Mathematical universe hypothesis

In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative 'theory of everything' (TOE) proposed by cosmologist Max Tegmark. In physics and cosmology, the mathematical universe hypothesis (MUH), also known as the ultimate ensemble theory, is a speculative 'theory of everything' (TOE) proposed by cosmologist Max Tegmark. Tegmark's MUH is: 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). 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 or Platonism in that it proposes the existence of mathematical entities; a form of mathematical monism in that it denies that anything exists except mathematical objects; and a formal expression of ontic structural realism. 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. 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. The MUH is related to Tegmark's categorization of four levels of the multiverse. This categorization posits a nested hierarchy of increasing diversity, with worlds corresponding to different sets of initial conditions (level 1), physical constants (level 2), quantum branches (level 3), and altogether different equations or mathematical structures (level 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'. Jürgen Schmidhuber 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, that is, computer programs; e.g., the Global Digital Mathematics Library and Digital Library of Mathematical Functions, 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 response, Tegmark notes (sec. V.E) that a constructive mathematics formalized measure of variations of physical 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'. 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.'