We discuss mathematical and physical arguments against continuity and in favor of discreteness, with particular emphasis on the ideas of Émile Borel.

The information-theoretic point of view proposed by Leibniz in 1686 and developed by algorithmic information theory (AIT) suggests that mathematics and physics are not that different. This will be a first-person account of some doubts and speculations about the nature of mathematics that I have entertained for the past three decades, and which have now been incorporated in a digital philosophy paradigm shift that is sweeping across the sciences.

In a famous lecture in 1900, David Hilbert listed 23 difficult problems he felt deserved the attention of mathematicians in the coming century. His conviction of the solvability of every mathematical problem was a powerful incentive to future generations: ``Wir müssen wissen. Wir werden wissen.'' (We must know. We will know.) Some of these problems were solved quickly, others might never be completed, but all have influenced mathematics. Later, Hilbert highlighted the need to clarify the methods of mathematical reasoning, using (...) a formal system of explicit assumptions, or axioms. Hilbert's vision was the culmination of 2,000 years of mathematics going back to Euclidean geometry. He stipulated that such a formal axiomatic system should be both `consistent' (free of contradictions) and `complete' (in that it represents all the truth). Hilbert also argued that any wellposed mathematical problem should be `decidable', in the sense that there exists a mechanical procedure, a computer program, for deciding whether something is true or not. Of course, the only problem with this inspiring project is that it turned out to be impossible. (shrink)

This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no ...

Hi everybody! It's a great pleasure for me to be back here at the new, improved Santa Fe Institute in this spectacular location. I guess this is my fourth visit and it's always very stimulating, so I'm always very happy to visit you guys. I'd like to tell you what I've been up to lately. First of all, let me say what algorithmic information theory is good for, before telling you about the new version of it I've got.

MATHEMATICS is a wonderful, mad subject, full of imagination, fantasy and creativity that is not limited by the petty details of the physical world, but only by the strength of our inner light. Does this sound familiar? Probably not from the mathematics classes you may have attended. But consider the work of three famous earlier mathematicians: Leonhard Euler, Georg Cantor and Srinivasa Ramanujan.

This article explores methodological issues central to the undertaking of joint Palestinian-Israeli research, work that is impacted by the violent conflict between the two peoples. Four issues are discussed: (a) collaborating under conflict, that is, how the conflict impacts relations between the researchers on either side of the border, (b) issues of power and equality, as they impact the research process, (c) relationships with participants, that is, how the conflict influences relations between the researcher and the research participants, and (d) (...) rethinking standards, that is, whether the normative standards of research quality are relevant for Palestinian-Israeli collaborative research. The article presents examples from joint research and offers preliminary ideas for dealing with these methodological issues. (shrink)