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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
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
James Martineau
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
Plato
Karl Popper
Porphyry
Huw Price
H.A.Prichard
Protagoras
Hilary Putnam
Willard van Orman Quine
Frank Ramsey
Ayn Rand
Michael Rea
Thomas Reid
Charles Renouvier
Nicholas Rescher
C.W.Rietdijk
Richard Rorty
Josiah Royce
Bertrand Russell
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
Galen Strawson
Peter Strawson
Eleonore Stump
Francisco Suárez
Richard Taylor
Kevin Timpe
Mark Twain
Peter Unger
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
Gregory Bateson
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
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
Lila Gatlin
Michael Gazzaniga
Nicholas Georgescu-Roegen
GianCarlo Ghirardi
J. Willard Gibbs
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
John-Dylan Haynes
Donald Hebb
Martin Heisenberg
Werner Heisenberg
John Herschel
Basil Hiley
Art Hobson
Jesper Hoffmeyer
Don Howard
William Stanley Jevons
Roman Jakobson
E. T. Jaynes
Pascual Jordan
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
David Layzer
Joseph LeDoux
Gilbert Lewis
Benjamin Libet
David Lindley
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
Owen Maroney
Humberto Maturana
James Clerk Maxwell
Ernst Mayr
John McCarthy
Warren McCulloch
N. David Mermin
George Miller
Stanley Miller
Ulrich Mohrhoff
Jacques Monod
Emmy Noether
Alexander Oparin
Abraham Pais
Howard Pattee
Wolfgang Pauli
Massimo Pauri
Roger Penrose
Steven Pinker
Colin Pittendrigh
Max Planck
Susan Pockett
Henri Poincaré
Daniel Pollen
Ilya Prigogine
Hans Primas
Henry Quastler
Adolphe Quételet
Lord Rayleigh
Jürgen Renn
Juan Roederer
Jerome Rothstein
David Ruelle
Tilman Sauer
Jürgen Schmidhuber
Erwin Schrödinger
Aaron Schurger
Sebastian Seung
Thomas Sebeok
Claude Shannon
David Shiang
Abner Shimony
Herbert Simon
Dean Keith Simonton
B. F. Skinner
Lee Smolin
Ray Solomonoff
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
Max Tegmark
Libb Thims
William Thomson (Kelvin)
Giulio Tononi
Peter Tse
Francisco Varela
Vlatko Vedral
Mikhail Volkenstein
Heinz von Foerster
Richard von Mises
John von Neumann
Jakob von Uexküll
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
Herman Weyl
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
Stephen Wolfram
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek
Konrad Zuse
Fritz Zwicky

Presentations

Biosemiotics
Free Will
Mental Causation
James Symposium
 
E. T. Jaynes

Edwin Thompson Jaynes extended statistical mechanics to connect it to probability theory, Claude Shannon's information theory, and Bayesian statistical inferences.

He championed the work of J. Willard Gibbs, contrasting it to the earlier work of Ludwig Boltzmann.

His 1957 "principle of maximum entropy" or "maxent" says that the probability distribution that best represents the current state of knowledge is the one with largest entropy.

In 1964, Jaynes examined the difference between the Boltzmann and Gibbs formulations of the entropy. They differ, he says, because of different treatments of "interparticle forces."

The status of the Gibbs and Boltzmann expressions for entropy has been a matter of some confusion in the literature. We show that:

(1) the Gibbs H-function yields the correct entropy as defined in phenomenological thermodynamics;

(2) the Boltzmann H yields an "entropy" that is in error by a nonnegligible amount whenever interparticle forces affect thermodynamic properties;

(3) Boltzmann's other interpretation of entropy, S = k log W, is consistent with the Gibbs H, and derivable from it;

(4) the Boltzmann H theorem does not constitute a demonstration of the second law for dilute gases;

(5) the dynamical invariance of the Gibbs H gives a simple proof of the second law for arbitrary interparticle forces;

(6) the second law is a special case of a general requirement for any macroscopic process to be experimentally reproducible.

Finally, the "anthropomorphic" nature of entropy, on both the statistical and phenomenological levels, is stressed.

Jaynes explains that Gibbs entropy is a conserved quantity, for the same reason as the Liouville theorem that conserves the hyper-volume in phase space of a cloud of particles as it traverses its trajectory.

Boltzmann entropy increases. We can show that this is a consequence of quantal interactions during particle collisions, which deny the claim of microscopic reversibility and erase the path information in the gas particles that would be needed to support Loschmidt's objection to the Boltzmann H-Theorem

In the writer's 1962 Brandeis lectures on statistical mechanics, the Gibbs and Boltzmann expressions for entropy were compared briefly, and it was stated that the Gibbs formula gives the correct entropy, as defined in phenomenological thermodynamics, while the Boltzmann H expression is correct only in the case of an ideal gas. However, there is a school of thought which holds that the Boltzmann expression is directly related to the entropy, and the Gibbs' one simply erroneous. This belief can be traced back to the famous Ehrenfest review article, which severely criticized Gibbs' methods.

While it takes very little thought to see that objections to the Gibbs II are immediately refuted by the fact that the Gibbs canonical ensemble does yield correct thermodynamic predictions, discussion with a number of physicists has disclosed a more subtle, but more widespread, misconception. The basic inequality of the Gibbs and Boltzmann H functions, to be derived in Sec. II, was accepted as mathematically correct; but it was thought that, in consequence of the "laws of large numbers" the difference between them would be practically negligible in the limit of large systems.

Now it is true that there are many different entropy expressions that go into substantially the same thing in this limit; several examples were given by Gibbs. However, the Boltzmann expression is not one of them; as we prove in Sec. Ill , the difference is a direct measure of the effect of interparticle forces on the potential energy and pressure, and increases proportionally to the size of the system.

Failure to recognize the fundamental role of the Gibbs H function is closely related to a much deeper confusion about entropy, probability, and irreversibility in general.

Gibbs' entropy is a constant because the loss of macroscopic order is conserved in the path information of the particles
For example, the Boltzmann H theorem is almost universally equated to a demonstration of the second law of thermodynamics for dilute gases, while ever since the Ehrenfest criticisms, it has been claimed repeatedly that the Gibbs H cannot be related to the entropy because it is constant in time.

Closer inspection reveals that the situation is very different. Merely to exhibit a mathematical quantity which tends to increase is not relevant to the second law unless one demonstrates that this quantity is related to the entropy as measured experimentally. But neither the Gibbs nor the Boltzmann H is so related for any distribution other than the equilibrium (i.e., canonical) one. Consequently, although Boltzmann's H theorem does show the tendency of a gas to go into a Maxwellian velocity distribution, this is not the same thing as the second law, which is a statement of experimental fact about the direction in which the observed macroscopic quantities (P,V,T) change.

The idea of classical coarse-graining takes on new significance with the minimal phase-space volumes of the quantum mechanical uncertainty principle.
Past attempts to demonstrate the second law for systems other than dilute gases have generally tried to retain the basic idea of the Boltzmann H theorem. Since the Gibbs H is dynamically constant, one has resorted to some kind of coarse-graining operation, resulting in a new quantity Ħ, which tends to decrease. Such attempts cannot achieve their purpose, because (a) mathematically, the decrease in Ħ is due only to the artificial coarse-graining operation and it cannot, therefore have any physical significance; (b) as in the Boltzmann H theorem, the quantity whose increase is demonstrated is not the same thing as the entropy. For the fine-grained and coarse-grained probability distributions lead to just the same predictions for the observed macroscopic quantities, which alone determine the experimental entropy; the difference between H and Ħ is characteristic, not of the macroscopic state, but of the particular way in which we choose to coarse-grain. Any really satisfactory demonstration of the second law must therefore be based on a different approach than coarse-graining.

Actually, a demonstration of the second law, in the rather specialized situation visualized in the aforementioned attempts, is much simpler than any H theorem. Once we accept the well-established proposition that the Gibbs canonical ensemble does yield the correct equilibrium thermodynamics, then there is logically no room for any assumption about which quantity represents entropy; it is a question of mathematically demonstrable fact. But as soon as we have understood the relation between Gibbs' H and the experimental entropy, Eq. (17) below, it is immediately obvious that the constancy of Gibbs' H, far from creating difficulties, is precisely the dynamical property we need for the proof.

It is interesting that, although this field has long been regarded as one of the most puzzling and controversial parts of physics, the difficulties have not been mathematical. Each of the above assertions is proved below or in the Brandeis lectures, using only a few lines of elementary mathematics, all of which was given by Gibbs. It is the enormous conceptual difficulty of this field which has retarded progress for so long. Readers not familiar with recent developments may, I hope, be pleasantly surprised to see how clear and basically simple these problems have now become, in several respects. However, as we will see, there are still many complications and unsolved problems.

Inspection of several statistical mechanics textbooks showed that, while most state the formal relations correctly, their full implications are never noted. Indeed, while all textbooks give extensive discussions of Boltzmann's H, some recent ones fail to mention even the existence of the Gibbs H. I was unable to find any explicit mathematical demonstration of their difference. It appeared, therefore, that the following note might be pedagogically useful.

As to the connections between entropy and information, in particular, "subjective human ignorance," Jaynes says,

The phase volume W0 therefore describes the full range of possible initial microstates; and not some arbitrary subset of them; this is the basic justification for using the canonical distribution to describe partial information.

On the "subjective" side, we can therefore say that W0 measures our degree of ignorance as to the true unknown microstate, when the only information we have consists of the macroscopic thermodynamic parameters; a remark first made by Boltzmann.

Gibbs Paradox
In Jaynes' article on the famous paradox, he writes...

Some important facts about thermodynamics have not been understood by others to this day, nearly as well as Gibbs understood them over 100 years ago. Other aspects of this “new” development have been reported elsewhere (Jaynes 1986, 1988, 1989). In the present note we consider the “Gibbs Paradox” about entropy of mixing and the logically inseparable topics of reversibility and the extensive property of entropy.

For 80 years it has seemed natural that, to find what Gibbs had to say about this, one should turn to his Statistical Mechanics. For 60 years, textbooks and teachers (including, regrettably, the present writer) have impressed upon students how remarkable it was that Gibbs, already in 1902, had been able to hit upon this paradox which foretold - and had its resolution only in - quantum theory with its lore about indistinguishable particles, Bose and Fermi statistics, etc.

It seems odd that Gibbs' original discussion of mixing gases, in his extensive monograph on the "Equilibrium of Heterogeneous Substances," should come as a "shock" to Jaynes. Most thermodynamic textbooks, since Lewis and Randall in 1923, mention the mixing problem as being discussed in what they called Gibbs' "great memoir." They also point out that Ludwig Boltzmann used a quote from the famous "paradox" passage to open volume 2 of his Lectures on Gas Theory (1898).

It was therefore a shock to discover that in the first Section of his earlier work (which must have been written by mid-1874 at the latest), Gibbs displays a full understanding of this problem, and disposes of it without a trace of that confusion over the “meaning of entropy” or “operational distinguishability of particles” on which later writers have stumbled. He goes straight to the heart of the matter as a simple technical detail, easily understood as soon as one has grasped the full meanings of the words “state” and “reversible” as they are used in thermodynamics. In short, quantum theory did not resolve any paradox, because there was no paradox.

References

Gibbs vs Boltzmann Entropies, American Journal of Physics 33, no. 5 (1965): 391-398.

The Gibbs Paradox, in Maximum Entropy and Bayesian Methods, pp. 1-21. Springer Netherlands, 1992.

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