From the Local Observer in QM to the Fixed-Point Observer in GR
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On the Universality of Memcomputing Machines Yan Ru Pei, Fabio L
1 On the Universality of Memcomputing Machines Yan Ru Pei, Fabio L. Traversa, and Massimiliano Di Ventra Abstract—Universal memcomputing machines (UMMs) [IEEE is related to the topology (or architecture) of the network of Trans. Neural Netw. Learn. Syst. 26, 2702 (2015)] represent memory units (memprocessors). It means the machine has a novel computational model in which memory (time non- access, at any given time, to more information (precisely locality) accomplishes both tasks of storing and processing of information. UMMs have been shown to be Turing-complete, originating from the topology of the network) than what is namely they can simulate any Turing machine. In this paper, we available if the memprocessors were not connected to each first introduce a novel set-theory approach to compare different other. Of course, this information overhead is not necessarily computational models, and use it to recover the previous results stored by the machine (the stored information is the Shannon on Turing completeness of UMMs. We then relate UMMs directly one) [4]. Nevertheless, with appropriate topologies, hence with to liquid-state machines (or “reservoir computing”) and quantum machines (“quantum computing”). We show that UMMs can appropriate information overhead, UMMs can solve complex simulate both types of machines, hence they are both “liquid-” or problems very efficiently [4]. Finally, functional polymor- “reservoir-complete” and “quantum-complete”. Of course, these phism means that the machine is able to compute different statements pertain only to the type of problems these machines functions by simply applying the appropriate input signals [4], can solve, and not to the amount of resources required for such without modifying the topology of the machine network. -
List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX. -
J\'Onsson Posets
JONSSON´ POSETS ROLAND ASSOUS AND MAURICE POUZET To the memory of Bjarni J´onsson Abstract. According to Kearnes and Oman (2013), an ordered set P is J´onsson if it is infinite and the cardinality of every proper initial segment of P is strictly less than the cardinaliy of P . We examine the structure of J´onsson posets. 1. Introduction An ordered set P is J´onsson if it is infinite and the cardinality of every proper initial segment of P is strictly less than the cardinaliy of P . This notion is due to Kearnes and Oman (2013). J´onsson posets, notably the uncountable one, appear in the study of algebras with the J´onnson property (algebras for which proper subalgebras have a cardinality strictly smaller than the algebra). The study of these algebras are the motivation of the paper by Kearnes and Oman [15] where the reader will find detailed information. Countable J´onsson posets occur naturally at the interface of the theory of relations and symbolic dynamic as Theorem 5 and Theorem 4 below illustrate. They were considered in the thesis of the second author [26], without bearing this name, and characterized in [27] under the name of minimal posets. This characterization involves first the notion of well-quasi-order(w.q.o. for short) – a poset P is w.q.o. if it is well founded, that is every non-empty subset A of P has at least a minimal element, and P has no infinite antichain –, next, the decomposition of a well founded poset into levels (for each ordinal α, the α-th level is defined by setting Pα ∶= Min(P ∖ ⋃β<α Pβ) so that P0 is the set Min(P ) of minimal elements of P ; each element x ∈ Pα has height α, denoted by hP (x); the height of P , denoted by h(P ), is the least ordinal α such that arXiv:1712.09442v1 [math.LO] 22 Dec 2017 Pα = ∅) and, finally, the notion of ideal of a poset (every non-empty initial segment which is up-directed). -
Codomain from Wikipedia, the Free Encyclopedia Contents
Codomain From Wikipedia, the free encyclopedia Contents 1 Algebra of sets 1 1.1 Fundamentals ............................................. 1 1.2 The fundamental laws of set algebra .................................. 1 1.3 The principle of duality ........................................ 2 1.4 Some additional laws for unions and intersections .......................... 2 1.5 Some additional laws for complements ................................ 3 1.6 The algebra of inclusion ........................................ 3 1.7 The algebra of relative complements ................................. 4 1.8 See also ................................................ 5 1.9 References ............................................... 5 1.10 External links ............................................. 5 2 Axiom of choice 6 2.1 Statement ............................................... 6 2.1.1 Nomenclature ZF, AC, and ZFC ............................... 7 2.1.2 Variants ............................................ 7 2.1.3 Restriction to finite sets .................................... 7 2.2 Usage ................................................. 8 2.3 Examples ............................................... 8 2.4 Criticism and acceptance ....................................... 8 2.5 In constructive mathematics ..................................... 9 2.6 Independence ............................................. 10 2.7 Stronger axioms ............................................ 10 2.8 Equivalents .............................................. 10 2.8.1 Category -
Transfinite Number in Wittgenstein's Tractatus
Journal for the History of Analytical Philosophy Transfinite Number in Wittgenstein’s Tractatus Volume 9, Number 2 James Connelly Editor in Chief Audrey Yap, University of Victoria In his highly perceptive, if underappreciated introduction to Editorial Board Wittgenstein’s Tractatus, Russell identifies a “lacuna” within Annalisa Coliva, UC Irvine Wittgenstein’s theory of number, relating specifically to the topic Henry Jackman, York University of transfinite number. The goal of this paper is two-fold. The Frederique Janssen-Lauret, University of Manchester first is to show that Russell’s concerns cannot be dismissed on Kevin C. Klement, University of Massachusetts the grounds that they are external to the Tractarian project, de- Consuelo Preti, The College of New Jersey riving, perhaps, from logicist ambitions harbored by Russell but Marcus Rossberg, University of Connecticut not shared by Wittgenstein. The extensibility of Wittgenstein’s Anthony Skelton, Western University theory of number to the case of transfinite cardinalities is, I shall Mark Textor, King’s College London argue, a desideratum generated by concerns intrinsic, and internal Richard Zach, University of Calgary to Wittgenstein’s logical and semantic framework. Second, I aim to show that Wittgenstein’s theory of number as espoused in the Editors for Special Issues Tractatus is consistent with Russell’s assessment, in that Wittgen- Sandra Lapointe, McMaster University stein meant to leave open the possibility that transfinite numbers Alexander Klein, McMaster University could be generated within his system, but did not show explic- Review Editors itly how to construct them. To that end, I show how one could Sean Morris, Metropolitan State University of Denver construct a transfinite number line using ingredients inherent in Sanford Shieh, Wesleyan University Wittgenstein’s system, and in accordance with his more general theories of number and of operations. -
Simple Inconsistency of Simple Realist Qm Interpretations and Related Mathematical Theories
Research and Communications in Mathematics and Mathematical Sciences Vol. 6, Issue 2, 2016, Pages 117-132 ISSN 2319-6939 Published Online on June 25, 2016 2016 Jyoti Academic Press http://jyotiacademicpress.net SIMPLE INCONSISTENCY OF SIMPLE REALIST QM INTERPRETATIONS AND RELATED MATHEMATICAL THEORIES YVON GAUTHIER Faculty of Arts and Sciences University of Montreal Montreal Canada e-mail: [email protected] Abstract A few realist theories in physics and mathematics are analyzed from a constructivist viewpoint. Technical notions like fixed-point theorems and homeomorphisms are employed to pinpoint the logical and mathematical import of the critique. 1. Introduction: Realism in Physics G. ‘t Hooft defends a realist interpretation of QM in his long working paper “The Cellular Automaton Interpretation of Quantum Mechanics” [9]. The author advocates a deterministic or superdeterministic view of the universal wave function for the Schrödinger equation that would not ramify as in Everett’s many-universes interpretation -- they are 2010 Mathematics Subject Classification: 03E30, 58D10, 81P10, 83F05. Keywords and phrases: quantum theory, relativity theory, cosmology, fixed point, homeomorphism. Communicated by Mansoor Saburov. Received May 23, 2016; Revised June 16, 2016 118 YVON GAUTHIER practically uncountable, as ‘t Hooft says in his informal idiom --, but would give rise to a unique universe, a fluctuating vacuum filled with « solid » quanta or « fluid » particles that would obey a law-like determinism evolving from a given fundamental field, such as a scalar field or a quantum (true or false) vacuum. The entire universe of the wave function has a real deterministic ontological basis in Hilbert space following ‘t Hooft. But ‘t Hooft does not say that such a basis must have a finite cardinality or at most an infinite countable cardinality ℵ0. -
Measurable Cardinal Wikipedia Contents
Measurable cardinal Wikipedia Contents 1 Atom (measure theory) 1 1.1 Definition ............................................... 1 1.2 Examples ............................................... 1 1.3 Non-atomic measures ......................................... 1 1.4 See also ................................................ 2 1.5 Notes ................................................. 2 1.6 References ............................................... 2 2 Axiom of determinacy 3 2.1 Types of game that are determined .................................. 3 2.2 Incompatibility of the axiom of determinacy with the axiom of choice ................ 3 2.3 Infinite logic and the axiom of determinacy .............................. 4 2.4 Large cardinals and the axiom of determinacy ............................ 4 2.5 See also ................................................ 4 2.6 References ............................................... 5 2.7 Further reading ............................................ 5 3 Cardinal number 6 3.1 History ................................................. 6 3.2 Motivation .............................................. 8 3.3 Formal definition ........................................... 9 3.4 Cardinal arithmetic .......................................... 10 3.4.1 Successor cardinal ...................................... 10 3.4.2 Cardinal addition ....................................... 10 3.4.3 Cardinal multiplication .................................... 11 3.4.4 Cardinal exponentiation ................................... 11 3.5 -
Da Costa on Ontology: a Naturalistic Interpretation
CDD: 149.946 DA COSTA ON ONTOLOGY: A NATURALISTIC INTERPRETATION ANTONIO MARIANO NOGUEIRA COELHO Departamento de Filosofia Universidade Federal de Santa Catarina Campus Trindade, 88010-970 Florianópolis, SC BRASIL [email protected] Abstract: da Costa’s conception of being modifies that of Quine to incorporate relativization to non-classical logics. A naturalistic view of this conception is discus- sed. This view tries to extend to logic some ideas of Maddy’s naturalism concerning mathematics. Keywords: da Costa. Ontology. Maddy’s naturalism. Da Costa’s conception of being, in his own words “may be sum- marized as follows: To be is to be the value of a variable in a specific language with a given underlying logic” (da Costa (2002, p. 279)). This stand, of course, is a variant of that of Quine, to be is to be the value of a variable, incorporating the possibility of relativization to some non-classical logic, that is, relativization to some logic other than the classical first order predicate calculus. For da Costa (2002) there are two kinds of relevance a subject can have to philosophy. These are: direct relevance and indirect relevance. According to him, logic does not have any direct relevance to philosophy; in particular, logic does not have any direct relevance to ontology. By this he means that logic is, in a certain sense, neutral. The logician’s research activities are independent of any philosophical doctrine, and in particular of any Manuscrito — Rev. Int. Fil., Campinas, v. 34, n. 1, p. 143-150, jan.-jun. 2011. 144 ANTONIO MARIANO NOGUEIRA COELHO ontological commitment. -
On the Number of Gods
On the Number of Gods Eric Steinhart Forthcoming in International Journal for the Philosophy of Religion. Online first at DOI: <10.1007/s11153-011-9325-2>. ABSTRACT: A god is a cosmic designer-creator. Atheism says the number of gods is 0. But it is hard to defeat the minimal thesis that some possible universe is actualized by some possible god. Monotheists say the number of gods is 1. Yet no degree of perfection can be coherently assigned to any unique god. Lewis says the number of gods is at least the second beth number. Yet polytheists cannot defend an arbitrary plural number of gods. An alternative is that, for every ordinal, there is a god whose perfection is proportional to it. The n-th god actualizes the best universe(s) in the n-th level of an axiological hierarchy of possible universes. Despite its unorthodoxy, ordinal polytheism has many metaphysically attractive features and merits more serious study. 1. Gods Although no definition of the common noun god is likely to satisfy everyone, it is plausible to start with the idea that a god is a cosmic designer-creator. One conventional way to refine this initial idea leads to the notion that godliness involves actualizing the best universe (or universes) in some set of possible universes. This is the definition of god used here: x is a god iff x actualizes the best universe(s) in some set of possible universes. The Platonic Demiurge and the God of Abraham both satisfy this definition. However, since Zeus is not depicted as a cosmic designer-creator, Zeus is not a god. -
Mathematical Symbols by Subject
List of mathematical symbols by subject This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub- regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents Guide Set theory Definition symbols Set construction Set operations Set relations Number sets Cardinality Arithmetic Arithmetic operators Equality signs Comparison Divisibility Intervals Elementary functions Complex numbers Mathematical constants Calculus Sequences and series Functions Limits Asymptotic behaviour Differential calculus Integral calculus Vector calculus Topology Functional analysis Linear algebra and geometry Elementary geometry Vectors and matrices Vector calculus Matrix calculus Vector spaces Algebra Relations Group theory Field theory Ring theory Combinatorics Stochastics Probability theory Statistics Logic Operators Quantifiers Deduction symbols See also References External links Guide The following information is provided for each mathematical symbol: Symbol The symbol as it is represented by LaTeX. If there are several typographic variants, only one of the variants is shown. -
Philosophical Status of the Continuum Hypothesis
Can we resolve the Continuum Hypothesis? Shivaram Lingamneni December 7, 2017 Abstract I argue that that contemporary set theory, as depicted in the 2011- 2012 EFI lecture series, lacks a program that promises to decide, in a genuinely realist fashion, the continuum hypothesis (CH) and related questions about the \width" of the universe. We can distinguish three possible objectives for a realist completion of set theory: maximizing structures, maximizing sets, and maximizing interpretive power. How- ever, none of these is allied to a program that can plausibly decide CH. I discuss the implications of this for set theory and other foundational programs. 1 Introduction The continuum hypothesis (CH) | the hypothesis or conjecture that 2@0 = @1 | is as old as set theory itself and has cast its long shadow over the discipline for the entirety of its history. As early as 1878, Cantor asked the question in its modern form: is every infinite X ⊆ R in bijection with either N or R? By 1900, the question was sufficiently well-established to be the first of the 23 Hilbert problems, unsolved questions that would guide the future of mathematical research for much of the 20th century. And its inclusion in the list did bear immediate fruit in shaping the evolution of descriptive set theory; for example, interest in the perfect set property was inspired by Cantor's search for subsets of R that could be counterexamples to CH [Kanamori, 2008]. Real progress on the original question, however, had to wait for G¨odel's 1938 identification of the constructible universe L, an inner model of any model of set theory which always satisfies the continuum hypothesis; this showed the equiconsistency of ZFC with ZFC + CH.