Gödel: From the Pure Theory of Gravitation to Newton’s Absolute1
p. 57-79
Texte intégral
1 On the relevance of cosmology for the understanding of Gödel’s philosophy
11.1 The long-term aim of the project Kurt Gödel Philosopher: from Logic to Cosmology is to disclose Gödel’s philosophy. Its short-term aim is to overcome some unjustified divisions or overly narrow views of fields of research, instituted by present-day academic customs, that prevent us from having a fair and integrated picture of Gödel’s philosophical views. The division between science and philosophy, or the idea that cosmology is alien to philosophy, are the most striking examples of what has to be overcome. Gödel himself said “When [he] entered the field of logic, there were 50 percent philosophy and 50 percent mathematics. There are now 99 percent mathematics and only 1 percent philosophy; even the 1 percent is bad philosophy”.2 The same thing could be said of 20th century cosmology, that in its Golden Age from 1917 to the forties, it was made of one third philosophy, one third mathematics and one third physics.
2Nonetheless, one could be reluctant to accept the definiens of the project. Several scholars, from different fields, have described Gödel’s contributions to relativity theory as a mere excursion3 and Gödel himself said that it was a digression4. This paper tries in some way to challenge this broad idea of the independence of cosmology from the Grundlagenforschung in Gödel’s works. But, beforehand, it is worth looking at some arguments that support such a working hypothesis.
3There are two ways to understand the definiendum of the project: a) an inner “relative” one, and b) an outer “absolute” one.
4a) If one feels concerned by Gödel’s philosophy, it is because he published masterworks that were quite influential on 20th century science and because these achievements blurred the distinction between science and philosophy. Gödel described the relation between science and philosophy in his works with a somewhat enigmatic formula:
5“My work is an application of a philosophy suggested outside of science and obtained on the occasion of thinking about science”. (Wang 1996: 9.9.2)
6According to Gödel’s own judgement about his works, made in 1968, five of them are important5. He mentioned among them the set made of his two papers on cosmology and his philosophical paper on time; he considered these three papers as a whole.6
7Thus, from the inner, relative (to Gödel), meaning of “Gödel Philosopher”, cosmology should belong to philosophy: his work on general relativity is an application of a philosophy suggested outside of general relativity and obtained on the occasion of thinking about it.
8One can draw a methodological consequence of the preceding statement. Gödel felt concerned by two philosophers, Leibniz and Husserl, and, in mathematics, he held constantly to Platonism. His last work was published in 1958. He began to read Husserl in 1959. Hence, to understand the content and nature of the “philosophy suggested outside of science” that Gödel applied to obtain these scientific results that everyone admires, one should not encumber the inquiry with the study of a possible influence of Husserl on Gödel’s philosophical views. Henceforward, “Gödel’s philosophy” always means philosophy in this restricted sense. In other words, we are concerned here in the first place with the “productive” philosophy of Gödel, the one that is the source of his scientific discoveries.
9There is a gulf between Gödel’s published views in his important works and his own philosophical opinions and convictions. A common feature of these five important works is that they criticize and correct the views of Gödel’s predecessors (Hilbert, Hanh, Bernays, to name some of them). So that they give, so to speak, a picture of Gödel’s philosophy in the negative and, in any case, an incomplete one. Thus, the understanding of Gödel’s works does not exhaust his philosophy. It just gives some of the clearest pieces of it and, there, we have nothing more than a first step towards its presentation. Could it be that, in the end, the issues about cosmology that Gödel touched upon are merely a collection of epistemological remarks, without any meaningful relevance to Gödel’s Weltanschauung? Yes, it could. But, in such a case what was called above the “absolute” meaning of “philosopher” would not be satisfied.
10b) By definition, a philosopher holds a philosophical system. And any philosophical system must give some coherent verdicts on invariant problems of philosophy, like the nature of the continuum or the nature of free will,7 as was the case from the time of Plato or Aristotle until the so-called “linguistic turn” to which Gödel is entirely alien.8 The problem of free will, namely the solution of the aporetic Master Argument, entails a conception of the Universe.9 It may be that Gödel’s philosophical views do not fit with the historically testified definition of a philosophical system. But, in such a case, this would simply show us that the conjectural definiendum “Gödel, philosopher” was a wrong hypothesis. Gödel just holds a Weltanschauung. And not every Weltanschauung is a philosophy, since there is no culture without a Weltanschauung, whereas a lot of cultures have no philosophy.
111.2. For now, we are far from such a delusive outcome because a lot of material evidence shows that Gödel felt constantly concerned by several cosmological questions.
12In the first place, in the notebook Max Phil X, written between March 1943 and January 1944, we found several remarks that Gödel labelled “Phil”, obviously for philosophy, where he expresses his views on subject matters that directly concern cosmology like inertia and the force of gravity (p. 4), the Newtonian concept of force (p. 5), the nature of a rigid body (p. 5-6), light (p. 4, 9, 19), physical theory (p. 19), time (p. 22, 23, 46, 58), motion (p. 44), and space and points of space (p. 66). This list is not exhaustive. What is worth noting is that the Bemerkungen are labelled differently according to their content: “Gr”, for Grammatik or Grundlagen10, “Phil”, “Theol” for Theologie, “Ps” or “Psych” for Psychologie and “Phys” for Physik. Hence, since Gödel makes a distinction between philosophical remarks and physical ones, the fact that he comments on the above notions in the Phil category, instead of the Phys one, means that he believed that the usual cosmological notions of force, inertia, motion, time and so on, are philosophical matters.
13We find further evidence of Gödel’s interest in cosmology in the letter he wrote to his mother on November 7th, 1947:
… I was asked to write a paper for a volume on the philosophical meaning of Einstein and his theory; of course, I could not very well refuse. I am also not sorry that I have accepted and chosen this theme [the relation of Kant to relativity theory], because the problem has always interested me and its fundamental investigation has in addition led to purely mathematical results… (Wang, 1987: 38; emphasis added)
14As a matter of fact, several elements allow us to suppose that the problem has always interested Gödel. He became acquainted with, and interested by, Kant’s philosophy in 1922.11 We know from Karl Menger (1994) that he was looking for exact solutions of Einstein’s equations of gravitation in the early thirties. His Nachlass shows that he had extensive knowledge of physics in his youth, both theoretical and experimental, and there is no reason to believe that this interest ceased or decreased with time: wasn’t he looking for integrals for Einstein’s equations in a period where his first concern was logic and set-theoretic foundations of mathematics?
15Some of the most dramatic evidence of Gödel’s concern with the group of problems on time and relativity theory is to be found in the two following remarks of Max Phil X
16(1) Bem<erkung> (Phil<osophie>): Welches ist die richtige Auffassung der Zeit: 1.) die Zeit verläuft„ objektiv “. Wirklich ist nur die Gegenwart, die vergangenen Ereignisse sind nichts (nicht wirklich). 2.) die„ Einstein-Kantische Auffassung “: das Vergehen der Zeit besteht in der Änderung unseres Gesichtspunkts, die vergangenen Ereignisse sind ebenso wirklich wie die gegenwärtigen. (p. 23) (This is just an excerpt of the Bemerkung)
17(2) Bem<erkung> (Phys<ik>): Zwei Auffassungen der vierdimensionalen Welt. Entweder 1. als etwas starr Existierendes <oder> 2. mit einer dreidimensionalen Ebene, die sich darin„ bewegt “(oder überhaupt nur dreidimensional) (p. 10)12.
18Scholars familiar with Gödel’s works can recognize immediately in the„ Einstein-Kantische Auffassung “mentioned in the philosophical remark the very subject of his paper for the volume Albert Einstein, Philosopher-Scientist. They can also see in the physics remark (the second of the two remarks above) the allusive description of Einstein’s static universe (and Gödel’s stationary universe) in the first Auffassung and Gödel’s rotating universes in the second one. All of this, some three years before Schilpp asked Gödel for a paper on Albert Einstein Philosopher-Scientist...
19Hence, on the whole, Gödel’s excursion into cosmology was preceded by lengthy preparation and this was not a journey into the unknown. Furthermore, his interest in cosmology did not cease with the publication of his three papers since “not long before he died, he called Freeman Dyson to inquire about the latest observational evidence for rotation; and when Dyson informed him there was none, he was unwilling to accept the conclusion”. (Dawson, 1997: 182).
20But the most compelling reason, which, indeed, is a philosophical one, to connect logic and cosmology is the nexus given by some remarks in the Max Phil notebooks. It is a problem relating to physical time13 that gave raise to Gödel’s published papers on general relativity. The need to clarify the meaning of physical time comes from two directions.
21The first one is the search for something like Leibniz’s scientia generalis. Gödel describes the way to reach this aim in the following remark.
22(3) Bem<erkung> (Phil<osophie>): Die Analogie zwischen Physik und Psychologie (wobei der physik<alische> Raum übergeht in den logischen Raum) ist dasjenige, was zur allgemeinen Theorie der Welt (sci<entia> gen<eralis>) führen muss. (Max Phil XI: 70). From another direction, in the first remark of Max Phil X, Gödel asks himself how to construct logic in order to teach it to someone who only knew some basic concepts (all, set, not, esti, and). After several queries that, here, do not concern us, he reaches the following conclusion:
Vergangenheit und Gegenwart entsprechen irgendwie: Objekt und Subjekt. Aber auch: Gegenstand und Präd<ikat>. Und schließlich: Ding und Modifikation. Das Vergehen der Zeit besteht darin, dass das Subjekt immer wieder <einen> anderen Inhalt hat, und der Zweck ist, dass wir etwas lernen, was nur dadurch geschehen kann, dass dieser Inhalt zum Objekt wird. Und„ Anwendung “ist tatsächlich die Grundoperation des Verknüpfens. Die Zeit ist die Form des Denkens? [Im Gegensatz zur Form der objektiv logischen Beziehungen.]
23These two remarks, taken together with remark (1) above, show that to uncover Gödel’s Weltanschaaung one cannot put aside his work on physical time (and space). Such a presentation goes largely beyond the scope of the present paper. Here I just want to give a general and, I hope, fair description of Gödel’s contributions to relativity theory. As a matter of fact, even if today we have clear, lucid and readable accounts of them,14 some essential aspects have still been neglected in the literature. In particular, two points should be emphasised: 1) these works allow us to recover a part of Gödel’s lost conversations with the “Philosopher-Scientist” Einstein, his beloved friend; and 2) in his set of papers and lecture on relativity theory, where he speaks in the first place to Einstein, Gödel put an end to the Newtonian pure theory of gravitation, in the sense that: a) he completed it, and b) he ruled out its contradictions.
24These are merely some of the first steps towards the short-term aim of the project “Kurt Gödel Philosopher”.
2. The reception of Gödel’s works
252.1. We saw above that the “fundamental investigations” of the relationship between Kant and the relativity theory led Gödel to purely mathematical results. Let’s recall briefly why it was so.
26Gödel’s first aim, in “A remark about the relationship between relativity theory and idealistic philosophy” (Gödel, 1949; subsequently cited here as “A remark”), was to show that “One of the most interesting aspects of relativity theory for the philosophical-minded consists in the fact that it gave new and surprising insights into the nature of time, of that mysterious and seemingly self-contradictory being which, on the other hand, seems to form the basis of the world and our own existence”.
27Since, says Gödel, the discovery of the relativity of simultaneity is the very starting point of special relativity, there can be no linear ordering of events but just a partial ordering of them. Hence, there can be no “layers of « now » which come into existence successively”. “Each observer has his own set of « nows », and none of these various systems of layers can claim the prerogative of representing the objective [one]”.
28Now, the notion of change depends of the notion of an objective lapse of time, which is an interval between two objective “nows”. Hence the notion of change is deprived of objective reality: it is an appearance or an illusion.
29In general relativity, the same situation prevails as in special relativity. They are still different systems of “nows” associated to each observer. But with relativistic cosmology the situation changes: the local times of each (physically representative) observer fit together into one world-time. So, one can conclude that we still have the intuitive idea of an absolute time lapsing objectively.
30This was a step backward with respect to the direct lessons of special relativity and with respect to the spirit of relativity theory and, of course, this step did not fit with Gödel’s scheme. Hence he had to show that there were other ways to develop relativistic cosmology, ways which do not lead to the conciliation of the various sets of “nows” into a unique, hence objective, set of “nows”. He had to show (and he did show), that, in general, in cosmological solutions of Einstein’s equations of the gravitational field, the local times of each observer cannot fit together into one world-time.
31These were the mathematical investigations to which Gödel was led. They are partly reported in the two papers:
- “An example of a new type of cosmological solutions of Einstein’s field equations of gravitation” (1949).
- “Rotating universes in general relativity theory” (1950) (henceforth abbreviated respectively as “An example” and “Rotating universes)”.
322.2. The reception of these papers was, and still is, very contrasting and even paradoxical. John Dawson (1997: 184) rightly compares it with the reception of the incompleteness theorems: “Both discoveries upset firmly held preconceptions. Both came from an unexpected quarter. Both were motivated by philosophical issues outside of the concerns of many in the scientific community. Both had an air of paradox about them that fostered dubiety. And both appeared to be theoretical curiosities, of little apparent relevance to mainstream work in physics or mathematics”. In the case of the papers on relativity theory, one can add two circumstances to explain this lack of understanding: 1) Gödel’s very sketchy style probably explains the hasty interpretations of his peers. 2) The papers came too late: they would have been probably considered as highly relevant in the early thirties when, after its rebirth, cosmology was looking for its way.
33It is methodologically instructive to have a glance at the landscape of opinions about Gödel’s works.
34At one extreme, some scholars, like Otto Heckmann and Engelbert Schücking, promptly exploited them15. At the other extreme, they were completely ignored, as was the case in the “Bible” of gravitation theory by Misner, Thorne and Wheeler (1973)16. In between these two extremes, the average opinion is that Gödel’s model of the universe given in “An example” is deprived of physical meaning. Besides, we also find some first-rank physicists who reproach Gödel, at least implicitly, for some mathematical or physical mistakes.
35It has repeatedly been said that the universe described in Gödel’s first paper was deprived of physical meaning for the following two reasons:
361) From any space-time point of this world, by making a round trip on a rocket ship in a sufficiently wide curve, it is possible to travel into any region of the past, present, and future, and back again; thus, it is said, the principle of causality would be violated because when the relation “A is before B” is satisfied, “B is before A” is also satisfied (I could travel into the past and kill my younger self).
372) It yields no red-shift for distant objects, in opposition to what is observed.
38Here, we consider the first point, the one about the possibility to travel into the past. We delay the examination of the second point to section 4, below.
39According to (Heckmann and Schücking 1962: 443) “that such strange situations might also be present in Minkowski space [namely, the space-time of special relativity, EA] if one changes its connectedness in the large. Here it could be possible, for instance, for a space traveller that his arrow of time had reverse with respect to his surroundings after he had made a round-trip through the universe. All these questions have not yet been discussed adequately since the problem of the space forms of Clifford and Klein has not yet been solved for spaces with indefinite metric”. Furthermore, one can add that as soon as 1917, Felix Klein pointed out that similar paradoxes occur in De Sitter’s universe.17 Hence, in relativity theory, the possibility to travel into the past is more a commonplace possibility rather than being an exception.
40What about the “violation of the principle of causality?” Joachim Pfarr (1981) observes that
“any anthropomorphic reference with respect to the famous two cosmic twins can only be regarded as an illustration for the topological properties of this space-time” [...] Suppose instead of the point [of the space-time] we take a box filled with a gas in non equilibrium state and we guide this box along a closed time-like world line. At the end of the travel we will expect two copies of the same gas in different states with different values of entropy, and there is no need to cite self-identity problems or logical paradoxes.
41Pfarr, to illustrate this point, has computed that, if Gödel’s rocket ship was the Earth as a whole, its radius at the end of the journey would be around 630 meters long. As a consequence, if a compound substance can travel into its own past, it will not be itself at the end of this journey.18
42To conclude with these short remarks, let’s emphasise that Gödel’s main point, namely the fact that the existence of universal time is an exception rather than a rule, is independent of the possibility to travel into the past. In the rotating universes, one cannot, in general, fit the various sets of “nows” in a common one.
43There is no conspiracy against the works of Gödel but the doubts about their scientific correctness expressed by some first-rank physicists did not help scholars to see their general relevance. The most known case is a paper by Chandrasekhar (Chandrasekhar and Wright, 1961) where it is proved that there can be no closed geodesics in Gödel’s universe, assuming, wrongly, that Gödel has claimed their existence.19
44This mistake has been in some sense “renewed” by Roger Penrose’s account in (Kreisel, 1980: 214). Here, referring to Gödel’s rocket ship, Penrose says: “He did not, however, consider the vastly ‘cheaper’ but equally paradoxical possibility of an observer merely sending a signal into his own past”. This objection is like Einstein’s answer to Gödel (Schilpp 1949: 687).20 Today, we know that Gödel had taken into consideration this kind of objection in his Lecture on rotating universes:
[...] a null line is the path of a possible light signal. It is not necessary the path of a light ray in vacuum because it need not be a geodesic line [...] But by means of a sufficient number of mirrors, you can force a light ray to go along any null lines approximating the given null line. If, in particular, you send a light signal along a null line running back into itself, this means that the light signal will come back into itself, at exactly the same moment at which it is sent. (CW III, pp. 284-85)
45Of course, in 1980, Penrose could not have known about the Lecture published in 1996. But, substantially, his objection is similar to the one by Chandrasekhar: Gödel never claimed that closed time-like lines in his universe are geodesics, neither not null nor null. Besides, it would be impossible to discriminate the received from the sent light signal. Furthermore, in the German translation of “A remark”, Gödel, thinking most probably about Einstein’s form of the objection, a complement to his footnote 11, on the possibility of “telegraphing a message into one’s own past: ”
[...] the practical difficulties in doing so would hardly seem to be trifling. Moreover, the boundary between difficulties in practice and difficulties in principle is not at all fixed. What was earlier a practical difficulty in atomic physics has today become an impossibility in principle, in consequence of the uncertainty principle: and the same could one day happen also for those difficulties that reside not in the domain of the “too small”, but of the “too large”. (CW II: 205)
46Thus, it is difficult to say what would be “cheaper”: to install, far away in outer spaces, a sufficient number of mirrors, or to send a material rocket ship along a sufficiently wide curve.
47One could write a whole book about the reception of Gödel’s works, but the considerations above are enough to show how it is difficult to rely on the divergent opinions of physicists in order to get a fair picture of Gödel’s achievements in relativity theory. The best option is to rely on one single opinion, for instance the one reported by Oskar Morgenstern: “Einstein told me, that Gödel’s papers were the most important ones on relativity theory since his own (Einstein’s) original paper appeared”.21 And, of course, to turn to Gödel himself.
3. Gödel’s “reasonable” models of the universe
48Gödel has given very accurate and informative statements about his works in the two letters he wrote to Carl Seelig in 1955.22 The most important passage in these letters is the following one:
My own work on the theory of relativity relates to the pure theory of gravitation published in 1916 which, I believe, was left, not only by Einstein himself but also by the whole generation of contemporary physicists, in its state of a torso, physically, mathematically, and with respect to its application in cosmology. (Emphasis added)
49Here “pure theory of gravitation” means two different things: not mixed and not applied. Not mixed like, for instance, G. Lemaître’s theory of the primordial atom, which is considered by cosmologists as an anticipation of present day standard cosmology (expansion of the universe from an initial singularity), where atomic physics is combined with general relativity. One can suppose that for Gödel such a mixture ought not to be done before the pure theory of gravitation was put in a better form than a torso, namely, before it was completed. Gödel, in this passage, makes a distinction between the pure theory and its application in cosmology. In his second letter to Seelig, after he has listed some important mathematical problems which were still not resolved in general relativity, he points out that he devoted himself “to a less difficult complex of questions from general relativity theory, namely cosmology”. Thus, it is somewhat misleading to say that the non-expanding universe described in “An example” belongs to the field of cosmology. It concerns the pure theory of gravitation.
50S. Hawking, in his introduction to both papers (CW II: 189-190), after he had recalled that there is a general agreement on the fact that Gödel’s non expanding universe cannot be our universe, says that Gödel’s second paper “describes more reasonable rotating cosmological models that are expanding and that do not have the possibility to travel into the past”.
51Although this leads us away from our questions, a few words should be said about Gödel’s achievements in “Rotating universes” because one would have a very pale idea of them in believing that here Gödel just proposed some more reasonable models of the universe.
52In cosmology, the material substratum of the universe, namely the galaxies, is often assimilated to a fluid. As (Heckmann, 1961: 600) put it, “Theoretical cosmology is a very rudimentary form of hydrodynamics”. The motion of this cosmic fluid is described by three different magnitudes23:
- a scalar of expansion
- a vorticity vector
- a shear24 tensor
53In the models discussed in “Rotating universes” none of these quantities vanish. In the universes of the “whole generation of physicists”, the second and the third vanish, in Einstein’s universe all of them vanish.25 In fact, for the last magnitude, the principal missing part is also the omission of rotation, because, according to a theorem proved by Gödel, every expanding rotating universe (provided that it does not contains time-like closed curves) can at no moment of time be rotationally symmetric around the vector of rotation. Hence, expansion plus rotation entails shear.
54From a more general point of view, Gödel gave up “half” of the so-called Cosmological Principle. According to this principle, the universe is isotropic (all spatial directions are the same at every point) and homogeneous (no feature distinguishes any point from any other).26 Isotropy entails homogeneity but the converse does not hold. Thus by weakening this principle, Gödel opened up the study to a wider class of models in accord with the fact that the isotropic models form, so to speak, a set of measure zero amongst the totality of all models permitted by the equations of general relativity. It is in this sense that Gödel says that the whole generation of physicists left in a state of a torso the application of general relativity to cosmology and, in fact, it was, and still is, a big blind spot.
55Now, from Gödel’s viewpoint, are the models described in “Rotating universes” more reasonable because they are expanding? The answer is no. The conclusion he gave to his description of the different classes of rotating universes is very explicit on this point: “Thus the problem arises of distinguishing, by properties of symmetry or simplicity, certain solutions in this vast manifold of solutions. E.g., one might try to require that the universe should expand from one point and contract to one point” (CW II: 215).27 Expansion, i.e. observed red shifts of distant objects, is seen from a small corner of the world. One cannot exclude that, from another corner of the world, blue shifts could be observed.
56From an epistemological point of view, the most important feature of rotating universes is that they avoid singularity and allow a smooth transition from contraction to expansion or vice-versa. It is true that there is a subclass of always expanding rotating universes (with shear, as said above), but the remark (2) from Max Phil X, cited above in section 1, shows that there is no room in Gödel’s worldview for the idea of a birth of the universe, in other words, for a Big-Bang. As Georg Kreisel put it, for Gödel, the universe just is! It is worth recording that at the end of the forties, when Gödel published his papers, the prevailing cosmological model was the Steady-State Theory, precisely because it avoids an initial singularity. But to do so, Hermann Bondi had to postulate a continuous creation of matter and to violate the classical principles of conservation. With the rotating universes, Gödel avoided singularity in keeping cosmology within the standard framework of relativistic cosmology and classical physics.
57Moreover, strictly speaking, we face here a question of terminology. When Lemaître proposed the hypothesis of the primordial atom, he rightly called his model a cosmogony, not a cosmology, as do present day cosmologists who follow this hypothesis. A hypothesis which, in some way, brings us back before the Ionians, and the birth of science and philosophy, who progressively substituted cosmology for cosmogony.
4. Gödel’s completion of the pure theory of gravitation
58We can come now to the physical meaning of the universe of “An example”.
59Beforehand, it should be stressed that this paper, like “A remark”, is aimed at Einstein. Several elements support this evidence: the paper was published in a special issue of Reviews of modern physics in homage to Einstein for his 70th birthday; it refers to Einstein’s universe in several places; there is a strict parallelism between the introductions of both Einstein’s Kosmologische Betrachtungen zur allgemeine Relativitätstheorie (Einstein 1917) and Gödel’s Lecture on rotating universes where both begin by a discussion about the problem of the potential of gravitation in Newtonian theory and its relationship with the potential in general relativity. This cannot be a mere coincidence and it is clear that Gödel, in that Lecture, has in the back of his mind Einstein’s attempt to complete his 1916 theory. But, independently of historical, textual or epistemological considerations, there is a drastic argument to believe that “An example” has something to do with Einstein’s universe, namely the theorem stated by Gödel:
Theorem: Gödel’s stationary universe and Einstein’s static universe are the only spatially homogeneous cosmological solutions with non-vanishing density of matter and equidistant world lines of matter.
60All of this shows that the hypothesis that Gödel’s concern was, in the first place, to show to Einstein what was wrong with his theory of general relativity as a pure theory of gravitation merits being taken into consideration. Indeed, the comparison of both conceptions of theory of gravitation is the best way to illustrate another passage of Gödel’s letters to Seelig: “I have often pondered why Einstein took pleasure in his conversations with me, and I believe one of the causes is to be found in the fact that I frequently was of the contrary opinion and made no secret of it”. As a matter of fact the study of Gödel’s contributions to relativity gives us a way to recover a part of the lost conversations between the two great philosopher-scientists. Such a study displays the following (open) list of topics where Gödel was in opposition to Einstein: the Cosmological Principle, the relationship between (local) physics and cosmology, fundamental physical objects, motion, the cosmological constant, inertia and gravitation, interpretation of Newton, Mach’s Principle, the relationship between mathematics and physics, time.
614.1. Einstein acknowledged the unfinished state of his 1916 theory and he published his 1917 Kosmologische Betrachtungen to fill what he believed to be an important gap in this theory. So, even if this paper is the birth of contemporary cosmology, because here Einstein proposed a cosmological model, he did not want so much to give a System of the World, as we are told in histories of cosmology. His aim was rather to overcome some difficulties of the 1916 theory and to give it complete Grundlagen. What were these difficulties? How did Einstein attempt to solve them? What was satisfying and unsatisfying in this attempt?
62These issues have been extensively studied in the literature and the sketchy description below is simply aimed at recalling the general context of the subject matter of Gödel’s reflexions.
634.1.1. According to Einstein, the 1916 theory is local. To deal with the problem of planets, he applied the equations of general relativity in the flat space-time of special relativity. Hence, in this theory, space-time is a kind of mixture, with pieces of general relativity immersed in the manifold of special relativity.
64Special relativity space-time shares a feature with Newton space. One can, in these frames, picture the material world of celestial bodies as a floating island in a void space. This picture is not very satisfying for the mind because it involves the idea of a centre of the world, which is no better that the Aristotelian view, according to Einstein. But, above all, it raises a physical difficulty. A part of the radiation of the celestial bodies that are on the edge of the “island” should get lost in the infinite void and even some of the celestial bodies could escape from the island. Thus this universe would become impoverished. The formal counterpart of this difficulty is that there is no satisfying way to substitute a potential function (Poisson‘s equation) to Newtonian distant action, so no satisfactory way to define the potential of gravitation at the boundaries of the island.
65Inertial mass, in the 1916 theory, is influenced (beeinflusst) by matter. This left Einstein dissatisfied because, among other things, according to the first principle of his general relativity, inertial mass and gravitational mass are identical (Principle of Equivalence). But gravitation is clearly an effect of matter; hence, inertia should be determined (bedingt) by matter. This is the “Mach’s Principle”, a principle that he added explicitly to the two principles of Equivalence and Covariance in (Einstein 1918).
664.1.2. To get rid of all these puzzling questions that are, of course, linked together, Einstein proposed his famous spherical cosmological model. Instead of the pseudo-Euclidean open space-time of 1916, we now have a space that is a 3-dimensionnal sphere. On this closed hyper surface, there are no more boundaries, hence no problem of the definition of the potential at the limits. Also, there is no more centre: every point of the sphere is identical with any other point. This universe cannot be impoverished because when a celestial body escapes from the gravitation caused by other ponderable bodies, it remains on the sphere. Last, but not least, from now on, inertia is determined by matter (more on this below, sec. 5.1.).
67Besides the fact that Einstein’s solution to this complex of problems seems more dialectical than physical, it also has two shortcomings.
68There is too much matter in this universe and it collapses in on itself under the effect of gravitation. Hence, Einstein had to add a new ad hoc term in the equations of the field of gravitation, the so-called cosmological λ-term,28 which acts as a pressure to balance the effect of gravitation. This modification was of no use. Einstein’s solution does not give a potential and it was quickly proved that his universe was unstable because the ad hoc repulsive force is proportional to distance whereas gravity is inversely proportional to the square of distance.
69The second shortcoming is worse: in this universe, there is an absolute time. Indeed, it is a double shortcoming. On one hand, this feature directly contradicts Einstein’s convictions on the nature of time.29 On the other hand, it violates the second principle of Einstein’s general relativity, the Principle of Covariance (or Principle of General Relativity), which says that the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion, because it restricts the possible substitutions on the time coordinate.
704.2. Gödel corrected these shortcomings, so to speak, in one move: rotation entails: a) the independence of Mach’s Principle from the other postulates of general relativity; and b) the impossibility, in general, to fit in one universal time the local times of different observers. Today, these features of his world model are well known.30 Yet, it is important to understand why his demonstrations are so cogent.
71Criticisms against Mach’s Principle and cosmic time were not new in 1949. When Einstein published his universe, in 1917, De Sitter reacted immediately against its shortcomings.31 He observed that the introduction of a cosmic time violates the Principle of Covariance and, like Gödel, he proposed a counterexample, a countermodel, to Einstein’s Machian solution in showing that a void universe is a solution of the field equations: no matter, hence no inertia determined by the matter. But, for Einstein, such a universe was deprived of physical meaning because we are not living in a void universe. The Einstein-De Sitter controversy generated a lot of discussions that gave rise to 20th century cosmology. Nonetheless, at that time, physicists have lost the question of the general covariance somewhere along the way. Gödel’s intervention should be understood as if it occurred in the context of these discussions. But with the supplementary condition that there are some agreements between him and Einstein: the universe is not void, the theory of gravitation must be exempt of any contradictions, the world has no centre, relativity theory entails inexistence of simultaneity. On the basis of these agreements, Gödel adds two things:
721) The method of inquiry must be philosophically neutral. Namely, the resources of group theory must command it. When he says that Einstein and the whole generation of physicists left the theory as a torso mathematically, he just means what we can read at the beginning of “An example”:
All cosmological solutions with non-vanishing density of matter known at present have the common property that, in a certain sense, they contain an “absolute” time coordinate, owing to the fact that there is a one parametric system of three-spaces everywhere orthogonal on the world lines of matter. It is easily seen that the non-existence of such a system of three-spaces is equivalent with a rotation of matter relative to the compass of inertia.32
732) The simplicity of the ways. Einstein’s universe has equidistant lines of matter, so it can be viewed as a rigid body, which is the simplest object of physics. But this universe is unstable. Can a universe be “als etwas starr Existierendes”, namely to be like a rigid body and be stable? Gravitation binds the parts (the galaxies) of the rigid body. This body would collapse into one point on the sole action of gravitation, but, if it rotates, centripetal forces can balance gravitation. Thus, the problem becomes a geometrical one: how can a body rotate without having an axis of rotation, without introducing a kind of centre of the world? Consultation of old treatises on Lie groups (Bianchi 1918), and Gödel’s ingenuity, gives the solution. Through each point of Gödel’s space-time there passes an axis of rotation.
74One of the requirements of Einstein’s work was that general relativity allowed us to recover Newton’s law of gravitation as a limit. Now, a rigid body can be studied through a sole magnitude: angular momentum; and the classical and the relativistic expressions of angular momentum are identical. The value of angular momentum increases when the angular velocity increases. The centrifugal forces increase when the angular velocity increases. Since gravitational forces are assimilated to the binding forces that hold together the particles of the rigid body, the value of the gravitational forces will increase to compensate centrifugal forces. We have Newton’s law of gravitation for a given value of the angular velocity and Einstein’s law for another (greater) value.33
5. Some philosophical issues
75Some philosophical issues are clinging to this train of thought. Among them we find time, the notion of rigid body and inertia.
765.1. Under the influence of Hermann Weyl, Howard Robertson, the sole cosmologist cited by Gödel, raised absolute time to the status of a postulate under the label “Principle of Cosmic Time”. Merleau-Ponty (1965: 78) observes rightly that this hypothesis is a rather confused one. It has “une grande diversité de significations: tantôt elle est donnée comme un simple corollaire de l’isotropie spatiale, tantôt comme un axiome justifié par sa seule simplicité [...], tantôt comme la traduction abstraite d’un fait empirique (les vitesses locales de la matière sont faibles), tantôt comme une sorte d’évidence ontologique—l’existence d’un contenu matériel de l’Univers entraînant d’elle-même la ségrégation de l’Espace et du Temps”.
77As such, it was a natural target for critical minds like Einstein34 and Gödel. Einstein can thank his friend for having corrected his own opinion.
78The “technical part” of Gödel’s papers is quite clear. The “philosophical part” of the set of papers, including the five preliminary drafts of “A remark”, is far from being clear. One cannot help but follow Howard Stein in his introductory note to these drafts: all that Gödel writes on interpretation of the role of observer35 in relativity theory, interpretation of Kant, similarities between Kant and the theory of relativity, is rather puzzling.
79If it is difficult to understand what his conclusions are, one can, at least, make a plausible hypothesis about Gödel’s former intention.
80Einstein has constantly expressed his reluctance about Kant. Gödel could not ignore it. Let’s suppose that he has shown that Einstein’s relativity theory and Kant share the same conception of time. Faced with such a demonstration, Einstein, who expresses his agreement with Gödel’s analysis “entirely aside from the relation of the theory of relativity to idealistic philosophy or to any philosophical formulation of questions” (Schillp 1949: 687), could have replied by saying what he had already said to Léon Brunschvicg, in 1922, in the discussions at Société française de philosophie: everybody has his own Kant. Hence, we can go one step further and ask: What conclusion did Gödel intend, anyway, to draw from such a similarity? Was it a vindication of Kant by a modern scientific theory or the proof of a defect, or a lacuna, of relativity theory? The Bermekung (1) partly quoted above (1.2.) shows that this alternative is sound for Gödel. A larger excerpt of the Bermekung shows towards which branch of the alternative, Gödel is tempted to go:
81Bem<erkung> (Phil<osophie>): Welches ist die richtige Auffassung der Zeit: 1.) die Zeit verläuft„ objektiv “. Wirklich ist nur die Gegenwart, die vergangenen Ereignisse sind nichts (nicht wirklich). 2.) die„ Einstein-Kantische Auffassung “: das Vergehen der Zeit besteht in der Änderung unseres Gesichtspunkts, die vergangenen Ereignisse sind ebenso wirklich wie die gegenwärtigen. Nach der zweiten Auffassung ist die „ Zeitlichkeit “nicht nur wirklich, sondern sogar ewig; nach der ersten nicht einmal wirklich, da sofort wieder vernichtet. Die Zweite <ist> inkonsistent, da wir auch ein Teil der objektiven Welt sind. Also muss man sagen: Unser Gesichtspunkt ändert sich nicht, sondern scheint sich zu ändern, und das ist ein Irrtum oder ich bin nicht ein Teil der Welt, [24] sondern„ über der Welt “[indem ich ja die Welt selbst mit erschaffe]. Nämlich: ich bin nicht Eines, sondern Vieles [in jedem Zeitpunkt zeigt sich ein anderes Wesen oder ein anderer Teil von mir]. Und die Aussage: „ Jetzt ist X wirklich “bedeutet bloß„ X ist mit einem bestimmten Teil meines Wesens seinsmässig verbunden “.
82The Einstein-Kantian conception of time is “inkonsistent” because it leads eventually to one of the following conclusions: either there are no modifications of our viewpoint or I am not one but several.
83It was impossible to conclude from the reading of the manuscripts that Gödel holds that the Einstein-Kantian conception of time is “inkonsistent”.
84It is in this way that Gödel’s “cosmology” leaves us on the threshold of philosophy, which, for us, is the threshold of the long-term aim of the project Kurt Gödel Philosopher: from Logic to Cosmology. The same situation occurs with rigid body and inertia.
855.2. “Rigid body” does not belong to the current language of cosmology. Cosmologists consider two possibilities for describing the material content of the universe: as a fluid or as a dust. Fluid, which is the material model considered above in section 3, is characterized by two properties, namely density and pressure, whereas the dust model is pressureless and characterized by the density of matter alone. These notions have more an analogical status than a physical one. In Gödel and Einstein’s universes the λ-terms have the same value but an opposite sign. Gödel, in “An example”, says of his λ-term that “it corresponds to a positive pressure”. He does so most probably because, on one hand, Einstein (1917), in his reasoning, relies on an analogy with kinetic theory of gases and uses the word “pressure” and, on another hand, the term became standard in cosmology. But the analogy with mechanics, introduced in the more relaxed context of the Lecture on rotating universes by the term “rigid body”, is clearer than the one with kinetic theory of gases and it is more fruitful as a heuristic principle. Furthermore, to this new opposition with Einstein, one could also add that, for him, the main progress of modern physics was the substitution of the concept of field to the classical concept of ponderable body.
86In the Max-Phil notebooks Gödel goes to the bother of finding a grounding for impenetrability and stability, the distinctive features of rigid bodies. In doing so, he speculates not logically, but analogically. Since analogical reasoning is also at work in explaining what inertia is, I will say a few words on that before quoting some illustrative pieces of the notebooks on the issues of analogy and rigid body.
875.3. Einstein believed that the Principle of Equivalence was the best idea of his life. The postulate of the equivalence of inertial and gravitational mass was also Einstein’s starting point in building up his theory of general relativity. The consequence of this postulate is that free falling bodies, like Newton’s apple, are at rest (see 6.1 below). Also, there are no longer any forces of attraction between ponderable bodies. The field of gravitation has taken the place of forces. Free falling bodies describe geodesics, namely they follow the shortest path between two points of space-time. Together with the principle of homogeneity, according to which the distribution of matter is uniform in the whole universe, the Principle of Equivalence leads eventually to a universe in which geometry is of constant curvature. The principle of general relativity is not involved in this geometry. It is needed to explain the local deformations of its constant curvature. But it is to be expected that this geometry does not contradict the principle of general relativity. We have already seen above that Einstein’s universe contradicts this principle, as it does for the main consequence of the principle of special relativity, namely the merging of space and time.
88In the sketchy reconstitution of Gödel’s reasoning given in 4.2, neither the Principle of Equivalence nor the Principle of Special Relativity are mentioned. The concepts involved were force, rigid body, motion plus an open-minded conception of geometry. Gödel deduced Einstein’s law of gravitation with this apparatus. This law is nothing more than a general isometric property of space-time. And in every point of Gödel’s geometry simultaneity is a relative notion, as required by the Principle of Special Relativity, space and time are merged as it was the case in Minkowski’s space-time. This is another facet of the physical meaning of Gödel’s universe, where this meaning focuses on method of physics: knowledge of local physical laws comes from knowledge of the whole universe. Knowledge of the parts comes from knowledge of the whole. We have here a new instance of this “virtuous” circle principle36.
89Incidentally, a point should be here emphasised. There are infinitely many exact solutions of Einstein’s equations of gravitation, infinitely many possible cosmologies. Faced with this fact, Edward Milne, in the early thirties, made fun of these relativists who were expecting that observational data would choose for them the structure of the Cosmos. Consequently, Milne proposed deducing the local physical laws from general cosmological postulates. This was also, in some way, the attitude of Bondi and Hoyle with the Steady-State theories. This is why Merleau-Ponty (1965) rightly divides the main cosmological models in two classes: inductive cosmologies, like the relativistic ones, that extend the physical local laws to the whole universe, and deductive cosmologies that do the opposite. Once again, we find that Gödel is a special case because his relativistic cosmologies are deductive37. Hence, Gödel also showed that to be deductive, as required by Milne, cosmology does not need to give up relativity theory, as Milne did.
90To see how concerned Gödel was by the notions of rigid body and inertia and to check that, for him, these are of a purely philosophical nature that should be analysed by the analogical method, the following extensive quotation of three consecutive “philosophical remarks” from Max Phil X will speak for itself. At first sight, these remarks, especially the first one, should discourage any rationalist, not to speak of the followers of the linguistic turn. But, besides that it is not the first time that Gödel upset preconceptions of scientists and philosophers, one should not lose sight of the facts that 1) these remarks belong to an intermediate stage of this developing thought that Gödel calls “a philosophy suggested outside of science” and that remained yet to be understood and 2) that they are a part of a large corpus. When related to the other parts, their meaning becomes clearer (and not so far from classical philosophy38), as it is the case if we put side by side the first remark on the analogy between physics and sociology and the one quoted below (6.3).
91Bem<erkung> (Phil<osophie>): Analogie zwischen Physik und Soziologie: Elektronen = Männer, Kerne = Frauen, elektrische Kraft = Geschlechtsliebe, magnetische [d.h. Kraft zwischen bewegten Elektronen] = Berufsleben, chemische = social life, Schwerkraft = Religion und Phil<osophie>, Molekül = Familie, Körper = politische Parteien (kleinere eventuelle Vereinigung wie Freimaurerei), Trägheit = Trägheit, d.h. die seelische Kraft, welche jeder Art von Betätigung entgegensteht. Analogie: Schwerkraft: Trägheit = Gott: Teufel. Die Schwerkraft regiert den Himmel und sucht alle Vielheit zu vernichten [sie würde erst aufhören, wenn alle Materie in einem Punkt vereinigt wäre (Der Unterschied zwischen einem und gar keinem Ding scheint aber schon sehr klein zu sein.)]. Beruf betrifft nur Männer. Es gibt viel weniger verschiedene Arten von Männern als von Frauen. Zu den chemischen Kräften gehört auch: Adh<äsion>, Coh<äsion>. Was ist Wärme? Eine in Trägheit bestehende „ Geschaftlhuabarei<?> “? Licht = eine Wirkung von Liebe und Beruf? Licht ist eigentlich nichts Physikalisches, denn die Farbengleichheit ist keine [5] physikalische Eigenschaft mehr.
92Bem<erkung> (Phil<osophie>): Der Newtonsche Kraftbegriff bedeutet irgendwie ein „sinnloses Streben“, indem meistens gar kein wirklicher Zustand existiert, in dem die Kraft aufhört, während z.B. die chemische Kraft aufhört, wenn die Verbindung sich hergestellt hat [d.h. eine gewisse„Vollkommenheit“erreicht ist].
93Bem<erkung> (Phil<osophie>): Vielleicht muss man, um den Zusammenhang starrer Körper zu erklären, annehmen, dass auch die Paare und Tripel Kraft ausüben. [Die Paare wären also so weit entfernt, bloße „Fiktionen“ zu sein, dass sie sogar eine physikalische Wirklichkeit hätten]. Die Paare würden umso stärker wirken, je näher die beiden Teile sind [d.h. je größere physikalische Wirklichkeit sie hätten], und zwar würde ein Paar im Allgemeinen abstoßend wirken. [Es widersetzt sich dem Eindringen eines [6] dritten, denn dieser würde es ja zerstören (also ein Selbsterhaltungstrieb der Paare.)] Vielleicht erklärt sich auf diese Weise das Vorhandensein von anziehenden und abstoßenden Kräften, welche ja für die Struktur des starren Körpers nötig sind. Vielleicht kann man auch solche Gesetze über die Selbsterhaltung von Struktur und Masse für„ Vollkommenheiten “der Struktur der chemischen Affinität erklären? Und vielleicht sogar die Quantenbedingungen <so erklären>, indem nur gewisse „vollkommene“ Strukturen [d.h. solche, in denen ein Max<imum> der Beziehung der Teile vorhanden ist] realisiert sind.
6. A new absolute
94To conclude I will consider a last opposition between Einstein and Gödel. It bears upon the nature of motion because, among the unsatisfactory features of the 1916 theory of gravitation, one also finds an unclear conception of motion.
956.1. Einstein’s building of general relativity can be viewed as a systematic and progressive destruction of the architecture of Newton’s Principia. A fact he expressed with his famous “Newton verzeih’ mir”.39 His criticisms of Newton’s conception of time and space are well known but they relate in fine on the nature of motion.
96Newton in the Author’s Preface to the Reader of the Principia, thinking obviously about Descartes, is quite explicit on the ontological character of motion when he expresses the precedence of mechanics on geometry:
For the description of straight lines and circles, which is the foundation of geometry, appertains to mechanics. Geometry does not teach how to describe these straight lines and circles, but postulates such a description. For geometry postulates that a beginner has learned to describe lines and circles exactly before he approaches the threshold of geometry, and then it teaches how problems are solved by these operations. To describe straight lines and to describe circles are problems, but not problems in geometry. Geometry postulates the solutions of these problems from mechanics and teaches the use of the problems thus solved. And geometry can boast that with so few principles obtained from other fields, it can do so much. Therefore geometry is founded on mechanical practice and is nothing other than that part of universal mechanics which reduces the art of measuring to exact proportions and demonstrations. (Cohen and Whitman 1999: 381-82)
97It could be believed that Newton is just speaking of the mathematical methods of physics, which is true, but, for him, phenomena are mathematical forms that fall under the scope of universal mechanics. And these mathematical forms, more precisely these primitive kinematics objects, pre-existed as such in the world just like, for Plato, numbers-ideas pre-existed to the participation of concrete numbers into actual computations.
98In his criticisms of Newton’s view, Einstein was led, in 1916, to a rather fuzzy notion of motion. Let’s quickly sketch out the main steps of the development of Einstein’s principle of relativity of motion.
99According to the Principle of Special Relativity, there can be no difference between inertial frames at rest and inertial frames in uniform rectilinear motion. Thus, the distinction introduced by Newton’s first axiom of motion (the so called “law of inertia”) between two kinds of kinematical states, namely “being at rest” and “moving uniformly straight forward”, which are both in state of dynamical rest, is abolished.
100According to the Principle of Equivalence, inertial and gravitational masses, which are, respectively, the coefficients that appear in the law of collision and in the law of attraction, are identical. The formulation of this principle does not explicitly concern motion but it has an immediate consequence on its definition. According to the Principle of Equivalence, illustrated by the Gedanken experiment of a free falling elevator, a system of coordinates uniformly accelerated is in a state of dynamical rest, as are the two Newtonian states of absolute rest and uniform rectilinear motion. Whilst the Principle of Special Relativity abolishes Newton’s definition of kinematical rest, the Principle of Equivalence abolishes the distinction between kinematical rest and dynamical rest.
101The next and last of Einstein’s steps to get a generalized principle of relativity of motion is the following postulate: the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion. With this last step, it is the distinction between dynamical rest and dynamical motion that vanishes. Every body can be at rest. One can choose, e.g. a flying butterfly as the centre of a system of reference to describe phenomena. But if everything can be at rest (or in motion), this means as well that nothing is at rest (or in motion). We are coming back to Descartes: “il n’y a point de lieu d’aucune chose au monde qui soit ferme et arrêté, sinon en tant que nous l’arrêtons en notre pensée”. (Principes de philosophie: part II, art. 13)
1026.2. Newton introduced an absolute, true and mathematical (namely not sensible) space precisely to overcome the difficulties related to the Cartesian generalized principle of relativity of motion, a principle identical to Einstein’s one. For Newton, it would have been impossible, e.g. to describe the motion of the Sun inside the Solar system without a true, absolute and mathematical space at rest. All scientists and philosophers have criticized this concept of space since the printing of the Principia. Most of the critics say that Newton’s space cannot be anything but a system of reference defined by distant objects at rest. This, of course, Newton would have rejected because distant objects also move under the attraction of the other bodies and, as such, are no more at rest than the Sun and the centre of mass of the solar system are. The true difficulty of Newton’s concept of space is elsewhere. It lies in what contemporary physicists called equi-locality, namely the fact that it is impossible, in a mathematical (void) space to know if two places are distinct or not.
103The fuzzy concept of motion of (Einstein 1916) becomes clearer in (Einstein 1917). This last paper, with its cosmos, put in physical form what was just a claim in §2 of (Einstein 1916). Two fluid spheres of the same size and nature are in a relative motion of rotation. One of the spheres has the shape of an ellipsoid while the other one has still the shape of a sphere. Newton theory explains the deformation of the rotating sphere by its motion with respect to space. For Einstein such an explanation relies on a fictitious cause. The deformation of the sphere “is partly conditioned, in quite essential respects, by distant masses”.
104On the static 3-dimensional sphere of Einstein’s cosmological model all the material points, namely all galaxies, are at rest relative to the others. Hence, to speak of the rotation of a body, like the fluid sphere, has an understandable meaning: it rotates relatively to the system of geodesics defined by the gravitational field caused by those galaxies that are at rest. Furthermore, as we saw above, the deformation of the liquid sphere is no longer “partly conditioned, in quite essential respects, by distant masses” but determined by them. Thus, Einstein’s completion of general relativity restores the distinction between motion and rest. But, here, the primitive notion is rest: the rest of the universe.
105It is the opposite with Gödel’s stationary universe where motion (rotation) is, as for Newton, a primitive concept.
1066.3. If we rely on the few data at our disposal, it seems that, from 1943 to 1949, Gödel does not demonstrate a particular interest in Newton. What’s more, his views, on the whole, are rather delusive. One remains with the feeling that he contends himself with the traditionnal scientist and oversimplified interpretation of the Principia. On the one hand, in his Max Phil, he speaks in several places about the Newtonian force, mostly to underline that it is insufficient to explain how the world is ruled, as if there was just one kind of force in the Principia.40 On the other hand, we see him criticizing more or less explicitily Newton’s concept of space from an inclined Leibnizian viewpoint. It is the case in the annotation in shorthand he added to the third version of the preparatory manuscript for “A remark”: “Correction: Space does not exist apart from and independent of matter in New. and Einst”.41
107It seems that, in the notebooks, there is just one place where Newton’s theory of gravitation is given an important role:
108Bem<erkung> (Phil<osophie>): Die scientia generalis des Leibniz ist offenbar etwas Ähnliches hinsichtlich des ganzen Gebiets der Erscheinung (d.h. aller Wissenschaften, inkl<usive> Math<ematik>) wie die Newtonsche Physik hinsichtlich der physikalischen Erscheinungen. Die „Cynosura notionum “ besteht dort aus Raumpunkt, Zeitpunkt, Maßpunkt, Lage, Kraft, Masse. Dadurch dass man alle physikalischen Erscheinungen auf dieses System „projiziert“, d.h. es durch sie zu „interpretieren“ sucht, werden die a priori bestehenden Möglichkeiten eingeschränkt, und es sind daher Voraussagen möglich. Daß die Newtonschen Begriffe selbst noch nicht das Gesuchte sind (was die Materialisten glauben), sieht man 1. aus der Mathematik, bei der überhaupt kein Verständnis durch sie möglich <ist>, 2. aus Psychologie und Soziologie, wo prinzipiell ein Verständnis möglich wäre, aber nicht praktisch. (Max Phil X: 67-68).
109This status of a model for the scientia generalis, given by Newtonian physics, sounds very familiar. We already know from Wang’s book (1974: 85), not to speak of the heavy insistence on this project in (Wang, 1996), that Gödel believed that “Philosophy as an exact theory should do for metaphysics as much as Newton did for physics”. The publicity of Gödel’s project came out of his lengthy conversations with Wang. If, on the basis of the present state of textual data, Gödel seems to have been mute on Newton in the forties, the situation was rather different in the seventies. Wang’s transcription of these conversations contained several statements on Newton. Putting together quotations 5.3.11, 9.2.36, 9.2.37, 9.3.13 and 9.5.2 and modifying the order given by Wang, we have:
The beginning of physics was Newton’s work of 1687, which needs only very simple primitives: force, mass, law. The Newtonian scheme was to a considerable extent obtained a priori. Proportionality, space, and time were a priori, while force, which produces acceleration, was empirical. Newton axiomatized physics and thereby made it into a science. Approximately speaking, Einstein filled some gaps in Newton’s scheme and introduced some modifications.
The Newtonian frame is a kind of absolute knowledge. It is a psychological phenomenon. In this sense absolute knowledge is the frame or backbone or axiom system of a good theory. The backbone of physics remains in Newtonism. Experience fills in the gaps after absolute knowledge is obtained. By pure thought, says the Newtonian scheme, we reach the frame.
110Indeed, we are far from Einstein’s “Newton verzeih’ mir”
1116.4. One more absolute
112In his Remarks before the Princeton bicentennial conference on problems in mathematics, Gödel (1946) compares the concepts of definability, demonstrability and computability. He observes that “With the concept of general recursiveness (or Turing computability) one has for the first time succeeded in giving an absolute definition of an interesting epistemological notion, i.e., one not depending of the formalism chosen”. He adds that it is “a kind of miracle” and “This, [he thinks] should encourage one to expect the same thing to be possible also in other cases (such as demonstrability or definability)”.
113Gödel, for whom, like Newton, motion is a primitive concept of physics, has perhaps succeeded in giving a new absolute definition of an interesting epistemological notion because “If the universe rotates, then the whole world moves with uniform motion. That would be the most striking confirmation of the absolutist view one can think of”. (Wang, 1996: 5.4.22).
114Gödel has always been a Platonist in mathematics. His investigations in general relativity, too, lead him to Platonism. This is what he obtained “on the occasion of thinking about science”.
115I have in no way meant that Platonism was the “philosophy suggested outside of science” he applied on that occasion.
Bibliographie
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Notes de bas de page
1 This work has been supported by a grant from Agence Nationale pour la Recherche for the project Kurt Gödel Philosopher: from Logic to Cosmology under the lead of Gabriella Crocco.
2 (Wang 1996: 82). This was in 1972. I do not dare to imagine what Gödel’s diagnostic would have been on today’s logic, nor to give a judgement on what cosmology is now.
3 E.g. (Kreisel 1980), (Dawson 1997), (Rindler 2012).
4 One could be tempted to add what Gödel says in his letter of November 18th, 1955 to Carl Seelig: these questions are remote from his area of specialization, logic and Grundlagenforschung. (CW V: 250). But, here, Gödel is speaking of some general problems that he introduced as a field of research for specialists in relativity theory.
5 Completeness of predicate logic, existence of undecidable propositions of mathematics, consistency of the continuum hypothesis, rotating universes, and interpretation of intuitionistic arithmetic. See (Wang 1996: 90-93).
6 Letter to P.A.Schilpp, December 21, 1949: “Shall I receive any reprints of my article? If so I should appreciate if you would send them as early as possible, so that I can send them out together with the reprints of my mathematical paper about the same subject” (CW V: 237; emphasis added).
7 For a clear and neutral description of the notion of philosophical systems, see (Vuillemin 1986).
8 “Kripke is, though not a positivist, still doing linguistic philosophy”. (Wang 1996: 4.3.2)
9 On this point see (Vuillemin 1984). There are several technical mistakes in Vuillemin’s logical analysis of the models of the universe implied by the solutions of the Master Argument. But when the wise man shows us the moon, should we wonder wether or not his fingernails are clean?
10 Gabriella Crocco, in private conversations, has argued convincingly in favour her choice of transcribing Gr as Grammatik. I have no definite opinion on this point.
11 See the drafts of Gödel’s answer to the Grandjean questionnaire in (Wang 1987: 19-20).
12 Howard Stein and David Malament, in their comments about the Gödel’s unpublished manuscripts in (CW III), have proposed implicitely the following historical sequence for the development of the idea of rotating universes: discovery of the stationary model, then discovery of the possibility to travel into the past, then generalisation of the concept of rotating universes. (Audureau 2005: 138) has rejected this interpretation and has proposed that the idea of rotating universes (with static or with dynamic metrics) came at once. Gödel’s remark settles the question.
For the same reason, Wolfgang Rindler’s suggestion (Rindler 2012: 189) that Gamow’s letter to Nature inspired to Gödel the idea of rotating universes should be disregarded.
13 The notion of “physical time” is not very clear. Here, it is to be undersood as “time as it is used in physics”.
14 For example (Rindler 2012).
15 For a very readable account on this point see Jacques Merleau-Ponty (1965: 283-90). To my knowledge, Merleau-Ponty’s book remains, up to now, the best introduction to Gödel’s works on cosmology and to 20th century cosmology tout court.
16 The authors met Gödel a couple of years before the publication of Gravitation. When Gödel asked them if they would speak of his results and received a negative answer, he was quite rightfully disappointed because, as we will see, his works relate to the pure theory of gravitation. Nonetheless, a paper by Gödel is quoted in the quite lengthy bibliography of this book: the one on the incompleteness theorem [...]
17 See Klein’s letter to De Sitter in (De Sitter 1917), a reference that Gödel could have known about. Note that these cases are different from the one described by Hermann Weyl, in Space-Time-Matter, p. 274, pointed out by Howard Stein in his introductive note to the drafts of “A remark” (CW III: 228). In this case, it is the density of matter that is responsible for the change in orientation of the arrow of time. Namely, the curvature of space-time is not constant, whereas it is for Minkowski, De Sitter and Gödel’s space-times.
18 In this short presentation, I leave aside Gödel’s convictions on these matters, namely that time is involved by the concept of cause (Wang, 1996: 9.1.18).
19 See (Merleau-Ponty 1965: 278), (Stein 1967), (Dawson 1997: 184-85).
20 This historical point seems to have escaped Penrose.
21 Quoted by (Ozsváth and Schücking 2001: 2243).
22 These letters were published in (Seelig 1960: 421-23) with Gödel’s agreement. So, they belong to the corpus of his published works. Hao Wang (1987: 155) translated large excerpts of them and the whole letters were eventually published in 2005 in (CW V: 248-53) with a new English translation. Thus, it is difficult to ignore their existence.
23 This description is oversimplified. For a precise account see (Raychaudhuri 1978), especially ch. 5.
24 The shear tensor measures any tendency of an initially fluid sphere to become distorted into a non spherical shape.
25 Present day cosmology follows the same line.
26 It is worth recording that in 1977 J.P. Vigier, relying on the works of astronomers, emphasized that in the present state of experimental knowledge, isotropy and universal proportionality of red shift and distance for all distant objects cannot be regarded as established facts.
27 He concludes the Lecture on rotating universes in the same way: “These rotating solutions have either a contraction or an expansion or both at the same time, namely a contraction in one direction of space and an expansion in another direction”. (CW III: 287).
28 The introduction of this cosmological constant generated a lot of somewhat confusing statements by Einstein (see his answer to Lemaître and to Gödel in (Schilpp 1949)) and several other cosmologists. It was proved by (Cartan 1922) that this term (vanishing or not) belongs necessarily to the field equations.
29 See, for example, Einstein’s letter to Besso from July 29th 1953 (Speziali 1972: 502).
30 See (Rindler, 2012); for a detailed explanation of the independence of Mach’s Principle, see the textbook (Adler, Bazin and Schiffer 1965: 367-377).
31 (Merleau-Ponty 1965: 44-65).
32 This is the mathematical counterpart of the big physical blind spot described above in section 3.
33 (Ozsváth and Schücking 2001: 2244) observe that, in his Lecture on rotating universes, Gödel made a mistake in the computing of the angular velocity of his Newtonian model (he forgot to take into account the cosmological λ-term). The authors “suspect that it was this discrepancy in the otherwise perfect match of Newtonian and relativistic theory that persuaded Gödel not to mention the Newtonian analogue of his model in the paper he published in the special issue of the Review of Modern Physics for Einstein’s 70th birthday”. This opinion does not seem to fit with the chronological data.
34 “Kurt Gödel’s essay constitutes, in my opinion, an important contribution to the general theory of relativity, especially to the analysis of the concept time. The problem here involved disturbed me already at the time of building up of the general theory of relativity, without my having succeeded in clarifying it”. (Schilpp 1949: 687).
35 In his July 25th 1946 letter to Schilpp, Gödel proposed to write about 3 pages under the title: “Some remarks about the relation between the theory of relativity and Kant”. He adds “I am doubtful, however, if such a contribution would be of any use, for, since this question touches the very essence of ‘relativity’, namely the role of the observer, I presumed it will be treated in one or more of the longer articles already”. (Emphasis added). The observer seems to have disappeared from the outcome of this project.
36 Expressed, among other places, in Russell’s mathematical logic: “Every sentence (of a given language) contains at least one relation word”. (CW II: 130).
37 Merleau-Ponty put Gödel’s case in a class of “Théories diverses”.
38 Namely close to the right, in the sense of (Gödel 1961) classification of philosophies.
39 “Newton, forgive me; you found the only way which, in your age, was just about possible for a man of highest thought - and creative power. The concepts, which you created, are even today still guiding our thinking in physics, although we know now that they will have to be replaced by others farther removed from the sphere of immediate experience, if we aim at a profounder understanding of relationships”. (Schilpp, 1949: 31).
40 See I. Bernard Cohen remarks on the varieties of Newton’s concepts of force in Principia (Cohen and Whitman 1999: 54). Gödel read Latin and he could be not mistaken by the different doubtful translations of Newton’s opus. No doubt, also, that he knew well Newton’s Opticks and, hence, Queries XXX and XXXI. Cf. the letter to his mother from 26 August 1946: “This Goethe book [published in 1912 by Houston Chamberlain] was also the beginning of my occupation with Goethe’s Farbenlehre and his dispute with Newton, which indirectly also contributed to my choice of vocation”. Quoted by (Wang 1987: 73). Hao Wang adds “Gödel’s brother Rudolf told me that Gödel had concluded from his study in favour of Newton’s position”.
41 CW III, 454. We find a similar remark in (Max Phil XII: 11) without reference to Newton: Bem<erkung> (Phil<osophie>): Ein gutes Beispiel der Substantialisierung eines Accid<enz> ist der Raum: Besteht zunächst in Relationen zwischen Körpern, dann wird er aber in „Raumpunkte“ transformiert und wird ein unabhängig existierendes „Wesen“.
Auteur
Is Researcher at CNRS (National Centre for Scientific Research) in philosophy of science affiliated with the Centre for Epistemology (CEPERC) at Aix-Marseille Université. His field of research is the relationship between science and philosophy during the third scientific revolution (1879-1936). His thesis entitled “Kurt Gödel, critique de la relativité générale” appeared in 2004. Together with Julien Bernard he has recently completed a commented and annotated edition of Hermann Weyl’s “Mathematische Analyse des Raumproblems” that is accompanied by a French translation, as well as a paper on “Poincaré, critique de Dedekind”.
Le texte seul est utilisable sous licence Licence OpenEdition Books. Les autres éléments (illustrations, fichiers annexes importés) sont « Tous droits réservés », sauf mention contraire.
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