This book traces the development of formal logic from its origins in ancient Greece to the present day. The authors first discuss the work of logicians from Aristotle to Frege, showing how they were influenced by the philosophical or mathematical ideas of their time. They then examine developments in the present century.
In Plato's Gorgias, Gorgias of Leontini, a famous teacher of rhetoric, has come to Athens to recruit students, promising to teach them how to become leaders in politics and business. A group has gathered at Callicles' house to hear Gorgias demonstrate the power of his art. This dialogue blends comic and serious discussion of the best human life, providing a penetrating examination of ethics, the foundations of knowledge, and the nature of the good.
The most perplexing aspect of Galileo's work in physics is without doubt the sharp distinction one can draw between his essentially dynamic studies in such juvenilia as De Motu and the consciously kinematical approach of his later output—particularly the Two New Sciences. Whether one chooses to call this a shift from the “why” of motion to the “how”, or, as I should prefer, a shift from dynamics to kinematics, there can be no denying its existence. The Galileo who wrote that (...) “The present does not seem to be the proper time to investigate the cause of the acceleration of natural motion …” is, on the face of it, a very different man from the one who had earlier written almost an entire treatise on precisely this topic. (shrink)
If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segré classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector (...) formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity. (shrink)
The lexicographic power ΔΓ of chains Δ and Γ is, roughly, the Cartesian power Πγ∈Γ Δ, totally ordered lexicographically from the left. Here the focus is on certain powers in which either Δ = R or Γ = R, with emphasis on when two such powers are isomorphic and on when ΔΓ is 2-homogeneous. The main results are: (1) For a countably infinite ordinal α, Rα* +α ≃ Rα. (2) RR ≄ RQ. (3) For Δ a countable ordinal ≥ 2. (...) ΔR, with its smallest element deleted, is 2-homogeneous. (shrink)
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