A SUMMARY IN ENGLISH [by Editor] Since Brouwer, Weyl, and, more recently, Lorenzen, KANT as an authority [in the philosophy of mathematics in general] has often been called in against some “uncritical” Formalism and Structuralism. The core of the present paper is to show, in the main, that the Denkmuster (patterns of thinking) of Kant is still very much alive in mathematics today to be in harmony with Formalism. The study is carried out in three parts, in the light of (...) the “concept of construction” of (i) Kant himself, (ii) Lorenzen's Operationalism, and (iii) Formalism in general. (shrink)

Even beginners and young graduate students will have something to learn from this book." (Andre Hautot, Physicalia, Vol. 57 (3), 2005)"All-in-all, this highly ...

Proceedings of the NATO Advanced Study Institute on Proof and Computation, held in Marktoberdorf, Germany, July 20 - August 1, 1993.

The theory of nonlinear complex systems has become a successful and widely used problem-solving approach in the natural sciences - from laser physics, quantum chaos and meteorology to molecular modeling in chemistry and computer simulations of cell growth in biology. In recent times it has been recognized that many of the social, ecological and political problems of mankind are also of a global, complex and nonlinear nature. And one of the most exciting topics of present scientific and public interest is (...) the idea that even the human mind is governed largely by the nonlinear dynamics of complex systems. In this wide-ranging but concise treatment Prof. Mainzer discusses, in nontechnical language, the common framework behind these endeavours. Special emphasis is given to the evolution of new structures in natural and cultural systems and it is seen clearly how the new integrative approach of complexity theory can give new insights that were not available using traditional reductionistic methods. (shrink)

How (Not) to Be Secular is what Jamie Smith calls "your hitchhiker's guide to the present" -- it is both a reading guide to Charles Taylor's monumental work A Secular Age and philosophical guidance on how we might learn to live in our times. Taylor's landmark book A Secular Age (2007) provides a monumental, incisive analysis of what it means to live in the post-Christian present -- a pluralist world of competing beliefs and growing unbelief. Jamie Smith's book is a (...) compact field guide to Taylor's insightful study of the secular, making that very significant but daunting work accessible to a wide array of readers. Even more, though, Smith's How (Not) to Be Secular is a practical philosophical guidebook, a kind of how-to manual on how to live in our secular age. It ultimately offers us an adventure in self-understanding and maps out a way to get our bearings in today's secular culture, no matter who "we" are -- whether believers or skeptics, devout or doubting, self-assured or puzzled and confused. This is a book for any thinking person to chew on. (shrink)

Klaus Mainzer legt in diesem essential dar, dass die Zukunft von KI und Digitalisierung eine nüchterne Analyse erfordert, die Grundlagenforschung mit Anwendung verbindet. Berechenbarkeits- und Beweistheorie können dazu beitragen, Big Data und Machine Learning sicherer zu bewältigen. Dabei zeigt sich, dass die komplexen Herausforderungen der digitalen und analogen Welt in Grundlagenfragen der Mathematik, Informatik und Philosophie tief verwurzelt sind.

Molecules have more or less symmetric and complex structures which can be defined in the mathematical framework of topology, group theory, dynamical systems theory, and quantum mechanics. But symmetry and complexity are by no means only theoretical concepts of research. Modern computer aided visualizations show real forms of matter which nevertheless depend on the technical standards of observation, computation, and representation. Furthermore, symmetry and complexity are fundamental interdisciplinary concepts of research inspiring the natural sciences since the antiquity.

Modern cognitive science cannot be understood without recent developments in computer science, artificial intelligence, robotics, neuroscience, biology, linguistics, and psychology. Classic analytic philosophy as well as traditional AI assumed that all kinds of knowledge must eplicitly be represented by formal or programming languages. This assumption is in contradiction to recent insights into the biology of evolution and developmental psychology of the human organism. Most of our knowledge is implicit and unconscious. It is not formally represented, but embodied knowledge which is (...) learnt by doing and understood by bodily interacting with ecological niches and social environments. That is true not only for low-level skills, but even for high-level domains of categorization, language, and abstract thinking. Embodied cognitive science, AI, and robotics try to build the embodied mind in an artificial evolution. From a philosophical point of view, it is amazing that the new ideas of embodied mind and robotics have deep roots in 20th-century philosophy. (shrink)

Mit P. Bernays geht S. Körner in der Nachfolge von I. Kant und J.F. Fries davon aus, "daß eine gewisse Art rein-anschaulicher Erkenntnis als Ausgangspunkt der Mathematik genommen werden muß." Andererseits betont Körner einen Wechsel z.B. der geometrischen Anschauung in den nicht-euklidischen Geometrien, der durch die Unabhängigkeitsbeweise für geometrische Axiome möglich wurde. Analog könnte man von einem Wechsel der mengentheoretischen Anschauung in nicht-cantorschen Mengenlehren sprechen, der durch Unabhängigkeitsbeweise mengentheoretischer Axiome eingeleitet wurde. In der Algebra werden Axiomensysteme untersucht, in denen nicht (...) mehr alle anschaulichen Rechengesetze der Zahlen gelten. Für die Analysis lassen sich nonstandard Modelle angeben. Angesichts dieses Pluralismus der Modelle und Axiomensysteme kann man nicht mehr von der einen anschaulichen Mathematik sprechen — wie in den Tagen von Euklid, Piaton, Leibniz und Kant. Es stellt sich daher die Aufgabe einer Erkenntnistheorie der Mathematik, deren Kategorien den modernen Problementwicklungen Rechnung tragen, aber auch ihre anschaulich-konstruktiven Grundlagen aufzeigen. (shrink)

Molecular models are typical topics of chemical research depending on the technical standards of observation, computation, and representation. Mathematically, molecular structures have been represented by means of graph theory, topology, differential equations, and numerical procedures. With the increasing capabilities of computer networks, computational models and computer-assisted visualization become an essential part of chemical research. Object-oriented programming languages create a virtual reality of chemical structures opening new avenues of exploration and collaboration in chemistry. From an epistemic point of view, virtual reality (...) is a new computer-assisted tool of human imagination and recognition. (shrink)

Die moderne Kognitionswissenschaft kann nicht verstanden werden ohne Einbeziehung der neuesten Errungenschaften aus der Computerwissenschaft, künstlichen Intelligenz , Robotik, Neurowissenschaft, Biologie, Linguistik und Psychologie. Die klassische analytische Philosophie, wie auch die traditionelle AI, setzten voraus, dass alle Arten des Wissens explizit durch formale oder Programmsprachen dargestellt werden müssen. Diese Annahme steht im Widerspruch zu den rezenten Einsichten in die Evolutionsbiologie und Entwicklungspsychologie des menschlichen Organismus. Der größte Teil unseres Wissens ist implizit und unbewusst. Es ist kein formal repräsentiertes, sondern ein (...) verkörpertes Wissen, das durch Handeln gelernt und durch körperliche Interaktion mit ökologischen Nischen und gesellschaftlichen Umgebungen verstanden wird. Dies gilt nicht nur für niedere Fertigkeiten, sondern auch für höher gestellte Domänen: Kategorisierung, Sprache und abstraktes Denken. Die verkörperte Erkenntniswissenschaft, AI und Robotik versuchen, den verkörperten Geist in einer artifiziellen Evolution zu bilden. Vom philosophischen Standpunkt gesehen ist es erstaunlich, wie tief die neuen Ideen des verkörperten Geistes und der Robotik in der Philosophie des 20. Jahrhunderts verankert sind. (shrink)