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Linie, Fläche, Raum – und Falte. Reflexionen eines Mathematikers / Line, surface, space – and fold. Reflections of a mathematician

Jochen Brüning


1. This essay will first briefly recall the standard procedure of mathematical reasoning, i.e. to deduce true statements from a set of axioms by the rules of formal logic, by reminding the reader that this method has its roots in explicit real-world applications and specific cultural techniques. However, every theory of this kind that describes a certain realm of phenomena and predicts their behavior has built-in limits, beyond which we encounter the ‘roughness of reality’. We will then explain why this essay conceives of lines as playing a fundamental role in explaining human orientation, perception and communication over centuries. 2. The underlying argument is that this history is part of an Evolutionary Heritage. This part will also call attention to the fact that the neurological mechanisms behind such a conception of lines are still not adequately understood. 3. This essay will then explore the historical development of the role of lines in mathematics, assuming this heritage as evolutionary. It will remind the reader that the beginnings of the formation of mathematical thinking are closely connected with philosophy: mathematics was a substantive part of Pre-Socratic thinking. Straight lines and circles played a fundamental role in explaining the cosmos and the universal phenomenon of motion, in particular the constellation of the planets. Furthermore, the formulation of logical rules was an essential step in the emancipation of mathematics from philosophy and in mathematics becoming a science of its own, as documented by Euclid’s Elements. 4. Along the way, lines were understood more generally as the traces of motions of bodies, allowing for curvature - like the examples of circles and conical sections - that could not be treated in the Euclidean way. We give a loose but suggestive general definition and point out that it opened up a huge field of discussion, illustrated, for example, by the work of the Renaissance painters. 5. Part 5 will turn to Newton’s invention of calculus and its ingenious application to mechanics that showed how a certain class of curved lines could be mastered by straight lines: differentiable lines that admit a well-defined tangent in each point. His law of mechanical motion also showed that only straight lines could describe a motion without a moving force, thus contradicting Aristotle fundamentally. The power of this invention will be shown with some applications to cosmology. 6. In a further section, we turn to the truly wild lines: the paths of Brownian motion which do not allow any individual classification. By introducing probability, we can say that, with near certainty, such a path is in fact a line but does not admit a tangent anywhere. This shows how rough the realm of non-differentiable lines is and how small the part of differentiable lines therein. But probability brings back differentiability, too, albeit on another level. 7. We then will turn to the issue of choosing a straight line in the Euclidean plane, which means that the plane then consists of the totality of all parallels to this line, just as the straight line consists of all its points. This brings to light the fundamental difficulty of comprehending the creation of a continuum from discrete parts and vice versa. We illustrate this with Claude Mellan’s fantastic Veil of Veronica (1647), and with Georg Cantor’s construction of the famous Cantor set (1874/1883). We then turn to the description of surfaces in space which, like the circular cone, can be made into a plane without bending, tearing, or cutting. Different from all other surfaces, these are characterized by having Gauss curvature 0, a rare phenomenon indeed. As a counter-example, we will describe the ‘doughnut’ or two-dimensional torus, but point out that an abstract version of it has curvature 0. 8. We finally turn to the notion of fold and folding and discuss first how its etymological roots indicate an interesting dichotomy between something that is safely and neatly stored away and something that is too complex to be (easily) understood. Folding begins with simple lines, and we recall the ingenious folding used by Gauss as a first-grader to add up the numbers from 1 to 100 within a minute. We proceed to the more familiar folding that occurs on surfaces, and indicate how the abstract torus can be embedded into Euclidean space such that, on the resulting surface, the longitudinal and the latitudinal circles have all the same length. This is a recent and complicated mathematical result based on folding the latitudinal circles ever more extremely; already the first approximations are mind-boggling, as can be checked on the relevant website. 9. In conclusion, we turn to the philosophical question: what is actually the goal of mathematics? Formally, the work of mathematicians consists in unfolding the axioms of their preferred theory. Yet we explain that this is not what happens in practice.

© 2017 Walter de Gruyter GmbH, Berlin/Munich/Boston
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