Online research in philosophy

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 (18th century), Georg Cantor (19th century) and Srinivasa Ramanujan (20th century).

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)

The paper concerns basic restrictions (and also simplifications and misinterpretations) which happens when one tries to explain mind processes (especially cognitive ones) by an analogy to formal, algorithmical and anticipated computation processes. The paper puts together the most important reasons why these attempts come to grief. The essence of computative reduction is shown among other things on the basic theorems of modern metamathematics. Especially it gives prominence epistemological consequences Gödel's theorems and recent discovers in metamathematics made by Gregory J. (...) Chaitin. But simplifications of computationisme (and partly of whole cognitivisme) don't mean that the other way is only postmodernism. (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)

This is a defense and extension of Stephen Yablo's claim that self-reference is completely inessential to the liar paradox. An infinite sequence of sentences of the form 'None of these subsequent sentences are true' generates the same instability in assigning truth values. I argue Yablo's technique of substituting infinity for self-reference applies to all so-called 'self-referential' paradoxes. A representative sample is provided which includes counterparts of the preface paradox, Pseudo-Scotus's validity paradox, the Knower, and other enigmas of the genre. I (...) rebut objections that Yablo's paradox is not a genuine liar by constructing a sequence of liars that blend into Yablo's paradox. I rebut objections that Yablo's liar has hidden self-reference with a distinction between attributive and referential self-reference and appeals to Gregory Chaitin's algorithmic information theory. The paper concludes with comments on the mystique of self-reference. (shrink)

Gregory Chaitin has done seminal work on the foundations of mathematics, especially the meaning of randomness and undecidability. In Meta Maths, he offers his ideas in a new popular version, which has a special interest because it comes directly from their originator. Long associated with the IBM Watson Research Center, Chaitin comes across as a kind of mathematical Richard Feynman, intuitive and high-spirited, irreverent and plain-spoken. Through this jovial persona, he presents many serious ideas in an engagingly (...) dishevelled way, mixing autobiography and reminiscence with exposition and example. He presents his insights so agreeably that it all seems a grand lark. (shrink)

"A person is not explainable in molecular, field-theoretical, or physiological terms alone." With that declaration, Nobel laureate Gerald M. Edelman goes straight to the heart of Nature's Imagination, a vibrant and important collection of essays by some of the world's foremost scientists. Ever since the Enlightenment, the authors write, science has pursued reductionism: the idea that the whole can be understood by examining and explaining each of its parts. But as this book shows, scientists in every discipline are reaching for (...) a new paradigm that accounts for the whole--from the individual person to the universe itself. Nature's Imagination gathers together the work of thirteen leading mathematicians, astronomers, neuroscientists, and philosophers, as they discuss the revolution sweeping the sciences. Here Roger Penrose, Oliver Sacks, John Barrow, Gregory Chaitin, Maragret Boden, and others explore how and why classic reductionism is falling by the wayside in their own fields. As Freeman Dyson writes in the introduction, science is an art form, not a philosophical method, and it is always in search of new tools. Reductionism has done its work, and scientists are in search of another. Roger Penrose offers a fascinating account of irreducibility in mathematics, starting with the example of an impossible triangle--a drawing of a triangular object twisted so that could not exist in three dimensions. He breaks the triangle into three parts, showing that each corner is physically possible; only in combination is the triangle impossible. Both Penrose and mathematician Gregory Chaitin explore Godel's incompleteness theorem--as does John Barrow, who explains that Chaitin's proof of the theorem shows that, if we ever arrive at a Theory of Everything, there may be a still deeper and simpler unifying theory beyond that. Other contributors discuss the changing thinking in neuroscience, and the limitations of a mechanical view of the mind: as Oliver Sacks writes, "if we are to have a model or theory of mind as this actually occurs in living creatures in the world, it may have to be radically different from anything like a computational one." In addition, this volume includes staunch defenders of the classic scientific approach, such as Peter Atkins ("The omnicompetence of science, and in particular the simplicity its reductionist insight reveals, should be accepted as a working hypothesis until, if ever, it is proved inadequate"). The advance of science has been so startlingly swift in the last century that it has begun to approach limits never dreamed of before. This remarkable volume captures the latest thinking on where we must turn if we are to truly understand oursevles and the universe we live in. (shrink)

Machine generated contents note: List of illustrations * Notes On Contributors * Introduction: B.Biebuyck, G.Buelens, O.de Graef, D.Hoens, S.Jttkandt * Who or What Decides: For Derrida: A Catastrophic Theory of Decision--J.Hillis Miller * Catastrophic Narratives and Why the Catastrophe to Catastrophe Might Have Already Happened--E.Vogt * Breath of Relief: Sloterdijk and the Politics of the Intimate--S.van Tuinen * Man is a swarm animal--J.Clemens * Notes on the Bird War: Biopolitics of the Visible (in the Era of Climate Change)--T.Cohen * Dialectical (...) Catastrophe: Hegels Allegory of Physiognomy and the Ethics of Survival--P.Moll * Catastrophe, Citationality and the Limits of Responsibility in Disgrace--G.Buelens * Unpredictable Inevitability and the Boundaries of Psychic Life; D.Nobus * Who is Nietzsche?--A.Badiou * Is Pleasure a Rotten Idea?--A.Schuster * Nationalist Ext(im)asy: Maurice Barrs and the Roots of Fascist Enjoyment--G.Chaitin * Topography of the Border: Derrida Rewriting Transcendental Aesthetics--J.Hodge * Index List of illustrations * Notes On Contributors * Introduction: B.Biebuyck, G.Buelens, O.de Graef, D.Hoens, S.Jttkandt * Who or What Decides: For Derrida: A Catastrophic Theory of Decision--J.Hillis Miller * Catastrophic Narratives and Why the Catastrophe to Catastrophe Might Have Already Happened--E.Vogt * Breath of Relief: Sloterdijk and the Politics of the Intimate--S.van Tuinen * Man is a swarm animal--J.Clemens * Notes on the Bird War: Biopolitics of the Visible (in the Era of Climate Change)--T.Cohen * Dialectical Catastrophe: Hegels Allegory of Physiognomy and the Ethics of Survival--P.Moll * Catastrophe, Citationality and the Limits of Responsibility in Disgrace--G.Buelens * Unpredictable Inevitability and the Boundaries of Psychic Life; D.Nobus * Who is Nietzsche?--A.Badiou * Is Pleasure a Rotten Idea?--A.Schuster * Nationalist Ext(im)asy: Maurice Barrs and the Roots of Fascist Enjoyment--G.Chaitin * Topography of the Border: Derrida Rewriting Transcendental Aesthetics--J.Hodge * Index. (shrink)