An important suggestion of objective Bayesians is that the maximum entropy principle can replace a principle which is known to get into paradoxical difficulties: the principle of indifference. No one has previously determined whether the maximum entropy principle is better able to solve Bertrand’s chord paradox than the principle of indifference. In this paper I show that it is not. Additionally, the course of the analysis brings to light a new paradox, (...) a revenge paradox of the chords, that is unique to the maximum entropy principle. (shrink)

This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. (...) objective interpretations that are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion. (shrink)

In recent decades, several theories have claimed to explain the teleological causality of organisms as a function of self-organising and self-producing processes. The most widely cited theories of this sort are variations of autopoiesis, originally introduced by Maturana and Varela. More recent modifications of autopoietic theory have focused on system organisation, closure of constraints and autonomy to account for organism teleology. This article argues that the treatment of teleology in autopoiesis and other organisation theories is inconclusive for three reasons: First, (...) non-living self-organising processes like autocatalysis meet the defining features of autopoiesis without being teleological; second, organisational approaches, whether defined in terms of the closure of constraints, self-determination or autonomy, are unable to specify teleological normativity, that is, the individuation of an ultimate beneficiary; third, all self-organised systems produce local order by maximising the throughput of energy and/or material (obeying the maximum entropy production (MEP) principle) and thereby are specifically organised to undermine their own critical boundary conditions. Despite these inadequacies, an alternative approach called teleodynamics accounts for teleology. This theory shows how multiple self-organising processes can be collectively linked so that they counter each other’s MEP principle tendencies to become codependent. Teleodynamics embraces – not ignoring – the difficulties of self-organisation, but reinstates teleology as a radical phase transition distinguishing systems embodying an orientation towards their own beneficial ends from those that lack normative character. (shrink)

Many religions and religious philosophies say that ultimate reality is a kind of primal energy. This energy is often described as a vital power animating living things, as a spiritual force directing the organization of matter, or as a divine creative power which generates all things. By refuting older conceptions of primal energy, modern science opens the door to new and more precise conceptions. Primal energy is referred to here as ‘spirit’. But spirit is a natural power. A naturalistic theory (...) of spirit is developed using ideas from information theory and thermodynamics, such as the maximum entropy production principle. Spirit drives the evolution of complexity at all levels of existence. (shrink)

In a preceding publication a fundamentally oriented and irreversible world was shown to be de- rivable from the important principle of least action. A consequence of such a paradigm change is avoidance of paradoxes within a “dynamic” quantum physics. This becomes essentially possible because fundamental irreversibility allows consideration of the “entropy” concept in elementary processes. For this reason, and for a compensation of entropy in the spread out energy of the wave, the duality of particle and wave (...) has to be mediated via an information self-image of matter. In this publication considerations are extended to irreversible thermodynamics, to gravitation and cos- mology with its dependence on quantum interpretations. The information self-image of matter around particles could be identified with gravitation. Because information can also impose an al- ways constant light velocity there is no need any more to attribute such a property to empty space, as done in relativity theory. In addition, the possibility is recognized to consider entropy genera- tion by expanding photon fields in the universe. Via a continuous activation of information on matter photons can generate entropy and release small energy packages without interacting with matter. This facilitates a new interpretation of galactic redshift, emphasizes an information link between quantum- and cosmological phenomena, and evidences an information-triggered origin of the universe. Self-organized processes approach maximum entropy production within their constraints. In a far from equilibrium world also information, with its energy content, can self- organize to a higher hierarchy of computation. It is here identified with consciousness. This ap- pears to explain evolution of spirit and intelligence on a materialistic basis. Also gravitation, here identified as information on matter, could, under special conditions, self-organize to act as a su- per-gravitation, offering an alternative to dark matter. Time is not an illusion, but has to be understood as flux of action, which is the ultimate reality of change. The concept of an irreversible physical world opens a route towards a rational understanding of complex contexts in nature. (shrink)

An indifference principle says that your credences should be distributed uniformly over each of the possibilities you recognise. A chance deference principle says that your credences should be aligned with the chances. My thesis is that, if we are anti-Humeans about chance, then these two principles are incompatible. Anti-Humeans think that it is possible for the actual frequencies to depart from the chances. So long as you recognise possibilities like this, you cannot both spread your credences evenly and (...) defer to the chances. I discuss some weaker forms of indifference which will allow anti-Humeans to defer to the chances. (shrink)

The principle of maximal entropy (further abbreviated as “MaxEnt”) can be founded on the formal mechanism, in which future transforms into past by the mediation of present. This allows of MaxEnt to be investigated by the theory of quantum information. MaxEnt can be considered as an inductive analog or generalization of “Occam’s razor”. It depends crucially on choice and thus on information just as all inductive methods of reasoning. The essence shared by Occam’s razor and MaxEnt is for (...) the relevant data known till now to be postulated as an enough fundament of conclusion. That axiom is the kind of choice grounding both principles. Popper’s falsifiability (1935) can be discussed as a complement to them: That axiom (or axiom scheme) is always sufficient but never necessary condition of conclusion therefore postulating the choice in the base of MaxEnt. Furthermore, the abstraction axiom (or axiom scheme) relevant to set theory (e.g. the axiom scheme of specification in ZFC) involves choice analogically. (shrink)

Background: Emergentism as an ontology of consciousness leaves unanswered the question as to its mechanism. Aim: I aim to solve the Body-Mind problem by explaining how conscious organisms emerged on an evolutionary basis at various times in accordance with an accepted scientific principle, through a mechanism that cannot be understood, in principle. Proposal: The reason for this cloak of secrecy is found in a seeming contradiction in the behaviour of information with respect to the first two laws of (...) thermodynamics. Information, the microstate of particles within an isolated system’s macrostate, can, like first law energy, be neither created nor destroyed, yet the information in the system, like second law entropy, will inevitably increase. To explain information increasing without being created, Laplace’s demon is invoked, able to predict where each particle is destined. This doesn’t work for emerging events like consciousness, which are unpredictable. This can be understood in terms of the derivation of entropy, and the emergence of classical physics, from the Relativistic Transactional Interpretation of Quantum Mechanics. I propose that the increased entropy in a time-irreversible, unpredictable (emergent) isolated system requires the simultaneous deletion of information concerning the steps, or calculations, involved. Conclusion: Thus, the steps leading to consciousness are immediately destroyed and must remain a mystery. Implications include that entropy, not panpsychism, is the Universal principle generative of consciousness, that our being conscious proves that we are not predetermined, and that consciousness requires assuming an “entropy debt” that can only be repaid by living organisms, prohibiting the emergence of conscious machines. (shrink)

The logical basis for information theory is the newly developed logic of partitions that is dual to the usual Boolean logic of subsets. The key concept is a "distinction" of a partition, an ordered pair of elements in distinct blocks of the partition. The logical concept of entropy based on partition logic is the normalized counting measure of the set of distinctions of a partition on a finite set--just as the usual logical notion of probability based on the Boolean (...) logic of subsets is the normalized counting measure of the subsets (events). Thus logical entropy is a measure on the set of ordered pairs, and all the compound notions of entropy (join entropy, conditional entropy, and mutual information) arise in the usual way from the measure (e.g., the inclusion-exclusion principle)--just like the corresponding notions of probability. The usual Shannon entropy of a partition is developed by replacing the normalized count of distinctions (dits) by the average number of binary partitions (bits) necessary to make all the distinctions of the partition. (shrink)

This article analyzes the role of entropy in Bayesian statistics, focusing on its use as a tool for detection, recognition and validation of eigen-solutions. “Objects as eigen-solutions” is a key metaphor of the cognitive constructivism epistemological framework developed by the philosopher Heinz von Foerster. Special attention is given to some objections to the concepts of probability, statistics and randomization posed by George Spencer-Brown, a figure of great influence in the field of radical constructivism.

We present a theory of cooperative-competitive intelligence (CCI), its measures, research program, and applications that stem from it. Within the framework of this theory, satisficing sub-optimal behavior is any behavior that does not promote a decrease in the prospective control of the functional action diversity/unpredictability (D/U) potential of the agent or team. This potential is defined as the entropy measure in multiple, context-dependent dimensions. We define the satisficing interval of behaviors as CCI. In order to manifest itself at individual (...) or team level, this capacity harnesses properties such as degeneracy, pleiotropy (pluri-potentiality), synergies, and metastability. Intelligence is embodied because intelligent behavior is deeply dependent on body functionalities, defined as entropy measures. We base our theory on three principles: (a) relativity of functional entropy/ information in agent (team)-environment systems, (b) tendency toward the satisficing level of D/U potential, and (c) tendency toward the non-decreasing D/U potential. The conjunction of these three principles provides existence of sub-optimal behaviors associated with CCI. First, we deal with the problem of how to reduce multidimensional behavior to a concept that accounts for the vast set of scenarios in which CCI is manifested. Secondly, we define and discuss the three interacting principles that underpin CCI behavior as well as providing an outline for a future CCI research program supported by agent￾based modeling and empirical research. Finally, we provide some preliminary practical issues that stem from the theory. (shrink)

This paper makes a novel linkage between the multiple-computations theorem in philosophy of mind and Landauer’s principle in physics. The multiple-computations theorem implies that certain physical systems implement simultaneously more than one computation. Landauer’s principle implies that the physical implementation of “logically irreversible” functions is accompanied by minimal entropy increase. We show that the multiple-computations theorem is incompatible with, or at least challenges, the universal validity of Landauer’s principle. To this end we provide accounts of both (...) ideas in terms of low-level fundamental concepts in statistical mechanics, thus providing a deeper understanding of these ideas than their standard formulations given in the high-level terms of thermodynamics and cognitive science. Since Landauer’s principle is pivotal in the attempts to derive the universal validity of the second law of thermodynamics in statistical mechanics, our result entails that the multiple-computations theorem has crucial implications with respect to the second law. Finally, our analysis contributes to the understanding of notions, such as “logical irreversibility,” “entropy increase,” “implementing a computation,” in terms of fundamental physics, and to resolving open questions in the literature of both fields, such as: what could it possibly mean that a certain physical process implements a certain computation. (shrink)

My objective of this paper is to suggest and workout a more credible form of the Principle of Beneficence from the common essential elements of the three major ethical theories (Deontology, Utilitarianism and Virtue Ethics) that will try to overcome the over-demanding objection of Utilitarianism and the rigorism of Kant’s Deontology. After analyzing these three moral systems, I find that beneficence lies within the very essence of humanity. Human beings are superior to other creatures in the world due to (...) rationality and humanity. From the humanitarian ground, a common goodness lies within every human. Beneficence, as a moral principle, is derived from this inner humanity of every individual. Despite their initial differences, utilitarianism, deontology and virtue ethics recognize this fundamental humanitarian disposition of doing good for all as a part of being a morally better person. The principle of beneficence as I suggest, is different from its consequential utilitarian notion suggested by Mill. This version of beneficence is more credible as it does not impose excessive demands upon an individual to develop any maximum beneficial outcome following utilitarian calculation of beneficence over cost, and it also strives to overcome the rigorous duty-based theory of Kantian deontology by appealing to the fundamental virtue of humanity. Finally, the credibility of this form of beneficence comes from the underlying transcendental humanism which is the chief feature of Indian tradition. (shrink)

Quantum theory brought an irreducible lawlessness in physics. This is accompanied by lack of specification of state of a system. We can not measure states even though they ever existed. We can measure only transition from one state into another. We deduce this lack of determination of state mathematically, and thus provide formalism for maximum precision of determination of mixed states. However, the results thus obtained show consistency with Heisenberg's uncertainty relations.

This paper presents a unified theory of reality, which integrates two interdependent frameworks: Cellular Primary Consciousness Theory (CPCT) and Open and Closed Systems Theory (OCST). Although CPCT and OCST can each stand as individual theories, they are, in this work, combined to form a cohesive explanation of both the mechanics and purpose of the universe. CPCT posits that consciousness is a fundamental aspect of all life, extending to even the simplest cells, rather than being an emergent property exclusive to complex (...) neural systems. Drawing upon evidence from quantum mechanics, cell biology, and systems theory, CPCT challenges traditional views on the origins of consciousness by suggesting that awareness is a universal property present from the cellular level. OCST, which forms the second component of this unified theory, examines the nature of the universe as a closed system subject to the laws of entropy. This theory explores how the universe is moving toward maximum entropy, where meaning and contrasts dissolve. However, beyond this closed system lies the realm of an open system—a domain of infinite potential that transcends the dualities and limitations of the physical universe. This open system offers a philosophical explanation for existence and the possibility of eternal meaning. Together, these two theories provide a novel and cohesive framework that addresses both the mechanics of consciousness and the ultimate purpose of the universe. The integration of CPCT and OCST bridges scientific evidence from quantum biology and systems theory with philosophical explorations of existence, presenting a comprehensive vision of reality as both a closed physical system and an open metaphysical one. This unified theory encourages a reexamination of our place in the cosmos, suggesting that beyond the observable universe lies an infinite realm of potential where consciousness and meaning endure. (shrink)

One of the most difficult problems in the foundations of physics is what gives rise to the arrow of time. Since the fundamental dynamical laws of physics are (essentially) symmetric in time, the explanation for time's arrow must come from elsewhere. A promising explanation introduces a special cosmological initial condition, now called the Past Hypothesis: the universe started in a low-entropy state. Unfortunately, in a universe where there are many copies of us (in the distant ''past'' or the distant (...) ''future''), the Past Hypothesis is not enough; we also need to postulate self-locating (de se) probabilities. However, I show that we can similarly use self-locating probabilities to strengthen its rival---the Fluctuation Hypothesis, leading to in-principle empirical underdetermination and radical epistemological skepticism. The underdetermination is robust in the sense that it is not resolved by the usual appeal to 'empirical coherence' or 'simplicity.' That is a serious problem for the vision of providing a completely scientific explanation of time's arrow. (shrink)

A new trend in deep learning, represented by Mutual Information Neural Estimation (MINE) and Information Noise Contrast Estimation (InfoNCE), is emerging. In this trend, similarity functions and Estimated Mutual Information (EMI) are used as learning and objective functions. Coincidentally, EMI is essentially the same as Semantic Mutual Information (SeMI) proposed by the author 30 years ago. This paper first reviews the evolutionary histories of semantic information measures and learning functions. Then, it briefly introduces the author’s semantic information G theory with (...) the rate-fidelity function R(G) (G denotes SeMI, and R(G) extends R(D)) and its applications to multi-label learning, the maximum Mutual Information (MI) classification, and mixture models. Then it discusses how we should understand the relationship between SeMI and Shannon’s MI, two generalized entropies (fuzzy entropy and coverage entropy), Autoencoders, Gibbs distributions, and partition functions from the perspective of the R(G) function or the G theory. An important conclusion is that mixture models and Restricted Boltzmann Machines converge because SeMI is maximized, and Shannon’s MI is minimized, making information efficiency G/R close to 1. A potential opportunity is to simplify deep learning by using Gaussian channel mixture models for pre-training deep neural networks’ latent layers without considering gradients. It also discusses how the SeMI measure is used as the reward function (reflecting purposiveness) for reinforcement learning. The G theory helps interpret deep learning but is far from enough. Combining semantic information theory and deep learning will accelerate their development. (shrink)

This paper considers questions about continuity and discontinuity between life and mind. It begins by examining such questions from the perspective of the free energy principle (FEP). The FEP is becoming increasingly influential in neuroscience and cognitive science. It says that organisms act to maintain themselves in their expected biological and cognitive states, and that they can do so only by minimizing their free energy given that the long-term average of free energy is entropy. The paper then argues (...) that there is no singular interpretation of the FEP for thinking about the relation between life and mind. Some FEP formulations express what we call an independence view of life and mind. One independence view is a cognitivist view of the FEP. It turns on information processing with semantic content, thus restricting the range of systems capable of exhibiting mentality. Other independence views exemplify what we call an overly generous non-cognitivist view of the FEP, and these appear to go in the opposite direction. That is, they imply that mentality is nearly everywhere. The paper proceeds to argue that non-cognitivist FEP, and its implications for thinking about the relation between life and mind, can be usefully constrained by key ideas in recent enactive approaches to cognitive science. We conclude that the most compelling account of the relationship between life and mind treats them as strongly continuous, and that this continuity is based on particular concepts of life (autopoiesis and adaptivity) and mind (basic and non-semantic). (shrink)

Many mechanisms, functions and structures of life have been unraveled. However, the fundamental driving force that propelled chemical evolution and led to life has remained obscure. The second law of thermodynamics, written as an equation of motion, reveals that elemental abiotic matter evolves from the equilibrium via chemical reactions that couple to external energy towards complex biotic non-equilibrium systems. Each time a new mechanism of energy transduction emerges, e.g., by random variation in syntheses, evolution prompts by punctuation and settles to (...) a stasis when the accessed free energy has been consumed. The evolutionary course towards an increasingly larger energy transduction system accumulates a diversity of energy transduction mechanisms, i.e. species. The rate of entropy increase is identified as the fitness criterion among the diverse mechanisms, which places the theory of evolution by natural selection on the fundamental thermodynamic principle with no demarcation line between inanimate and animate. (shrink)

The concept of time is examined using the second law of thermodynamics that was recently formulated as an equation of motion. According to the statistical notion of increasing entropy, flows of energy diminish differences between energy densities that form space. The flow of energy is identified with the flow of time. The non-Euclidean energy landscape, i.e. the curved space–time, is in evolution when energy is flowing down along gradients and levelling the density differences. The flows along the steepest descents, (...) i.e. geodesics are obtained from the principle of least action for mechanics, electrodynamics and quantum mechanics. The arrow of time, associated with the expansion of the Universe, identifies with grand dispersal of energy when high-energy densities transform by various mechanisms to lower densities in energy and eventually to ever-diluting electromagnetic radiation. Likewise, time in a quantum system takes an increment forwards in the detection-associated dissipative transformation when the stationary-state system begins to evolve pictured as the wave function collapse. The energy dispersal is understood to underlie causality so that an energy gradient is a cause and the resulting energy flow is an effect. The account on causality by the concepts of physics does not imply determinism; on the contrary, evolution of space–time as a causal chain of events is non-deterministic. (shrink)

The principle of Information Conservation or Determinism is a governing assumption of physical theory. Determinism has counterfactual consequences. It entails that if the present were different, then the future would be different. But determinism is temporally symmetric: it entails that if the present were different, the past would also have to be different. This runs contrary to our commonsense intuition that what has happened in the future depends on the past in a way the past does not depend on (...) the future. To understand how this can be so we observe that while the truth of some counterfactuals is guaranteed by the laws of logic or the laws of nature, some are not. It is among the latter contingent, counterfactuals that we find temporal asymmetry. It is this asymmetry that gives causation a temporal direction. The temporal asymmetry of these counterfactuals is explained by the fact that the dynamical laws of nature are logically irreversible functions from partial states of the world onto other partial states. (Logical reversibility is not to be confused, though it too often is, with time-reversal invariance). Though these irreversible laws are locally indeterministic, they can sum to give a globally deterministic description of the world. This combination of global determinism and local indeterminism gives rise to contingent counterfactual dependence and gives that dependence a direction. That direction is independent of the direction of entropy. The direction of contingent counterfactual dependence is time's arrow. (shrink)

The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the rest of reality as we know it, and the implications of the existence of this structure for quantum theory. Understanding how this structure could relate to space and to the rest of reality requires, I (...) believe, that we consider how space itself relates to reality, and how other so-called "spaces" used in physics relate to reality. In chapter 2, I compare space and space-time to other spaces used in physics, such as configuration space, phase space and Hilbert space. I support what is known as the "property view" of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that all these spaces are property spaces. After examining the relationships of these spaces to causality, I argue that configuration space has, due to its role in quantum mechanics, a special status in the microscopic world similar to the status of position space in the macroscopic world. In chapter 3, prespace itself is considered. One way of approaching this structure is through the comparison of the prespace structure with a computational system, in particular to a cellular automaton, in which space or space-time and all other physical quantities are broken down into discrete units. I suggest that one way open for a prespace metaphysics can be found if physics is made fully discrete in this way. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized, adapting a heuristic principle of this type proposed by Feynman. In chapter 4, some of the ideas of the previous chapters are applied in an examination of the physics and metaphysics of quantum theory. I first discuss the "measurement problem" of quantum mechanics: this problem and its proposed solution are the primary subjects of chapter 4. It turns out that considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be modified to give rise to a solution to the measurement problem. The computational heuristic principle reinforces the same solution. I call the modified quantum mechanics Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with some of the other proposed solutions to the measurement problem, in particular the spontaneous localization model of Ghirardi, Rimini and Weber. Finally, in chapters 5 and 6, I argue that the measure of complexity of quantum mechanical states I introduce in CCQM also provides a new definition of entropy for quantum mechanics, and suggests a solution to the problem of providing an objective foundation for statistical mechanics, thermodynamics, and the arrow of time. (shrink)

In the article “A Tension in the Strong Program: The Relation between the Rational and the Social,” I stated that David Bloor, citing the principle of symmetry, expresses that rational and irrational beliefs must be explained in the same way, that is, by causes of the same kind. On this wise, he rejects the methodology of traditional philosophers and historians of science as asymmetrical; since they explain evidence-based beliefs with epistemic reasons and unreasonable beliefs—e.g. beliefs based on indoctrination, propaganda, (...) ideology, and superstition—citing social factors. On the other hand, Bloor argues that the rational is made entirely of the social and, therefore, is itself a kind of social factor. But by admitting this maximum conception of social constructivism, Bloor can no longer consider the rational and the social as two separate identities and accuse the traditional view of asymmetry. However, Professor Finn Collin says that Bloor’s critique of the traditional model never commits him to the duality of the rational and the social. (shrink)

I explain in what sense the structure of space and time is probably vague or indefinite, a notion I define. This leads to the mathematical representation of location in space and time by a vague interval. From this, a principle of complementary inaccuracy between spatial location and velocity is derived, and its relation to the Uncertainty Principle discussed. In addition, even if the laws of nature are deterministic, the behaviour of systems will be random to some degree. These (...) and other considerations draw classical physics closer to Quantum Mechanics. An arrow of entropy is also derived, given an arrow of time. Lastly, chaos is given an additional, objective meaning. (shrink)

The black hole information loss paradox is a catch-all term for a family of puzzles related to black hole evaporation. For almost 50 years, the quest to elucidate the implications of black hole evaporation has not only sustained momentum, but has also become increasingly populated with proposals that seem to generate more questions than they purport to answer. Scholars often neglect to acknowledge ongoing discussions within black hole thermodynamics and statistical mechanics when analyzing the paradox, including the interpretation of Bekenstein-Hawking (...) entropy, which is far from settled. To remedy the dialectical gridlock, I have formulated an overarching, unified framework, which I call ``Black Hole Paradoxes'', that integrates the debates and taxonomizes the relevant `camps' or philosophical positions. -/- I demonstrate that black hole evaporation within Hawking's semi-classical framework insinuates how late-time Hawking radiation is an entangled global system, a contradiction in terms. The relevant forms of information loss are associated with a decrease in maximal Boltzmann entropy and an increase in global von Neumann entropy respectively, which engender what I've branded the ``paradox of phantom entanglement''. Prospective solutions are then tasked with demonstrating how late-time Hawking radiation is either exclusively an entangled subsystem, in which a black hole remnant lingers as an information safehouse, or exclusively an unentangled global system, in which information is evacuated to the exterior. -/- The disagreement between safehouse and evacuation solutions boils down to the statistical interpretation of thermodynamic black hole entropy, i.e., Bekenstein-Hawking entropy. Safehouse solutions attribute Bekenstein-Hawking entropy to a minority of black hole degrees of freedom, those that are associated with the horizon. Evacuation solutions, in contrast, attribute Bekenstein-Hawking entropy to all black hole degrees of freedom. I argue that the interpretation of Bekenstein-Hawking entropy is the litmus test to vet the overpopulated proposal space. So long as any proposal rejecting Hawking's original calculation independently derives black hole evaporation, globally conserves degrees of freedom and entanglement, preserves a version of semi-classical gravity at sub-Planckian scales, and describes black hole thermodynamics in statistical terms, then it counts as a genuine solution to the paradox. (shrink)

FDE, LP and K3 are closely related to each other and admit of an intuitive informational interpretation. However, all these logics are co-NP complete, and so idealized models of how an agent can think. We address this issue by shifting to signed formulae, where the signs express imprecise values associated with two bipartitions of the corresponding set of standard values. We present proof systems whose operational rules are all linear and have only two structural branching rules that express a generalized (...) Principle of Bivalence. Each of these systems leads to defining an infinite hierarchy of tractable approximations to the respective logic, in terms of the maximum number of allowed nested applications of the two branching rules. Further, each resulting hierarchy admits of an intuitive 5-valued non-deterministic semantics. (shrink)

All reasoners described in the most widespread models of a rational reasoner exhibit logical omniscience, which is impossible for finite reasoners (real reasoners). The most common strategy for dealing with the problem of logical omniscience is to interpret the models using a notion of beliefs different from explicit beliefs. For example, the models could be interpreted as describing the beliefs that the reasoner would hold if the reasoner were able reason indefinitely (stable beliefs). Then the models would describe maximum (...) rationality, which a finite reasoner can only approach in the limit of a reasoning sequence. This strategy has important consequences for epistemology. If a finite reasoner can only approach maximum rationality in the limit of a reasoning sequence, then the efficiency of reasoning is epistemically (and not only pragmatically) relevant. In this paper, I present an argument to this conclusion and discuss its consequences, as, for example, the vindication of the principle 'no rationality through brute-force'. (shrink)

One of the criteria to a strong principle in natural sciences is simplicity. This paper claims that the Free Energy Principle (FEP), by virtue of unifying particles with mind, is the simplest. Motivated by Hilbert’s 24th problem of simplicity, the argument is made that the FEP takes a seemingly mathematical complex domain and reduces it to something simple. More specifically, it is attempted to show that every ‘thing’, from particles to mind, can be partitioned into systemic states by (...) virtue of self-organising symmetry break, i.e. self-entropy in terms of the balance between risk and ambiguity to achieve epistemic gain. By virtue of its explanatory reach, the FEP becomes the simplest principle under quantum, statistical and classical mechanics conditions. (shrink)

To become more useful and efficient in sustaining the Earth's health, economics must undergo a paradigm shift in its thinking. From a humanistic perspective, humans should be the center of everything. However, from the standpoint of physics and the universe, this is not the case. As a species, having a planet among the millions in the universe where humans can survive and thrive is already a great fortune. Through this book, we also try to answer one of our long-standing questions: (...) why has economics learned so little from physics, the oldest and most rigorous academic discipline that studies nature? -/- Learning from the quantum and information theories, we also provide a new definition of value, which is expected to help economists relax assumptions and loosen established economic principles in order to change and evolve. The new definition is also expected to enable social scientists to address newly arising phenomena or events that are not accounted for by existing value systems, such as environmental crises, artificial intelligence (AI), and interdisciplinary information, more proactively and productively. (shrink)

We review some of the main implications of the free-energy principle (FEP) for the study of the self-organization of living systems – and how the FEP can help us to understand (and model) biotic self-organization across the many temporal and spatial scales over which life exists. In order to maintain its integrity as a bounded system, any biological system - from single cells to complex organisms and societies - has to limit the disorder or dispersion (i.e., the long-run (...) class='Hi'>entropy) of its constituent states. We review how this can be achieved by living systems that minimize their variational free energy. Variational free energy is an information theoretic construct, originally introduced into theoretical neuroscience and biology to explain perception, action, and learning. It has since been extended to explain the evolution, development, form, and function of entire organisms, providing a principled model of biotic self-organization and autopoiesis. It has provided insights into biological systems across spatiotemporal scales, ranging from microscales (e.g., sub- and multicellular dynamics), to intermediate scales (e.g., groups of interacting animals and culture), through to macroscale phenomena (the evolution of entire species). A crucial corollary of the FEP is that an organism just is (i.e., embodies or entails) an implicit model of its environment. As such, organisms come to embody causal relationships of their ecological niche, which, in turn, is influenced by their resulting behaviors. Crucially, free-energy minimization can be shown to be equivalent to the maximization of Bayesian model evidence. This allows us to cast natural selection in terms of Bayesian model selection, providing a robust theoretical account of how organisms come to match or accommodate the spatiotemporal complexity of their surrounding niche. In line with the theme of this volume; namely, biological complexity and self-organization, this chapter will examine a variational approach to self-organization across multiple dynamical scales. (shrink)

"Procedural Justice" offers a theory of procedural fairness for civil dispute resolution. The core idea behind the theory is the procedural legitimacy thesis: participation rights are essential for the legitimacy of adjudicatory procedures. The theory yields two principles of procedural justice: the accuracy principle and the participation principle. The two principles require a system of procedure to aim at accuracy and to afford reasonable rights of participation qualified by a practicability constraint. The Article begins in Part I, Introduction, (...) with two observations. First, the function of procedure is to particularize general substantive norms so that they can guide action. Second, the hard problem of procedural justice corresponds to the following question: How can we regard ourselves as obligated by legitimate authority to comply with a judgment that we believe (or even know) to be in error with respect to the substantive merits? The theory of procedural justice is developed in several stages, beginning with some preliminary questions and problems. The first question - what is procedure? - is the most difficult and requires an extensive answer: Part II, Substance and Procedure, defines the subject of the inquiry by offering a new theory of the distinction between substance and procedure that acknowledges the entanglement of the action-guiding roles of substantive and procedural rules while preserving the distinction between two ideal types of rules. The key to the development of this account of the nature of procedure is a thought experiment, in which we imagine a world with the maximum possible acoustic separation between substance and procedure. Part III, The Foundations of Procedural Justice, lays out the premises of general jurisprudence that ground the theory and answers a series of objections to the notion that the search for a theory of procedural justice is a worthwhile enterprise. Sections II and III set the stage for the more difficult work of constructing a theory of procedural legitimacy. Part IV, Views of Procedural Justice, investigates the theories of procedural fairness found explicitly or implicitly in case law and commentary. After a preliminary inquiry that distinguishes procedural justice from other forms of justice, Part IV focuses on three models or theories. The first, the accuracy model, assumes that the aim of civil dispute resolution is correct application of the law to the facts. The second, the balancing model, assumes that the aim of civil procedure is to strike a fair balance between the costs and benefits of adjudication. The third, the participation model, assumes that the very idea of a correct outcome must be understood as a function of process that guarantees fair and equal participation. Part IV demonstrates that none of these models provides the basis for a fully adequate theory of procedural justice. In Part V, The Value of Participation, the lessons learned from analysis and critique of the three models are then applied to the question whether a right of participation can be justified for reasons that are not reducible to either its effect on the accuracy or its effect on the cost of adjudication. The most important result of Part V is the Participatory Legitimacy Thesis: it is (usually) a condition for the fairness of a procedure that those who are to be finally bound shall have a reasonable opportunity to participate in the proceedings. The central normative thrust of Procedural Justice is developed in Part VI, Principles of Procedural Justice. The first principle, the Participation Principle, stipulates a minimum (and minimal) right of participation, in the form of notice and an opportunity to be heard, that must be satisfied (if feasible) in order for a procedure to be considered fair. The second principle, the Accuracy Principle, specifies the achievement of legally correct outcomes as the criterion for measuring procedural fairness, subject to four provisos, each of which sets out circumstances under which a departure from the goal of accuracy is justified by procedural fairness itself. In Part VII, The Problem of Aggregation, the Participation Principle and the Accuracy Principle are applied to the central problem of contemporary civil procedure - the aggregation of claims in mass litigation. Part VIII offers some concluding observations about the point and significance of Procedural Justice. (shrink)

To naturalize the concept of teleological causality in biology it is not enough to avoid assuming backward causation or positing the existence of an inscrutable te- leological essence like the élan vital. We must also specify how the causality of or- ganisms is distinct from the causality of designed artifacts like thermostats or asym- metrically oriented processes like the ubiquitous increase of entropy. Historically, the concept of teleological causality in biology has been based on an analogy to the familiar (...) experience of purposeful action. This is experienced by us as a disposition to achieve a general type of end that is represented in advance, and which regulates the selection of efficient means to achieve it. Inspired by this analogy, to bridge the gap between biology and human agency we describe a simple molecular pro- cess called autogenesis that shows how two linked complementary self-organizing processes can give rise to higher-order relations that resemble purposeful disposi- tions, though expressed in terms of constraints on molecular processes. Because the autogenic model is described in sufficient detail to be empirically realizable, it provides a proof of principle demonstrating a simple form of teleological causality. (shrink)

Most philosophers of religion subscribe to some variety of religious realism: they believe that religious statements aim at capturing a mind-independent reality and are true precisely if they successfully do so. Curiously, most religious realists also believe that at least some of our religious beliefs are rationally justified. In this paper, I argue that these positions are actually at odds with each other. Religious realists should rather be religious skeptics. I first argue that realism always implies the possibility of our (...) beliefs being false. Then, I develop a Copernican argument to show that we have no reason to believe that we are in an epistemically privileged position where we could be confident that our particular belief is justified while all competing beliefs are not. This seems impossible for two reasons: (a) From a probabilistic point of view (adopting a principle of mediocrity), it is highly unlikely that current humans have the maximum of epistemic capacities possible. Rather, our ability to understand reality is limited, so we should expect at least some parts of religious reality to remain unknown to us. (b) From a deep-time perspective, humanity has not pondered religious questions for long. We are probably still in the early stages of our cognitive evolution, and so we should treat our religious beliefs as provisional and open to revision. (shrink)

In this work I propose a theory of evolution as a process of unfolding. This theory is based on four logically concatenated principles. The principle of evolutionary order establishes that the more complex cannot be generated from the simpler. The principle of origin establishes that there must be a maximum complexity that originates the others by logical deduction. Finally, the principle of unfolding and the principle of actualization guarantee the development of the evolutionary process from (...) the simplest to the most complex. These logical principles determine the existence of a virtual ideological matrix that contains the sequence of the preformed and folded morphogenetic fields. In this manner, the evolutionary process consists of the sequential unfolding and actualization of these fields, which is motorized by a process of teleologization carried out by the opening consciousness of the forms included in the fields of the ideological matrix. This theory leads to a radical change of perspective regarding the materialist worldview, and places life at the center of the evolutionary process as an activity carried out by a consciousness that seeks to fulfill a purpose by actualizing its own potentialities. (shrink)

Many philosophers sympathetic with a Humean understanding of laws of nature have thought that, in the final analysis, the fundamental laws will include not only the traditional dynamical equations, but also two additional principles: the Past Hypothesis and the Statistical Postulate. The former says that the universe began in a particular very-low-entropy macrostate M(0), and the latter posits a uniform probability distribution over the microstates compatible with M(0). Such a view is arguably vindicated by the orthodox Humean Best System (...) Account (BSA). However, I argue here that recent developments of the BSA render the Past Hypothesis otiose. In particular, the trend among Humeans toward a more pragmatic view of laws — according to which the best system is the one that is maximally effective at helping creatures like us amplify our information about the world — does not support the idea that the Past Hypothesis is a law of nature. (shrink)

Gilabert argues that the humanist conception of duties of global justice and the principle of cosmopolitan justifiability will lead us to accept an egalitarian definition of individual autonomy. Gilabert further argues that realizing conditions of individual autonomy can serve as the cut-off point to duties of global justice. I investigate his idea of autonomy, arguing that in order to make sense of this claim, we need a concept of autonomy. I propose 4 possible definitions of autonomy, none of which (...) seem to necessitate Gilabert’s duties of egalitarian global justice. Instead, I propose that he may have in mind Autonomy 5, which requires that individuals have access to a maximum number of options and not simply a sufficient range of options to choose from. I criticize this premise as too demanding in the global world characterized by fundamental inequality. Second, I argue that if we were to endorse the preconditions for Autonomy 5, we would have to accept that Gilabert’s theory of global justice doesn’t provide for a cut-off point of duties of global justice. (shrink)

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and Klein bottle (...) logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

This insight to art came from chess composition concentrating art in a very dense form. To identify and mathematically assess the uniqueness is the key applicable to other areas eg. computer programming. Maximization of uniqueness is minimization of entropy that coincides as well as goes beyond Information Theory (Shannon, 1948). The reusage of logic as a universal principle to minimize entropy, requires simplified architecture and abstraction. Any structures (e.g. plugins) duplicating or dividing functionality increase entropy and (...) so unreliability (eg. British Airways IT system). The ideas here were verified by my chess compositions, art works and complex information system as an author of each co uk, and were presented at conferences in Santorini, Adelaide, Geneva, Daejon and virtually. (shrink)

The algebra of transactions as fundamental measurements is constructed on the basis of the analysis of their properties and represents an expansion of the Boolean algebra. The notion of the generalized economic measurements of the economic “quantity” and “quality” of objects of transactions is introduced. It has been shown that the vector space of economic states constructed on the basis of these measurements is relativistic. The laws of kinematics of economic objects in this space have been analyzed and the stages (...) of constructing the dynamics have been formulated. In particular, the “principle of maximum benefit”, which represents an economic analog of the principle of least action in the classical mechanics, and the principle of relativity as the principle of equality of all possible consumer preferences have been formulated. The notion of economic interval between two economic objects invariant to the selection of the vector of consumer preferences has been introduced. Methods of experimental verification of the principle of relativity in the space of economic states have been proposed. (shrink)

Unlike other classical arguments for the existence of God, Pascal’s Wager provides a pragmatic rationale for theistic belief. Its most popular version says that it is rationally mandatory to choose a way of life that seeks to cultivate belief in God because this is the option of maximum expected utility. Despite its initial attractiveness, this long-standing argument has been subject to various criticisms by many philosophers. What is less discussed, however, is the rationality of this choice in situations where (...) the decision-makers are confronted with greater uncertainty. In this paper, I examine the imprecise version of Pascal’s Wager: those scenarios where an agent’s credence that God exists is imprecise or vague rather than precise. After introducing some technical background on imprecise probabilities, I apply five different principles for decision-making to two cases of state uncertainty. In the final part of the paper, I argue that it is not rationally permitted to include zero as the lower probability of God’s existence. Although the conditions for what makes an act uniquely optimal vary significantly across those principles, I also show how the option of wagering for God can defeat any mixed strategy under two distinct interpretations of salvation. (shrink)

Probability can be used to measure degree of belief in two ways: objectively and subjectively. The objective measure is a measure of the rational degree of belief in a proposition given a set of evidential propositions. The subjective measure is the measure of a particular subject’s dispositions to decide between options. In both measures, certainty is a degree of belief 1. I will show, however, that there can be cases where one belief is stronger than another yet both beliefs are (...) plausibly measurable as objectively and subjectively certain. In ordinary language, we can say that while both beliefs are certain, one belief is more certain than the other. I will then propose second, non probabilistic dimension of measurement, which tracks this variation in certainty in such cases where the probability is 1. A general principle of rationality is that one’s subjective degree of belief should match the rational degree of belief given the evidence available. In this paper I hope to show that it is also a rational principle that the maximum stake size at which one should remain certain should match the rational weight of certainty given the evidence available. Neither objective nor subjective measures of certainty conform to the axioms of probability, but instead are measured in utility. This has the consequence that, although it is often rational to be certain to some degree, there is no such thing as absolute certainty. (shrink)

Susan Okin read Robert Nozick as taking it to be fundamental to his Libertarianism that people own themselves, and that they can acquire entitlement to other things by making them. But she thinks that, since mothers make people, all people must then be owned by their mothers, a consequence Okin finds absurd. She sees no way for Nozick to make a principled exception to the idea that people own what they make when what they make is people, concluding that Nozick’s (...) theory of entitlement must be false, and that entitlement must instead be rooted in people’s needs. I say Okin misreads Nozick’s Libertarianism. In fact, its fundamental principle is that, simply by being persons, people are entitled to the maximum negative liberty compatible with a like liberty for all persons. Further, Nozick, and Jan Narveson, who has taken on the advocacy of Libertarian ideas, analyze liberty as freedom to interact with things, and analyze being entitled to or having property in something, as freedom to interact with it, to determine what may be done with it. People therefore have such freedom to do what they want with themselves, and such freedom to do what they want with other things, as is compatible with all persons having similar freedom. The former is what self-ownership amounts to, the latter, ownership of other things. Libertarianism’s fundamental principle therefore both grounds and delimits entitlements in ways entailing that mothers don’t own persons by dint of making them. Otherwise, since it would then be the prerogative of mothers to determine what shall be done with the persons they made, the persons made would lack equal liberty, this violating the fundamental principle. (shrink)

The remarkable connections between gravity and thermodynamics seem to imply that gravity is not fundamental but emergent, and in particular, as Verlinde suggested, gravity is probably an entropic force. In this paper, we will argue that the idea of gravity as an entropic force is debatable. It is shown that there is no convincing analogy between gravity and entropic force in Verlinde’s example. Neither holographic screen nor test particle satisfies all requirements for the existence of entropic force in a thermodynamics (...) system. As a result, there is no entropic force in the gravity system. Furthermore, we show that the entropy increase of the screen is not caused by its statistical tendency to increase entropy as required by the existence of entropic force, but in fact caused by gravity. Therefore, Verlinde’s argument for the entropic origin of gravity is problematic. In addition, we argue that the existence of a minimum size of spacetime, together with the Heisenberg uncertainty principle in quantum theory, may imply the fundamental existence of gravity as a geometric property of spacetime. This provides a further support for the conclusion that gravity is not an entropic force. (shrink)

In 2013, I've transformed the each_co_uk server side system (aspx / c#) to the client system of a few javascript files ≈400KB potentially webassembly, without plugins / 3rd party / css.. The system maximally reuses functions & data to apply: 1) Linguistic principle of arbitrariness of the sign maximizing repetitions, 2) Multi layer logic of the chess composition, 3) Information Theory minimizing entropy. Some of these ideas were presented at conferences in Santorini, Adelaide, Geneva, Daejon and virtually. Philosophically, (...) any system only organizes the existing reality (information), but can't create it. Its fragmentation easily leads to illusions (entropy): e.g. Zeno's Achilles can't overtake the slower tortoise because as he gets closer, the tortoise gets a bit further, so he gets closer and closer but never reaches her. So the more are logic and data fragmented in various layers (server, javascript, css, sql, frameworks..), the less capable system is. (shrink)

The article considers the problem of ensuring the economic security of trade enterprises by forming an optimal pricing policy. The methodology of formation the minimum and maximum selling prices of trade enterprise, maintenance of margin of economic security, which is based on research of turnover costs and working capital of trade enterprise is offered. Based on statistical data of trade enterprises, the types of prices by product range are determined, which form a stable economic situation and ensure economic security (...) of trade enterprise. The necessity of forming the optimal price policy and selling prices of trade enterprises with the use of Pareto efficiency tools and construction of single-criteria and multi-criteria tasks, indicators of quality of commodity resources of trade enterprises is substantiated. Using the Excel software processor and the «Regression» function, economic-mathematical models of optimal prices for the product range of trade enterprises are built. The results testified to the effectiveness of the proposed model of the optimal pricing policy of trade enterprises, as the obtained values are within the minimum selling prices, which provides economic security of trade enterprises. The process of forming the pricing policy of trade enterprises to ensure economic security is proposed, which is based on the principles of pricing, appropriate methodological tools and monitoring of market environment factors. Organizational, economic, legal, social and market (marketing) mechanisms for the formation of effective pricing policy of trade enterprises aimed at ensuring economic security are identified. This study is practically interesting for personnel of trade enterprises, regardless of organizationallegal forms of ownership and activities, and theoretically – for researchers dealing with pricing. (shrink)

In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we ask the (...) question, whether at least the variational principle for Haus- dor® dimension holds, which means that there is a sequence of ergodic measures such that their Hausdor® dimension approximates the Hausdor® dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal di®eomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two di®erent rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of de- velopment of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the ¯rst step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. A±ne dynamical systems represent simple examples of non conformal systems. They are easy to de¯ne, but studying their dimensional theoretical properties does never- theless provide challenging mathematical problems and exemplify interesting phe- nomena. We consider here a special class of self-a±ne repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise a±ne maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as gener- alized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of di- mension theory of a±ne dynamical systems and de¯ne the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and ¯nd representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see the- orem 4.1.). Then in chapter ¯ve we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to ques- tions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the iden- tity of box-counting and Hausdor® dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdor® dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdor® dimension for the repellers. On the other hand the vari- ational principle for Hausdor® dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will ¯nd a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdor® dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical excep- tions to our generic results. The starting point of our considerations in section nine are results of ErdÄos [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for in¯nite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdor® and box-counting dimension for all of our systems. In the ¯rst section of chapter ten we ¯nd parameter values such that the variational principle for Hausdor® dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the ¯rst known examples of dynamical systems for which the variational principle for Hausdor® dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdor® dimension; the question if the variational principle for Hausdor® dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdor® and box-counting dimension can drops because there are number theoretical pecu- liarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will ¯nd two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor JÄorg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFÄoG der Freien UniversitÄat Berlin". (shrink)

Due to the shortage comprehensive study about the ethics of belief in the Quran, This study will provide a detailed understanding based on the concepts of the Quran. Numerous books and articles have been written criticizing and defending the ethics of belief, these are three categories: critical for the basics, criticism of the method and reasoning process and criticism of the results. Some of these criticisms like fideism and irrational religious beliefs are not accepted. Maximum rationality, that Clifford believes, (...) is not feasible and practical because it contains all of the partial and simple beliefs. Also, rational argument is not provable for everyone. Clifford and the holy Quran collectively believe that blind acceptance of traditions and beliefs of fathers or ancestors are wrong. In the viewpoint of Islam (Shia), investigation of the religion principles has been emphasized. Another similar point of view between them is criticism of credulity. The advantage of this article is research on the ethics of belief in the Quran. (shrink)

Leibniz said that the universe, if God-created, would exist at a unique, conjoint, physical maximum: Of all possible worlds, it would be richest in phenomena, but its richness would arise from the simplest physical laws and initial conditions. Using concepts of ‘‘variety’’ and algorithmic informational complexity, Leibniz’ claim can be reframed as a testable theory. This theory predicts that the laws and conditions of the actual universe should be simpler, and the universe richer in phenomena, than the presence of (...) observers would require. Tegmark has shown that inhabitants of an infinite multiverse would likely observe simple laws and conditions, but also phenomenal richness just great enough to explain their existence. Empirical observations fit the claim of divine choice better than the claim of an infinite multiverse. The future of the universe, including its future information-processing capacity, is predicted to be endless. -/- . (shrink)

The Turing machine halting problem can be explained by several factors, including arithmetic logic irreversibility and memory erasure, which contribute to computational uncertainty due to information loss during computation. Essentially, this means that an algorithm can only preserve information about an input, rather than generate new information. This uncertainty arises from characteristics such as arithmetic logical irreversibility, Landauer's principle, and memory erasure, which ultimately lead to a loss of information and an increase in entropy. To measure this uncertainty (...) and loss of information, the concept of arithmetic logical entropy can be used. The Turing machine and its equivalent, general recursive functions can be understood through the λ calculus and the Turing/Church thesis. However, there are certain recursive functions that cannot be fully understood or predicted by other algorithms due to the loss of information during logical-arithmetic operations. In other words, the behaviour of these functions cannot be completely determined at every stage of the computation due to a lack of information in their definition. While there are some cases where the behaviour of recursive functions is highly predictable, the lack of information in most cases makes it impossible for algorithms to determine if a program will halt or not. This inability to predict the outcome of the computation is the essence of the halting problem of the Turing machine. Even in cases where more information is available about the program, it is still difficult to determine with certainty if the program will halt or not. This also highlights the importance of the Turing oracle machine, which introduces information from outside the computation to compensate for the lack of information and ultimately decide the result of the computation. (shrink)

Leibniz proposed the ‘Most Determined Path Principle’ in seventeenth century. According to it, ‘ease’ of travel is the end purpose of motion. Using this principle and his calculus method he demonstrated Snell’s Laws of reflection and refraction. This method shows that light follows extremal (local minimum or maximum) time path in going from one point to another, either directly along a straight line path or along a broken line path when it undergoes reflection or refraction at plane (...) or spherical (concave or convex) surfaces. The extremal time path avoided the criticism that Fermat’s least time path was subjected to, by Cartesians who cited examples of reflections at spherical surfaces where light took the path of longest time. Thereby it became the standard method of demonstration of Snell’s Laws. Ptolemy’s theorem is a fundamental theorem in geometry. A special case of it offers a method of finding the minimum sum of the two distances of a point from two given fixed points. We show in this paper that Leibniz’s calculus proof of Snell’s Laws violates Ptolemy’s theorem, whereby Leibniz’s proof becomes invalid. (shrink)