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Theory of everything
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==Arguments against== In parallel to the intense search for a theory of everything, various scholars have debated the possibility of its discovery. ===Gödel's incompleteness theorem=== A number of scholars claim that [[Gödel's incompleteness theorem]] suggests that attempts to construct a theory of everything are bound to fail. Gödel's theorem, informally stated, asserts that any formal theory sufficient to express elementary arithmetical facts and strong enough for them to be proved is either inconsistent (both a statement and its denial can be derived from its axioms) or incomplete, in the sense that there is a true statement that can't be derived in the formal theory. [[Stanley Jaki]], in his 1966 book ''The Relevance of Physics'', pointed out that, because a "theory of everything" will certainly be a consistent non-trivial mathematical theory, it must be incomplete. He claims that this dooms searches for a deterministic theory of everything.<ref> {{cite book |last=Jaki |first=S.L. |date=1966 |pages=127–130 |title=The Relevance of Physics |publisher=Chicago Press }}</ref> [[Freeman Dyson]] has stated that "Gödel's theorem implies that pure mathematics is inexhaustible. No matter how many problems we solve, there will always be other problems that cannot be solved within the existing rules. […] Because of Gödel's theorem, physics is inexhaustible too. The laws of physics are a finite set of rules, and include the rules for doing mathematics, so that Gödel's theorem applies to them."<ref>Freeman Dyson, NYRB, May 13, 2004</ref> [[Stephen Hawking]] was originally a believer in the Theory of Everything, but after considering Gödel's Theorem, he concluded that one was not obtainable. "Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind."<ref>Stephen Hawking, [http://www.hawking.org.uk/godel-and-the-end-of-physics.html Gödel and the end of physics] {{Webarchive|url=https://web.archive.org/web/20200529232552/http://www.hawking.org.uk/godel-and-the-end-of-physics.html |date=2020-05-29 }}, July 20, 2002</ref> [[Jürgen Schmidhuber]] (1997) has argued against this view; he asserts that Gödel's theorems are irrelevant for [[computable]] physics.<ref>{{cite book |last=Schmidhuber |first=Jürgen |date=1997 |title=A Computer Scientist's View of Life, the Universe, and Everything. Lecture Notes in Computer Science |volume=1337 |url=http://www.idsia.ch/~juergen/everything/ |pages=201–208 |publisher=[[Springer (publisher)|Springer]] |isbn=978-3-540-63746-2 |doi=10.1007/BFb0052071 |citeseerx=10.1.1.580.1970 |s2cid=21317070 |access-date=2008-03-26 |archive-date=2014-02-27 |archive-url=https://web.archive.org/web/20140227162350/http://www.idsia.ch/~juergen/everything/ |url-status=live }}</ref> In 2000, Schmidhuber explicitly constructed limit-computable, deterministic universes whose [[pseudo-randomness]] based on [[undecidable problem|undecidable]], Gödel-like [[halting problem]]s is extremely hard to detect but does not prevent formal theories of everything describable by very few bits of information.<ref> {{Cite journal |author=Schmidhuber, Jürgen |date=2002 |title=Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit |journal=International Journal of Foundations of Computer Science |volume=13 |issue=4 |pages=587–612 |arxiv=quant-ph/0011122 |bibcode=2000quant.ph.11122S |doi=10.1142/s0129054102001291 }}</ref> Related critique was offered by [[Solomon Feferman]]<ref>{{cite web |last=Feferman |first=Solomon |url=http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf |title=The nature and significance of Gödel's incompleteness theorems |publisher=[[Institute for Advanced Study]] |date=17 November 2006 |access-date=2009-01-12 |archive-date=2008-12-17 |archive-url=https://web.archive.org/web/20081217035530/http://math.stanford.edu/~feferman/papers/Godel-IAS.pdf |url-status=live }}</ref> and others. Douglas S. Robertson offers [[Conway's game of life]] as an example:<ref> {{cite journal |last=Robertson |first=Douglas S. |date=2007 |title=Goedel's Theorem, the Theory of Everything, and the Future of Science and Mathematics |journal=[[Complexity (journal)|Complexity]] |volume=5 |pages=22–27 |doi=10.1002/1099-0526(200005/06)5:5<22::AID-CPLX4>3.0.CO;2-0 |issue=5 |bibcode=2000Cmplx...5e..22R}}</ref> The underlying rules are simple and complete, but there are formally undecidable questions about the game's behaviors. Analogously, it may (or may not) be possible to completely state the underlying rules of physics with a finite number of well-defined laws, but there is little doubt that there are questions about the behavior of physical systems which are formally undecidable on the basis of those underlying laws. Since most physicists would consider the statement of the underlying rules to suffice as the definition of a "theory of everything", most physicists argue that Gödel's Theorem does ''not'' mean that a theory of everything cannot exist.{{Citation needed|reason=Precarious wording, unclear significance, and any relevance of the purported affidavits necessitate scrutiny|date=August 2021}} On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of [[prime number]]s into a physical question.<ref>{{cite web |last=Hawking |first=Stephen |date=20 July 2002 |title=Gödel and the end of physics |url=http://www.damtp.cam.ac.uk/strings02/dirac/hawking/ |access-date=2009-12-01 |archive-date=2011-05-21 |archive-url=https://web.archive.org/web/20110521123113/http://www.damtp.cam.ac.uk/strings02/dirac/hawking/ }}</ref> This definitional discrepancy may explain some of the disagreement among researchers. ===Fundamental limits in accuracy=== No physical theory to date is believed to be precisely accurate. Instead, physics has proceeded by a series of "successive approximations" allowing more and more accurate predictions over a wider and wider range of phenomena. Some physicists believe that it is therefore a mistake to confuse theoretical models with the true nature of reality, and hold that the series of approximations will never terminate in the "truth".<ref>{{cite book|title=The New Cosmic Onion: Quarks and the Nature of the Universe|first=search|last=results|date=17 December 2006|publisher=CRC Press|isbn = 978-1-58488-798-0}}</ref> Einstein himself expressed this view on occasions.<ref>Einstein, letter to Felix Klein, 1917. (On determinism and approximations.) Quoted in Pais (1982), Ch. 17.</ref> ===Definition of fundamental laws=== There is a philosophical debate within the physics community as to whether a theory of everything deserves to be called ''the'' fundamental law of the universe.<ref>Weinberg (1993), Ch 2.</ref> One view is the hard [[reductionist]] position that the theory of everything is the fundamental law and that all other theories that apply within the universe are a consequence of the theory of everything. Another view is that [[emergence|emergent]] laws, which govern the behavior of [[complex system]]s, should be seen as equally fundamental. Examples of emergent laws are the [[second law of thermodynamics]] and the theory of [[natural selection]]. The advocates of emergence argue that emergent laws, especially those describing complex or living systems are independent of the low-level, microscopic laws. In this view, emergent laws are as fundamental as a theory of everything. A well-known debate over this took place between Steven Weinberg and [[Philip Warren Anderson|Philip Anderson]].<ref>{{Cite book|title=Superstrings, P-branes and M-theory|page=7}}</ref> ====Impossibility of calculation==== Weinberg<ref>Weinberg (1993) p. 5</ref> points out that calculating the precise motion of an actual projectile in the Earth's atmosphere is impossible. So how can we know we have an adequate theory for describing the motion of projectiles? Weinberg suggests that we know ''principles'' (Newton's laws of motion and gravitation) that work "well enough" for simple examples, like the motion of planets in empty space. These principles have worked so well on simple examples that we can be reasonably confident they will work for more complex examples. For example, although [[general relativity]] includes equations that do not have exact solutions, it is widely accepted as a valid theory because all of its equations with exact solutions have been experimentally verified. Likewise, a theory of everything must work for a wide range of simple examples in such a way that we can be reasonably confident it will work for every situation in physics. Difficulties in creating a theory of everything often begin to appear when combining [[quantum mechanics]] with the theory of [[general relativity]], as the equations of quantum mechanics begin to falter when the force of gravity is applied to them.
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