snail or the shell line, introduces a new construction of an ellipse, and recognises that the ellipse, parabola, and hyperbola are specific conic sections. He even developed approximate solutions for the three problems of classical antiquity: divide an angle into three parts, transform a circle to a square, and double the volume of a cube.
For more trivia, see: www.um.edu.mt/think.