Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

While physical objects exist, they occupy space. We have developed a representation for the spaces that objects can occupy that meets the following criteria.

While physical objects exist, they occupy space. We have developed a representation for the spaces that objects can occupy that meets the following criteria.

Absolutely no one still believes that every physical interactionconsists of material bodies bumping into each other. Those who have tried to work out a completely mechanistic physics have been unable to explain common phenomena like liquidity, gravitation and magnetism. In fact, there is great reason to doubt that such a physics could ever account for attractive forces in general.

Absolutely no one still believes that every physical interactionconsists of material bodies bumping into each other. Those who have tried to work out a completely mechanistic physics have been unable to explain common phenomena like liquidity, gravitation and magnetism. In fact, there is great reason to doubt that such a physics could ever account for attractive forces in general.

_Paradoxes from A to Z, Third edition_ is the essential guide to paradoxes, and takes the reader on a lively tour of puzzles that have taxed thinkers from Zeno to Galileo, and Lewis Carroll to Bertrand Russell. Michael Clark uncovers an array of conundrums, such as Achilles and the Tortoise, Theseus’ Ship, and the Prisoner’s Dilemma, taking in subjects as diverse as knowledge, science, art and politics. Clark discusses each paradox in non-technical terms, considering its significance and looking at likely (...) solutions. This third edition is revised throughout, and adds nine new paradoxes that have important bearings in areas such as law, logic, ethics and probability. Paradoxes from A to Z, Third edition is an ideal starting point for those interested not just in philosophical puzzles and conundrums, but anyone seeking to hone their thinking skills. (shrink)

This book combines physics, history, and philosophy in a radical new approach to introducing the philosophy of physics.

Determinism is a perennial topic of philosophical discussion. Very little acquaintance with the philosophical literature is needed to reveal the Tower of ...