In the first half of the 19th century geometry changed radically, and withina century it helped to revolutionize both mathematics and physics. It also put the epistemology and the philosophy of science on a new footing. In this volume a sound overview of this development is given by leading mathematicians, physicists, philosophers, and historians of science. This interdisciplinary approach gives this collection a unique character. It can be used by scientists and students, but it also addresses a general readership.

Beyond their remarkable technical accomplishments, the new directions taken by the sciences in recent decades call for renewal of their epistemological basis. The purpose of this book is to show that Husserl s transcendental phenomenology, if properly re-examined, provides the required framework for such an epistemology. This re-examination is both critical and constructive. (i) The absolute subjectivization or the full naturalization of consciousness must be rejected. (ii) The necessarily transcendental character of phenomenology is put to work in the search for a systematic connection between the modes of theoretical objectivation and the apprehension of the phenomenal world by intentional consciousness. A new look at some of the fundamental issues opened up by Husserl is thus suggested by recent advances in the theory of perception, attention, and the will; foundations of mathematics and formal logic; space-time or quantum physics."

This book proposes a new phenomenological analysis of the questions of perception and cognition which are of paramount importance for a better understanding of those processes which underlies the formation of knowledge and consciousness. It presents many clear arguments showing how a phenomenological perspective helps to deeply interpret most fundamental findings of current research in neurosciences and also in mathematical and physical sciences.

This interdisciplinary volume collects contributions from experts in their respective fields with as common theme diagrams. Diagrams play a fundamental role in the mathematical visualization and philosophical analysis of forms in space. Some of the most interesting and profound recent developments in contemporary sciences, whether in topology, geometry, dynamic systems theory, quantum field theory or string theory, have been made possible by the introduction of new types of diagrams, which, in addition to their essential role in the discovery of new classes of spaces and phenomena, have contributed to enriching and clarifying the meaning of the operations, structures and properties that are at the heart of these spaces and phenomena. The volume gives a closer look at the scope and the nature of diagrams as constituents of mathematical and physical thought, their function in contemporary artistic work, and appraise, in particular, the actual importance of the diagrams of knots, of braids, of fields, of interaction, of strings in topology and geometry, in quantum physics and in cosmology, but also in theory of perception, in plastic arts and in philosophy. The editors carefully curated this volume to be an inspiration to students and researchers in philosophy, phenomenology, mathematics and the sciences, as well as artists, musicians and the general interested audience.

In questo libro matematica e arte si incontrano, confrontando ed esplorando tre oggetti-concetti: nodi, buchi e spazi. Oggetti fisici e concetti astratti allo stesso tempo, i nodi e i buchi sono strumenti di creazione e generatori di forme: si intrecciano e si compenetrano su piu piani, ridefinendo le configurazioni e le dinamiche dello spazio.

Anche se generalmente non ne siamo consapevoli, nodi e buchi sono onnipresenti, pervadono la nostra vita quotidiana e noi stessi. Queste forme danno luogo a una rigenerazione continua della realta e partecipano al senso del vero e del bello che avvertiamo nella ricerca incessante per capire il mondo che ci circonda. Nodi e buchi costituiscono elementi fondamentali dei molteplici spazi studiati dai matematici e dai fisici teorici. Eppure non e facile, neanche per un matematico, pensare un nodo, un buco, gli spazi a n-dimensioni o l infinito.

Accanto a riflessioni matematiche, filosofiche e letterarie, il libro propone e indaga opere d arte di diverse epoche. In particolare, due grandi artisti Jorge Eielson e Lucio Fontana sono riusciti nell intento di pensare l impossibile, di rendere visibile l invisibile, di immaginare l infinito al di la delle immagini finite e apparenti.

Il nodo esprime dunque un percorso interiore, dove analisi e intuizione si incontrano e si fondono in un atto di creazione, ma rappresenta anche un percorso concreto che si svolge nello spazio e nel tempo e lungo il quale culture, saperi e persone si legano per costruire un dialogo infinito.

Questo libro e un invito a entrare nel mondo meraviglioso dei nodi e dei buchi, sorgenti di conoscenza, principi di vita, fonti di desiderio, ragioni di eresia.

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This volume focuses on the interactions between mathematics, physics, biology and neuroscience by exploring new geometrical and topological modeling in these fields. Among the highlights are the central roles played by multilevel and scale-change approaches in these disciplines.

The integration of mathematics with physics, molecular and cell biology, and the neurosciences, will constitute the new frontier and challenge for 21st century science, where breakthroughs are more likely to span across traditional disciplines.