Citation for this page in APA citation style.           Close


Mortimer Adler
Rogers Albritton
Alexander of Aphrodisias
Samuel Alexander
William Alston
Louise Antony
Thomas Aquinas
David Armstrong
Harald Atmanspacher
Robert Audi
Alexander Bain
Mark Balaguer
Jeffrey 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
Michael Burke
Joseph Keim Campbell
Rudolf Carnap
Ernst Cassirer
David Chalmers
Roderick Chisholm
Randolph Clarke
Samuel Clarke
Anthony Collins
Antonella Corradini
Diodorus Cronus
Jonathan Dancy
Donald Davidson
Mario De Caro
Daniel Dennett
Jacques Derrida
René Descartes
Richard Double
Fred Dretske
John Dupré
John Earman
Laura Waddell Ekstrom
Herbert Feigl
John Martin Fischer
Owen Flanagan
Luciano Floridi
Philippa Foot
Alfred Fouilleé
Harry Frankfurt
Richard L. Franklin
Michael Frede
Gottlob Frege
Peter Geach
Edmund Gettier
Carl Ginet
Alvin Goldman
Nicholas St. John Green
H.Paul Grice
Ian Hacking
Ishtiyaque Haji
Stuart Hampshire
Sam Harris
William Hasker
Georg W.F. Hegel
Martin Heidegger
Thomas Hobbes
David Hodgson
Shadsworth Hodgson
Baron d'Holbach
Ted Honderich
Pamela Huby
David Hume
Ferenc Huoranszki
William James
Lord Kames
Robert Kane
Immanuel Kant
Tomis Kapitan
Jaegwon Kim
William King
Hilary Kornblith
Christine Korsgaard
Saul Kripke
Andrea Lavazza
Keith Lehrer
Gottfried Leibniz
Michael Levin
George Henry Lewes
David Lewis
Peter Lipton
C. Lloyd Morgan
John Locke
Michael Lockwood
E. Jonathan Lowe
John R. Lucas
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
Thomas Nagel
Friedrich Nietzsche
John Norton
Robert Nozick
William of Ockham
Timothy O'Connor
David F. Pears
Charles Sanders Peirce
Derk Pereboom
Steven Pinker
Karl Popper
Huw Price
Hilary Putnam
Willard van Orman Quine
Frank Ramsey
Ayn Rand
Michael Rea
Thomas Reid
Charles Renouvier
Nicholas Rescher
Richard Rorty
Josiah Royce
Bertrand Russell
Paul Russell
Gilbert Ryle
Jean-Paul Sartre
Kenneth Sayre
Moritz Schlick
Arthur Schopenhauer
John Searle
Wilfrid Sellars
Alan Sidelle
Ted Sider
Henry Sidgwick
Walter Sinnott-Armstrong
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
G.H. von Wright
David Foster Wallace
R. Jay Wallace
Ted Warfield
Roy Weatherford
William Whewell
Alfred North Whitehead
David Widerker
David Wiggins
Bernard Williams
Timothy Williamson
Ludwig Wittgenstein
Susan Wolf


Michael Arbib
Bernard Baars
Gregory Bateson
John S. Bell
Charles Bennett
Ludwig von Bertalanffy
Susan Blackmore
Margaret Boden
David Bohm
Niels Bohr
Ludwig Boltzmann
Emile Borel
Max Born
Satyendra Nath Bose
Walther Bothe
Hans Briegel
Leon Brillouin
Stephen Brush
Henry Thomas Buckle
S. H. Burbury
Donald Campbell
Anthony Cashmore
Eric Chaisson
Jean-Pierre Changeux
Arthur Holly Compton
John Conway
John Cramer
E. P. Culverwell
Charles Darwin
Terrence Deacon
Louis de Broglie
Max Delbrück
Abraham de Moivre
Paul Dirac
Hans Driesch
John Eccles
Arthur Stanley Eddington
Paul Ehrenfest
Albert Einstein
Hugh Everett, III
Franz Exner
Richard Feynman
R. A. Fisher
Joseph Fourier
Lila Gatlin
Michael Gazzaniga
GianCarlo Ghirardi
J. Willard Gibbs
Nicolas Gisin
Paul Glimcher
Thomas Gold
Brian Goodwin
Joshua Greene
Jacques Hadamard
Patrick Haggard
Stuart Hameroff
Augustin Hamon
Sam Harris
Hyman Hartman
John-Dylan Haynes
Martin Heisenberg
Werner Heisenberg
John Herschel
Jesper Hoffmeyer
E. T. Jaynes
William Stanley Jevons
Roman Jakobson
Pascual Jordan
Ruth E. Kastner
Stuart Kauffman
Simon Kochen
Stephen Kosslyn
Ladislav Kovàč
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
David Layzer
Benjamin Libet
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
James Clerk Maxwell
Ernst Mayr
Ulrich Mohrhoff
Jacques Monod
Emmy Noether
Howard Pattee
Wolfgang Pauli
Massimo Pauri
Roger Penrose
Steven Pinker
Colin Pittendrigh
Max Planck
Susan Pockett
Henri Poincaré
Daniel Pollen
Ilya Prigogine
Hans Primas
Adolphe Quételet
Juan Roederer
Jerome Rothstein
David Ruelle
Erwin Schrödinger
Aaron Schurger
Claude Shannon
David Shiang
Herbert Simon
Dean Keith Simonton
B. F. Skinner
Roger Sperry
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
William Thomson (Kelvin)
Peter Tse
Heinz von Foerster
John von Neumann
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek


Free Will
Mental Causation
James Symposium
Jacques Hadamard
Jacques Hadamard was a great mathematician who studied his thought processes in solving mathematical problems. He interviewed many other leading mathematicians and scientists, including Henri Poincaré, many of whom shared Hadamard's experience that solutions to problems often came suddenly and completely, generally after long reflections on the problem.

In his 1945 book Psychology of Invention in the Mathematical Field, Hadamard described the Synthèse conference in Paris in 1936 to study creativity. In Chapter III, Hadamard described how the combination of random ideas could lead to a choice of the best combination. Chance alone is not enough.

Combination of Ideas. What we just observed concerning the unconscious in general will be seen again from another angle, when speaking of its relations with discovery. We shall see a little later that the possibility of imputing discovery to pure chance is already excluded by Poincaré's observations, when more attentively considered.

On the contrary, that there is an intervention of chance but also a necessary work of unconsciousness, the latter implying and not contradicting the former, appears, as Poincaré shows, when we take account not merely of the results of introspection, but of the very nature of the question.

Indeed, it is obvious that invention or discovery, be it in mathematics or anywhere else, takes place by combining ideas.1 Now, there is an extremely great number of such combinations, most of which are devoid of interest, while, on the contrary, very few of them can be fruitful. Which ones does our mind — I mean our conscious mind — perceive? Only the fruitful ones, or exceptionally, some which could be fruitful.

However, to find these, it has been necessary to construct the very numerous possible combinations, among which the useful ones are to be found.

It cannot be avoided that this first operation take place, to a certain extent, at random, so that the role of chance is hardly doubtful in this first step of the mental process. But we see that that intervention of chance occurs inside the unconscious: for most of these combinations — more exactly, all those which are useless — remain unknown to us.

Moreover, this shows us again the manifold character of the unconscious, which is necessary to construct those numerous combinations and to compare them with each other.

The Following Step. It is obvious that this first process, this building up of numerous combinations, is only the beginning of creation, even, as we should say, preliminary to it. As we just saw, and as Poincaré observes, to create consists precisely in not making useless combinations and in examining only those which are useful and which are only a small minority. Invention is discernment, choice.

To Invent Is to Choose. This very remarkable conclusion appears the more striking if we compare it with what Paul Valéry writes in the Nouvelle Revue Française: "It takes two to invent anything. The one makes up combinations; the other one chooses, recognizes what he wishes and what is important to him in the mass of the things which the former has imparted to him.

"What we call genius is much less the work of the first one than the readiness of the second one to grasp the value of what has been laid before him and to choose it."

We see how beautifully the mathematician and the poet agree in that fundamental view of invention consisting of a choice.

Esthetics in Invention. How can such a choice be made? The rules which must guide it "are extremely fine and delicate. It is almost impossible to state them precisely; they are felt rather than formulated. Under these conditions, how can we imagine a sieve capable of applying them mechanically?"

Though we do not directly see this sieve at work, we can answer the question, because we are aware of the results it affords, i.e., the combinations of ideas which are perceived by our conscious mind. This result is not doubtful. "The privileged unconscious phenomena, those susceptible of becoming conscious, are those which, directly or indirectly, affect most profoundly our emotional sensibility.

"It may be surprising to see emotional sensibility invoked à propos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility."

That an affective element is an essential part in every discovery or invention is only too evident, and has been insisted upon by several thinkers; indeed, it is clear that no significant discovery or invention can take place without the will of finding. But with Poincaré, we see something else, the intervention of the sense of beauty playing its part as an indispensable means of finding. We have reached the double conclusion:

that invention is choice
that this choice is imperatively governed by the sense of scientific beauty.

Hadamard described Poincaré as the source of the basic idea, but he credited Paul Valéry with the idea that there are two stages in creativity, perhaps even two entities - one to generate random alternative_possibilities, and the other to select or choose the best alternative.

These suggestions of Hadamard's were a major influence on Daniel Dennett's 1978 two-stages model of decision making which were later dubbed "Valerian." Dennett quotes the Valéry "It takes two to invent anything...," and then imagines the two stages in one mind.

Hadamard quoted Mozart to show that the first stage involves ideas that just "come to us" freely.

When I feel well and in a good humour, or when I am taking a drive or walking after a good meal, or in the night when I cannot sleep, thoughts crowd into my mind as easily as you could wish. Whence and how do they come? I do not know and I have nothing to do with it. Those which please me I keep in my head and hum them; at least others have told me that I do so....Then my soul is on fire with inspiration.
Hadamard and Poincaré both describe ideas that "present themselves" as William James described it.

Hadamard and Irreversibility
In 1906 Hadamard wrote a review of Josiah Willard Gibbs' Elementary Principles of Statistical Mechanics. (Bulletin of the American Mathematical Society, 12, p.194-210) He called it pure mathematics, applying the calculus of probabilities (of Laplace and others) to mechanics.

He wrote:

It remains to address the most important and most delicate that raises the study of the distribution phase. What happens to this distribution in the course of the movement, when part of any state: it tends, for example, to move closer to the canonical distribution or any distribution with similar properties?

This is, in short, the vital issue for kinetic theories. The paradox related and which seems at first to determine in advance any theory of this kind is as follows. How can the equations of dynamics, which are all reversible, lead to irreversible laws, to the growth of entropy? (p.201)

[Il reste à traiter la question la plus importante et la plus délicate que soulève cette étude de la distribution en phase. Que devient cette distribution au cours du mouvement, lorsqu'on part d'un état quelconque: tend-elle, par exemple, à se rapprocher de la distribution canonique ou d'une distribution présentant des propriétés analogues?

C'est, en somme, la question vitale pour les théories cinétiques. Le paradoxe qui s'y rattache et qui semble, au premier abord, miner par avance toute théorie de cette nature est eu effet le suivant. Comment, eu partant d'équations de la dynamique, toutes réversibles, parviendra-t-on à des lois irréversibles, à la croissance de l'entropie?]

We develop an answer in our treatment of micro-reversibility.

Normal | Teacher | Scholar