Philosophy

See	discussions,	stats,	and	author	profiles	for	this	publication	at:	https://www.researchgate.net/publication/233532260 From	the	Closed	Classical	Algorithmic	Universe to	an	Open	World	of	Algorithmic	Constellations Article	*	October	2012 DOI:	10.1007/978-3-642-37225-4_16	*	Source:	arXiv CITATIONS 9 READS 16 2	authors: Some	of	the	authors	of	this	publication	are	also	working	on	these	related	projects: Morphological	Computing	in	Cognitive	Systems	View	project Cloud	Agnostic	Distributed	Computing	and	Interoperable	Clouds	View	project Mark	Burgin University	of	California,	Los	Angeles 286	PUBLICATIONS 1,072	CITATIONS SEE	PROFILE Gordana	Dodig	Crnkovic University	of	Gothenburg 105	PUBLICATIONS 471	CITATIONS SEE	PROFILE All	content	following	this	page	was	uploaded	by	Gordana	Dodig	Crnkovic	on	13	March	2017. The	user	has	requested	enhancement	of	the	downloaded	file.	All	in-text	references	underlined	in	blue	are	added	to	the	original	document and	are	linked	to	publications	on	ResearchGate,	letting	you	access	and	read	them	immediately. From the Closed Classical Algorithmic Universe to an Open World of Algorithmic Constellations Mark Burgin1 and Gordana Dodig-Crnkovic2 1 Dept. of Mathematics, UCLA, Los Angeles, USA. E-mail: email@example.com 2 Mälardalen University, Department of Computer Science and Networks, School of Innovation, Design and Engineering, Västerås, Sweden; E-mail: firstname.lastname@example.com Abstract In this paper we analyze methodological and philosophical implications of algorithmic aspects of unconventional computation. At first, we describe how the classical algorithmic universe developed and analyze why it became closed in the conventional approach to computation. Then we explain how new models of algorithms turned the classical closed algorithmic universe into the open world of algorithmic constellations, allowing higher flexibility and expressive power, supporting constructivism and creativity in mathematical modeling. As Gödel's undecidability theorems demonstrate, the closed algorithmic universe restricts essential forms of mathematical cognition. In contrast, the open algorithmic universe, and even more the open world of algorithmic constellations, remove such restrictions and enable new, richer understanding of computation. Keywords: Unconventional algorithms, unconventional computing, algorithmic constellations, Computing beyond Turing machine model. Introduction Te development of various systems is characterized by a tension between forces of conservation (tradition) and change (innovation). Tradi2 tion sustains system and its parts, while innovation moves it forward advancing some segments and weakening the others. Efficient functioning of a system depends on the equilibrium between tradition and innovation. When there is no equilibrium, system declines; too much tradition brings stagnation and often collapse under the pressure of inner or/and outer forces, while too much innovation leads to instability and frequently in rupture. The same is true of the development of different areas and aspects of social systems, such as science and technology. In this article we are interested in computation, which has become increasingly important for society as the basic aspect of information technology. Tradition in computation is represented by conventional computation and classical algorithms, while unconventional computation stands for the far-reaching innovation. It is possible to distinguish three areas in which computation can be unconventional: 1. Novel hardware (e.g. quantum systems) provides material realization for unconventional computation. 2. Novel algorithms (e.g. super-recursive algorithms) provide operational realization for unconventional computation. 3. Novel organization (e.g. evolutionary computation or selfoptimizing computation) provides structural realization for unconventional computation. Here we focus on algorithmic aspects of unconventional computation and analyze methodological and philosophical problems related to it, 3 making a distinction between three classes of algorithms: recursive, subrecursive, and super-recursive algorithms. Each type of recursive algorithms form a class in which it is possible to compute exactly the same functions that are computable by Turing machines. Examples of recursive algorithms are partial recursive functions, RAM, von Neumann automata, Kolmogorov algorithms, and Minsky machines. Each type of subrecursive algorithms forms a class that has less computational power than the class of all Turing machines. Examples of subrecursive algorithms are finite automata, primitive recursive functions and recursive functions. Each type of super-recursive algorithms forms a class that has more computational power than the class of all Turing machines. Examples of super-recursive algorithms are inductive and limit Turing machines, limit partial recursive functions and limit recursive functions. The main problem is that conventional types and models of algorithms make the algorithmic universe, i.e., the world of all existing and possible algorithms, closed because there is a rigid boundary in this universe formed by recursive algorithms, such as Turing machines, and described by the Church-Turing Thesis. This closed system has been overtly dominated by discouraging incompleteness results, such as Gödel incompleteness theorems. Contrary to this, super-recursive algorithms controlling and directing unconventional computations break this boundary leading to an open algorithmic multiverse – world of unrestricted creativity. 4 The paper is organized as follows. First, we summarize how the closed algorithmic universe was created and what are advantages and disadvantages of living inside such a closed universe. Next, we describe the breakthrough brought about by the creation of super-recursive algorithms. In Section 4, we analyze super-recursive algorithms as cognitive tools. The main effect is the immense growth of cognitive possibilities and computational power that enables corresponding growth of information processing devices. The Closed Universe of Turing Machines and other Recursive Algorithms Historically, after having an extensive experience of problem solving, mathematicians understood that problem solutions were based on various algorithms. Construction algorithms and deduction algorithms have been the main tools of mathematical research. When they repeatedly encountered problems they were not able to solve, mathematicians, and especially experts in mathematical logic, came to the conclusion that it was necessary to develop a rigorous mathematical concept of algorithm and to prove that some problems are indeed unsolvable. Consequently, a diversity of exact mathematical models of algorithm as a general concept was proposed. The first models were λ-calculus developed by Church in 1931 – 1933, general recursive functions introduced by Gödel in 1934, ordinary Turing machines constructed by Turing in 1936 and in a less explicit form by Post in 1936, and partial recursive functions built by Kleene in 1936. Creating λ-calculus, Church was developing a logical theory of functions and suggested a formalization of the notion of com5 putability by means of λ-definability. In 1936, Kleene demonstrated that λ-definability is computationally equivalent to general recursive functions. In 1937, Turing showed that λ-definability is computationally equivalent to Turing machines. Church was so impressed by these results that he suggested what was later called the Church-Turing thesis. Turing formulated a similar conjecture in the Ph.D. thesis that he wrote under Church's supervision. It is interesting to know that the theory of Frege [1] actually contains λ-calculus. So, there were chances to develop a theory of algorithms and computability in the 19th century. However, at that time, the mathematical community did not feel a need of such a theory and probably, would not accept it if somebody created it. The Church-Turing thesis explicitly mark out a rigid boundary for the algorithmic universe, making this universe closed by Turing machines. Any algorithm from this universe was inside that boundary. After the first breakthrough, other mathematical models of algorithms were suggested. They include a variety of Turing machines: multihead, multitape Turing machines, Turing machines with n-dimensional tapes, nondeterministic, probabilistic, alternating and reflexive Turing machines, Turing machines with oracles, Las Vegas Turing machines, etc.; neural networks of various types – fixed-weights, unsupervised, supervised, feedforward, and recurrent neural networks; von Neumann automata and general cellular automata; Kolmogorov algorithms finite automata of different forms – automata without memory, autonomous automata, automata without output or accepting automata, deterministic, nondeterministic, probabilistic automata, etc.; Minsky machines; 6 Storage Modification Machines or simply, Shönhage machines; Random Access Machines (RAM) and their modifications Random Access Machines with the Stored Program (RASP), Parallel Random Access Machines (PRAM); Petri nets of various types – ordinary and ordinary with restrictions, regular, free, colored, and self-modifying Petri nets, etc.; vector machines; array machines; multidimensional structured model of computation and computing systems; systolic arrays; hardware modification machines; Post productions; normal Markov algorithms; formal grammars of many forms – regular, context-free, context-sensitive, phrase-structure, etc.; and so on. As a result, the theory of algorithms, automata and computation has become one of the foundations of computer science. In spite of all differences between and diversity of algorithms, there is a unity in the system of algorithms. While new models of algorithm appeared, it was proved that no one of them could compute more functions than the simplest Turing machine with a one-dimensional tape. All this give more and more evidence to validity of the Church-Turing Thesis. Even more, all attempts to find mathematical models of algorithms that were stronger than Turing machines were fruitless. Equivalence with Turing machines has been proved for many models of algorithms. That is why the majority of mathematicians and computer scientists have believed that the Church-Turing Thesis was true. Many logicians assume that the Thesis is an axiom that does not need any proof. Few believe that it is possible to prove this Thesis utilizing some evident axioms. More accurate researchers consider this conjecture as a law of the theory of algorithms, which is similar to the laws of nature that might be sup7 ported by more and more evidence or refuted by a counter-example but cannot be proved. Besides, the Church-Turing Thesis is extensively utilized in the theory of algorithms, as well as in the methodological context of computer science. It has become almost an axiom. Some researchers even consider this Thesis as a unique absolute law of computer science. Thus, we can see that the initial aim of mathematicians was to build a closed algorithmic universe, in which a universal model of algorithm provided a firm foundation and as it was found later, a rigid boundary for this universe supposed to contain all of mathematics. It is possible to see the following advantages and disadvantages of the closed algorithmic universe. Advantages: 1. Turing machines and partial recursive functions are feasible mathematical models. 2. These and other recursive models of algorithms provide an efficient possibility to apply mathematical techniques. 3. The closed algorithmic universe allowed mathematicians to build beautiful theories of Turing machines, partial recursive functions and some other recursive and subrecursive algorithms. 4. The closed algorithmic universe provides sufficiently exact boundaries for knowing what is possible to achieve with algorithms and what is impossible. 5. The closed algorithmic universe provides a common formal language for researchers. 8 6. For computer science and its applications, the closed algorithmic universe provides a diversity of mathematical models with the same computing power. Disadvantages: 1. The main disadvantage of this universe is that its main principle the Church-Turing Thesis is not true. 2. The closed algorithmic universe restricts applications and in particular, mathematical models of cognition. 3. The closed algorithmic universe does not correctly reflect the existing computing practice. The Open World of Super-Recursive Algorithms and Algorithmic Constellations Contrary to the general opinion, some researchers expressed their concern for the Church-Turing Thesis. As Nelson writes [2], "Although Church-Turing Thesis has been central to the theory of effective decidability for fifty years, the question of its epistemological status is still an open one." There were also researchers who directly suggested arguments against validity of the Church-Turing Thesis. For instance, Kalmar [3] raised intuitionistic objections, while Lucas and Benacerraf discussed objections to mechanism based on theorems of Gödel that indirectly threaten the Church-Turing Thesis. In 1972, Gödel's observation entitled "A philosophical error in Turing's work" was published where he declared that: "Turing in his 1937, p. 250 (1965, p. 136), gives an argument which is supposed to show that mental procedures cannot go beyond mechanical procedures. However, this argument is inconclu9 sive. What Turing disregards completely is the fact that mind, in its use, is not static, but constantly developing, i.e., that we understand abstract terms more and more precisely as we go on using them, and that more and more abstract terms enter the sphere of our understanding. There may exist systematic methods of actualizing this development, which could form part of the procedure. Therefore, although at each stage the number and precision of the abstract terms at our disposal may be finite, both (and, therefore, also Turing's number of distinguishable states of mind) may converge toward infinity in the course of the application of the procedure." [4] Thus, pointing that Turing disregarded completely the fact that mind, in its use, is not static, but constantly developing, Gödel predicted necessity for super-recursive algorithms that realize inductive and topological computations [5]. Recently, Sloman [6] explained why recursive models of algorithms, such as Turing machines, are irrelevant for artificial intelligence. Even if we abandon theoretical considerations and ask the practical question whether recursive algorithms provide an adequate model of modern computers, we will find that people do not see correctly how computers are functioning. An analysis demonstrates that while recursive algorithms gave a correct theoretical representation for computers at the beginning of the "computer era", super-recursive algorithms are more adequate for modern computers. Indeed, at the beginning, when computers appeared and were utilized for some time, it was necessary to print out data produced by computer to get a result. After printing, the computer stopped functioning or began to solve another problem. Now 10 people are working with displays and computers produce their results mostly on the screen of a monitor. These results on the screen exist there only if the computer functions. If this computer halts, then the result on its screen disappears. This is opposite to the basic condition on ordinary (recursive) algorithms that implies halting for giving a result. Such big networks as Internet give another important example of a situation in which conventional algorithms are not adequate. Algorithms embodied in a multiplicity of different programs organize network functions. It is generally assumed that any computer program is a conventional, that is, recursive algorithm. However, a recursive algorithm has to stop to give a result, but if a network shuts down, then something is wrong and it gives no results. Consequently, recursive algorithms turn out to be too weak for the network representation, modeling and study. Even more, no computer works without an operating system. Any operating system is a program and any computer program is an algorithm according to the general understanding. While a recursive algorithm has to halt to give a result, we cannot say that a result of functioning of operating system is obtained when computer stops functioning. To the contrary, when the operating system does not work, it does not give an expected result. Looking at the history of unconventional computations and superrecursive algorithms we see that Turing was the first who went beyond the "Turing" computation that is bounded by the Church-Turing Thesis. In his 1938 doctoral dissertation, Turing introduced the concept of a Turing machine with an oracle. This, work was subsequently published in 1939. Another approach that went beyond the Turing-Church Thesis was 11 developed by Shannon [7], who introduced the differential analyzer, a device that was able to perform continuous operations with real numbers such as operation of differentiation. However, mathematical community did not accept operations with real numbers as tractable because irrational numbers do not have finite numerical representations. In 1957, Grzegorczyk introduced a number of equivalent definitions of computable real functions. Three of Grzegorczyk's constructions have been extended and elaborated independently to super-recursive methodologies: the domain approach [8,9], type 2 theory of effectivity or type 2 recursion theory [10,11], and the polynomial approximation approach [12]. In 1963, Scarpellini introduced the class M1 of functions that are built with the help of five operations. The first three are elementary: substitutions, sums and products of functions. The two remaining operations are performed with real numbers: integration over finite intervals and taking solutions of Fredholm integral equations of the second kind. Yet another type of super-recursive algorithms was introduced in 1965 by Gold and Putnam, who brought in concepts of limiting recursive function and limiting partial recursive function. In 1967, Gold produced a new version of limiting recursion, also called inductive inference, and applied it to problems of learning. Now inductive inference is a fruitful direction in machine learning and artificial intelligence. One more direction in the theory of super-recursive algorithms emerged in 1967 when Zadeh introduced fuzzy algorithms. It is interesting that limiting recursive function and limiting partial recursive function were not considered as valid models of algorithms even by their authors. A proof that fuzzy algorithms are more powerful than Turing 12 machines was obtained much later (Wiedermann, 2004). Thus, in spite of the existence of super-recursive algorithms, researchers continued to believe in the Church-Turing Thesis as an absolute law of computer science. After the first types of super-recursive models had been studied, a lot of other super-recursive algorithmic models have been created: inductive Turing machines, limit Turing machines, infinite time Turing machines, general Turing machines, accelerating Turing machines, type 2 Turing machines, mathematical machines, δ-Q-machines, general dynamical systems, hybrid systems, finite dimensional machines over real numbers, R-recursive functions and so on. To organize the diverse variety of algorithmic models, we introduce the concept of an algorithmic constellation. Namely, an algorithmic constellation is a system of algorithmic models that have the same type. Some algorithmic constellations are disjoint, while other algorithmic constellations intersect. There are algorithmic constellations that are parts of other algorithmic constellations. Below some of algorithmic constellations are described. The sequential algorithmic constellation consists of models of sequential algorithms. This constellation includes such models as deterministic finite automata, deterministic pushdown automata with one stack, evolutionary finite automata, Turing machines with one head and one tape, Post productions, partial recursive functions, normal Markov algorithms, formal grammars, inductive Turing machines with one head and one tape, limit Turing machines with one head and one tape, reflexive Turing machines with one head and one tape, infinite time Turing machines, 13 general Turing machines with one head and one tape, evolutionary Turing machines with one head and one tape, accelerating Turing machines with one head and one tape, type 2 Turing machines with one head and one tape, Turing machines with oracles. The concurrent algorithmic constellation consists of models of concurrent algorithms. This constellation includes such models as Petri nets, artificial neural networks, nondeterministic Turing machines, probabilistic Turing machines, alternating Turing machines, Communicating Sequential Processes (CSP) of Hoare, Actor model, Calculus of Communicating Systems (CCS) of Milner, Kahn process networks, dataflow process networks, discrete event simulators, View-Centric Reasoning (VCR) model of Smith, event-signal-process (ESP) model of Lee and Sangiovanni-Vincentelli, extended view-centric reasoning (EVCR) model of Burgin and Smith, labeled transition systems, Algebra of Communicating Processes (ACP) of Bergstra and Klop, event-actionprocess (EAP) model of Burgin and Smith, synchronization trees, and grid automata. The parallel algorithmic constellation consists of models of parallel algorithms and is a part of the concurrent algorithmic constellation. This constellation includes such models as pushdown automata with several stacks, Turing machines with several heads and one or several tapes, Parallel Random Access Machines, Kolmogorov algorithms, formal grammars with prohibition, inductive Turing machines with several heads and one or several tapes, limit Turing machines with several heads and one or several tapes, reflexive Turing machines with several heads and one or several tapes, general Turing machines with several heads 14 and one or several tapes, accelerating Turing machines with several heads and one or several tapes, type 2 Turing machines with several heads and one or several tapes. The discrete algorithmic constellation consists of models of algorithms that work with discrete data, such as words of formal language. This constellation includes such models as finite automata, Turing machines, partial recursive functions, formal grammars, inductive Turing machines and Turing machines with oracles. The topological algorithmic constellation consists of models of algorithms that work with data that belong to a topological space, such as real numbers. This constellation includes such models as the differential analyzer of Shannon, limit Turing machines, finite dimensional and general machines of Blum, Shub, and Smale, fixed point models, topological algorithms, neural networks with real number parameters. Although several models of super-recursive algorithms already existed in 1980s, the first publication where it was explicitly stated and proved that there are algorithms more powerful than Turing machines was [13]. In this work, among others, relations between Gödel's incompleteness results and super-recursive algorithms were discussed. Super-recursive algorithms have different computing and accepting power. The closest to conventional algorithms are inductive Turing machines of the first order because they work with constructive objects, all steps of their computation are the same as the steps of conventional Turing machines and the result is obtained in a finite time. In spite of these similarities, inductive Turing machines of the first order can compute much more than conventional Turing machines [14, 5]. 15 Inductive Turing machines of the first order form only the lowest level of super-recursive algorithms. There are infinitely more levels and as a result, the algorithmic universe grows into the algorithmic multiverse becoming open and amenable. Taking into consideration algorithmic schemas, which go beyond super-recursive algorithms, we come to an open world of information processing, which includes the algorithmic multiverse with its algorithmic constellations. Openness of this world has many implications for human cognition in general and mathematical cognition in particular. For instance, it is possible to demonstrate that not only computers but also the brain can work not only in the recursive mode but also in the inductive mode, which is essentially more powerful and efficient. Some of the examples are considered in the next section. Absolute Prohibition in The Closed Universe and Infinite Opportunities in The Open World To provide sound and secure foundations for mathematics, David Hilbert proposed an ambitious and wide-ranging program in the philosophy and foundations of mathematics. His approach formulated in 1921 stipulated two stages. At first, it was necessary to formalize classical mathematics as an axiomatic system. Then, using only restricted, "finitary" means, it was necessary to give proofs of the consistency of this axiomatic system. Achieving a definite progress in this direction, Hilbert became very optimistic. As a response to the Latin dictum: "Ignoramus et ignorabimus" or "We do not know, we cannot know", in his speech in Königsberg in 1930, he made a famous statement: Wir müssen wissen. Wir werden wissen. (We must know. We will know.) 16 Next year the Gödel undecidability theorems were published [15]. They undermined Hilbert's statement and his whole program. Indeed, the first Gödel undecidability theorem states that it is impossible to validate truth for all true statements about objects in an axiomatic theory that includes formal arithmetic. This is a consequence of the fact that it is impossible to build all sets from the arithmetical hierarchy by Turing machines. In such a way, the closed Algorithmic Universe imposed restriction on the mathematical exploration. Indeed, rigorous mathematical proofs are done in formal mathematical systems. As it is demonstrated (cf., for example, [16]), such systems are equivalent to Turing machines as they are built by means of Post productions. Thus, as Turing machines can model proofs in formal systems, it is possible to assume that proofs are performed by Turing machines. The second Gödel undecidability theorem states that for an effectively generated consistent axiomatic theory T that includes formal arithmetic and has means for formal deduction, it is impossible to prove consistency of T using these means. From the very beginning, Gödel undecidability theorems have been comprehended as absolute restrictions for scientific cognition. That is why Gödel undecidability theorems were so discouraging that many mathematicians consciously or unconsciously disregarded them. For instance, the influential group of mostly French mathematicians who wrote under the name Bourbaki completely ignored results of Gödel [17]. However, later researchers came to the conclusion that these theorems have such drastic implications only for formalized cognition based on rigorous mathematical tools. For instance, in the 1964 postscript, Gödel 17 wrote that undecidability theorems "do not establish any bounds for the powers of human reason, but rather for the potentialities of pure formalism in mathematics." Discovery of super-recursive algorithms and acquisition of the knowledge of their abilities drastically changed understanding of the Gödel's results. Being a consequence of the closed nature of the closed algorithmic universe, these undecidability results lose their fatality in the open algorithmic universe. They become relativistic being dependent on the tools used for cognition. For instance, the first undecidability theorem is equivalent to the statement that it is impossible to compute by Turing machines or other recursive algorithms all levels of the Arithmetical Hierarchy [18]. However, as it is demonstrated in [19], there is a hierarchy of inductive Turing machines so that all levels of the Arithmetical Hierarchy are computable and even decidable by these inductive Turing machines. Complete proofs of these results were published only in 2003 due to the active opposition of the proponents of the Church-Turing Thesis [14]. In spite of the fast development of computer technology and computer science, the research community in these areas is rather conservative although more and more researchers understand that the Church-Turing Thesis is not correct. The possibility to use inductive proofs makes the Gödel's results relative to the means used for proving mathematical statements because decidability of the Arithmetical Hierarchy implies decidability of the formal arithmetic. For instance, the first Gödel undecidability theorem is true when recursive algorithms are used for proofs but it becomes false when inductive algorithms, such as inductive Turing machines, are uti18 lized. History of mathematics also gives supportive evidence for this conclusion. For instance, in 1936 by Gentzen, who in contrast to the second Gödel undecidability theorem, proved consistency of the formal arithmetic using ordinal induction. The hierarchy of inductive Turing machines also explains why the brain of people is more powerful than Turing machines, supporting the conjecture of Roger Penrose [20]. Besides, this hierarchy allows researchers to eliminate restrictions of recursive models of algorithms in artificial intelligence described by Sloman [6]. It is important to remark that limit Turing machines and other topological algorithms [21] open even broader perspectives for information processing technology and artificial intelligence than inductive Turing machines. The Open World of Knowledge and The Internet The open world, or more exactly, the open world of knowledge, is an important concept for the knowledge society and its knowledge economy. According to Rossini [12], it emerges from a world of pre-Internet political systems, but it has come to encompass an entire worldview based on the transformative potential of open, shared, and connected technological systems. The idea of an open world synthesizes much of the social and political discourse around modern education and scientific endeavor and is at the core of the Open Access (OA) and Open Educational Resources (OER) movements. While the term open society comes from international relations, where it was developed to describe the transition from political oppression into a more democratic society, it is now 19 being appropriated into a broader concept of an open world connected via technology [22]. The idea of openness in access to knowledge and education is a reaction to the potential afforded by the global networks, but is inspired by the sociopolitical concept of the open society. Open Access (OA) is a knowledge-distribution model by which scholarly, peer-reviewed journal articles and other scientific publications are made freely available to anyone, anywhere over the Internet. It is the foundation for the open world of scientific knowledge, and thus, a principal component of the open world of knowledge as a whole. In the era of print, open access was economically and physically impossible. Indeed, the lack of physical access implied the lack of knowledge access if one did not have physical access to a well-stocked library, knowledge access was impossible. The Internet has changed all of that, and OA is a movement that recognizes the full potential of an open world metaphor for the network. In OA, the old tradition of publishing for the sake of inquiry, knowledge, and peer acclaim and the new technology of the Internet have converged to make possible an unprecedented public good: "the world-wide electronic distribution of the peer-reviewed journal literature" [23]. The open world of knowledge is based on the Internet, while the Internet is based on computations that go beyond Turing machines. One of the basic principles of the Internet is that it is always on, always available. Without these features, the Internet cannot provide the necessary support for the open world of knowledge because ubiquitous availability of knowledge resources demands non-stopping work of the Internet. At 20 the same time, classical models of algorithms, such as Turing machines, stop after giving that result. This contradicts the main principles of the Internet. In contrast to classical models of computation, as it is demonstrated in [5], if an automatic system, e.g., a computer or computer network, works without halting, gives results in this mode and can simulate any operation of a universal Turing machine, then this automatic (computer) system is more powerful than any Turing machine. This means that this automatic (computer) system, in particular, the Internet, performs unconventional computations and is controlled by super-recursive algorithms. As it is explained in [5], attempts to reduce some of these systems, e.g., the Internet, to the recursive mode, which allows modeling by Turing machines, make these systems irrelevant. Conclusions This paper shows how the universe (the world) of algorithms became open with the discovery of super-recursive algorithms, providing more powerful tools for computational cognition and artificial intelligence. Here we considered only some of the consequences of the open world environment of unconventional algorithms and algorithmic constellations for mathematical (computation-theoretical) cognition. It would be interesting to study other consequences of current break through into an open world of unconventional algorithms and computation. It is known that not all quantum mechanical events are Turingcomputable. So, it would be interesting to find a class of super-recursive algorithms that compute all such events or to prove that such a class does not exist. 21 It might be methodologically and philosophically interesting to contemplate relations between the Open World of Algorithmic Constellations and the Open Science in the sense of Nielsen [24]. For instance, one of the pivotal features of the Open Science is accessibility of research results on the Internet. At the same time, as it is demonstrated in [5], the Internet and other big networks of computers are always working in the inductive mode or some other super-recursive mode. Moreover, actual accessibility depends on such modes of functioning. One more interesting problem is to explore relations between the Open World of Algorithmic Constellations with the theoretical framework of Info-computationalism, a synthesis of Pancomputationalism (Naturalist Computationalism) with Informational Structural Realism – the model of a universe as a network of computational processes on informational structures. Info-computationalism connects algorithms with interactive computing in natural (physical) systems [25,26][28]. Connecting new unconventional models of super-recursive algorithms and Algorithmic Constellations with unconventional computations performed by natural systems opens new possibilities for the development of innovative models of physical computation with "Trans-Turing" algorithms and "NonVon" computing architectures. [27]. 22 Acknowledgements The authors would like to thank Andree Ehresmann, Hector Zenil and Marcin Schroeder for useful and constructive comments on the previous version of this work. References 1 G. Frege, Grundgesetze der Arithmetik, Begriffschriftlich Abgeleitet, Viena (1893/1903) 2 R. J. Nelson, Church's thesis and cognitive science, Notre Dame J. of Formal Logic, v. 28, no. 4, 581-614 (1987) 3 L. Kalmar, An argument against the plausibility of Church's thesis, in Constructivity in mathematics, North-Holland Publishing Co., Amsterdam, pp. 72-80 (1959) 4 K. Gödel, Some Remarks on the Undecidability Results, in G odel, K. (1986–1995), Collected Works, v. 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Richards, Computability in Analysis and Physics. Perspectives in Mathematical Logic, Vol. 1. Berlin: Springer. (1989) 23 13 M. Burgin, The Notion of Algorithm and the Turing-Church Thesis, In Proceedings of the VIII International Congress on Logic, Methodology and Philosophy of Science, Moscow, v. 5, part 1, pp. 138-140 (1987) 14 M. Burgin, Nonlinear Phenomena in Spaces of Algorithms, International Journal of Computer Mathematics, v. 80, No. 12, pp. 1449-1476 (2003) 15 K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematics und verwandter System I, Monatshefte für Mathematik und Physik, b. 38, s.173-198 (1931) 16 R.M. Smullian, Theory of Formal Systems, Princeton University Press (1962) 17 A.R.D. Mathias, The Ignorance of Bourbaki, Physis Riv. Internaz. Storia Sci (N.S.) 28, pp. 887-904 (1991) 18 H. Rogers, Theory of Recursive Functions and Effective Computability, MIT Press, Cambridge Massachusetts (1987) 19 M. Burgin, Arithmetic Hierarchy and Inductive Turing Machines, Notices of the Russian Academy of Sciences, v. 299, No. 3, pp. 390-393 (1988) 20 Penrose, R. The Emperor's New Mind, Oxford University Press, Oxford (1989) 21 Burgin, M. Topological Algorithms, in Proceedings of the ISCA 16th International Conference "Computers and their Applications", ISCA, Seattle, Washington, pp. 61-64 (2001) 22 C. Rossini, Access to Knowledge as a Foundation for an Open World, EDUCAUSE Review, v. 45, No. 4, pp. 60–68 (2010) 23 Budapest Open Access Initiative: <http://www.soros.org/openaccess/read.shtml>. 24 M. Nielsen, Reinventing Discovery: The New Era of Networked Science, Princeton University Press, Princeton and Oxford (2012) 25 Dodig-Crnkovic, G. and Muller, V.C. A Dialogue Concerning Two World Systems, in Information and Computation, World Scientific, New York/Singapore, pp. 107-148 (2011) 26 G. Rozenberg, T.H.W. Bäck & J.N. Kok (Eds.): Handbook of Natural Computing. Heidelberg, Germany, Springer (2012) 27 Dodig Crnkovic G. and Burgin M., Unconventional Algorithms: Complementarity of Axiomatics and Construction, Entropy, Special issue "Selected Papers from the Symposium on Natural/Unconventional Computing and its Philosophical Significance" http://www.mdpi.com/journal/entropy/special_issues/unconvent_computing, forthcoming 2012 24 28 Dodig-Crnkovic G., Significance of Models of Computation from Turing Model to Natural Computation. Minds and Machines, ( R. Turner and A. Eden guest eds.) Volume 21, Issue 2 (2011), Page 301. All the links accessed at 08 06 2012 View publication stats


Anomalies of General Relativity

Philosophy

Over time, the general theory of relativity has accumulated several anomalies and discrepancies, indicating the need for a better theory about gravity or other approaches. The ad-hoc hypotheses introduced in general relativity to explain gravitational singularities based on energy conditions are not very efficient. More detailed assumptions on the content of the subject are needed. Many scientists and philosophers have come to the conclusion that singularities must be associated with reaching the limits of the physical validity of general relativity, and a new theory of quantum gravity needs to be developed.
          DOI: 10.13140/RG.2.2.32579.55848

Nicolae Sfetcu: Anomalies of General Relativity Anomalies of General Relativity Nicolae Sfetcu 12.10.2019 Sfetcu, Nicolae, "Anomalies of General Relativity", SetThings (October 12, 2019), URL = https://www.setthings.com/en/anomalies-of-general-relativity/ Email: firstname.lastname@example.com This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/4.0/. A partial translation of Sfetcu, Nicolae, "Epistemologia gravitației experimentale – Raționalitatea științifică", SetThings (1 august 2019), MultiMedia Publishing (ed.), ISBN: 978-606-033-234-3, DOI: 10.13140/RG.2.2.15421.61925, URL = https://www.setthings.com/ro/e-books/epistemologiagravitatiei-experimentale-rationalitatea-stiintifica/ THE SATURATION POINT OF GENERAL RELATIVITY ......................................... 4 BIBLIOGRAPHY ............................................................................................................ 11 Nicolae Sfetcu: Anomalies of General Relativity 2 Over time, the general theory of relativity has accumulated several anomalies and discrepancies, indicating the need for a better theory about gravity or other approaches: • Stars in galaxies have a distribution of increasing speeds from the center to the periphery, with a greater variation than predicted. The same is true of galaxies in galaxy clusters. The hypothesis of dark matter, which would interact by gravity but not electromagnetically, could explain the discrepancy. There are also various changes in Newtonian dynamics that can explain this anomaly, such as the MOND theory. • Spacecrafts experienced greater acceleration than predicted during gravitational maneuvers. • The metric expansion of space seems to be accelerating. Dark energy was introduced as a hypothesis to explain this. A recent explanation is that space geometry is not homogeneous due to galaxy clusters, but this hypothesis is challenged. 1 • Recent measurements show that planetary orbits grow faster than predicted by the loss of the Sun's mass by radiative energy. • Photons from cosmic radiation should gain energy and then lose it on the way, but in reality, they gain twice as much energy as predicted by theory. One hypothesis would be that gravity decreases faster than the inverse square at certain distances. • Extra massive hydrogen clouds: Lyman-alpha spectral lines suggest that hydrogen clouds are more crowded at certain scales than expected and, like dark flux, may indicate that gravity is slower than inverse squares at certain distances. 2 The ad-hoc hypotheses introduced in general relativity to explain gravitational singularities based on energy conditions are not very efficient. More detailed assumptions on the content of the subject are needed. 3 Many scientists and philosophers have come to the conclusion that singularities must be associated with reaching the limits of the physical validity of general relativity, and a new theory of quantum gravity needs to be developed. A singularity that can causally influence certain parts of the space is called naked singularity. Penrose proposed the elimination of naked singularities using the cosmic censorship hypothesis. 4 Demonstrating the cosmic censorship hypothesis is one of the central mathematical problems of general relativity. 1 Anil Ananthaswamy, "Swiss-Cheese Model of the Cosmos Is Full of Holes," New Scientist, 2008, https://www.newscientist.com/article/mg20026783-800-swiss-cheese-model-of-the-cosmos-is-full-of-holes/. 2 Marcus Chown, "Gravity May Venture Where Matter Fears to Tread," New Scientist, accessed May 3, 2019, https://www.newscientist.com/article/mg20126990-400-gravity-may-venture-where-matter-fears-to-tread/. 3 Alan D. Rendall, "The Nature of Spacetime Singularities," ArXiv:Gr-Qc/0503112, November 2005, 76–92, https://doi.org/10.1142/9789812700988_0003. 4 R. Penrose, "Singularities and Time-Asymmetry.," 1979, 581–638, http://adsabs.harvard.edu/abs/1979grec.conf..581P. Nicolae Sfetcu: Anomalies of General Relativity 3 According to some scientists, general relativity contains the germs of its own destruction, since the theory is incapable of predicting physics on the Planck scale, and problems such as nonrenormalizability and singularities are "known unknown." 5 5 David Tong, String Theory (University of Cambridge, 2009), http://www.damtp.cam.ac.uk/user/tong/string/string.pdf. Nicolae Sfetcu: Anomalies of General Relativity 4 The saturation point of general relativity According to the methodology of Lakatos research programs, general relativity can be divided into several periods: the initial period, the stagnation period, the maturity period, and the saturation point. The initial period ("Genesis", 1887-1919) includes the two great experiments of relativistic physics, the Michelson-Morley experiment and the Eotvos experiment, and the two confirmations, the deformation of light and the perihelion advance of Mercury. There followed a period of stagnation ("Hibernation", 1920-1960) during which the theory took it before technological and experimental possibilities, the theory being even removed from the textbooks of physics and astronomy. The maturity of the theory begins in 1960, when the astronomical discoveries and new experiments drew attention to the GR. This period (1960 1980) was named by Will as a "Golden Era",6 in which observable predictions of GR were systematized, compared to other alternative theories, and new experiments were proposed for testing. The first experiment of this period was developed to confirm the change in gravitational frequency of light (1960) and ended with the confirmation of the GR prediction of the energy loss of gravitational waves (1979) by observing the Hulse-Taylor binary pulsar. From 1980 the saturation zone of the GR began, called by Will as "Quest for Strong Gravity". Some of the new predictions of the theory are now insignificant and difficult to verify, in some cases requiring still undeveloped technologies. The theory began to be attacked by new experimental theories or techniques, such as using laser-cooled ion atoms and traps to perform ultra-precise tests, proposing a "fifth" force, or additional dimensions to test the inverse square law of gravity. Increased attention has begun to be paid to the effects of strong gravitational fields, near the horizon of the event of a non-rotating black hole, in neutron stars or, for the extended universe, associated gravitational fields on the Planck scale. In Einstein's equations of classical general relativity, there remains a fundamental asymmetry between gravitational and non-gravitational fields: on the left, a geometrical object (gμν, the Einstein tensor), representing the curvature of spacetime, is identical with the phenomenological but non geometrical representation of tensor of the matter on the right side. (1) Gμν = kTμν, where Gμν ≡ Rμν - (1/2)gμνR Thus, in his lecture for the Nobel Prize in July 1923, Einstein stated: "The mind striving after unification of the theory cannot be satisfied that two fields should exist which, by their nature, are quite independent. A mathematically unified field theory is sought in which the gravitational field and the electromagnetic field are interpreted as only different components or manifestations of the same uniform field, ... The gravitational theory, considered 6 Clifford M. Will, "The Confrontation between General Relativity and Experiment," Living Reviews in Relativity 17, no. 1 (December 2014): 4, https://doi.org/10.12942/lrr-2014-4. Nicolae Sfetcu: Anomalies of General Relativity 5 in terms of mathematical formalism, i.e., Riemannian geometry, should be generalized so that it includes the laws of the electromagnetic field." 7 There is theoretical evidence that disregards Einstein equivalence principle (EEP) in certain cases, through quantum gravitational effects, effects derived from string theory, or through other undetected interactions so far. In string theory there are such fields that violate EEP, but the theory is not yet mature enough to materialize such a refutation. Clifford M. Will states that observing effects that appear to violate the EEP is, to some extent, semantic. The fields involved in string theory can be long distances and can mimic gravitational fields, but no way has been found to do so. 8 The idea of using EEP tests in this way appeared in the 1980s, in search of a "fifth" force9 as a force of about a percentage of gravity but with a range of several hundred meters, implying a deviation from the inverse-square law of Newtonian gravity. The idea came about when using gravity profile measurements from deep mines in Australia and from new ideas in particle physics, suggesting the possible presence of very small gravity particles. Numerous experiments have sought evidence of this force by measuring acceleration differences by composition, but the results have not been conclusive, the consensus being that there is no credible experimental evidence for a fifth force. 10 The possibility that the inverse square law would be violated at very short intervals in laboratory tests11 provided that some of the extra spatial dimensions in string theory could extend beyond macroscopic scales. On a small scale, gravity deviates from the known law. Many methods of high precision and low noise have been developed, adapted for laboratory tests. No deviations from the inverse square law were found. 12 Gravitational singularities are considered to be a spacetime limit. General relativity allows for the existence of singularities, but it cannot say anything about what is happening inside them, and scientists have not yet agreed on a definition of them, also considering that without a geometry in accordance with the laws of physics cannot exist a spacetime location. In conclusion, they say, one cannot speak of singularities, but rather of singular spacetimes, although in principle these terms 7 Albert Einstein, Fundamental Ideas and Problems of the Theory of Relativity: Lecture Delivered to the Nordic Assembly of Naturalists at Göteborg on July 11, 1923 (Nobel Museum, 2009), 489. 8 Will, "The Confrontation between General Relativity and Experiment." 9 E. Fischbach et al., "Reanalysis of the Eotvos Experiment," Physical Review Letters 56 (January 1, 1986): 56, 3–6, https://doi.org/10.1103/PhysRevLett.56.3. 10 C. M. Will, "Twilight Time for the Fifth Force?," 1990, 80, 472–479, http://connection.ebscohost.com/c/articles/9011192203/twilight-time-fifth-force. 11 Lisa Randall and Raman Sundrum, "An Alternative to Compactification," Physical Review Letters 83, no. 23 (December 6, 1999): 83, 4690–4693, https://doi.org/10.1103/PhysRevLett.83.4690. 12 Joshua C. Long et al., "Upper Limits to Submillimetre-Range Forces from Extra Space-Time Dimensions," Nature 421, no. 6926 (February 2003): 421, 922–925, https://doi.org/10.1038/nature01432. Nicolae Sfetcu: Anomalies of General Relativity 6 are equivalent. 13 Clarke14 and Earman, 15 as well as Geroch, Can-bin and Wald16 and Curiel, 17 argue that a precise, rigorous and univocal definition of singularity is needed for a better approach to them and for a more accurate modeling of spacetime aspects. 18 It is common assertion that general relativity, considering spacetime as singular, predicts its own inability to limit the singularities of black holes and the Big Bang, negating their reality. It is hoped that a more fundamental theory, possibly quantum gravity, will solve this problem. 19 The black holes appear, according to general relativity, when the cosmic body collapsed under the so-called Schwarzschild ray, proportional to the body mass. The "event horizon" of a black hole is the point where there is no turning back, within which the gravitational attraction is greater than any attempt to exit this area, including for light.20 (1) For a standard black hole (uncharged, nonrotating), the horizon of the event is within the Schwarzschild radius. From the point of view of a person outside the event horizon, the time near a black hole is delayed due to the strong gravity, until the time intervals reach infinitely large within the horizons of events. From the perspective of the person entering the horizon of events, nothing unusual happens. Time is running the same way and he does not realize that he has entered the horizon of events. Relativistic black holes are purely gravitational entities. They are solutions to the "vacuum" of Einstein's field equations. In the context of general relativity, Erik Curiel states that gravity is given up and a curved geometry of spacetime is postulated that produces all the effects of gravity, the black hole being no more a "thing" in space, but a feature of spacetime itself. 21 The matter of the collapsing star disappears in the singularity of the black hole, leaving only the geometrical properties 13 Erik Curiel, "Singularities and Black Holes," in The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta, Spring 2019 (Metaphysics Research Lab, Stanford University, 2019), https://plato.stanford.edu/archives/spr2019/entries/spacetime-singularities/. 14 C. J. S. Clarke, "The Analysis of Space-Time Singularities by C. J. S. Clarke," Cambridge Core, May 1994, https://doi.org/10.1017/CBO9780511608155. 15 John Earman, "Bangs, Crunches, Whimpers and Shrieks: Singularities and Acausalities in Relativistic Spacetimes," British Journal for the Philosophy of Science 49, no. 2 (1998): 338–347. 16 Robert Geroch, Liang Can‐bin, and Robert M. Wald, "Singular Boundaries of Space–Times," Journal of Mathematical Physics 23, no. 3 (March 1, 1982): 23(3): 432–435, https://doi.org/10.1063/1.525365. 17 Erik Curiel, "The Analysis of Singular Spacetimes," Philosophy of Science 66, no. 3 (1999): 66(S1): S119–S145. 18 Nicolae Sfetcu, Singularitățile ca limite ontologice ale relativității generale (MultiMedia Publishing, 2018), http://doi.org/10.13140/RG.2.2.17470.18242. 19 Abhay Ashtekar and Martin Bojowald, "Quantum Geometry and the Schwarzschild Singularity," Classical and Quantum Gravity 23, no. 2 (January 21, 2006): 23(2): 391–411, https://doi.org/10.1088/0264-9381/23/2/008. 20 A more accurate description distinguishes other types of horizon, such as apparent horizons, cf. HawkingStephen W. Hawking et al., The Large Scale Structure of Space-Time, New Ed edition (Cambridge: Cambridge University Press, 1975), 312–20. 21 Curiel, "Singularities and Black Holes." Nicolae Sfetcu: Anomalies of General Relativity 7 of the black hole (mass, charge and angular momentum), according to theorems called "no-hair", regardless of the previous physical properties of objects that collapse into a black hole. A "naked" singularity does not have a horizon of events. This implies a fundamental break in the structure of spacetime. 22 23 A version of a naked singularity is the "white hole", a black hole reversed in time, from which matter and objects from nothing could appear. Since the equations of the field of general relativity do not select a preferred direction of time, and the formation of a black hole is allowed, then white holes will be allowed by these laws. 24 Roger Penrose asserts that naked singularities will never be formed, introducing an ad-hoc hypothesis called the "cosmic censorship hypothesis": a singularity will always be in a black hole that is surrounded by the horizon of events. The former was abandoned in time. Several alternative hypotheses have been proposed to eliminate the possibility of naked singularities that violate the principle of causality, 25 26 but none is considered satisfactory to date. Black holes provide an essential testing ground for the conceptual problems underlying quantum gravity and general relativity, regarding the violation of energy conservation and micro-causality, and the paradox of information loss. Quantum gravity seems to be the best candidate for modeling these phenomena. In 1971, Hawking introduced the conjecture that the total surface of event horizons in any group of black holes does not decrease, even if they unite (the second law of black hole mechanics, by similarity to entropy in thermodynamics). 27 To prevent the black holes from having zero entropy, Bekenstein proposed that a black hole would have an entropy proportional to the area of its horizon. 28 Hawking discovered that quantum field theory predicts that a black hole behaves like a black body radiating at a constant temperature, thus violating the second law of black hole mechanics due to energy loss and hence shrinkage. But radiation removes also entropy, and so the amount of entropy of matter is increasing. This allows the formulation of the first law of black hole mechanics similar to the first law of thermodynamics, with the mass acting as energy, the gravity 22 Sfetcu, Singularitățile ca limite ontologice ale relativității generale. 23 Earman, "Bangs, Crunches, Whimpers and Shrieks," 65–66. 24 Curiel, "Singularities and Black Holes." 25 Pankaj S. Joshi, "Cosmic Censorship: A Current Perspective," ArXiv:Gr-Qc/0206087, June 28, 2002, 17(15): 1067–1079, http://arxiv.org/abs/gr-qc/0206087. 26 Earman, "Bangs, Crunches, Whimpers and Shrieks," chap. 3. 27 S. W. Hawking, "Gravitational Radiation from Colliding Black Holes," Physical Review Letters 26, no. 21 (May 24, 1971): 26 (21): 1344–1346, https://doi.org/10.1103/PhysRevLett.26.1344. 28 Robert M. Wald, "The Thermodynamics of Black Holes," Living Reviews in Relativity 4, no. 1 (July 9, 2001): 4 (1): 6, https://doi.org/10.12942/lrr-2001-6. Nicolae Sfetcu: Anomalies of General Relativity 8 of the surface as temperature, and the area as entropy. 29 In this interpretation of the black hole, general relativity is unsatisfactory, and a better theory of quantum gravity is needed. 30 A black hole only holds information about the total mass, charge and angular momentum. The theory of stable black holes states that this loss is not a problem, because the information can be considered as being present in the black hole, inaccessible from the outside but represented on the horizon of the event in accordance with the holographic principle. But in the theory that black holes slowly evaporate through Hawking radiation, information about the matter that formed the black hole is irretrievably lost. In quantum mechanics, the loss of information corresponds to the violation of unity, related to the conservation of probability, resulting in the violation of energy conservation. 31 The latest studies show that information and unity are nevertheless preserved in a quantum treatment of the problem. 32 In the case of a body falling into a black hole, the theory of the quantum field in curved space involves quantities of Hawking radiation, including only a finite amount of information encoded in Hawking radiation. But the inseparability of the particle at the exit of all the Hawking radiation that the black hole has previously emitted creates a paradox called "the monogamy of inseparability." 33 To solve the paradox, one of the three theories tested over time should be discarded: the principle of Einstein's equivalence, unitarity, or the existing theory of the quantum field. The renunciation of the principle of equivalence implies a "firewall" that destroys the particles that enter in the horizon of the event. 34 The 2016 LIGO data shows possible echo signals due to a fuzzy horizon of events, possible in fuzzball theories, but impossible in general classical relativity. 35 29 Wald, 4 (1): 6. 30 S. Carlip, "Black Hole Thermodynamics and Statistical Mechanics," in Physics of Black Holes: A Guided Tour, ed. Eleftherios Papantonopoulos, Lecture Notes in Physics (Berlin, Heidelberg: Springer Berlin Heidelberg, 2009), 769: 89–123, https://doi.org/10.1007/978-3-540-88460-6_3. 31 Steven B. Giddings, "The Black Hole Information Paradox," ArXiv:Hep-Th/9508151, August 28, 1995, http://arxiv.org/abs/hep-th/9508151. 32 Samir D. Mathur, "The Information Paradox: Conflicts and Resolutions," Pramana 79, no. 5 (November 1, 2012): 1059–73, https://doi.org/10.1007/s12043-012-0417-z. 33 Zeeya Merali, "Astrophysics: Fire in the Hole!," Nature News 496, no. 7443 (April 4, 2013): 20–23, https://doi.org/10.1038/496020a. 34 Jennifer Ouellette, "Black Hole Firewalls Confound Theoretical Physicists," Scientific American, 2012, https://www.scientificamerican.com/article/black-hole-firewalls-confound-theoretical-physicists/. 35 Zeeya Merali, "LIGO Black Hole Echoes Hint at General-Relativity Breakdown," Nature News, 2016, 540, https://doi.org/10.1038/nature.2016.21135. Nicolae Sfetcu: Anomalies of General Relativity 9 The need for consistency between quantum theory and general relativity, 36 and the existence of singularities, require the emergence of a complete theory of quantum gravity. 37 So far, such a complete and consistent theory has failed to develop, although there are several candidates. 38 The generalization of quantum field theory from the elementary particle physics to include gravity has failed. 39 At low energies the theory is acceptable, but at very high energies, the results are very divergent and lead to models without predictive power. 40 An attempt to eliminate these limitations is string theory, a quantum theory. The theory promises a unification of gravity with the other forces, supplementing the three spatial dimensions with another six. 41 A newer version of the theory, the superstring theory, is trying to unify general relativity and supersymmetry, under the name of supergravity, 42 and a hypothetical unifying model with eleven dimensions known as M-theory. 43 Another approach uses the canonical quantization of quantum theory in which, starting from general relativity, one reaches the Wheeler-deWitt equation, an analogue of the Schrödinger 36 S. Carlip, "Quantum Gravity: A Progress Report," Reports on Progress in Physics 64, no. 8 (August 1, 2001): sec. 2, https://doi.org/10.1088/0034-4885/64/8/301. 37 Bernard Schutz, "Gravity from the Ground Up by Bernard Schutz," Cambridge Core, December 2003, 407, https://doi.org/10.1017/CBO9780511807800. 38 Herbert W. Hamber, Quantum Gravitation: The Feynman Path Integral Approach (Berlin Heidelberg: SpringerVerlag, 2009), https://www.springer.com/gp/book/9783540852926. 39 G. 't Hooft and M. Veltman, "One-Loop Divergencies in the Theory of Gravitation," Annales de L'Institut Henri Poincare Section (A) Physique Theorique 20 (1974): 20 (1): 69, http://adsabs.harvard.edu/abs/1974AIHPA..20...69T. 40 Steven Weinberg, The Quantum Theory of Fields, Volume 2: Modern Applications, 1 edition (Cambridge: Cambridge University Press, 2005). 41 M. B. Green, J. H. Schwarz, and E. Witten, "Superstring Theory. Vol. 1: Introduction," ZAMM Journal of Applied Mathematics and Mechanics / Zeitschrift Für Angewandte Mathematik Und Mechanik 68, no. 6 (1988): 258–258, https://doi.org/10.1002/zamm.19880680630. 42 Steven Weinberg, The Quantum Theory of Fields, Volume 3: Supersymmetry, 1st Edition edition (Cambridge: Cambridge University Press, 2005). 43 P. K. Townsend, "Four Lectures on M-Theory," ArXiv:Hep-Th/9612121, December 11, 1996, 13: 385, http://arxiv.org/abs/hep-th/9612121. Nicolae Sfetcu: Anomalies of General Relativity 10 equation, but which has been wrongly defined. 44 By introducing Ashtekar ad-hoc (variable) hypotheses, it was developed the theory of loop quantum gravity. 45 There are numerous other attempts to arrive at a viable theory of quantum gravity, based on the Feynman approach and the Regge calculation, dynamic triangulations, causality sets, twistor models46 or the models based on integrals of paths of the quantum cosmology. 47 All candidate theories still have major formal and conceptual problems that are difficult to overcome so far, including the impossibility of verifying predictions through experimental tests. 48 44 Karel Kuchař, "Canonical Quantization of Gravity," Relativity, Astrophysics and Cosmology, 1973, 237–288, https://doi.org/10.1007/978-94-010-2639-0_5. 45 Abhay Ashtekar and Jerzy Lewandowski, "Background Independent Quantum Gravity: A Status Report," Classical and Quantum Gravity 21, no. 15 (August 7, 2004): 21 (15): R53–R152, https://doi.org/10.1088/02649381/21/15/R01. 46 Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Reprint edition (New York: Vintage, 2007). 47 S. W. Hawking and W. Israel, Quantum Cosmology, in Three Hundred Years of Gravitation (Cambridge University Press, 1989), 631–651. 48 John H. Schwarz, "String Theory: Progress and Problems," Progress of Theoretical Physics Supplement 170 (May 1, 2007): 170: 214–226, https://doi.org/10.1143/PTPS.170.214. Nicolae Sfetcu: Anomalies of General Relativity 11 Bibliography Ananthaswamy, Anil. "Swiss-Cheese Model of the Cosmos Is Full of Holes." New Scientist, 2008. https://www.newscientist.com/article/mg20026783-800-swiss-cheese-model-of-thecosmos-is-full-of-holes/. Ashtekar, Abhay, and Martin Bojowald. "Quantum Geometry and the Schwarzschild Singularity." Classical and Quantum Gravity 23, no. 2 (January 21, 2006): 391–411. https://doi.org/10.1088/0264-9381/23/2/008. Ashtekar, Abhay, and Jerzy Lewandowski. "Background Independent Quantum Gravity: A Status Report." Classical and Quantum Gravity 21, no. 15 (August 7, 2004): R53–152. https://doi.org/10.1088/0264-9381/21/15/R01. Carlip, S. 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