A.1) that "The notion of a mathematical structure is rigorously defined in any book on Model Theory and that non-human mathematics would only differ from our own "because we are uncovering a different part of what is in fact. 222) that Tegmark's solution to this problem, the assigning of lower "weights" to the more complex structures ( 6 citation needed sec. In the end the metaphysics just demands that we use a different language for saying what we already knew." Tegmark responds (sec. However, Jannes 13 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 animals as well: "Tegmark argues that. The idea that math is 'out there' is incompatible with the idea that it consists of formal systems." Tegmark's response in 10 (sec. 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". Schmidhuber (2000) " Algorithmic Theories of Everything. Hill and Wang, New York.
V.E) that "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." Consistency with our "simple universe" edit Alexander Vilenkin comments 16 (Ch. In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world". 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". Thus, he reasons, it is preferred over other theories-of-everything. Coexistence of all mathematical structures edit Don Page has argued 15 (sec 4) that "At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all possible worlds. Cosmological theory, in physics and cosmology, the mathematical universe hypothesis mUH also known as the ultimate ensemble theory and struogony (from mathematical structure, Latin: stru is a speculative " theory of everything " (TOE) proposed by cosmologist, max Tegmark. Brian Greene (. . A.1) is to offer a new hypothesis "that only Gdel-complete ( fully decidable ) mathematical structures have physical existence. A) 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. Consistency with Gdel's theorem edit See also: Consistency and Gdel's completeness theorem It has also been suggested that the MUH is inconsistent with Gdel's incompleteness theorem.
3, that is, the physical universe is not merely described by mathematics, but is mathematics (specifically, a mathematical structure ). Vilenkin (2006) Many Worlds in One: The Search for Other Universes. "Towards a Theory of Universes: Structure Theory and the Mathematical Universe Hypothesis". Ellis 12 (p29) specifically criticizes the MUH, stating that an infinite ensemble of completely disconnected universes is "completely untestable, despite hopeful remarks sometimes made, see,.g., Tegmark (1998)." Tegmark maintains that MUH is testable, stating that it predicts (a) that "physics research. A b Tegmark, Max (2003). "Is the 'theory of everything' merely the ultimate ensemble theory?". 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 only universe representations describable by constructive. 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 the undecidability of the halting problem.
VII) 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 Gdel's theorem does not require them to contain any undecidable or uncomputable theorems. " Book Review: 'Our Mathematical Universe' by Max Tegmark The Wall Street Journal. Occam's razor edit Tegmark has been criticized as misunderstanding the nature and application of Occam's razor ; Massimo Pigliucci reminds that "Occam's razor is just a useful heuristic, it should never be used as the final arbiter to decide which theory is to be favored". International Journal of Foundations of Computer Science. However, none of the non-human intelligent beings that we know of confirm the status of (advanced) mathematics as an objective language." In the paper "On Math, Matter and Mind" 10 the secularist viewpoint examined argues (sec. "Is "the Theory of Everything" Merely the Ultimate Ensemble Theory?". Plausibility of radical Platonism edit See also: Philosophy of mathematics Mathematicism The MUH is based on the Radical Platonist view that math is an external reality ( 3 sec.C). Observers, including humans, are "self-aware substructures (SASs. Contents, description edit, tegmark's MUH is: Our external physical reality is a mathematical structure.
4, the theory can be considered a form. 203) that "the number of mathematical structures increases with increasing complexity, suggesting that 'typical' structures should be horrendously large and cumbersome. Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. 8 9 In response, Tegmark notes 3 citation needed (sec. 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. V.B) seems arbitrary Who determines the weights? Lecture Notes in Computer Science, Springer:. . Ellis, "83 years of general relativity and cosmology: Progress and problems Class.
He adds, "non-human intelligent beings should exist that understand the language of advanced mathematics. 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 Gdel-undecidable. Observability edit Stoeger, Ellis, and Kircher 11 (sec. Tegmark 2014, " Our Mathematical Universe Knopf a b c d e f Tegmark, Max (February 2008). M/article/10.1007/s y External links edit.
This lack of any causal connection in such multiverses really places them beyond any scientific support". 299) argues similarly: "The deepest description of the universe should not require concepts whose meaning relies on human experience or interpretation. In Barrow,.D.; Davies,.C.W.; Harper,.L. Chown, Markus (June 1998). Page, " Predictions and Tests of Multiverse Theories. "Science and Ultimate Reality: From Quantum to Cosmos" honoring John Wheeler's 90th birthday.
This seems to be in conflict with the beauty and simplicity of the theories describing our world". (1997) " A Computer Scientist's View of Life, the Universe, and Everything ". Sources edit Further reading edit Schmidhuber,. 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 animal intelligence with respect. "Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit". 39, 397-406, 2009 arXiv:0904.0867.
Tegmark also considers augmenting the MUH with a second assumption, the computable universe hypothesis ( CUH which says that the mathematical structure that is our external physical reality is defined by computable functions. Tegmark claims that the hypothesis has no free parameters and is not observationally ruled out. Kirchner (2006) " Multiverses and Cosmology: Philosophical Issues. 5 The MUH is related to Tegmark's categorization of four levels of the multiverse. 7 Criticisms and responses edit Definition of the ensemble edit Jrgen Schmidhuber 8 argues that "Although Tegmark suggests that '. 17 See also edit References edit Tegmark, Max (November 1998). In a three-way debate between Tegmark and fellow physicists Piet Hut and Mark Alford, 10 the "secularist" (Alford) states that "the methods allowed by formalists cannot prove all the theorems in a sufficiently powerful system. Reception edit Andreas Albrecht of Imperial College in London, called it a "provocative" solution to one of the central problems facing physics.
Several animals have also passed the mirror test of self-consciousness. 16, A37-A75, 1999 Gil Jannes, "Some comments on 'The Mathematical Universe Found. V.E) that a constructive mathematics formalized measure of free parameter variations of physical dimensions, constants, and laws over all universes has not yet been constructed for the string theory landscape either, so this should not be regarded as a "show-stopper". "On Math, Matter and Mind". Freksa,., Foundations of Computer Science: Potential - Theory - Cognition. He goes on to note (footnote 8,. .