Hermann Weyl was one of the twentieth century's most important mathematicians, as well as a seminal figure in the development of quantum physics and general relativity. He was also an eloquent writer with a lifelong interest in the philosophical implications of the startling new scientific developments with which he was so involved. Mind and Nature is a collection of Weyl's most important general writings on philosophy, mathematics, and physics, including pieces that have never before been published in any language or (...) translated into English, or that have long been out of print. Complete with Peter Pesic's introduction, notes, and bibliography, these writings reveal an unjustly neglected dimension of a complex and fascinating thinker. In addition, the book includes more than twenty photographs of Weyl and his family and colleagues, many of which are previously unpublished. Included here are Weyl's exposition of his important synthesis of electromagnetism and gravitation, which Einstein at first hailed as "a first-class stroke of genius"; two little-known letters by Weyl and Einstein from 1922 that give their contrasting views on the philosophical implications of modern physics; and an essay on time that contains Weyl's argument that the past is never completed and the present is not a point. Also included are two book-length series of lectures, The Open World and Mind and Nature, each a masterly exposition of Weyl's views on a range of topics from modern physics and mathematics. Finally, four retrospective essays from Weyl's last decade give his final thoughts on the interrelations among mathematics, philosophy, and physics, intertwined with reflections on the course of his rich life. (shrink)

An exploration of the relationship between quantum theory and concepts of individuality and identity from ancient Greece to the present.

Gregory Chaitin has done seminal work on the foundations of mathematics, especially the meaning of randomness and undecidability. In Meta Maths, he offers his ideas in a new popular version, which has a special interest because it comes directly from their originator. Long associated with the IBM Watson Research Center, Chaitin comes across as a kind of mathematical Richard Feynman, intuitive and high-spirited, irreverent and plain-spoken. Through this jovial persona, he presents many serious ideas in an engagingly dishevelled way, mixing (...) autobiography and reminiscence with exposition and example. He presents his insights so agreeably that it all seems a grand lark. (shrink)

ABSTRACT Because the modern concept of number emerged within a quadrivium that included music alongside arithmetic, geometry, and astronomy, musical considerations affected mathematical developments. Michael Stifel embedded the then‐paradoxical term “irrational numbers” (numerici irrationales) in a musical context (1544), though his philosophical aversion to the “cloud of infinity” surrounding such numbers finally outweighed his musical arguments in their favor. Girolamo Cardano gave the same status to irrational and rational quantities in his algebra (1545), for which his contemporaneous work on music (...) suggested parallels and empirical examples. Nicola Vicentino's attempt to revive ancient “enharmonic” music (1555) required and hence defended the use of “irrational proportions” (proportiones inrationales) as if they were numbers. These developments emerged in richly interactive social and cultural milieus whose participants interwove musical and mathematical interests so closely that their intense controversies about ancient Greek music had repercussions for mathematics as well. The musical interests of Stifel, Cardano, and Vicentino influenced their respective treatments of “irrational numbers.” Practical as well as theoretical music both invited and opened the way for the recognition of a radically new concept of number, even in the teeth of paradox. (shrink)

ABSTRACT Because the modern concept of number emerged within a quadrivium that included music alongside arithmetic, geometry, and astronomy, musical considerations affected mathematical developments. Michael Stifel embedded the then‐paradoxical term “irrational numbers” (numerici irrationales) in a musical context (1544), though his philosophical aversion to the “cloud of infinity” surrounding such numbers finally outweighed his musical arguments in their favor. Girolamo Cardano gave the same status to irrational and rational quantities in his algebra (1545), for which his contemporaneous work on music (...) suggested parallels and empirical examples. Nicola Vicentino's attempt to revive ancient “enharmonic” music (1555) required and hence defended the use of “irrational proportions” (proportiones inrationales) as if they were numbers. These developments emerged in richly interactive social and cultural milieus whose participants interwove musical and mathematical interests so closely that their intense controversies about ancient Greek music had repercussions for mathematics as well. The musical interests of Stifel, Cardano, and Vicentino influenced their respective treatments of “irrational numbers.” Practical as well as theoretical music both invited and opened the way for the recognition of a radically new concept of number, even in the teeth of paradox. (shrink)

Aiming to provide an accessible introduction to critical changes in the biosphere that have occurred since the origins of life, this text presents an integrated view of our planet's evolution. Fifteen scientists reflect on major events in the history of the Earth.

Leibniz diskutierte den Nutzen der gerichtlichen Folter, obwohl er von ihr abgestoßen war. Er betrachtete Folter als eine Ermittlungsmethode, die den höchsten Grad an Sicherheit erreichen kann und nutzte sie metaphorisch, um analoge Praktiken in der experimentellen Wissenschaft zu beschreiben. Er befiirwortete jedoch nicht den Mißbrauch der Natur, sondern eher eine unübertreffliche Methode, die die Natur dazu bringt, zu 'gestehen', ohne sie dauerhaft zu schädigen.

Greek mathematics did not know the zero. Without denying this commonplace observation, I would like to explore ways in which Plato’s Theaetetus and Sophist may be read as reaching towards the concept of the zero, if not towards anything like its Hindu-Arabic symbol. This may help connect Plato’s reflections on mathematics with his arguments about Nonbeing.

We reexamine Plato’s use of visual metaphors by considering his own treatment of light and sight, which differed from the later view that the eye is purely passive. Instead, he considered the eye to be active, sending out beams that contact the “outer fire” and then return to the seer. This essential activity of the act of seeing changes the way we should read many passages in Plato based on metaphors of vision, in order to bring forward the fundamental activity (...) of the soul in knowing. Understood in this way, Plato’s words eidos and idea may refer not only to self-subsistent, transcendent “forms” but also to the active process of “seeing” that connects us with them. (shrink)