|Sans les mathématiques on ne pénètre point au fond de la philosophie. |
Sans la philosophie on ne pénètre point au fond des mathématiques.
Sans les deux on ne pénètre au fond de rien. — Leibniz
[Without mathematics we cannot penetrate deeply into philosophy.
Without philosophy we cannot penetrate deeply into mathematics.
Without both we cannot penetrate deeply into anything.]
Tribute to Leibniz: Essay on Leibniz, Complexity and Incompleteness
METABIOLOGY: a field parallel to biology, dealing with the random evolution of artificial software (computer programs) rather than natural software (DNA), and simple enough that it is possible to prove rigorous theorems or formulate heuristic arguments at the same high level of precision that is common in theoretical physics. For more information about this new field, click here and here.
Ursula Molter, Gregory Chaitin and Hernán Lombardi opening the Buenos Aires Mathematics Festival (Argentina, May 2009)
G J Chaitin Home Page
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This website contains most of Chaitin's published papers, many book chapters, and the LISP, Java, C, and Mathematica software for Chaitin's Springer-Verlag trilogy. It also contains interviews and reviews of Chaitin's books.
Hector Zenil, Stephen Wolfram, Paul Davies, Ugo Pagallo, Gregory Chaitin, Cristian Calude, Karl Svozil, Gordana Dodig-Crnkovic and John Casti (Photo by Sally McCay, Burlington, VT, July 2007)
- Website: http://www.umcs.maine.edu/~chaitin
- Mirror: http://www.cs.auckland.ac.nz/~chaitin
- Affiliation: IBM Thomas J. Watson Research Center (Emeritus); lifetime honorary visiting professor, Computer Science Department, University of Auckland (New Zealand); honorary professor, University of Buenos Aires (Argentina); honorary doctorate, University of Maine (Orono); honorary doctorate, University of Cordoba (Argentina).
- Also associated with: International Academy of the Philosophy of Science(Belgium), Brazilian Academy of Philosophy, Foundational Questions Institute FQXi, Valparaiso Complex Systems Institute, DIMACS (Rutgers),Frege Centre for Structural Sciences (Jena), Santa Fe Institute Complexitymagazine, International Journal of Bifurcation and Chaos, Advances in Applied Mathematics.
- Mailing address:
G. J. Chaitin
IBM T. J. Watson Research Center
P. O. Box 218
Yorktown Heights, NY 10598
- Mathematics, Complexity & Philosophy: Three Lectures (Draft, November 2009)
- The Search for the Perfect Language (Draft, November 2009)
- How Real are Real Numbers? (Draft, November 2009)
- Algorithmic Information as a Fundamental Concept in Physics, Mathematics and Biology (Draft, November 2009)
- Metaphysics, Metamathematics and Metabiology (Course, August 2009)
A mathematical analysis of the scientific method, the axiomatic method,
and Darwin's theory of evolution.
- Mathematics, Biology and Metabiology (Foils, July 2009)
- Conversations on Truth (Interview, June 2009)
- Arturo Sangalli, Pythagoras' Revenge (May 2009)
A mystery novel on the philosophy of mathematics
in which the halting probability Ω plays a role.
- MetaMat! Em Busca do Ômega (April 2009)
- Evolution of Mutating Software (February 2009)
- Hasard et Complexité en Mathématiques (January 2009)
- Randomness: 5 Questions (Interview)
- The Search for the Perfect Language (Foils)
Hebrew University, Jerusalem, 17 Sept 2008
- Leibniz, Complexity and Incompleteness (Video)
Hebrew University, Jerusalem, 13 Sept 2008
- Leibniz, Complexity and Incompleteness (Draft)
University of Rome "Tor Vergata," 6 Jun 2008
- The Information Economy (Draft)
- Simply Gödel (Interview)
- Robert Epstein et al., Parsing the Turing Test (Book Review)
- Mario Livio, Is God a Mathematician? (Book Review)
- Rebecca Goldstein, Incompleteness (Book Review)
- William Byers, How Mathematicians Think (Book Review)
- Dangerous Knowledge: Cantor, Boltzmann, Gödel, Turing (Video)
Order DVD of this BBC TV program — Get post-production script
- 60th Birthday Celebrations: Vermont, Vienna, New York
Flyer for Two Festschrift Volumes
- The Limits of Reason (Scientific American)
- Randomness in Arithmetic (Russian Scientific American)
- L'Univers est-il intelligible? (La Recherche)
- Computers, Paradoxes and the Foundations of Mathematics (American Scientist)
- Ordenadores, paradojas y fundamentos de las matemáticas (Investigación y Ciencia)
- Grenzen der Berechenbarkeit (Spektrum der Wissenschaft)
- Komputery, Paradoksy i Podstawy Matematyki (Polish Translation)
Books with LISP Software
Collections of Interviews
Collections of Technical Papers
The Leibniz/Chaitin Medal
|"Dieu a choisi celuy qui est... le plus simple en hypotheses et le plus riche en phenomenes" |
[God has chosen that which is the most simple in hypotheses and the most rich in phenomena]
"Mais quand une regle est fort composée, ce qui luy est conforme, passe pour irrégulier"
[But when a rule is extremely complex, that which conforms to it passes for random]
— Leibniz, Discours de métaphysique, VI, 1686
[The Discours is also available online from Gallica; see pp. 32, 33 for the above texts.]
Tribute to Leibniz: Essay on Leibniz, Complexity and Incompleteness
Medallion commemorating Leibniz's discovery of binary arithmetic:
Medallion presented by Stephen Wolfram to Gregory Chaitin, 15 July 2007:
History of the Leibniz/Chaitin medal Story of the Latin translation
Gian-Carlo Rota: Problem Solvers and TheorizersMathematicians can be subdivided into two types: problem solvers and theorizers. Most mathematicians are a mixture of the two although it is easy to find extreme examples of both types.
To the problem solver, the supreme achievement in mathematics is the solution to a problem that had been given up as hopeless. It matters little that the solution may be clumsy; all that counts is that it should be the first and that the proof be correct. Once the problem solver finds the solution, he will permanently lose interest in it, and will listen to new and simplified proofs with an air of condescension suffused with boredom.
The problem solver is a conservative at heart. For him, mathematics consists of a sequence of challenges to be met, an obstacle course of problems. The mathematical concepts required to state mathematical problems are tacitly assumed to be eternal and immutable.
Mathematical exposition is regarded as an inferior undertaking. New theories are viewed with deep suspicion, as intruders who must prove their worth by posing challenging problems before they can gain attention. The problem solver resents generalizations, especially those that may succeed in trivializing the solution of one of his problems.
The problem solver is the role model for budding young mathematicians. When we describe to the public the conquests of mathematics, our shining heroes are the problem solvers.
To the theorizer, the supreme achievement of mathematics is a theory that sheds sudden light on some incomprehensible phenomenon. Success in mathematics does not lie in solving problems but in their trivialization. The moment of glory comes with the discovery of a new theory that does not solve any of the old problems but renders them irrelevant.
The theorizer is a revolutionary at heart. Mathematical concepts received from the past are regarded as imperfect instances of more general ones yet to be discovered. Mathematical exposition is considered a more difficult undertaking than mathematical research.
To the theorizer, the only mathematics that will survive are the definitions. Great definitions are what mathematics contributes to the world. Theorems are tolerated as a necessary evil since they play a supporting role — or rather, as the theorizer will reluctantly admit, an essential role — in the understanding of definitions.
Theorizers often have trouble being recognized by the community of mathematicians. Their consolation is the certainty, which may or may not be borne out by history, that their theories will survive long after the problems of the day have been forgotten.
If I were a space engineer looking for a mathematician to help me send a rocket into space, I would chose a problem solver. But if I were looking for a mathematician to give a good education to my child, I would unhesitatingly prefer a theorizer.
[From Gian-Carlo Rota, Indiscrete Thoughts, Birkhäuser, Boston, 1997, pp. 45-46.]