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Book Review A World Without Time: The Forgotten Legacy of Gödel and Einstein Reviewed by John Stachel A World Without Time: The Forgotten Legacy of Gödel and Einstein Palle Yourgrau Hardcover, Basic Books,

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Book Review A World Without Time: The Forgotten Legacy of Gödel and Einstein Reviewed by John Stachel A World Without Time: The Forgotten Legacy of Gödel and Einstein Palle Yourgrau Hardcover, Basic Books, 2004, US$24.00, 224 pages ISBN-13: Paperback, Perseus Publishing, 2006 US$15.00, 224 pages ISBN-13: The book s title suggests its three main themes: 1) A World Without Time : Gödel s argument, based on his interpretation of the theories of relativity (both special and general), for the unreality of time. At a generous estimate, no more than forty of the book s 180-odd pages are devoted to this theme (essentially the last part of Chapter 6, and Chapter 7). 2) Gödel and Einstein : An attempt to draw parallels between the lives and views of its two protagonists. An account of Gödel s life in Vienna (Chapters 3 5) includes a lengthy excursus into his seminal contributions to logic (Chapter 4). The account of the relationship between the two after Gödel s permanent move to the Institute for Advanced Study in 1940 (Chapters 1, 6, and 8; the last also discusses Gödel s final years) includes brief glimpses of Einstein s pre-princeton years. 3) The Forgotten Legacy : Yourgrau s polemic against what he sees as the neglect by the analytically-oriented American philosophical establishment of Gödel s significant contributions to metaphysics (the last part of Chapter 8 and Chapter 9). Insofar as Einstein is presumed to share Gödel s German Bias for Metaphysics (the title of John Stachel is professor of physics at Boston University. He was the founding editor of The Collected Papers of Albert Einstein. His address is bu.edu. Chapter 2), he is also portrayed as a victim of this Conspiracy of Silence (the title of Chapter 1). The Forgotten Legacy Yourgrau, himself a philosopher, has been urging recognition of Gödel as an important philosopher of mathematics and of space and time (p. 181) for almost two decades. He regards the dialectic of the formal and the intuitive as the leitmotif of Gödel s lifework (p. 124), seeing both continuity and contrast in this work. There is continuity in method: Overarching much of his research in philosophy and logic was the Gödel program, the investigation of the limits of formal methods in capturing intuitive concepts (p. 182; see also pp. 114, 127). The contrast lies in the conclusions Gödel drew from the existence of these limits: In mathematics, he concluded from the incompleteness of Hilbert s proof-theoretic system for arithmetic that the Platonic realm of numbers cannot be fully captured by the formal structures of logic. For Gödel, the devices of formal proof are too weak to capture all that is true in the world of numbers, not to say in mathematics as a whole. (p. 136) 1. But in physics: When it came to relativistic cosmology, however, he took the opposite tack [R]elativity is just fine, whereas time in the intuitive sense is an illusion. Relativity does not capture the essence 1 Feferman 2006 points out that: The incompleteness theorems in and of themselves do not support mathematical Platonism, as Gödel admitted in 1951: Of course I do not claim that the foregoing considerations amount to a real proof of this view of the nature of mathematics. The most I could assert would be to have disproved the nominalistic view, which considers mathematics to consist solely in syntactic conventions and their consequences (Gödel 1995, pp ). Raatikainen 2005 discusses various philosophical interpretations of the incompleteness theorems. August 2007 Notices of the AMS 861 of intuitive time, because when it comes to time, our intuitions betray us (pp ). So far, so good: The argument seems clear enough. Yourgrau explains that Gödel was at once a mathematical realist, who believed in the reality of the Platonic realm of numbers, and a temporal idealist because time in the intuitive sense is an illusion. He speaks of the nonexistence of time in the actual world (p. 139), presumably because it corresponds to nothing in the realm of Platonic ideas. Yourgrau s World Without Time is a world of real, objective concepts (p. 171) that does not include time. How are we to square Yourgrau s words on pp with his account thirty pages later of Gödel s views on time and intuition? Time, for example, in relation to being, Gödel considered one of the basic concepts [of metaphysics], but he believed that in the attempt to discover what is fundamental about our thinking about time we can receive no assistance from physics, which, he argued, combines concepts without analyzing them. Instead, we must reconstruct the original nature of our thinking For this, he turned not to Einstein but to Husserl and phenomenology Gödel saw phenomenology as an attempt to reconstruct our original use of basic ideas on what we meant in the first place by our most fundamental acts of thought [B]oth Gödel and Husserl (in his later period) were conceptual realists (pp ). It seems to follow from these quotations that, for Gödel, time is a basic metaphysical concept, one of the fundamental concepts that underlie reality, about the nature of which we can receive no help from physics. 2 Instead, one must use selfreflection to grasp this real, objective concept. Remember, the Yourgrau of pp , also expounding Gödel, had assured us of the nonexistence of time in the actual world. The disparity between the two Yourgraus left this reader unable to answer a basic question raised by the book: What is the Yourgrau-Gödelian concept of time, which must be grasped by self-reflection but is not based on an intuition of time that is illusory? In the final section of the review, I shall return to the question of what relativistic physics (pace the Yourgrau of pp ) does tell us about the nature of time and what is perhaps best left forgotten in Gödel s Forgotten Legacy. The neglect of Gödel s philosophical views by analytic philosophers is mainly due to his affiliation with their bête noir : The metaphysical tradition associated with Plato, Leibniz, and the later Husserl, to name some of Gödel s favorites. Concepts have an objective existence Gödel wrote in a notebook entry on My Philosophical Viewpoint 2 On p. 105, Yourgrau cites Gödel s list of the fundamental concepts that underlie reality, which includes time. (quoted on p. 104), and his conceptual realism is more or less the same as what other philosophers call objective idealism. 3 Indeed, while having problems with the Kantian philosophy, which is strong in epistemology but weak in ontology (weak that is for [conceptual JS] realists like Gödel, Frege, and Husserl) (p. 175), Gödel was an admirer of Hegel (see pp. 157, 182), and the method used in the Gödel program in logic has interesting parallels with Hegel s dialectical method of subverting a philosophical system from within. 4 Starting from the system s own premises, one demonstrates its inability to reach its own goals by exposing some contradiction between premises and goals. These contradictions are then sublated 5 by synthesis in some higher, more advanced system. Hilbert s formalist program started from some set of axioms and syntactic rules of deduction with the goal of proving the completeness and consistency of arithmetic. Gödel subverted the program from within: using a newly developed formal technique (Gödel numbering), he proved the impossibility of reaching this goal. One might even say that he did so by sublating the syntactic concept of provability within a formal system in the semantic concept of truth in some model of that system (see the sidebar Gödel s Theorems ). Yourgrau writes of Gödel s dialectical dance with intuitive and formal time in the theory of relativity (p. 128) ; similarly there is a dialectical dance with semantics and syntax in his logic. Gödel and Einstein Both Gödel and Einstein are described in over-thetop superlatives: Gödel is the greatest logician of all time, a beacon in the intellectual landscape of the last thousand years (p. 1). Einstein is the most famous scientist of all time (p. 2), the greatest scientist since Newton (p. 31). Together with another German-speaking theorist, Werner Heisenberg, they were the authors of the three most fundamental scientific results of the century. Each man s discovery, moreover, established a profound and disturbing limitation (p. 2). Even limiting 3 Terminological confusion abounds here since different philosophers attach opposing senses to the terms real and realism, and ideal and idealism. For advocates of conceptual realism, the adjective ideal is pejorative: It implies that the noun it modifies does not have an objective conceptual counterpart. 4 The similarity is in method, not motivation. Gödel s original intent was to contribute to Hilbert s program, and only years later did he realize that he had subverted it (see, e.g., Feferman 2006). 5 Sublation is the best English equivalent for Hegel s das Aufheben, which means simultaneously to preserve, destroy, and raise to a higher level (see the entry Sublation in Inwood 1992, pp ). 862 Notices of the AMS Volume 54, Number 7 oneself to limitations, one might well argue, for example, that Bell s theorem beats Heisenberg s uncertainty principle hands down. 6 But are such claims necessary? Isn t a sober statement of the results and their profound implications sufficient? All is not rosy in Yourgrau s picture of Einstein: After he arrived at the institute [for Advanced Study in 1933] never again would he enjoy the intellectual camaraderie that had formed a cloak against all the ugliness that beset his years in Berlin (p. 148). There is no mention of Walter Mayer, Peter Bergmann, Valentine Bargmann, Nathan Rosen, Leopold Infeld, Bruria Kaufman, Ernst Straus, all of them Einstein s scientific collaborators in Princeton; he remained close to many of them, both intellectually and personally, long after their collaborations ended. Nor is there mention of visits or longer stays at the institute by such scientific colleagues as Niels Bohr, Abraham Pais, Wolfgang Pauli, H. P. Robertson; nor of his close contact with fellow-expatriates such as the historian Erich Kahler and his wife Lily, the writer Hermann Broch; the philosopher Paul Oppenheim and his wife Gaby; art historian Erwin Panofsky and Princeton librarian Johanna Fantova; not to mention various romantic liaisons, such as that with Margarita Konenkova, a Russian woman recently accused of being a spy. Nor was he isolated at the institute: Batterson 2006 describes the important role Einstein played in its affairs from its formative years until his retirement. Yourgrau s picture of Gödel as a social isolate in Princeton, with few friends except Einstein, and of his tragic descent into paranoia and death by selfstarvation, is duly accurate. But to say together they remained isolated and alone at the institute (p. 4) is simply to overlook the profound difference between the personalities of the two. Reliability of the Book In contrast to his earlier book on the topic (Yourgrau 1999), intended primarily for philosophers this one [is] accessible to normal readers (p. vii). Presumably, he means non-scholars, i.e., that the book is intended for a popular audience. 7 The 6 For a discussion of Bell s theorem that there is an upper limit to the correlation of distant events, if one just assumes the validity of the principle of local causes (Peres 1993, p. 160), and the profound significance of its violation by quantum phenomena, see ibid, Chapter 6. 7 But even scholarly readers, let alone normal ones, will often find it rough going: The physicist s prophetic idea of describing a physical system by locating it in a logical framework in various dimensions of physical significance would have not only a profound effect on the future of quantum mechanics but on the Bible of the Schlick circle (p. 38). That the physicist is Boltzmann and the Bible is Wittgenstein s Tractatus is clear from the context, but otherwise I can t make sense of the sentence, perhaps because I am not normal. writer of such a book has a particularly great responsibility, because its readers often take the author s word for factual and technical assertions not substantiated in the text. So if anything, popular books should be held to even higher standards of sobriety and accuracy than books addressed to other experts, capable of forming independent judgments on such matters. This book often falls short of such standards. I have already given some examples of lack of sobriety and, unfortunately, it is not hard to find examples of inaccuracy. Contradictory assertions occur within a few pages: Further separating Einstein from Gödel was the fact that Einstein never fully resolved his native suspicion of mathematics. [T]he physicist remained forever wary of being led by the nose by mathematicians (p. 15). Einstein and Gödel, in turn, each in his own way, approached the world mathematically. For both, mathematics was a window onto ultimate reality, not, as for many of their scientific colleagues, a mere tool for intellectual bookkeeping. (p. 17). Sometimes one of the two statements is so downright silly that it can only be ascribed to carelessness: On p. 44, Yourgrau speaks of rational numbers as infinite sequences of natural numbers, and irrational numbers as infinite sequences of rational numbers [my emphasis JS]. Three pages later he describes irrational numbers [as] those that cannot be expressed as ratios of two natural numbers [my emphasis JS] correctly implying that all positive rationals can be so defined. Yourgrau s comment on Einstein: Never too concerned with consistency unlike his logician companion [Gödel] (p. 14) applies to many passages in this book! Confusion even creeps into one of the best parts of the book: the account in Chapter 4 of Gödel s results in logic. Yourgrau s definition of ω-consistency (p. 67) is actually the definition of ω-incompleteness 8. Conflating the two concepts is particularly unfortunate at this point, since the discussion concerns precisely Gödel s proof that ω-consistency implies ω-incompleteness. The book also has its share of historical blunders. I cite just two related examples, the Schwarzschild and desitter solutions of the Einstein equations, treated on pp : When Karl Schwarzschild discovered in 1916 that if a star began an extreme gravitational collapse into itself, its mass would eventually reach a critical point after which space-time would be so severely curved that nothing inside (what is now known as) the event horizon, including light, would be able to escape, Einstein dismissed the Schwarzschild 8 I thank Martin Davis for pointing this out to me (personal communication, December 25, 2005). Davis 2001 includes an excellent chapter on Gödel s contributions to logic. August 2007 Notices of the AMS 863 What Did Einstein Know and When Did He Know It? Yourgrau is not alone in propagating the myth that Einstein was taken by surprise when presented with Gödel s results. Stephen Hawking states: It was therefore a great shock to Einstein when, in 1949, Kurt Gödel discovered a solution that represented a universe full of rotating matter, with closed time-like curves through every point (Hawking 2002, p. 90). Actually in 1914, almost as soon as Einstein realized the need to introduce a non-flat, dynamical space-time metrical structure, and well before he arrived at the final form of his field equations, he worried in print about the problem of closed time-like world-lines. Since his words seem little known, perhaps it is worthwhile to present here what Einstein wrote then: I shall now raise an even deeper-reaching question of fundamental significance, which I am not able to answer. In the ordinary [i.e., special JS] theory of relativity, every line that can describe the motion of a material point, i.e., every line consisting only of time-like elements, is necessarily non-closed, for such a line never contains elements for which d x 4 vanishes. An analogous statement cannot be claimed for the theory developed here. Therefore a priori a point motion is conceivable, for which the four-dimensional path of the point would be an almost closed one. In this case one and the same material point could be present in an arbitrarily small space-time region in several seemingly mutually independent exemplars. This runs counter to my physical imagination most vividly. However, I am not able to demonstrate that the theory developed here excludes the occurrence of such paths (I have modified the translation of Einstein 1914, p. 1079, given in Einstein 1997, pp ). J. S. singularity as a mathematical anomaly with no physical significance (p. 116). What Schwarzschild actually did soon before his untimely death in 1916 was to find the unique spherically-symmetric solution to the vacuum Einstein field equations (i.e., outside any source) and show that the solution is static in this region. No discussion of gravitational collapse of a spherically symmetric source beyond the Schwarzschild radius was published until the late 1930s, and the interpretation of this radius as an event horizon came even later (see below) [see for example Stachel 1995]. Later in 1917, the Dutch Astronomer Wellem [sic!] de Sitter proposed a cosmological model for general relativity in which the universe was not static, as Einstein believed it to be, but rather expanding (p. 117). De Sitter s original interpretation of his solution was similar to that of the Schwarzschild solution: a static model with a singularity. This interpretation was accepted by Einstein and others, and debate raged over the interpretation of both of these presumed singularities. It was not until , when Lanczos found a singularity-free but nonstatic form of the de Sitter metric, that it began to be interpreted as an expanding universe (see for example Kerszberg 1989). In 1924, Eddington similarly found out how to remove the Schwarzschild singularity, but this did not become common knowledge among relativists until Finkelstein rediscovered it in A World Without Time? Yourgrau s views on the impact of relativity theory on the concept of time often clash directly with Einstein s. Following Gödel, Yourgrau identifies the concept of time with that of global simultaneity ( simultaneity, and thus time ). They proceed to reject the reality of time because there exist cosmological models (such as the Gödel universe), in which no such concept of cosmic or global time can be defined. Einstein, on the other hand, in his Autobiographical Notes (in Schilpp 1949, the same volume as Gödel s article), lists the insights of a definitive nature 9 that physics owes to the special theory of relativity [my emphasis JS]. He gives pride of place to the insight: There is no such thing as simultaneity of distant events (Es gibt keine Gleichzeitigkeit distanter Ereignisse) (translation from Einstein 1979). If this is Einstein s view of special relativity, Yourgrau s assertion is surely wrong that the father of relativity was shocked (p. 7) by Gödel s demonstration that there are cosmological models in general relativity, for which no global definition of distant simultaneity is even possible. Indeed, Einstein took this so much for granted that he does not even mention it in his comments on Gödel (Einstein 1949). Since the exclusive identification of the concept of time with that of global simultaneity is the crux of Gödel s argument for the unreality of time, let us pause for further discussion of this point. Surely, we all have some intuitive concept of time. Does it embrace the concept of a unique cosmic or global time, marching forward in lock step throughout the entire universe? The only intuitive concept of time that I have is a purely local one, associated with my progress through the universe. And I seriously doubt that, without a good deal of education, anyone has an intuition that the march of his or her local time mu

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