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Philosophers
Mortimer Adler Rogers Albritton Alexander of Aphrodisias G.E.M.Anscombe Anselm Thomas Aquinas Aristotle David Armstrong Augustine J.L.Austin A.J.Ayer Alexander Bain Mark Balaguer William Belsham Henri Bergson Isaiah Berlin Bernard Berofsky Susanne Bobzien Emil du Bois-Reymond George Boole Émile Boutroux F.H.Bradley C.D.Broad C.A.Campbell Joseph Keim Campbell Carneades Ernst Cassirer Roderick Chisholm Chrysippus Cicero Randolph Clarke Samuel Clarke Anthony Collins Diodorus Cronus Donald Davidson Democritus Daniel Dennett René Descartes Richard Double Fred Dretske John Earman Laura Waddell Ekstrom Epictetus Epicurus Herbert Feigl John Martin Fischer Owen Flanagan Luciano Floridi Philippa Foot Alfred Fouilleé Harry Frankfurt Richard L. Franklin Michael Frede Carl Ginet Nicholas St. John Green H.Paul Grice Ian Hacking Ishtiyaque Haji Stuart Hampshire W.F.R.Hardie R.M.Hare Georg W.F. Hegel Martin Heidegger R.E.Hobart Thomas Hobbes David Hodgson Shadsworth Hodgson Ted Honderich Pamela Huby David Hume Ferenc Huoranszki William James Lord Kames Robert Kane Immanuel Kant Tomis Kapitan William King Christine Korsgaard Keith Lehrer Gottfried Leibniz Leucippus Michael Levin C.I.Lewis David Lewis Peter Lipton John Locke Michael Lockwood John R. Lucas Lucretius James Martineau Hugh McCann Colin McGinn Michael McKenna Paul E. Meehl Alfred Mele John Stuart Mill Dickinson Miller G.E.Moore Thomas Nagel Friedrich Nietzsche P.H.Nowell-Smith Robert Nozick William of Ockham Timothy O'Connor David F. Pears Charles Sanders Peirce Derk Pereboom Steven Pinker Plato Karl Popper H.A.Prichard Hilary Putnam Willard van Orman Quine Frank Ramsey Ayn Rand Thomas Reid Charles Renouvier Nicholas Rescher C.W.Rietdijk Josiah Royce Bertrand Russell Paul Russell Gilbert Ryle T.M.Scanlon Moritz Schlick Arthur Schopenhauer John Searle Wilfrid Sellars Henry Sidgwick Walter Sinnott-Armstrong J.J.C.Smart Saul Smilansky Michael Smith L. Susan Stebbing George F. Stout Galen Strawson Peter Strawson Eleonore Stump Richard Taylor Kevin Timpe Peter van Inwagen Manuel Vargas John Venn Kadri Vihvelin Voltaire G.H. von Wright David Foster Wallace R. Jay Wallace W.G.Ward Ted Warfield Roy Weatherford Alfred North Whitehead David Widerker David Wiggins Bernard Williams Ludwig Wittgenstein Susan Wolf Scientists Michael Arbib Bernard Baars John S. Bell Charles Bennett Margaret Boden David Bohm Neils Bohr Ludwig Boltzmann Emile Borel Max Born Leon Brillouin Stephen Brush Henry Thomas Buckle Donald Campbell Anthony Cashmore Eric Chaisson Jean-Pierre Changeux Arthur Holly Compton John Conway E. H. Culverwell Charles Darwin Abraham de Moivre Paul Dirac John Eccles Arthur Stanley Eddington Paul Ehrenfest Albert Einstein Richard Feynman Joseph Fourier Michael Gazzaniga GianCarlo Ghirardi Nicolas Gisin Thomas Gold A.O.Gomes Joshua Greene Jacques Hadamard Patrick Haggard Augustin Hamon Sam Harris Martin Heisenberg Werner Heisenberg William Stanley Jevons Pascual Jordan Simon Kochen Stephen Kosslyn Rolf Landauer Alfred Landé Pierre-Simon Laplace David Layzer Benjamin Libet Josef Loschmidt Ernst Mach Henry Margenau James Clerk Maxwell Ernst Mayr Jacques Monod Roger Penrose Steven Pinker Max Planck Henri Poincaré Adolphe Quételet Jerome Rothstein Erwin Schrödinger Claude Shannon Herbert Simon Dean Keith Simonton B. F. Skinner Henry Stapp Antoine Suarez Leo Szilard William Thomson (Kelvin) John von Neumann Daniel Wegner Steven Weinberg Norbert Wiener Eugene Wigner E. O. Wilson Ernst Zermelo |
The Recurrence Problem
The idea that the macroscopic conditions in the world will repeat after some interval of time is an ancient idea, but it plays a vital role in modern physics as well.
Ancient middle eastern civilizations called it the Great Year. They calculated it as the time after which the planets would realign themselves in identical positions in the sky.
The Great Year should not be confused with the time that the precession of the equinoxes takes to return the equinoxes to the same position along the Zodiac - although this time (about 26,000 years) is of the same order of magnitude as one famous number given by Babylonian astronomers for the Great Year (36,000 years).
Many societies have the concept of the Great Year, but none did calculations as carefully as the Babylonians. But since the planets orbital periods are not really commensurate, they kept increasing the time for the Great Year in the search for a better recurrence time.
The Greek and Roman Stoics thought the Great Year was a sign of the rule of law in nature and the God of reason that lay behind nature.
Nietsche's Eternal Return
In modern philosophy, Friedrich Nietzsche described an eternal return in his Also Sprach Zarathustra.
Zermelo's Paradox
Zermelo's paradox was a criticism of Ludwig Boltzmann's H-Theorem, the attempt to derive the increasing entropy required by the second law of thermodynamics from basic classical dynamics.
It was the second "paradox" attack on Boltzmann. The first was Josef Loschmidt's claim that entropy would be reduced if time were reversed. This is the problem of microscopic reversibility.
Ernst Zermelo was an extraordinary mathematician. He was (in 1908) the founder of axiomatic set theory, which with the addition of the axiom of choice (also by Zermelo, in 1904) is the most common foundation of mathematics. The axiom of choice says that given any collection of sets, one can find a way to unambiguously select one object from each set, even if the number of sets is infinite.
Before this amazing work, Zermelo was a young associate of Max Planck in Berlin, one of many German physicists who opposed the work of Boltzmann to establish the existence of atoms.
Zermelo's criticism was based on the work of Henri Poincaré, an expert in the three-body problem, which, unlike the problem of two particles, has no exact analytic solution.
Poincaré had been able to establish limits or bounds on the possible configurations of the three bodies from conservation laws. Planck and Zermelo applied some of Poincaré's thinking to the n particles in a gas. They argued that given a long enough time, the particles would return to a distribution in "phase space" (a 6n dimensional space of possible velocities and positions) that would be indistinguishable from the original distribution.
Thus, they argued, Boltzmann's formula for the entropy would at some future time go back down, vitiating Boltzmann's claim that his measure of entropy always increases - as the second law of thermodynamics requires.
Boltzmann replied that his argument was statistical. He only claimed that entropy increase was overwhelmingly more probable than Zermelo's predicted decrease. Boltzmann calculated the probability of a decrease of a very small gas of only a few hundred particles and found the time needed to realize such a decrease was many orders of magnitude larger than the presumed age of the universe.
The idea that a macroscopic system can return to exactly the same physical conditions is closely related to the idea that an agent may face "exactly the same circumstances in making a decision. Determinists maintain that given the "fixed past" and the "laws of nature" that the agent would have to make exactly the same decision again.
The Extreme Improbabilty of Perfect Recurrence
In a classical universe, such as that of Laplace, where information is constant, Zermelo's recurrence is mathematically possible. Given enough time, the universe can return to the exact circumstance of any earlier instant of time, because it contains the same amount of matter, energy, and information.
But, in the real universe, information (and the material content of the universe) expands from a minimum at the origin, to ever larger amounts of information. Consequently, it is statistically and realistically improbable (if not impossible) for the universe as a whole to return to exactly the same circumstance of any earlier time.
By accepting the theory of the expanding universe we are relieved of one conclusion which we had felt to be intrinsically absurd. It was argued that every possible configuration of atoms must repeat itself at some distant date. But that was on the assumption that the atoms will have only the same choice of configurations in the future that they have now. In an expanding space any particular congruence becomes more and more improbable. The expansion of the universe creates new possibilities of distribution faster than the atoms can work through them, and there is no longer any likelihood of a particular distribution being repeated. If we continue shuffling a pack of cards we are bound sometime to bring them into their standard order — but not if the conditions are that every morning one more card is added to the pack.H. Dieter Zeh also saw that the age of the universe being much less than the Poincaré recurrence time may invalidate the recurrence objection. Another argument against the statistical interpretation of irreversibility, the recurrence objection (or Wiederkehreinwand), was raised much later by Ernst Friedrich Zermelo, a collaborator of Max Planck at a time when the latter still opposed atomism, and instead supported the 'energeticists', who attempted to understand energy and entropy as fundamental 'substances'. This argument is based on a mathematical theorem due to Henri Poincaré, which states that every bounded mechanical system will return as close as one wishes to its initial state within a sufficiently large time. The entropy of a closed system would therefore have to return to its former value, provided only the function F(z) is continuous. This is a special case of the quasiergodic theorem which asserts that every system will corne arbitrarily close to any point on the hypersurface of fixed energy (and possibly with other fixed analytical constants of the motion) within finite time. While all these theorems are mathematically correct, the recurrence objection fails to apply to reality for quantitative reasons. The age of our Universe is much smaller than the Poincaré recurrence times even for a gas consisting of no more than a few tens of particles. Their recurrence to the vicinity of their initial states (or their coming close to any other similarly specific state) can therefore be excluded in practice. Nonetheless, some .'foundations' of irreversible thermodynamics in the literature rely on formal idealizations that would lead to strictly infinite Poincaré recurrence times (for example the 'thermodynamical limit' of infinite particle number). Such assumptions are not required in our Universe of finite age, and they would not invalidate the reversibility objection (or the equilibrium expectation, mentioned above). However, all foundations of irreversible behavior have to presume some very improbable initial conditions...In order to reverse the thermodynamical arrow of time in a bounded system, it would not therefore suffice to "go ahead and reverse all momenta" in the system itself, as ironically suggested by Boltzmann as an answer to Loschmidt. In an interacting Laplacean universe, the Poincaré cycles of its subsystems could in general only be those of the whole Universe, since their exact Hamiltonians must always depend on their time-dependent environment.
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