Erfahrung,Weltbild und Erkenntnis bei Nikolaus Cusanus

To explain the interaction of stillness and motion of thought, Nicholas Cusanus formulated his renowned comparison with a cosmographer, which through five gateways, corresponding to the five senses, receives information about the world in the form of messages. What follows therefrom is not directly an analysis of the world but of the Creator, whom the philosopher mirrors in himself as a creator of scientific symbols. Cusanus was repeatedly suspected of Pantheism. What is crucial, however, for the critique of reasonning is the parallelism, that God's omnipresence in his creation corresponds to a universal capacity of the human mind to perceive everything by means of a hypothetical otherness (alteritate coniecturali). Therefrom proceeds the general projection that everything can be seen in mathematical terms. Mathematical calculating, working with figures, reducing to units, leads Cusanus to God's creative power as much as to the functioning of the intellect. However, his renowned mental experiments on the minimum and maximum were purely in pursuit of the goal of describing the fluid frontiers of defined thought. This is also true of his cosmology. Cusanus argued mathematically in order to prove the non-mathematical and the non-realistic.     

Präzision versus Exaktheit: Konfligierende Ideale der angewandten mathematischen Forschung: Das Beispiel der Tragflügeltheorie

It is often supposed that mathematical argument provides a model of precision for the sciences. In contrast to this view, the present article proposes to distinguish between mathematical exactness as a (historically variable) ideal regulating the inner standards of mathematical argumentation and precision as a (historically variable) norm governing the relation between products of mathematical reasoning in scientific contexts and empirical or practical data. By discussing a major achievement in the mathematization of flight, Ludwig Prandtl's lifting line theory of wings, it is shown that exact reasoning does not necessarily lead to scientific precision, and that the achievement of precision may even require to loosen existing standards of exactness. It is argued that the main contribution of mathematical argument to generating precision in science lies in its capacity to provide sophisticated tools for the production of data, rather than in its adherence to an ideal of exactness.     

Mathematik und Religion in der frühen Neuzeit

Some protestant Mathematicians had a strong preoccupation with the Day of Judgement. Stifel, Faulhaber, Napier and Newton made calculations in order to determine the date of the end of the world. Craig gave mathematical rules for a decline in the reliability of Christian tradition; in order to prevent a reliability of nearly zero, the Day of Judgement must come before. Furthermore, some conflicts between theology and mathematics are discussed. The Council of Konstanz condemned Wyclif's theory of the continuum. It seems that the relation between part and whole was a crucial point between the Aristotelians and the “moderns”.     

Die Entwicklung der Sozialgeschichte der modernen Mathematik und Naturwissenschaft und die Frage nach dem sozialen Raum zwischen Disziplin und Wissenschaftler

The use of the concept of social history of science is sketched in the Anglo‐American and the German discussions from the mid 1970s up to recent work. By presenting a ‘social map’ of a selected scientific community it is argued that between the categories of discipline and single scientist there exists a wide ‘social space’ of groups within which science is pursued. In adopting a milieu theoretic approach an ecology of science is proposed as a suitable extension of the social history of science.     

Analogie und mathematisches Denken

This article deals with six aspects of analogical thinking in mathematics: 1. Platonism and continuity principle or the “geometric voices of analogy” (as Kepler put it), 2. analogies and the surpassing of limits, 3. analogies and rule stretching, 4. analogies and concept stretching, 5. language and the art of inventing, 6. translation, or constructions instead of discovery. It takes especially into account the works of Kepler, Wallis, Leibniz, Euler, and Laplace who all underlined the importance of analogy in finding out new mathematical truth. But the meaning of analogy varies with the different authors. Isomorphic structures are interpreted as an outcome of analogical thinking.