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Andreas Albrecht of [[Imperial College]] in London called it a "provocative" solution to one of the central problems facing physics. Although he "wouldn't dare" go so far as to say he believes it, he noted that "it's actually quite difficult to construct a theory where everything we see is all there is".<ref>{{cite journal|last=Chown |first=Markus |month=June |year=1998 |title=Anything goes |journal=New Scientist |volume=158 |number=2157 url=http://space.mit.edu/home/tegmark/toe_press.html}}</ref> |
Andreas Albrecht of [[Imperial College]] in London called it a "provocative" solution to one of the central problems facing physics. Although he "wouldn't dare" go so far as to say he believes it, he noted that "it's actually quite difficult to construct a theory where everything we see is all there is".<ref>{{cite journal|last=Chown |first=Markus |month=June |year=1998 |title=Anything goes |journal=New Scientist |volume=158 |number=2157 url=http://space.mit.edu/home/tegmark/toe_press.html}}</ref> |
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==Criticisms and responses== |
==Criticisms and responses== |
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===Definition of the Ensemble=== |
===Definition of the Ensemble=== |
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[[Jürgen Schmidhuber]] <ref>[[Jürgen Schmidhuber|J. Schmidhuber]] (2000) "[http://arxiv.org/abs/quant-ph/0011122 Algorithmic Theories of Everything.]"</ref> argues that "Although Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight', there is no way of assigning equal nonvanishing probability to all (infinitely many) mathematical structures". Schmidhuber puts forward a more restricted ensemble which 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]] (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>. |
[[Jürgen Schmidhuber]] <ref>[[Jürgen Schmidhuber|J. Schmidhuber]] (2000) "[http://arxiv.org/abs/quant-ph/0011122 Algorithmic Theories of Everything.]"</ref> argues that "Although Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight', there is no way of assigning equal nonvanishing probability to all (infinitely many) mathematical structures". Schmidhuber puts forward a more restricted ensemble which 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]] (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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In response, Tegmark notes <ref name="Tegmark2008">"[http://arxiv.org/abs/0704.0646 The Mathematical Universe,]" ''Foundations of Physics'' 38: 101-50.</ref> (sec. V.E) that the measure over all universes has not yet been constructed for the [[String theory landscape]] either, so this should not be regarded as a "show-stopper". |
In response, Tegmark notes <ref name="Tegmark2008">"[http://arxiv.org/abs/0704.0646 The Mathematical Universe,]" ''Foundations of Physics'' 38: 101-50.</ref> (sec. V.E) that the measure over all universes has not yet been constructed for the [[String theory landscape]] either, so this should not be regarded as a "show-stopper". |
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===Consistency with Gödel's theorem=== |
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It has also been suggested that the MUH is inconsistent with [[Gödel's incompleteness theorem]]. In a three-way debate between Tegmark and fellow physicists [[Piet Hut]] and [[Mark Alford]] <ref name='HAT'>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>, the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system... The idea that math is "out there" is incompatible with the idea that it consists of formal systems." |
It has also been suggested that the MUH is inconsistent with [[Gödel's incompleteness theorem]]. In a three-way debate between Tegmark and fellow physicists [[Piet Hut]] and [[Mark Alford]] <ref name='HAT'>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>, the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system... The idea that math is "out there" is incompatible with the idea that it consists of formal systems." |
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Tegmark's response <ref name="Tegmark2008"/> (sec. VII) is to suggest "drastically limiting the number of mathematical structures to be considered", by modifying the MUH to a much more restricted "Computable Universe Hypothesis" (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecideable/uncomputable theorems. Tegmark admits that this approach faces "serious challeges", including (a) it excludes much of the mathematical landscape; (b) the measure on the space of allowed theories may itself be uncomputable; and (c) "virtually all historically successful theories of physics violate the CUH". |
Tegmark's response <ref name="Tegmark2008"/> (sec. VII) is to suggest "drastically limiting the number of mathematical structures to be considered", by modifying the MUH to a much more restricted "Computable Universe Hypothesis" (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecideable/uncomputable theorems. Tegmark admits that this approach faces "serious challeges", including (a) it excludes much of the mathematical landscape; (b) the measure on the space of allowed theories may itself be uncomputable; and (c) "virtually all historically successful theories of physics violate the CUH". |
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===Observability=== |
===Observability=== |
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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 |
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 multiverse theories in general <ref>W. R. Stoeger, [[G. F. R. Ellis]], U. Kirchner (2006) "[http://arxiv.org/abs/astro-ph/0407329 Multiverses and Cosmology: Philosophical Issues.]"</ref> and the MUH in particular <ref name='HAT'/> are not empirically testable, and therefore do not constitute scientific theories. |
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Tegmark maintains that MUH is testable, stating that predicts (a) that "physics research will uncover mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start testing multiverse predictions by assessing how typical our universe is" (<ref name="Tegmark2008"/>, sec. VIII.C). |
Tegmark maintains that MUH is testable, stating that it predicts (a) that "physics research will uncover mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start testing multiverse predictions by assessing how typical our universe is" (<ref name="Tegmark2008"/>, sec. VIII.C). |
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===Coexistence of contradictory structures=== |
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===Self-contradictoriness=== |
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[[Don Page (mathematician)|Don Page]] has argued that the MUH is self-contradictory because one cannot subsume all possible (partly contradictory) mathematical structures into one structure.<ref>D. Page, "[http://arxiv.org/abs/hep-th/0610101 Predictions and Tests of Multiverse Theories.]"</ref> Tegmark responds (<ref name="Tegmark2008"/>, sec. V.E) that different inconsistent mathematical structures correspond to different parallel universes. |
[[Don Page (mathematician)|Don Page]] has argued that the MUH is self-contradictory because one cannot subsume all possible (partly contradictory) mathematical structures into one structure.<ref>D. Page, "[http://arxiv.org/abs/hep-th/0610101 Predictions and Tests of Multiverse Theories.]"</ref> Tegmark responds (<ref name="Tegmark2008"/>, sec. V.E) that different inconsistent mathematical structures correspond to different parallel universes. |
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===Consistency with our "simple universe"=== |
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[[Alexander Vilenkin]] notes<ref name="Vilenkin2006">A. Vilenkin (2006) ''Many Worlds in One: The Search for Other Universes''. Hill and Wang, New York.</ref> that philosopher [[Keith Ward]] 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> Vilenkin remarked that Tegmark's initial solution, an assigned "weight" as a means of selection filtering, seems too arbitrary for physics.<ref name="Vilenkin2006"/> |
[[Alexander Vilenkin]] notes<ref name="Vilenkin2006">A. Vilenkin (2006) ''Many Worlds in One: The Search for Other Universes''. Hill and Wang, New York.</ref> that philosopher [[Keith Ward]] 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> Vilenkin remarked that Tegmark's initial solution, an assigned "weight" as a means of selection filtering, seems too arbitrary for physics.<ref name="Vilenkin2006"/> |
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Revision as of 01:28, 8 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][3] 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 and to Tegmark's categorization of theories of the multiverse.[4]
Andreas Albrecht of Imperial College in London called it a "provocative" solution to one of the central problems facing physics. Although he "wouldn't dare" go so far as to say he believes it, he noted that "it's actually quite difficult to construct a theory where everything we see is all there is".[5]
Criticisms and responses
Definition of the Ensemble
Jürgen Schmidhuber [6] argues that "Although Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight', there is no way of assigning equal nonvanishing probability to all (infinitely many) mathematical structures". Schmidhuber puts forward a more restricted ensemble which 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. [7]. In response, Tegmark notes [2] (sec. V.E) that the measure over all universes has not yet been constructed for the String theory landscape either, so this should not be regarded as a "show-stopper".
Consistency with Gödel's theorem
It has also been suggested that the MUH is inconsistent with Gödel's incompleteness theorem. In a three-way debate between Tegmark and fellow physicists Piet Hut and Mark Alford [8], the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system... The idea that math is "out there" is incompatible with the idea that it consists of formal systems." Tegmark's response [2] (sec. VII) is to suggest "drastically limiting the number of mathematical structures to be considered", by modifying the MUH to a much more restricted "Computable Universe Hypothesis" (CUH) which only includes mathematical structures that are simple enough that Gödel's theorem does not require them to contain any undecideable/uncomputable theorems. Tegmark admits that this approach faces "serious challeges", including (a) it excludes much of the mathematical landscape; (b) the measure on the space of allowed theories may itself be uncomputable; and (c) "virtually all historically successful theories of physics violate the CUH".
Observability
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 multiverse theories in general [9] and the MUH in particular [8] are not empirically testable, and therefore do not constitute scientific theories. Tegmark maintains that MUH is testable, stating that it predicts (a) that "physics research will uncover mathematical regularities in nature", and (b) by assuming that we occupy a typical member of the multiverse of mathematical structures, one could "start testing multiverse predictions by assessing how typical our universe is" ([2], sec. VIII.C).
Coexistence of contradictory structures
Don Page has argued that the MUH is self-contradictory because one cannot subsume all possible (partly contradictory) mathematical structures into one structure.[10] Tegmark responds ([2], sec. V.E) that different inconsistent mathematical structures correspond to different parallel universes.
Consistency with our "simple universe"
Alexander Vilenkin notes[11] that philosopher Keith Ward 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.[12] Vilenkin remarked that Tegmark's initial solution, an assigned "weight" as a means of selection filtering, seems too arbitrary for physics.[11]
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?". Annals of Physics. 270 (1): 1–51. doi:10.1006/aphy.1998.5855.
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- ^ Tegmark, Max (2003). "Parallel Universes". In Barrow, J.D.; Davies, P.C.W.' & Harper, C.L. (ed.). “Science and Ultimate Reality: From Quantum to Cosmos” honoring John Wheeler's 90th birthday. Cambridge University Press.
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: CS1 maint: multiple names: editors list (link) - ^ Chown, Markus (1998). "Anything goes". New Scientist. 158 (2157 url=http://space.mit.edu/home/tegmark/toe_press.html).
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ignored (help) - ^ 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.
- ^ W. R. Stoeger, G. F. R. Ellis, U. Kirchner (2006) "Multiverses and Cosmology: Philosophical Issues."
- ^ D. Page, "Predictions and Tests of Multiverse Theories."
- ^ a b A. Vilenkin (2006) Many Worlds in One: The Search for Other Universes. Hill and Wang, New York.
- ^ Keith Ward (1996) God, Chance & Necessity. Oxford: Oneworld.
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.