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349:." In he gives a more detailed response, proposing as an alternative to MUH the 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 undecidable or uncomputable theorems. Tegmark admits that this approach faces "serious challenges", 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".
400:
language." In the paper "On Math, Matter and Mind" the secularist viewpoint examined argues that math is evolving over time, there is "no reason to think it is converging to a definite structure, with fixed questions and established ways to address them", and also that "The
Radical Platonist position is just another metaphysical theory like solipsism... In the end the metaphysics just demands that we use a different language for saying what we already knew." Tegmark responds that "The notion of a mathematical structure is rigorously defined in any book on
341:) mathematical structures have physical existence. This drastically shrinks the Level IV multiverse, essentially placing an upper limit on complexity, and may have the attractive side effect of explaining the relative simplicity of our universe." Tegmark goes on to note that although conventional theories in physics are Gödel-undecidable, the actual mathematical structure describing our world could still be Gödel-complete, and "could in principle contain observers capable of thinking about Gödel-incomplete mathematics, just as
404:", and that non-human mathematics would only differ from our own "because we are uncovering a different part of what is in fact a consistent and unified picture, so math is converging in this sense." In his 2014 book on the MUH, Tegmark argues that the resolution is not that we invent the language of mathematics, but that we discover the structure of mathematics.
36:
179:. Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world".
432:
Tegmark's solution to this problem, the assigning of lower "weights" to the more complex structures seems arbitrary ("Who determines the weights?") and may not be logically consistent ("It seems to introduce an additional mathematical structure, but all of them are supposed to be already included in the set").
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with respect to mathematics are limited to basic counting abilities. He adds, "non-human intelligent beings should exist that understand the language of advanced mathematics. However, none of the non-human intelligent beings that we know of confirm the status of (advanced) mathematics as an objective
259:
argues that "Although
Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight,' there is no way of assigning equal non-vanishing probability to all (infinitely many) mathematical structures." Schmidhuber puts forward a more restricted ensemble which admits
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fundamental theorems intended to serve as building blocks for additional mathematical results. 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
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or at least our own, there must be one unique mathematical structure that describes ultimate reality. So I think it is logical nonsense to talk of Level 4 in the sense of the co-existence of all mathematical structures." This means there can only be one mathematical corpus. Tegmark responds that
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comments that "The number of mathematical structures increases with increasing complexity, suggesting that 'typical' structures should be horrendously large and cumbersome. This seems to be in conflict with the beauty and simplicity of the theories describing our world". He goes on to note that
362:, 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".
357:
Stoeger, Ellis, and
Kircher note that in a true multiverse theory, "the universes are then completely disjoint and nothing that happens in any one of them is causally linked to what happens in any other one. This lack of any causal connection in such multiverses really places them beyond any
247:
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".
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and Mark Alford, 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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scientific support". Ellis specifically criticizes the MUH, stating that an infinite ensemble of completely disconnected universes is "completely untestable, despite hopeful remarks sometimes made, see, e.g., Tegmark (1998)." Tegmark maintains that MUH is
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argues similarly: "The deepest description of the universe should not require concepts whose meaning relies on human experience or interpretation. Reality transcends our existence and so shouldn't, in any fundamental way, depend on ideas of our making."
376:
The MUH is based on the radical
Platonist view that math is an external reality. However, Jannes argues that "mathematics is at least in part a human construction", on the basis that if it is an external reality, then it should be found in some other
395:. But a few surprising examples of mathematical abstraction notwithstanding (for example, chimpanzees can be trained to carry out symbolic addition with digits, or the report of a parrot understanding a "zero-like concept"), all examples of
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as well: "Tegmark argues that, if we want to give a complete description of reality, then we will need a language independent of us humans, understandable for non-human sentient entities, such as aliens and future supercomputers".
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However, there are many non-human entities, plenty of which are intelligent, and many of which can apprehend, memorise, compare and even approximately add numerical quantities. Several animals have also passed the
420:"This is less inconsistent with Level IV than it may sound, since many mathematical structures decompose into unrelated substructures, and separate ones can be unified."
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Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Thus, he reasons, it is preferred over other theories-of-everything by
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Tegmark's MUH is the hypothesis that our external physical reality is a mathematical structure. That is, the physical universe is not merely
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measure of free parameter variations of physical dimensions, constants, and laws over all universes has not yet been constructed for the
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has argued that "At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all
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in and of itself. Tegmark extends this idea to hypothesize that all mathematical objects exist, which he describes as a form of
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1004:, "83 Years of General Relativity and Cosmology Progress and Problems", Classical and Quantum Gravity 16, A37-A75, 1999.
224:. This categorization posits a nested hierarchy of increasing diversity, with worlds corresponding to different sets of
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549:
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Hamlin, Colin (2017). "Towards a Theory of
Universes: Structure Theory and the Mathematical Universe Hypothesis".
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874:"Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit"
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232:(level 2), quantum branches (level 3), and altogether different equations or mathematical structures (level 4).
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Stoeger, W. R.; Ellis, G. F. R.; Kirchner, U. (2006-01-19). "Multiverses and
Cosmology: Philosophical Issues".
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Tegmark, Max (November 1998). "Is "the Theory of
Everything" Merely the Ultimate Ensemble Theory?".
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argues that it is not possible to assign an equal weight or probability to all mathematical objects
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Gil Jannes, "Some comments on 'The
Mathematical Universe'", Found. Phys. 39, 397-406, 2009
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Tegmark and Harris discuss efficacy of mathematics, multiverses, artificial intelligence.
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Tegmark has been criticized as misunderstanding the nature and application of
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Page, Don N. (2006-10-09). "Predictions and Tests of
Multiverse Theories".
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Tegmark's response is to offer a new hypothesis "that only Gödel-complete (
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Tegmark replies that not only is the universe mathematical, but it is also
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205:. Tegmark also considers augmenting the MUH with a second assumption, the
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Schmidhuber, Juergen (2000-12-20). "Algorithmic
Theories of Everything".
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1301:" (and archives). Discusses the idea that all possible universes exist.
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and published on January 7, 2014, this book describes Tegmark's theory.
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Our Mathematical Universe: My Quest for the Ultimate Nature of Reality
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can prove certain theorems about Gödel-incomplete formal systems like
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in that it proposes the existence of mathematical entities; a form of
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Hut, P.; Alford, M.; Tegmark, M. (2006). "On Math, Matter and Mind".
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The Unreasonable Effectiveness of Mathematics in the Natural Sciences
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The MUH is related to Tegmark's categorization of four levels of the
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The ensemble of universes describable by constructive mathematics.
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A Computer Scientist's View of Life, the Universe, and Everything
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Foundations of Computer Science: Potential - Theory - Cognition
1328:"Is the Universe made of math?" Excerpt in Scientific American
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Collection of interviews with Max Tegmark in 'Closer to truth"
1132:. Lecture Notes in Computer Science, Springer: p. 201-08.
329:. In a three-way debate between Tegmark and fellow physicists
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It has also been suggested that the MUH is inconsistent with
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either, so this should not be regarded as a "show-stopper".
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https://link.springer.com/article/10.1007/s11229-015-0959-y
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Tegmark, Max (February 2008). "The Mathematical Universe".
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International Journal of Foundations of Computer Science
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Book Review: 'Our Mathematical Universe' by Max Tegmark
1186:Tegmark, Max (2008). "The Mathematical Universe".
1065:Many Worlds in One: The Search for Other Universes
708:Tegmark, Max (2008). "The Mathematical Universe".
144:have suggested that the idea is incompatible with
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105:. According to the hypothesis, the universe
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296:In response, Tegmark notes that a
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393:mirror test of self-consciousness
366:Plausibility of radical Platonism
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311:Consistency with Gödel's theorem
101:" (TOE) proposed by cosmologist
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343:finite-state digital computers
327:Gödel's incompleteness theorem
207:computable universe hypothesis
58:provide an accessible overview
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1067:. Hill and Wang, New York.
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252:Definition of the ensemble
1310:Interview with Sam Harris
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1104:Our Mathematical Universe
1081:. Folsom, CA: Science 2.0
960:10.1007/s10701-006-9048-x
908:10.1142/S0129054102001291
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136:many of them. Physicists
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262:constructive mathematics
236:Criticisms and responses
196:ontic structural realism
95:ultimate ensemble theory
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1189:Foundations of Physics
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177:mathematical structure
1234:Tegmark, Max (2014),
1128:" in C. Freksa, ed.,
698:Tegmark (1998), p. 1.
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93:), also known as the
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1251:(17 January 2014), "
489:Church–Turing thesis
285:program, due to the
215:computable functions
99:theory of everything
97:, is a speculative "
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1163:1998AnPhy.270....1T
1063:A. Vilenkin (2006)
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732:2008FoPh...38..101T
670:2008FoPh...38..101T
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397:animal intelligence
280:representations of
132:due to there being
111:mathematical object
27:Cosmological theory
18:Ultimate multiverse
1363:Physical cosmology
1315:2017-08-25 at the
1282:Jürgen Schmidhuber
1028:The Hidden Reality
621:M. Tegmark 2014, "
429:Alexander Vilenkin
257:Jürgen Schmidhuber
230:physical constants
226:initial conditions
126:Jürgen Schmidhuber
1244:978-0-307-59980-3
1141:Annals of Physics
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574:
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530:
439:
427:
411:
402:Model Theory
389:
384:Brian Greene
375:
356:
336:
324:
295:
268:; e.g., the
255:
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219:
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200:
181:
172:
171:mathematics
168:
165:described by
164:
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150:
129:
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94:
90:
86:
76:
63:
47:
45:lead section
1147:(1): 1–51.
1109:Max Tegmark
577:(1): 1–51.
317:Consistency
264:, that is,
228:(level 1),
159:Description
142:Mark Alford
103:Max Tegmark
1353:Multiverse
1337:Categories
557:References
370:See also:
315:See also:
301:formalized
282:formalized
222:multiverse
153:computable
134:infinitely
1203:0704.0646
723:0704.0646
661:0704.0646
625:", Knopf.
450:heuristic
188:Platonism
115:Platonism
83:cosmology
50:summarize
1358:Ontology
1313:Archived
1266:Synthese
1249:Woit, P.
1179:41548734
1124:(1997) "
968:17559900
872:(2002).
804:12701329
609:41548734
526:Ontology
456:See also
413:Don Page
360:testable
331:Piet Hut
138:Piet Hut
130:a priori
1228:9890455
1208:Bibcode
1159:Bibcode
1097:Sources
948:Bibcode
896:Bibcode
834:(2157).
784:Bibcode
748:9890455
728:Bibcode
686:9890455
666:Bibcode
589:Bibcode
379:animals
289:of the
79:physics
1242:
1226:
1177:
966:
802:
746:
684:
607:
85:, the
1224:S2CID
1198:arXiv
1175:S2CID
1149:arXiv
1085:7 May
1042:arXiv
983:arXiv
964:S2CID
938:arXiv
886:arXiv
850:arXiv
774:arXiv
744:S2CID
718:arXiv
682:S2CID
656:arXiv
605:S2CID
579:arXiv
1240:ISBN
1087:2024
800:PMID
319:and
272:and
155:.
140:and
81:and
1255:",
1216:doi
1167:doi
1145:270
956:doi
904:doi
832:158
792:doi
770:288
736:doi
674:doi
597:doi
575:270
243:of
211:CUH
186:or
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117:or
91:MUH
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