Europolis. 3. Europa, Ökonomie, Wissenschaft und Erfindungen

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Werner DePauli-Schimanovich, 2005 - 536 Seiten
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Seite 239 - ... minds are superior to computers because we can often construct meta-systems, with intuitively reasonable axioms, that will decide such questions? This thesis was first strongly defended by the Oxford philosopher J. Anthony Lucas and more recently by Oxford's mathematical physicist Roger Penrose, among others. Almost all artificial-intelligence experts disagree. They are convinced that computers of the kind we know how to build — that is, computers made of wires and switches — are capable...
Seite 238 - Logic, I should maintain, must no more admit a unicorn than zoology can; for logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.
Seite 238 - Some of our colleagues in particle physics," writes physicist Freeman Dyson in Infinite in All Directions, "think that they are coming to a complete understanding of the basic laws of nature. . . . But I hope [this] . . . will prove as illusory as the notion of a formal decision process for all of mathematics. If it should turn out that the whole of physical reality can be described by a finite set of equations, I would be disappointed. I would feel that the Creator had been uncharacteristically...
Seite 347 - Mein teurer Freund, ich rat' Euch drum Zuerst Collegium Logicum. Da wird der Geist Euch wohl dressiert, In Spanische Stiefeln eingeschnürt, Daß er bedächtiger so fortan Hinschleiche die Gedankenbahn, Und nicht etwa die Kreuz und Quer Irrlichteliere hin und her.
Seite 238 - Platonist, and apparently believed an eternal 'not' was laid up in heaven, where virtuous logicians hope to meet it hereafter." Here is how Gödel replied in an unsent letter: As far as the passage about me [in Russell's autobiography] is concerned, I have to say first (for the sake of truth) that I am not a Jew (even though I don't think this question is of any importance), 2.) that the passage gives the wrong impression that I had many discussions with Russell, which was by no means the case (I...
Seite 238 - Russell, which was by no means the case (I remember only one). 3.) Concerning my "unadulterated" Platonism, it is no more "unadulterated" than Russell's own in 1921 when in the Introduction [to Mathematical Philosophy} he said "[Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features". At that time evidently Russell had met the "not" even in this world, but later on under the influence of Wittgenstein he chose to overlook it.
Seite 239 - The offer is promotion for a Greek novel by Apostolos Doxiadis with the English title Uncle Petros and Goldbach's Conjecture. It tells how Petros Papachristos was driven mad by his vain efforts to prove Goldbach's conjecture before he decided it was undecidable. No proof was found, so the prize was not given. It is easy to show that if Goldbach's theorem is undecidable it must be true. Assume it undecidable and false.
Seite 239 - Assume it undecidable and false. If so, there will be a counterexample, an even number not the sum of two primes, which a computer could find in a finite number of steps. This makes the theorem decidable, thus contradicting the assumption that the conjecture is undecidable and false. Therefore it must be undecidable and true. Casti and DePauli take up a question which is currently controversial.
Seite 236 - ... Godel: A Life of Logic, by the American science writer John Casti and the mathematician Werner DePauli of the University of Vienna, is a splendid nontechnical account of the Godelian revolution and at the same time a sketch of Godel's eccentric life and its tragic ending. The authors begin with Godel's birth and childhood in Brno, a city now part of the Czech Republic, where he became fluent in German, French, and English. In 1924, he settled in Vienna. There his philosophical interests put him...
Seite 236 - Truth," as the authors of this new book capsule it, "is larger than proof." Statements that are true but unprovable inside a formal system are called "Godel undecidable." Like consistency, they can be shown true by a meta-system, but the larger system is sure to contain its own unprovable statements. There is no escape from the endless regress of meta-systems, each with undecidable theorems. Many books and thousands of technical papers have dealt with Godel's epochal bombshell and its implications....

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