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 see Proving Darwin and the material on metabiology at the Virginia Chaitin home page.

Ursula Molter, Gregory Chaitin and Hernán Lombardi opening the Buenos Aires Mathematics Festival (Argentina, May 2009)

G J Chaitin Home Page

He is a professor at the Federal University of Rio de Janeiro and an honorary professor at the University of Buenos Aires, and has honorary doctorates from the University of Cordoba in Argentina and the University of Maine in the United States. He is also a member of the Académie Internationale de Philosophie des Sciences (Belgium).

Carnival in Rio de Janeiro, 1970. Photo by Peter Albrecht

• "Bold, provocative, and potentially revolutionary" — SCIENTIFIC AMERICAN BOOK CLUB

• "[Chaitin] makes the point that his model reflects the creativity and open-endedness of nature,
  in parallel with the incompleteness and open-endedness of mathematics itself" — KIRKUS REVIEWS

• (Dis)Proving Darwin: Podcast #004 — FROM ALPHA TO OMEGA - CRITICAL THINKING & IN DEPTH INTERVIEWS

     

• Algorithmic Information Theory, Cambridge University Press, 1987.

• Information, Randomness and Incompleteness: Papers on Algorithmic Information Theory, World Scientific, 1987, 2nd edition, 1990.

• Information-Theoretic Incompleteness, World Scientific, 1992.

• The Limits of Mathematics: A Course on Information Theory and the Limits of Formal Reasoning, Springer, 1998. Also in Japanese.

• The Unknowable, Springer, 1999. Also in Japanese.

• Exploring Randomness, Springer, 2001.

• Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer, 2002. Also in Portuguese and Japanese.

• From Philosophy to Program Size: Key Ideas and Methods. Lecture Notes on Algorithmic Information Theory from the 8th Estonian Winter School in Computer Science, EWSCS '03, Tallinn Institute of Cybernetics, 2003.

• Meta Math! The Quest for Omega, Pantheon, 2005. Also UK, French, Italian, Portuguese, Japanese and Greek editions.

• With Ugo Pagallo, Teoria algoritmica della complessità, Giappichelli, 2006.

• Thinking about Gödel and Turing: Essays on Complexity, 1970-2007, World Scientific, 2007.

• Matemáticas, Complejidad y Filosofía / Mathematics, Complexity and Philosophy (bilingual Spanish/English edition), Midas, 2011.

• With Newton da Costa and Francisco Antonio Doria, Gödel's Way: Exploits into an Undecidable World, CRC Press, 2012.

• Proving Darwin: Making Biology Mathematical, Pantheon, 2012. (Forthcoming in Spanish and Japanese.)

• Life as Evolving Software
  Universidade Federal do Rio Grande do Sul, 2 May 2011

• Hacia una teoría matemática de la evolución y de la creatividad biológica
  Instituto de Sistemas Complejos de Valparaíso, 30 de noviembre 2010

• Leibniz, Complexity and Incompleteness
  Hebrew University, Jerusalem, 13 September 2008

• Dangerous Knowledge: Cantor, Boltzmann, Gödel, Turing
  BBC TV Program

• Day 1—Introduction: Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Part 8

• Day 2—Conceptual: Part 1 Part 2 Part 3 Part 4 Part 5 Part 6 Part 7 Part 8

• Day 3—Technical: Part 1 Part 2 Part 3 Part 4 ? Part 6 Part 7 Part 8 Part 9 Part 10 Part 11

• Day 4—Philosophical: Part 1 Part 2 ? Part 4 Part 5 Part 6 Part 7

• Day 1, Day 2, Day 3, Day 4

• Life as Evolving Software
  To appear in Zenil, A Computable Universe, World Scientific

  

• Complexity, Randomness
  Chapter 2, Chaitin, da Costa, Doria, Gödel's Way, CRC Press

• Forays into Uncharted Landscapes
  Excerpts from Chapter 6, Chaitin, da Costa, Doria, Gödel's Way, CRC Press

• On Turing, Creativity and Metabiology
  Excerpts from S. Barry Cooper, J. van Leeuwen, Alan Turing: His Work and Impact, Elsevier

• 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)

• Low-Energy Nuclear Reactions / Andrea Rossi E-Cat

  Chaitin Interview in El Mercurio de Valparaíso

  "And it ought to be remembered that there is nothing more difficult to take in hand, more perilous to conduct, or more uncertain in its success, than to take the lead in the introduction of a new order of things. Because the innovator has for enemies all those who have done well under the old conditions, and lukewarm defenders in those who may do well under the new. This coolness arises partly from fear of the opponents, who have the laws on their side, and partly from the incredulity of men, who do not readily believe in new things until they have had a long experience of them. Thus it happens that whenever those who are hostile have the opportunity to attack they do it like partisans, whilst the others defend lukewarmly, in such wise that the prince is endangered along with them." — Machiavelli, The Prince, Chapter VI

  ECAT.com — Andrea Rossi's Energy Catalyzer, E-Cat - ECAT
  Buy one megawatt industrial "e-cat" heating units here.

  Defkalion - Green Technologies
  Buy 5 to 45 kilowatt domestic "hyperion" heating units here.
  Engineering specs for hyperion products

  Brillouin Energy Corporation
  Product = Brillouin Boiler

  Presentations at September 2011 NASA LENR Workshop:
  Bushnell, Nelson, Zawodny

  Ny Teknik (New Technology) articles on "Cold Fusion"
  Best reporting on Andrea Rossi and related developments

  "Cold Fusion" Poised to Become an Industrial Reality!
  Article in Science Reporter (India, July 2011)

  Pathological Disbelief
  Lecture by Nobel Laureate Brian D. Josephson

  Andrea Rossi's 'E-cat' nuclear reactor: a video FAQ
  YouTube Video by Nobel Laureate Brian D. Josephson

  Large excess heat production in Ni-H systems
  S. Focardi et al., IL NUOVO CIMENTO, Novembre 1998

• 3D Printers / Printers for Objects / Self-Reproducing Printers

  3DPrintReport Blog
  Exploring 3D Printing: Manufacturing of the Future

  RepRap Wiki
  RepRap is a free desktop 3D printer capable of printing plastic objects. Since many parts of RepRap are made from plastic and RepRap can print those parts, RepRap is a self-replicating machine - one that anyone can build given time and materials. It also means that - if you've got a RepRap - you can print lots of useful stuff, and you can print another RepRap for a friend...

• New Directions in Biology / Maurício Abdalla, Máximo Sandín

  Maurício Abdalla, La crisis latente del Darwinismo (Paper)

  Maurício Abdalla, La crisis latente del Darwinismo (Book)

  Página Oficial de Máximo Sandín / Máximo Sandín Home-Page
  "Tejiendo la Red de la Vida" / "Weaving the Web of Life"
  http://www.somosbacteriasyvirus.com / (We are bacteria and viruses)

• An Imaginative New Approach in Set Theory / Patrick St-Amant

  Deciding the Continuum Hypothesis with the Inverse Powerset

• Energy Transport Using Entanglement and Maxwell's Demon / Masahiro Hotta

• On Quantum Effects in a Theory of Biological Evolution / Miguel Angel Martin-Delgado

• Questões em Metabiologia / Felipe Sobreira Abrahão

History of the Leibniz/Chaitin medal       Story of the Latin translation

1987 photo by Alan Haywood

• LISP interpreter as Java applet
• LISP interpreter in C
• LISP interpreter in Mathematica (Original Mma v2 version)
• LISP interpreter in Mathematica (Mma v4.1 version contributed by Joe Yoon)
• LISP Tutorial (from The Unknowable)
• LISP Manual (from Exploring Randomness)
• LISP Manual—Advanced Features (from Exploring Randomness)

The Limits of Mathematics (1998)

LISP Code

• "Manual" for our LISP (alternate version)
• You can't prove that an S-expression is elegant! (commented version)
• The universal Turing machine used to define our complexity measure H (alternate version)
• You can't prove lower bounds on the complexity H! (commented version)
• Ω in the limit from below—stupid version (with additional output)
• Ω in the limit from below—elegant version (with additional output)
• Ω in the limit from below—fast version
• Ω is algorithmically incompressible/random (commented version)
• You can't prove what Ω's bits are! (commented version)

LISP Runs

• "Manual" for our LISP (alternate version)
• You can't prove that an S-expression is elegant! (commented version)
• The universal Turing machine used to define our complexity measure H (alternate version)
• You can't prove lower bounds on the complexity H! (commented version)
• Ω in the limit from below—stupid version (with additional output)
• Ω in the limit from below—elegant version (with additional output)
• Ω in the limit from below—fast version
• Ω is algorithmically incompressible/random (commented version)
• You can't prove what Ω's bits are! (commented version)

The Unknowable (1999)

LISP Code

• Elementary set theory in LISP
• A self-reproducing LISP expression
• Gödel's proof of his incompleteness theorem
• Turing's proof of the unsolvability of the halting problem
• My proof that you can't show that a LISP expression is elegant

LISP Runs

• Elementary set theory in LISP
• A self-reproducing LISP expression
• Gödel's proof of his incompleteness theorem
• Turing's proof of the unsolvability of the halting problem
• My proof that you can't show that a LISP expression is elegant

Exploring Randomness (2001)

• Historical introduction—A century of controversy over the foundations of mathematics

• What is LISP? Why do I like it?

• How to program my universal Turing machine in LISP
  utm2 code, utm2 run

• A self-delimiting Turing machine considered as a set of (program, output) pairs
  exec code, exec run

• How to construct self-delimiting Turing machines: the Kraft inequality
  kraft code, kraft run

• The connection between program-size complexity and algorithmic probability: $H(x) = -\log_2 P(x) + O(1)$.
  Occam's razor: there are few minimum-size programs
  occam code, occam run

• The basic result on relative complexity: $H(y|x) = H(x, y) - H(x) + O(1)$
  decomp code, decomp run, lemma code, lemma run

• Theoretical interlude—What is randomness? My definitions

• Proof that Martin-Löf randomness is equivalent to Chaitin randomness
  martin-lof code, martin-lof run, martin-lof2 code, martin-lof2 run

• Proof that Solovay randomness is equivalent to Martin-Löf randomness
  solovay code, solovay run

• Proof that Solovay randomness is equivalent to strong Chaitin randomness
  chaitin code, chaitin run, chaitin2 code, chaitin2 run

$$\Omega = \sum_{\text{program $P$ halts}} 2^{-(\text{the size in bits of $P$})}$$