Machine Elf 1735 (talk | contribs) <ref name="EUH"/> |
Dark Formal (talk | contribs) There is no reason to reframe all criticisms in terms of Tegmark's response to them! Please, let's discuss on the talk page and then rewrite the article once consensus is achieved. |
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The hypothesis is related to the [[anthropic principle]], to Tegmark's categorization of theories of the [[Multiverse (science)|multiverse]], and to [[Jürgen Schmidhuber]]'s ultimate ensemble of all computable universes.<ref>[[Jürgen Schmidhuber]] (1997) "[http://www.idsia.ch/~juergen/everything/ A Computer Scientist's View of Life, the Universe, and Everything]" in C. Freksa, ed., ''Foundations of Computer Science: Potential - Theory - Cognition''. Lecture Notes in Computer Science. Springer: 201-08.</ref> |
The hypothesis is related to the [[anthropic principle]], to Tegmark's categorization of theories of the [[Multiverse (science)|multiverse]], and to [[Jürgen Schmidhuber]]'s ultimate ensemble of all computable universes.<ref>[[Jürgen Schmidhuber]] (1997) "[http://www.idsia.ch/~juergen/everything/ A Computer Scientist's View of Life, the Universe, and Everything]" in C. Freksa, ed., ''Foundations of Computer Science: Potential - Theory - Cognition''. Lecture Notes in Computer Science. Springer: 201-08.</ref> |
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==Criticisms== |
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As is the case with other multiverse theories, the MUH has been forcefully criticized by some scientists and philosophers. |
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⚫ | [[Jürgen Schmidhuber]] noted that there is no such thing as a [[Uniform distribution|uniform]] [[prior distribution]] over infinitely many mathematical structures, as suggested by Tegmark. Schmidhuber's more restricted ensemble admits only universe representations describable by [[constructive mathematics]], that is, [[computer program]]s. He explicitly includes universe representations describable by non-[[halting problem|halting programs]] whose output bits converge after finite time, although the convergence time itself may not be predictable by a halting program, due to [[Kurt Gödel]]'s limitations.<ref>[[Jürgen Schmidhuber|J. Schmidhuber]] (2000) "[http://arxiv.org/abs/quant-ph/0011122 Algorithmic Theories of Everything.]"</ref><ref>[[Jürgen Schmidhuber|J. Schmidhuber]] (2002) "[http://www.idsia.ch/~juergen/kolmogorov.html Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit,]" ''International Journal of Foundations of Computer Science'' 13(4): 587–612.</ref>. |
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This question is also vigorously discussed in a three-way debate between Tegmark and fellow physicists [[Piet Hut]] and [[Mark Alford]] <ref name='HTA'>Hut, P., Alford, M., Tegmark, M. (2006) "[http://arxiv.org/abs/physics/0510188 On Math, Matter and Mind.]" ''Foundations of Physics'' 36: 765-94.</ref>. Hut and Alford point out that Tegmark offers a formalist (axiomatic) recipe for building his Platonic ensemble, but that it is known from [[Gödel's incompleteness theorem]] that [[Formalism (mathematics)|formalism]] and [[Platonism]] are incompatible: the axiomatic method can only encompass the simplest mathematics systems. Other issues arising from Gödel's theorem have also been raised by physicist [[Alexander Vilenkin]] <ref>A. Vilenkin (2006) ''Many Worlds in One: The Search for Other Universes''. Hill and Wang, New York.</ref>. |
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⚫ | Since the other "universes" in the ensemble are unobservable, and there is no independent evidence for the scheme that predicts them, many scientists have raised the point that the MUH is not empirically testable, and therefore does not constitute a scientific theory.<ref name='HTA'/> <ref>W. R. Stoeger, [[G. F. R. Ellis]], U. Kirchner (2006) "[http://arxiv.org/abs/astro-ph/0407329 Multiverses and Cosmology: Philosophical Issues.]"</ref> |
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===Other criticisms=== |
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⚫ | Since the other "universes" in the ensemble are unobservable, and there is no independent evidence for the scheme that predicts them, many scientists have raised the point that the MUH is not empirically testable, and therefore does not constitute a scientific theory.<ref name='HTA'/> <ref>W. R. Stoeger, [[G. F. R. Ellis]], U. Kirchner (2006) "[http://arxiv.org/abs/astro-ph/0407329 Multiverses and Cosmology: Philosophical Issues.]"</ref> |
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⚫ | Philosopher [[Keith Ward]] has argued that if MUH or any similar highly profligate view of universes is true, then the vast majority of these universes will be very complex. On the other hand, the universe in which we find ourselves is very simple, as evidenced by its intellegibility and elegance. So the MUH fails to explain a fundamental property of our universe unless we assume that by an enormous coincidence we exist within the vanishingly small proportion of simple universes.<ref>[[Keith Ward]] (1996) ''God, Chance & Necessity''. Oxford: Oneworld.</ref> The problem with this view is the following {{fact|date=July 2010|reason=Who has made this argument?}}: One can ask "Doesn't the notion of high probability of us being in a complex universe depend on there being a probability density function for the collection of all universes?". So far, there is no such function defined for all logically possible universes. |
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⚫ | Tegmark<ref |
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===Response=== |
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Tegmark, along with physicists [[Piet Hut]] and Mark Alford, coauthored a journal article in the format of a fictional debate which erupts between three characters who each approach physics from one angle, of either: Math, Matter, or Mind. In this [[Penrose triangle]]: {{nowrap|Math explains Matter,}} {{nowrap|Matter explains Mind, and}} {{nowrap|Mind explains Math.}} The authors explain that their intention was to caution the non-physicist, to beware of any claim that modern physics leads to ''any'' particular resolution of this circularity.<ref name='HTA'>{{cite journal |last1=Hut |first1=Piet |last2=Alford |first2=Mark |last3=Tegmark |first3=Max |year=2006 |title=On Math, Matter and Mind |journal=Foundations of Physics |volume=36 |issue=6 |pages=765-794 |doi=10.1007/s10701-006-9048-x |href=http://arxiv.org/abs/physics/0510188}}</ref> |
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⚫ | Tegmark<ref>[[Max Tegmark]] (2008) “[http://arxiv.org/abs/0704.0646 The Mathematical Universe,]” ''Foundations of Physics'' 38: 101-50.</ref> has replied to some of these criticisms by positing an External Reality Hypothesis (ERH), stating that an external physical reality exists independently of humans. Tegmark argues that given a sufficiently broad definition of mathematics, the ERH implies the MUH. He then formalizes the Ultimate Ensemble as the "Level IV Multiverse."<ref>http://ocw.mit.edu/NR/rdonlyres/hs/tbq/LectureNotes/lec3.pdf</ref> |
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==See also== |
==See also== |
Revision as of 03:01, 1 December 2010
In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the Ultimate Ensemble, is a speculative theory of everything (TOE) proposed by Max Tegmark.[1]
Description
Tegmark's sole postulate is: All structures that exist mathematically also exist physically. That is, in the sense that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world". [2] The hypothesis suggests that worlds corresponding to different sets of initial conditions, physical constants, or altogether different equations should be considered real.
Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Thus, he reasons, it's more preferable than other theories-of-everything per Occam's Razor. He suggests conscious experience would take the form of mathematical "self-aware substructures" that exist in a physically "'real'" world.
The hypothesis is related to the anthropic principle, to Tegmark's categorization of theories of the multiverse, and to Jürgen Schmidhuber's ultimate ensemble of all computable universes.[3]
Criticisms
As is the case with other multiverse theories, the MUH has been forcefully criticized by some scientists and philosophers.
Definition of the ensemble
Some argue that the "set of all mathematical structures" is not well-defined.
Jürgen Schmidhuber noted that there is no such thing as a uniform prior distribution over infinitely many mathematical structures, as suggested by Tegmark. Schmidhuber's more restricted ensemble admits only universe representations describable by constructive mathematics, that is, computer programs. He explicitly includes universe representations describable by non-halting programs whose output bits converge after finite time, although the convergence time itself may not be predictable by a halting program, due to Kurt Gödel's limitations.[4][5].
This question is also vigorously discussed in a three-way debate between Tegmark and fellow physicists Piet Hut and Mark Alford [6]. Hut and Alford point out that Tegmark offers a formalist (axiomatic) recipe for building his Platonic ensemble, but that it is known from Gödel's incompleteness theorem that formalism and Platonism are incompatible: the axiomatic method can only encompass the simplest mathematics systems. Other issues arising from Gödel's theorem have also been raised by physicist Alexander Vilenkin [7].
Don Page has argued that the MUH is self-contradictory because one cannot subsume all possible (partly contradictory) mathematical structures into one structure.[8]
Other criticisms
Since the other "universes" in the ensemble are unobservable, and there is no independent evidence for the scheme that predicts them, many scientists have raised the point that the MUH is not empirically testable, and therefore does not constitute a scientific theory.[6] [9]
Philosopher Keith Ward has argued that if MUH or any similar highly profligate view of universes is true, then the vast majority of these universes will be very complex. On the other hand, the universe in which we find ourselves is very simple, as evidenced by its intellegibility and elegance. So the MUH fails to explain a fundamental property of our universe unless we assume that by an enormous coincidence we exist within the vanishingly small proportion of simple universes.[10] The problem with this view is the following [citation needed]: One can ask "Doesn't the notion of high probability of us being in a complex universe depend on there being a probability density function for the collection of all universes?". So far, there is no such function defined for all logically possible universes.
Response
Tegmark[11] has replied to some of these criticisms by positing an External Reality Hypothesis (ERH), stating that an external physical reality exists independently of humans. Tegmark argues that given a sufficiently broad definition of mathematics, the ERH implies the MUH. He then formalizes the Ultimate Ensemble as the "Level IV Multiverse."[12]
See also
- Cosmology
- Digital physics
- Impossible world
- Modal realism
- Multiverse
- Ontology
- String theory
- Theory of everything
- The Unreasonable Effectiveness of Mathematics in the Natural Sciences
References
- ^ Tegmark, Max (1998) "Is 'the theory of everything' merely the ultimate ensemble theory?"
- ^ Tegmark (1998), p. 1.
- ^ Jürgen Schmidhuber (1997) "A Computer Scientist's View of Life, the Universe, and Everything" in C. Freksa, ed., Foundations of Computer Science: Potential - Theory - Cognition. Lecture Notes in Computer Science. Springer: 201-08.
- ^ J. Schmidhuber (2000) "Algorithmic Theories of Everything."
- ^ J. Schmidhuber (2002) "Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit," International Journal of Foundations of Computer Science 13(4): 587–612.
- ^ a b Hut, P., Alford, M., Tegmark, M. (2006) "On Math, Matter and Mind." Foundations of Physics 36: 765-94.
- ^ A. Vilenkin (2006) Many Worlds in One: The Search for Other Universes. Hill and Wang, New York.
- ^ D. Page, "Predictions and Tests of Multiverse Theories."
- ^ W. R. Stoeger, G. F. R. Ellis, U. Kirchner (2006) "Multiverses and Cosmology: Philosophical Issues."
- ^ Keith Ward (1996) God, Chance & Necessity. Oxford: Oneworld.
- ^ Max Tegmark (2008) “The Mathematical Universe,” Foundations of Physics 38: 101-50.
- ^ http://ocw.mit.edu/NR/rdonlyres/hs/tbq/LectureNotes/lec3.pdf
Further reading
- Jürgen Schmidhuber (1997) "A Computer Scientist's View of Life, the Universe, and Everything" in C. Freksa, ed., Foundations of Computer Science: Potential - Theory - Cognition. Lecture Notes in Computer Science. Springer: 201-08.
- Max Tegmark (1998) “Is the ‘theory of everything’ merely the ultimate ensemble theory?” Annals of Physics 270: 1-51.
- -------- (2008) “The Mathematical Universe,” Foundations of Physics 38: 101-50.
External links
- Jürgen Schmidhuber "The ensemble of universes describable by constructive mathematics."
- Page maintained by Max Tegmark with links to his technical and popular writings.
- "The 'Everything' mailing list" (and archives). Discusses the idea that all possible universes exist.
- "Is the universe actually made of math?" Interview with Max Tegmark in Discover Magazine.