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Philosophers

Mortimer Adler
Rogers Albritton
Alexander of Aphrodisias
Samuel Alexander
William Alston
Anaximander
G.E.M.Anscombe
Anselm
Louise Antony
Thomas Aquinas
Aristotle
David Armstrong
Harald Atmanspacher
Robert Audi
Augustine
J.L.Austin
A.J.Ayer
Alexander Bain
Mark Balaguer
Jeffrey Barrett
William Barrett
William Belsham
Henri Bergson
George Berkeley
Isaiah Berlin
Richard J. Bernstein
Bernard Berofsky
Robert Bishop
Max Black
Susanne Bobzien
Emil du Bois-Reymond
Hilary Bok
Laurence BonJour
George Boole
Émile Boutroux
Daniel Boyd
F.H.Bradley
C.D.Broad
Michael Burke
Lawrence Cahoone
C.A.Campbell
Joseph Keim Campbell
Rudolf Carnap
Carneades
Nancy Cartwright
Gregg Caruso
Ernst Cassirer
David Chalmers
Roderick Chisholm
Chrysippus
Cicero
Randolph Clarke
Samuel Clarke
Anthony Collins
Antonella Corradini
Diodorus Cronus
Jonathan Dancy
Donald Davidson
Mario De Caro
Democritus
Daniel Dennett
Jacques Derrida
René Descartes
Richard Double
Fred Dretske
John Dupré
John Earman
Laura Waddell Ekstrom
Epictetus
Epicurus
Austin Farrer
Herbert Feigl
Arthur Fine
John Martin Fischer
Frederic Fitch
Owen Flanagan
Luciano Floridi
Philippa Foot
Alfred Fouilleé
Harry Frankfurt
Richard L. Franklin
Bas van Fraassen
Michael Frede
Gottlob Frege
Peter Geach
Edmund Gettier
Carl Ginet
Alvin Goldman
Gorgias
Nicholas St. John Green
H.Paul Grice
Ian Hacking
Ishtiyaque Haji
Stuart Hampshire
W.F.R.Hardie
Sam Harris
William Hasker
R.M.Hare
Georg W.F. Hegel
Martin Heidegger
Heraclitus
R.E.Hobart
Thomas Hobbes
David Hodgson
Shadsworth Hodgson
Baron d'Holbach
Ted Honderich
Pamela Huby
David Hume
Ferenc Huoranszki
Frank Jackson
William James
Lord Kames
Robert Kane
Immanuel Kant
Tomis Kapitan
Walter Kaufmann
Jaegwon Kim
William King
Hilary Kornblith
Christine Korsgaard
Saul Kripke
Thomas Kuhn
Andrea Lavazza
Christoph Lehner
Keith Lehrer
Gottfried Leibniz
Jules Lequyer
Leucippus
Michael Levin
Joseph Levine
George Henry Lewes
C.I.Lewis
David Lewis
Peter Lipton
C. Lloyd Morgan
John Locke
Michael Lockwood
Arthur O. Lovejoy
E. Jonathan Lowe
John R. Lucas
Lucretius
Alasdair MacIntyre
Ruth Barcan Marcus
Tim Maudlin
James Martineau
Nicholas Maxwell
Storrs McCall
Hugh McCann
Colin McGinn
Michael McKenna
Brian McLaughlin
John McTaggart
Paul E. Meehl
Uwe Meixner
Alfred Mele
Trenton Merricks
John Stuart Mill
Dickinson Miller
G.E.Moore
Thomas Nagel
Otto Neurath
Friedrich Nietzsche
John Norton
P.H.Nowell-Smith
Robert Nozick
William of Ockham
Timothy O'Connor
Parmenides
David F. Pears
Charles Sanders Peirce
Derk Pereboom
Steven Pinker
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Karl Popper
Porphyry
Huw Price
H.A.Prichard
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Willard van Orman Quine
Frank Ramsey
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Michael Rea
Thomas Reid
Charles Renouvier
Nicholas Rescher
C.W.Rietdijk
Richard Rorty
Josiah Royce
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Paul Russell
Gilbert Ryle
Jean-Paul Sartre
Kenneth Sayre
T.M.Scanlon
Moritz Schlick
Arthur Schopenhauer
John Searle
Wilfrid Sellars
Alan Sidelle
Ted Sider
Henry Sidgwick
Walter Sinnott-Armstrong
J.J.C.Smart
Saul Smilansky
Michael Smith
Baruch Spinoza
L. Susan Stebbing
Isabelle Stengers
George F. Stout
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Peter Strawson
Eleonore Stump
Francisco Suárez
Richard Taylor
Kevin Timpe
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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
C.F. von Weizsäcker
William Whewell
Alfred North Whitehead
David Widerker
David Wiggins
Bernard Williams
Timothy Williamson
Ludwig Wittgenstein
Susan Wolf

Scientists

David Albert
Michael Arbib
Walter Baade
Bernard Baars
Jeffrey Bada
Leslie Ballentine
Marcello Barbieri
Gregory Bateson
Horace Barlow
John S. Bell
Mara Beller
Charles Bennett
Ludwig von Bertalanffy
Susan Blackmore
Margaret Boden
David Bohm
Niels Bohr
Ludwig Boltzmann
Emile Borel
Max Born
Satyendra Nath Bose
Walther Bothe
Jean Bricmont
Hans Briegel
Leon Brillouin
Stephen Brush
Henry Thomas Buckle
S. H. Burbury
Melvin Calvin
Donald Campbell
Sadi Carnot
Anthony Cashmore
Eric Chaisson
Gregory Chaitin
Jean-Pierre Changeux
Rudolf Clausius
Arthur Holly Compton
John Conway
Jerry Coyne
John Cramer
Francis Crick
E. P. Culverwell
Antonio Damasio
Olivier Darrigol
Charles Darwin
Richard Dawkins
Terrence Deacon
Lüder Deecke
Richard Dedekind
Louis de Broglie
Stanislas Dehaene
Max Delbrück
Abraham de Moivre
Bernard d'Espagnat
Paul Dirac
Hans Driesch
John Eccles
Arthur Stanley Eddington
Gerald Edelman
Paul Ehrenfest
Manfred Eigen
Albert Einstein
George F. R. Ellis
Hugh Everett, III
Franz Exner
Richard Feynman
R. A. Fisher
David Foster
Joseph Fourier
Philipp Frank
Steven Frautschi
Edward Fredkin
Benjamin Gal-Or
Howard Gardner
Lila Gatlin
Michael Gazzaniga
Nicholas Georgescu-Roegen
GianCarlo Ghirardi
J. Willard Gibbs
James J. Gibson
Nicolas Gisin
Paul Glimcher
Thomas Gold
A. O. Gomes
Brian Goodwin
Joshua Greene
Dirk ter Haar
Jacques Hadamard
Mark Hadley
Patrick Haggard
J. B. S. Haldane
Stuart Hameroff
Augustin Hamon
Sam Harris
Ralph Hartley
Hyman Hartman
Jeff Hawkins
John-Dylan Haynes
Donald Hebb
Martin Heisenberg
Werner Heisenberg
John Herschel
Basil Hiley
Art Hobson
Jesper Hoffmeyer
Don Howard
John H. Jackson
William Stanley Jevons
Roman Jakobson
E. T. Jaynes
Pascual Jordan
Eric Kandel
Ruth E. Kastner
Stuart Kauffman
Martin J. Klein
William R. Klemm
Christof Koch
Simon Kochen
Hans Kornhuber
Stephen Kosslyn
Daniel Koshland
Ladislav Kovàč
Leopold Kronecker
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
Karl Lashley
David Layzer
Joseph LeDoux
Gerald Lettvin
Gilbert Lewis
Benjamin Libet
David Lindley
Seth Lloyd
Hendrik Lorentz
Werner Loewenstein
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
Owen Maroney
David Marr
Humberto Maturana
James Clerk Maxwell
Ernst Mayr
John McCarthy
Warren McCulloch
N. David Mermin
George Miller
Stanley Miller
Ulrich Mohrhoff
Jacques Monod
Vernon Mountcastle
Emmy Noether
Donald Norman
Alexander Oparin
Abraham Pais
Howard Pattee
Wolfgang Pauli
Massimo Pauri
Wilder Penfield
Roger Penrose
Steven Pinker
Colin Pittendrigh
Walter Pitts
Max Planck
Susan Pockett
Henri Poincaré
Daniel Pollen
Ilya Prigogine
Hans Primas
Zenon Pylyshyn
Henry Quastler
Adolphe Quételet
Pasco Rakic
Nicolas Rashevsky
Lord Rayleigh
Frederick Reif
Jürgen Renn
Giacomo Rizzolati
Emil Roduner
Juan Roederer
Jerome Rothstein
David Ruelle
David Rumelhart
Tilman Sauer
Ferdinand de Saussure
Jürgen Schmidhuber
Erwin Schrödinger
Aaron Schurger
Sebastian Seung
Thomas Sebeok
Franco Selleri
Claude Shannon
Charles Sherrington
David Shiang
Abner Shimony
Herbert Simon
Dean Keith Simonton
Edmund Sinnott
B. F. Skinner
Lee Smolin
Ray Solomonoff
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
Max Tegmark
Teilhard de Chardin
Libb Thims
William Thomson (Kelvin)
Richard Tolman
Giulio Tononi
Peter Tse
Alan Turing
Francisco Varela
Vlatko Vedral
Mikhail Volkenstein
Heinz von Foerster
Richard von Mises
John von Neumann
Jakob von Uexküll
C. S. Unnikrishnan
C. H. Waddington
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
Herman Weyl
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
Günther Witzany
Stephen Wolfram
H. Dieter Zeh
Semir Zeki
Ernst Zermelo
Wojciech Zurek
Konrad Zuse
Fritz Zwicky

Presentations

Biosemiotics
Free Will
Mental Causation
James Symposium
 
The Ergodic Hypothesis
Ludwig Boltzmann was criticized for his 1872 attempt to prove his H-theorem (that entropy always increases) by a dynamical analysis of molecular collisions. Josef Loschmidt and others pointed out that if the molecular velocities were to be reversed at an instant, Boltzmann's work would show that the entropy should decrease. This was the reversibility objection.

Entropy as Lost Information about Molecular Positions
Entropy increase can be easily understood as the loss of information as a system moves from an initially ordered state to a final disordered state. Ludwig Boltzmann was the first to describe entropy as "missing information."
Dr. Shannon's work roots back, as von Neumann has pointed out, to Boltzmann's observation, in some of his work on statistical physics (1894), that entropy is related to "missing information," inasmuch as it is related to the number of alternatives which remain possible to a physical system after all the macroscopically observable information concerning it has been recorded. L. Szilard (Zsch. f. Phys. Vol. 53, 1925) extended this idea to a general discussion of information in physics, and von Neumann (Math. Foundation of Quantum Mechanics, Berlin, 1932, Chap. V) treated information in quantum mechanics and particle physics.
(The Mathematical Theory of Information, Claude Shannon and Warren Weaver, p. 3n)

Although the physical dimensions of thermodynamic entropy (joules/ºK) are not the same as (dimensionless) mathematical information, apart from units they share the same famous formula.

S = ∑ pi ln pi
To see this very simply, let's consider the well-known example of a bottle of perfume in the corner of a room. We can represent the room as a grid of 64 squares. Suppose the air is filled with molecules moving randomly at room temperature (blue circles). In the lower left corner a small number of (red) perfume molecules will be released when we open the bottle (when you start the demonstration animation below).

What is the quantity of information we have about the perfume molecules? At the start we know their location in the lower left square, a bit less than 1/64th of the container. The quantity of information is determined by the minimum number of yes/no questions it takes to locate them. The best questions are those that split the locations evenly (a binary tree).

For example:

  • Are they in the upper half of the container? No.
  • Are they in the left half of the container? Yes.
  • Are they in the upper half of the lower left quadrant? No.
  • Are they in the left half of the lower left quadrant? Yes.
  • Are they in the upper half of the lower left octant? No.
  • Are they in the left half of the lower left octant? Yes.
Answers to these six optimized questions give us six bits of information for each molecule, locating it to 1/64th of the container. This is the amount of information that will be lost for each molecule if it is allowed to escape and diffuse fully into the room. The thermodynamic entropy increase is Boltzmann's constant k multiplied by the number of bits.

If the room had no air, the perfume would rapidly reach an equilibrium state, since the molecular velocity at room temperature is about 400 meters/second. Collisions with air molecules prevent the perfume from dissipating quickly. This lets us see the approach to equilibrium. When the perfume has diffused to one-sixteenth of the room, the entropy will have risen 2 bits for each molecule, to one-quarter of the room, four bits, etc.

Click here to start a computer visualization of the equilibration process in a new window.

Entropy as Evolution to the Most Probable Macrostate

In 1877, Boltzmann simply ignored classical dynamics and instead made the assumption that all phase space cells were equally probable. Classical dynamics could not prove that the path of the system in phase space would move through all the cells, let alone spend equal time in all cells. Boltzmann described a system he called "ergode," later called the canonical ensemble by J. Willard Gibbs. Equal a priori probabilities for all the phase space cells came to be called the ergodic hypothesis.

Paul and Tatiana Ehrenfest made the ergodic hypothesis the central question in statistical mechanics. Mathematicians took up the problem of ergodicity in continuous mathematics, which has questionable relevance for problems in discrete particle physics.

In modern quantum statistical mechanics, the same ergodic hypothesis (equiprobability of phase space cells) shows up in an assumption about transition probabilities between phase space cells. The transition probability for any microstate A to jump to microstate B is assumed to be the same as the reverse quantum jump from B to A.

The matrix element for the A - B transition is the complex conjugate of the reverse transition B - A. This is called Fermi's Golden Rule, although it was first derived by Paul Dirac.

We can see how any system with equal transition probabilities to and from any other state will quickly establish equilibrium populations. If 1000 systems are in state A and none in B, the early transitions will overwhelmingly be from A to B. An equal number of transitions back from B to A is not likely until the populations of A and B are about the same.

That is the basic idea behind the statistical formulation of Boltzmann's H-theorem. When all phase space cells are equally populated, the number of ways this can be achieved (the number of microstates) is at its maximum. Although cell populations will fluctuate away from this equilbrium condition, it corresponds to the maximum entropy.

Number of systems
Number of cycles
Transition Probability

The initial distribution of 500 systems in the upper left corner evolves rapidly to the normal distribution function for occupation numbers
Normal | Teacher | Scholar