From philosophy to program size by Chaitin G.

By Chaitin G.

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G. Chaitin, Exploring Randomness, Springer-Verlag, London, 2001. • Day IV—LISP Implementation of AIT – G. Chaitin, “Elegant LISP programs,” in C. S. , People and Ideas in Theoretical Computer Science, pp. 32–52. Springer-Verlag, Singapore, 1999. – G. Chaitin, “An invitation to algorithmic information theory,” in D. S. Bridges, C. S. Calude, J. Gibbons, S. Reeves, I. H. , Combinatorics, Complexity & Logic: Proc. of DMTCS ’96, pp. 1–23. Springer-Verlag, Singapore, 1997. – [The above two papers are both chapters in] G.

Time complexity depends polynomially on the computational model. But in the case of program-size complexity the dependence on U is only an additive constant. So the unfashionable field of AIT depends less on the choice of computer than the fashionable field of time complexity does! So much for fashion! • Remember, I do not think that AIT is a practical theory for practical applications. 1 All elegant theories are simplified models of the horrendous complexity of the chaotic real world. Nevertheless, they give us insight.

When dealing with infinite computations in AIT, in some cases it is not even clear whether the definitions that were chosen are the right ones! ] 56 CHAITIN, FROM PHILOSOPHY TO PROGRAM SIZE • As I stated above, meta-mathematics and the incompleteness results I’ve presented in this course are a static framework in which one studies the power of a fixed set of axioms and rules of inference. How about a dynamic theory that has something to say about the emergence of new mathematical concepts? 3 • I would like to see an abstract mathematical theory defining “life” and predicting its emergence and evolution under very general circumstances.

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