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Philosophers

Mortimer Adler
Rogers Albritton
Alexander of Aphrodisias
Samuel Alexander
William Alston
Anaximander
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Anselm
Louise Antony
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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
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Daniel Dennett
Jacques Derrida
René Descartes
Richard Double
Fred Dretske
John Dupré
John Earman
Laura Waddell Ekstrom
Epictetus
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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
William James
Lord Kames
Robert Kane
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Tomis Kapitan
Walter Kaufmann
Jaegwon Kim
William King
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Christine Korsgaard
Saul Kripke
Thomas Kuhn
Andrea Lavazza
Christoph Lehner
Keith Lehrer
Gottfried Leibniz
Jules Lequyer
Leucippus
Michael Levin
George Henry Lewes
C.I.Lewis
David Lewis
Peter Lipton
C. Lloyd Morgan
John Locke
Michael Lockwood
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
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Karl Popper
Porphyry
Huw Price
H.A.Prichard
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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
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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
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Kadri Vihvelin
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G.H. von Wright
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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
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Pascual Jordan
Ruth E. Kastner
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Stephen Kosslyn
Daniel Koshland
Ladislav Kovàč
Leopold Kronecker
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
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Joseph LeDoux
Gilbert Lewis
Benjamin Libet
David Lindley
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Hendrik Lorentz
Josef Loschmidt
Ernst Mach
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Humberto Maturana
James Clerk Maxwell
Ernst Mayr
John McCarthy
Warren McCulloch
N. David Mermin
George Miller
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Ulrich Mohrhoff
Jacques Monod
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Colin Pittendrigh
Max Planck
Susan Pockett
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Hans Primas
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Lord Rayleigh
Jürgen Renn
Juan Roederer
Jerome Rothstein
David Ruelle
Tilman Sauer
Jürgen Schmidhuber
Erwin Schrödinger
Aaron Schurger
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Thomas Sebeok
Claude Shannon
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Herbert Simon
Dean Keith Simonton
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Ray Solomonoff
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John Stachel
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Giulio Tononi
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Francisco Varela
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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

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The Schrödinger Equation

1. The Schrōdinger Equation.

The fundamental equation of motion in quantum mechanics is Erwin Schrōdinger's famous wave equation that describes the evolution in time of his wave function ψ.

iℏ δψ / δt = H ψ         (1)

Max Born interpreted the square of the absolute value of Schrōdinger's wave function |ψn |2 (or < ψn | ψn > in Dirac notation) as providing the probability of finding a quantum system in a particular state n. As long as this absolute value (in Dirac bra-ket notation) is finite,

< ψn | ψn > ≡ ψ* (q) ψ (q) dq < ,         (2)

then ψ can be normalized, so that the probability of finding a particle somewhere < ψ | ψ > = 1, which is necessary for its interpretation as a probability. The normalized wave function can then be used to calculate "observables" like the energy, momentum, etc. For example, the probable or expectation value for the position r of the system, in con figuration space q, is

< ψ | r | ψ > = ψ* (q) r ψ (q) dq.         (3)

2. The Principle of Superposition.

The Schrōdinger equation (1) is a linear equation. It has no quadratic or higher power terms, and this introduces a profound - and for many scientists and philosophers a disturbing - feature of quantum mechanics, one that is impossible in classical physics, namely the principle of superposition of quantum states. If ψa and ψb are both solutions of equation (1), then an arbitrary linear combination of these,

| ψ > = ca | ψa > + cb | ψb >,         (4)

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

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.

H ψn = En ψn,         (5)

The discrete spectrum energy eigenvalues En limit interactions (for example, with photons) to specifi c energy diff erences En - Em.

In the old quantum theory, Bohr postulated that electrons in atoms would be in "stationary states" of energy En, and that energy differences would be of the form En - Em = , where ν is the frequency of the observed spectral line.

Einstein, in 1916, derived these two Bohr postulates from basic physical principles in his paper on the emission and absorption processes of atoms. What for Bohr were assumptions, Einstein grounded in quantum physics, though virtually no one appreciated his foundational work at the time, and few appreciate it today, his work eclipsed by the Copenhagen physicists.

The eigenfunctions ψn are orthogonal to each other

< ψn | ψm > = δnm         (6)

where the "delta function"

δnm = 1, if n = m, and = 0, if n ≠ m.         (7)

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 >.         (8)

The expansion coefficients are

cn = < ψn | φ >.         (9)

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.

2.1 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.

3. The Axiom of Measurement.

The axiom of measurement depends on the idea of "observables," physical quantities that can be measured in experiments. A physical observable is represented as an operator A that is "Hermitean" (one that is "self-adjoint" - equal to its complex conjugate, A* = A). The diagonal n, n elements of the operator's matrix,

< ψn | A | ψn > = ∫ ∫ ψ* (q) A (q) ψ (q) dq,         (11)

are interpreted as giving the expectation value for An (when we make a measurement).

The molecule suffers a recoil in the amount of hν/c during this elementary process of emission of radiation; the direction of the recoil is, at the present state of theory, determined by "chance"...

The weakness of the theory is, on the one hand, that it does not bring us closer to a link-up with the wave theory; on the other hand, it also leaves time of occurrence and direction of the elementary processes a matter of "chance."

It speaks in favor of the theory that the statistical law assumed for [spontaneous] emission is nothing but the Rutherford law of radioactive decay.
Albert Einstein, 1916

The off -diagonal n, m elements describe the uniquely quantum property of interference between wave functions and provide a measure of the probabilities for transitions between states n and m.

It is the intrinsic quantum probabilities that provide the ultimate source of indeterminism, and consequently of irreducible irreversibility, as we shall see.

Transitions between states are irreducibly random, like the decay of a radioactive nucleus (discovered by Rutherford in 1901) or the emission of a photon by an electron transitioning to a lower energy level in an atom (explained by Einstein in 1916).

The axiom of measurement is the formalization of Bohr's 1913 postulate that atomic electrons will be found in stationary states with energies En. In 1913, Bohr visualized them as orbiting the nucleus. Later, he said they could not be visualized, but chemists routinely visualize them as clouds of probability amplitude with easily calculated shapes that correctly predict chemical bonding.

The off-diagonal transition probabilities are the formalism of Bohr's "quantum jumps" between his stationary states, emitting or absorbing energy = En - Em. Einstein explained clearly in 1916 that the jumps are accompanied by his discrete light quanta (photons), but Bohr continued to insist that the radiation was classical for another ten years, deliberately ignoring Einstein's foundational efforts in what Bohr might have felt was his area of expertise (quantum mechanics).

The axiom of measurement asserts that a large number of measurements of the observable A, known to have eigenvalues An, will result in the number of measurements with value An that is proportional to the probability of fi nding the system in eigenstate ψn.

Quantum mechanics is a probabilistic and statistical theory. The probabilities are theories about what experiments will show. Experiments provide the statistics (the frequency of outcomes) that confirm the predictions of quantum theory - with the highest accuracy of any theory ever discovered!

4. The Projection Postulate.

The third novel idea of quantum theory is often considered the most radical. It has certainly produced some of the most radical ideas ever to appear in physics, in attempts by various "interpretations" to deny it.

The projection postulate is actually very simple, and arguably intuitive as well. It says that when a measurement is made, the system of interest will be found in (will instantly "collapse" into) one of the possible eigenstates of the measured observable.

We have several possibilities for eigenvalues. We can calculate the probabilities for each eigenvalue. Measurement simply makes one of these actual, and it does so, said Max Born, in proportion to the absolute square of the probability amplitude wave function ψn.

Note that Einstein saw the chance in quantum theory at least ten years before Born

In this way, ontological chance enters physics, and it is partly this fact of quantum randomness that bothered Einstein ("God does not play dice") and Schrōdinger (whose equation of motion for the probability-amplitude wave function is deterministic).

The projection postulate, or collapse of the wave function, is the element of quantum mechanics most often denied by various "interpretations." The sudden discrete and discontinuous "quantum jumps" are considered so non-intuitive that interpreters have replaced them with the most outlandish (literally) alternatives. The famous "many-worlds interpretation" substitutes a "splitting" of the entire universe into two equally large universes, massively violating the most fundamental conservation principles of physics, rather than allow a diagonal photon arriving at a polarizer to suddenly "collapse" into a horizontal or vertical state.

4.1 An example of projection.

Given a quantum system in an initial state | φ >, we can expand it in a linear combination of the eigenstates of our measurement apparatus, the | ψn >.

| φ > = n = 0 n = ∞ cn | ψn >.         (8)

In the case of Dirac's polarized photons, the diagonal state | d > is a linear combination of the horizontal and vertical states of the measurement apparatus, | v > and | h >. When we square the (1/√2) coefficients, we see there is a 50% chance of measuring the photon as either horizontal or vertically polarized.

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

4.2 Visualizing projection.

When a photon is prepared in a vertically polarized state | v >, its interaction with a vertical polarizer is easy to visualize. We can picture the state vector of the whole photon simply passing through the polarizer unchanged.

The same is true of a photon prepared in a horizontally polarized state | h > going through a horizontal polarizer. And the interaction of a horizontal photon with a vertical polarizer is easy to understand. The vertical polarizer will absorb the horizontal photon completely.

The diagonally polarized photon | d >, however, fully reveals the non-intuitive nature of quantum physics. We can visualize quantum indeterminacy, its statistical nature, and we can dramatically visualize the process of collapse, as a state vector aligned in one direction must rotate instantaneously into another vector direction.

As we saw above (Figure 2.1), the vector projection of | d > onto | v >, with length (1/√2), gives us the probability 1/2 for photons to emerge from the vertical polarizer. But this is only a statistical statement about the expected probability for large numbers of identically prepared photons.

When we have only one photon at a time, we never get one-half of a photon coming through the polarizer. Critics of standard quantum theory sometimes say that it tells us nothing about individual particles, only ensembles of identical experiments. There is truth in this, but nothing stops us from imagining the strange process of a single diagonally polarized photon interacting with the vertical polarizer.

There are two possibilities. We either get a whole photon coming through (which means that it "collapsed" or the diagonal vector was "reduced to" a vertical vector) or we get no photon at all. This is the entire meaning of "collapse." It is the same as an atom "jumping" discontinuously and suddenly from one energy level to another. It is the same as the photon in a two-slit experiment suddenly appearing at one spot on the photographic plate, where an instant earlier it might have appeared anywhere.

We can even visualize what happens when no photon appears. We can imagine that the diagonal photon was reduced to a horizontally polarized photon and was completely absorbed.

Why can we see the statistical nature and the indeterminacy? First, statistically, in the case of many identical photons, we can say that half will pass through and half will be absorbed.

The indeterminacy is that in the case of one photon, we have no ability to know which it will be. This is just as we cannot predict the time when a radioactive nucleus will decay, or the time and direction of an atom emitting a photon.

This indeterminacy is a consequence of our diagonal photon state vector being "represented" (transformed) into a linear superposition of vertical and horizontal photon state vectors. Thus the principle of superposition together with the projection postulate provides us with indeterminacy, statistics, and a way to "visualize" the collapse of a superposition of quantum states into one of the basis states.

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