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.
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.
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)
One recursively enumerable real α dominates another one β if there are nondecreasing recursive sequences of rational numbers (a[n] : n ∈ ω) approximating α and (b[n] : n ∈ ω) approximating β and a positive constant C such that for all n, C(α − a[n]) ≥ (β − b[n]). See [R. M. Solovay, Draft of a Paper (or Series of Papers) on Chaitin’s Work, manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1974, p. 215] and [G. (...) J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350–359]. We show that every recursively enumerable random real dominates all other recursively enumerable reals. We conclude that the recursively enumerable random reals are exactly the Ω-numbers [G. J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350–359]. Second, we show that the sets in a universal Martin-Lof test for randomness have random measure, and every recursively enumerable random number is the sum of the measures represented in a universal Martin-Lof test. (shrink)
The fourthDiscrete Mathematics andTheoreticalComputer Science Conference was jointly organized by the Centre for Discrete Mathematics and Theoretical Computer Science of the University of Auckland and the University of Bourgogne in Dijon, France, and took place in Dijon from 7 to12 July2003.Thepreviousconferenceswereheld inAuckland,NewZealand and Constan ̧ ta, Romania. The?ve invited speakers of the conference were: G.J. Chaitin, C. Ding, S. Istrail, M. Margenstein, and T. Walsh. The Programme Committee, consisting of V. Berthe, S. Boza- lidis,C.S.Calude,V.E.Cazanescu, F. Cucker, M. Deza, J. (...) Diaz, M.J. D- neen,B.Durand,L.Hemaspaandra, P. Hertling, J. Kohlas, G. Markowski, M. Mitrovic, A. Salomaa, L. Staiger, D. Skordev, G. Slutzki, I. Tomescu, M. Yasugi, and V. Vajnovszki, selected 18 papers to be presented as regular contributions and 1 5 other special CDMTCS papers. (shrink)
The question that I want to debate a little in this paper could be put in this way: what, and how much, empirical information is required for, or relevant to, moral philosophy? That question may well strike one as somewhat vague and woolly. Rightly so. What is needed to get rather clearer about its answer or possible answers is chiefly, I believe, to get clearer about its sense.
The familiar issue of corporate social responsibility takes on a new topic. Added to the list of concerns from affirmative action and environmental integrity is their growing contributions to education. At first glance, the efforts may appear to be ordinary gestures of communal good will in terms of providing computers, sponsoring book covers, and interactive materials provided by Scholastic Magazine. A closer view reveals a targeted market of student life who are vulnerable to commercials placed in these formats. Among the (...) most effective corporate intervention is Channel One News. It offers a newsworthy show but with mandatory commercial viewing. This increasing trend of corporations intervening to assist schools that need more money and/or equipment is disingenuous.In this essay, I present the background of this commercialization of education and demonstrate the violations against student autonomy and integrity. Although there may be utilitarian merits to some interventions, I argue that these infringe upon the moral value of personhood. Advertising in schools in its current practice is immoral on deontological grounds. (shrink)
This book offers the first historical treatment and analytic analysis of the problem of the criterion. It provides analyses of the ancient and modern characterizations of the problem and a resolution of each. My purpose is to show that there are at least two versions of the problem, one posed by a Pyrrhonian sceptic and one by a dogmatic sceptic. I show that both versions have a dissolution. Then, by examining the presuppositions of the dogmatic sceptic, I demonstrate that the (...) sceptic's position is self-undermining. I argue that meta-epistemological scepticism is much less reasonable than some forms of meta-epistemic cognitivism. (shrink)