physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

and

cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...

, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative "

theory of everything A theory of everything (TOE), final theory, ultimate theory, unified field theory, or master theory is a hypothetical singular, all-encompassing, coherent theoretical physics, theoretical framework of physics that fully explains and links togeth ...

" (TOE) proposed by cosmologist

Max Tegmark Max Erik Tegmark (born 5 May 1967) is a Swedish-American physicist, machine learning researcher and author. He is best known for his book ''Life 3.0'' about what the world might look like as artificial intelligence continues to improve. Tegmark i ...

. According to the hypothesis, the universe ''is'' a

mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...

in and of itself. Tegmark extends this idea to hypothesize that all mathematical objects exist, which he describes as a form of

Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundam ...

Modal realism Modal realism is the view propounded by the philosopher David Lewis that all possible worlds are real in the same way as is the actual world: they are "of a kind with this world of ours." It states that possible worlds exist, possible worlds are ...

. The hypothesis has proven controversial.

Jürgen Schmidhuber Jürgen Schmidhuber (born 17 January 1963) is a German computer scientist noted for his work in the field of artificial intelligence, specifically artificial neural networks. He is a scientific director of the Dalle Molle Institute for Artifici ...

argues that it is not possible to assign an equal weight or probability to all mathematical objects ''a priori'' due to there being infinitely many of them. Physicists Piet Hut and Mark Alford have suggested that the idea is incompatible with Gödel's first incompleteness theorem. Tegmark replies that not only is the universe mathematical, but it is also

computable Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is cl ...

. In 2014, Tegmark published a popular science book about the topic, titled ''

Our Mathematical Universe ''Our Mathematical Universe: My Quest for the Ultimate Nature of Reality'' is a 2014 non-fiction book by the Swedish-American cosmologist Max Tegmark. Written in popular science format, the book interweaves what a ''New York Times'' reviewer call ...

''.

Description

Tegmark's MUH is the hypothesis that our external physical reality is a mathematical structure. That is, the physical universe is not merely ''described by'' mathematics, but ''is'' mathematics ''—'' specifically, a

mathematical structure In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...

. 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". The theory can be considered a form of

Pythagoreanism Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the Ancient Greece, ancient Greek co ...

in that it proposes the existence of mathematical entities; a form of mathematicism in that it denies that anything exists except mathematical objects; and a formal expression of

ontic structural realism In the philosophy of science, structuralism (also known as scientific structuralism or as the structuralistic theory-concept) asserts that all aspects of reality are best understood in terms of empirical scientific constructs of entities and thei ...

. 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

Occam's Razor In philosophy, Occam's razor (also spelled Ockham's razor or Ocham's razor; ) is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements. It is also known as the principle o ...

. 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 function Computable functions are the basic objects of study in computability theory. Informally, a function is ''computable'' if there is an algorithm that computes the value of the function for every value of its argument. Because of the lack of a precis ...

s. The MUH is related to Tegmark's categorization of four levels of the

multiverse The multiverse is the hypothetical set of all universes. Together, these universes are presumed to comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describ ...

. This categorization posits a nested hierarchy of increasing diversity, with worlds corresponding to different sets of

initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). Fo ...

(level 1),

physical constants A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a ...

(level 2), quantum branches (level 3), and altogether different equations or mathematical structures (level 4).

Criticisms and responses

Andreas Albrecht when at

Imperial College Imperial College London, also known as Imperial, is a public research university in London, England. Its history began with Prince Albert, husband of Queen Victoria, who envisioned a cultural district in South Kensington that included museums ...

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".

Definition of the ensemble

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 only universe representations describable by

constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...

, that is,

computer program A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. It is one component of software, which also includes software documentation, documentation and other intangibl ...

s; e.g., the Global Digital Mathematics Library and

Digital Library of Mathematical Functions Digital usually refers to something using discrete digits, often binary digits. Businesses *Digital bank, a form of financial institution *Digital Equipment Corporation (DEC) or Digital, a computer company *Digital Research (DR or DRI), a software ...

, linked open data representations of formalized 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 program, due to the undecidability of the

halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ...

. In response, Tegmark notes that a

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 In string theory, the string theory landscape (or landscape of vacua) is the collection of possible false vacua,The number of metastable vacua is not known exactly, but commonly quoted estimates are of the order 10500. See M. Douglas, "The stat ...

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, 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 is to offer a new hypothesis "that only Gödel-complete ( fully decidable) 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 finite-state digital computers can prove certain theorems about Gödel-incomplete formal systems like

Peano arithmetic In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nea ...

." 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".

Observability

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 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

testable Testability is a primary aspect of science and the scientific method. There are two components to testability: #Falsifiability or defeasibility, which means that counterexamples to the hypothesis are logically possible. #The practical feasibilit ...

, 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".

Plausibility of radical Platonism

The MUH is based on the radical Platonist view that math is an external reality. However, JannesGil Jannes, "Some comments on 'The Mathematical Universe'", Found. Phys. 39, 397-406, 200
arXiv:0904.0867
/ref> 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 Animals are multicellular, eukaryotic organisms in the biological kingdom Animalia (). With few exceptions, animals consume organic material, breathe oxygen, have myocytes and are able to move, can reproduce sexually, and grow from a ...

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".

Brian Greene Brian Randolph Greene (born February 9, 1963) is an American physicist known for his research on string theory. He is a professor of physics and mathematics at Columbia University, director of its center for theoretical physics, and the cha ...

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." 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 mirror test of self-consciousness. 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 Animal cognition encompasses the mental capacities of non-human animals, including insect cognition. The study of animal conditioning and learning used in this field was developed from comparative psychology. It has also been strongly influ ...

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 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

Model Theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

", 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.

Coexistence of all mathematical structures

Don Page has argued that "At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all

possible world A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic. Their met ...

s 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 "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"

Alexander Vilenkin Alexander Vilenkin (; ; born 13 May 1949) is the Leonard Jane Holmes Bernstein Professor of Evolutionary Science and Director of the Institute of Cosmology at Tufts University. A theoretical physicist who has been working in the field of cosmolo ...

commentsA. Vilenkin (2006) ''Many Worlds in One: The Search for Other Universes''. Hill and Wang, New York. 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 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").

Occam's razor

Tegmark has been criticized as misunderstanding the nature and application of

Occam's razor In philosophy, Occam's razor (also spelled Ockham's razor or Ocham's razor; ) is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements. It is also known as the principle o ...

;

Massimo Pigliucci Massimo Pigliucci (; born January 16, 1964) is an American philosopher and biologist who is professor of philosophy at the City College of New York, former co-host of the '' Rationally Speaking Podcast'', and former editor in chief for the online ...

reminds that "Occam's razor is just a useful

heuristic A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...

, it should never be used as the final arbiter to decide which theory is to be favored".

References

Sources

*''

'': written by

and published on January 7, 2014, this book describes Tegmark's theory.

External links

The ensemble of universes describable by constructive mathematics.

with links to his technical and popular writings.

(and archives). Discusses the idea that all possible universes exist.
Interview with Sam Harris
{{Webarchive, url=https://web.archive.org/web/20170825190356/https://www.samharris.org/podcast/item/the-multiverse-you-you-you-you , date=2017-08-25 Tegmark and Harris discuss efficacy of mathematics, multiverses, artificial intelligence.
Collection of interviews with Max Tegmark in 'Closer to truth""Is the Universe made of math?" Excerpt in Scientific American
Abstract object theory Computability theory Mathematical objects Max Tegmark Metaphysical realism Multiverse Ontology Physical cosmology Mathematical Platonism Pythagoreanism Thought experiments in physics