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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
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
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

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James Symposium
 
The Principle of Superposition
The Schrōdinger equation is
H ψn = Enψn,         (1)

where H is the Hamiltonian operator, ψn is the wave function for state n, and the En are the energy eigenvalues for the states.

The eigenfunctions ψn are orthogonal to each other

< ψn | ψm > = δnm         (2)

where the "delta function"

δnm = 1, if n = m, and = 0, if n ≠ m.         (3)

Once they are normalized, the ψn form an orthonormal set of functions (or vectors) which can serve as a basis for the expansion of an arbitrary wave function φ 

| φ > = n = 0 n = ∞ cn | ψn >.         (4)

The expansion coefficients are

cn = < ψn | φ >.         (5)

In the abstract Hilbert space, < ψn | φ > is the "projection" of the vector φ onto the orthogonal axes ψn of the ψn "basis" vector set.

The Schrōdinger equation is a linear equation. It has no quadratic or higher power terms, and this introduces the principle of superposition of quantum states, a profound - and for many scientists and philosophers a disturbing - feature of quantum mechanics, one that is impossible in classical physics.

If ψa and ψb are both solutions of equation (1), then an arbitrary linear combination of these,

| ψ > = ca | ψa > + cb | ψb >,         (6)

with complex coefficients ca and cb, is also a solution.

Born's rule, that waves are probabilities, was actually first suggested by Einstein
Together with Born's probabilistic (statistical) interpretation of the wave function, the principle of superposition accounts for the major mysteries of quantum theory, some of which we hope to resolve, or at least reduce, with an objective (observer-independent) explanation of irreversible information creation during quantum processes.

Observable information is critically necessary for measurements, though observers can come along anytime after the information comes into existence as a consequence of the interaction of a quantum system and a measuring apparatus.

The quantum (discrete) nature of physical systems results from there generally being a large number of solutions ψn (called eigenfunctions) of equation (1) in its time independent form, with energy eigenvalues En.

An example of superposition.
Dirac tells us that a diagonally polarized photon can be represented as a superposition of vertical and horizontal states, with complex number coefficients that represent "probability amplitudes." Horizontal and vertical polarization eigenstates are the only "possibilities," if the measurement apparatus is designed to measure for horizontal or vertical polarization.

Thus,

| d > = ( 1/√2) | v > + ( 1/√2) | h >          (10)

The vectors (wave functions) v and h are the appropriate choice of basis vectors, the vector lengths are normalized to unity, and the sum of the squares of the probability amplitudes is also unity. This is the orthonormality condition needed to interpret the (squares of the) wave functions as probabilities.

When these (in general complex) number coefficients (1/√2) are squared (actually when they are multiplied by their complex conjugates to produce positive real numbers), the numbers (1/2) represent the probabilities of finding the photon in one or the other state, should a measurement be made on an initial state that is diagonally polarized.

Note that if the initial state of the photon had been vertical, its projection along the vertical basis vector would be unity, its projection along the horizontal vector would be zero. Our probability predictions then would be - vertical = 1 (certainty), and horizontal = 0 (also certainty). Quantum physics is not always uncertain, despite its reputation.

For Teachers
For Scholars

Chapter 6.8 - Reason Chapter 6.10 - Triads
Part Five - Problems Part Seven - Afterword
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