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

This website contains Greek letters and other mathematical symbols. If "Ω" isn't a capital Greek letter Omega, you should switch to another browser, for example, Safari and Chrome.

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)

Contents

• Metabiology     • Lectures in Canada & Argentina

• Latest News     • Recent Books     • LISP Applet

• Books with LISP Software     • Collections of Interviews

• Essays on Leibniz & Other Non-Technical Articles

• Collections of Technical Papers     • Leibniz Medal

• Photo Gallery     • Short Autobiographical Essay

• Research Time-Line     • List of Publications

• Computing Lower Bounds on Ω     • Explanation

• Diophantine Equation for Bits of Ω     • Explanation

• Gian-Carlo Rota: Problem Solvers and Theorizers

Latest News

Recent Books

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 Theorizers

Mathematicians 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.]