Freely falling point-like objects converge toward the center of the Earth. Hence the gravitational field of the Earth is inhomogeneous, and possesses a tidal component. The free fall of an extended quantum mechanical object such as a hydrogen atom prepared in a high principal-quantum-number state, i.e. a circular Rydberg atom, is predicted to fall more slowly than a classical point-like object, when both objects are dropped from the same height above the Earth’s surface. This indicates that, apart from transitions between (...) quantum states, the atom exhibits a kind of quantum mechanical incompressibility during free fall in inhomogeneous, tidal gravitational fields like those of the Earth.A superconducting ring-like system with a persistent current circulating around it behaves like the circular Rydberg atom during free fall. Like the electronic wavefunction of the freely falling atom, the Cooper-pair wavefunction is quantum mechanically incompressible. The ions in the lattice of the superconductor, however, are not incompressible, since they do not possess a globally coherent quantum phase. The resulting difference during free fall in the response of the nonlocalizable Cooper pairs of electrons and the localizable ions to inhomogeneous gravitational fields is predicted to lead to a charge separation effect, which in turn leads to a large Coulomb force that opposes the convergence caused by the tidal gravitational force on the superconducting system.A “Cavendish-like” experiment is proposed for observing the charge separation effect induced by inhomogeneous gravitational fields in a superconducting circuit. The charge separation effect is determined to be limited by a pair-breaking process that occurs when low frequency gravitational perturbations are present. (shrink)

Weak emergence is the view that a system’s macro properties can be explained by its micro properties but only in an especially complicated way. This paper explains a version of weak emergence based on the notion of explanatory incompressibility and “crawling the causal web.” Then it examines three reasons why weak emergence might be thought to be just in the mind. The first reason is based on contrasting mere epistemological emergence with a form of ontological emergence that involves irreducible (...) downward causation. The second reason is based on the idea that attributions of emergence are always a reflection of our ignorance of non-emergent explanations. The third reason is based on the charge that complex explanations are anthropocentric. Rather than being just in the mind, weak emergence is seen to involve a distinctive kind of complex, macro-pattern in the mind-independent objective micro-causal structure that exists in nature. The paper ends by addressing two further questions. One concerns whether weak emergence applies only or mainly to computer simulations and computational systems. The other concerns the respect in which weak emergence is dynamic rather than static. (shrink)

As regards cognitive capacities, the point of view of classical Artificial Intelligence has been much challenged by the so-called emergentist point of view. This paper attempts to outline,on the basis of logical considerations dealing with practical feasibility, a general theory of incompressible unfolding that is consonant with an old Leibnizian stance rather with the contemporary theory of complexity. I defend a variant of emergentism according to which any process that leads to endow a system with cognitive capacities is such an (...) incompressible unfolding. (shrink)

In this paper I discuss the kinds of idealisations invoked in normative theories—logic, epistemology, and decision theory. I argue that very often the so-called norms of rationality are in fact mere idealisations invoked to make life easier. As such, these idealisations are not too different from various idealisations employed in scientific modelling. Examples of the latter include: fluids are incompressible (in fluid mechanics), growth rates are constant (in population ecology), and the gravitational influence of distant bodies can be ignored (in (...) celestial mechanics). Thinking of logic, epistemology, and decision theory as normative models employing various idealisations of these kinds, changes the way we approach the justification of the models in question. (shrink)

We revisit the analogy suggested by Madelung between a non-relativistic time-dependent quantum particle, to a fluid system which is pseudo-barotropic, irrotational and inviscid. We first discuss the hydrodynamical properties of the Madelung description in general, and extract a pressure like term from the Bohm potential. We show that the existence of a pressure gradient force in the fluid description, does not violate Ehrenfest’s theorem since its expectation value is zero. We also point out that incompressibility of the fluid implies (...) conservation of density along a fluid parcel trajectory and in 1D this immediately results in the non-spreading property of wave packets, as the sum of Bohm potential and an exterior potential must be either constant or linear in space. Next we relate to the hydrodynamic description a thermodynamic counterpart, taking the classical behavior of an adiabatic barotopric flow as a reference. We show that while the Bohm potential is not a positive definite quantity, as is expected from internal energy, its expectation value is proportional to the Fisher information whose integrand is positive definite. Moreover, this integrand is exactly equal to half of the square of the imaginary part of the momentum, as the integrand of the kinetic energy is equal to half of the square of the real part of the momentum. This suggests a relation between the Fisher information and the thermodynamic like internal energy of the Madelung fluid. Furthermore, it provides a physical linkage between the inverse of the Fisher information and the measure of disorder in quantum systems—in spontaneous adiabatic gas expansion the amount of disorder increases while the internal energy decreases. (shrink)

I investigate the relationship between adaptation, as defined in evolutionary theory through natural selection, and the concept of emergence. I argue that there is an essential correlation between the former, and “emergence” defined in the field of algorithmic simulations. I first show that the computational concept of emergence (in terms of incompressible simulation) can be correlated with a causal criterion of emergence (in terms of the specificity of the explanation of global patterns). On this ground, I argue that emergence in (...) general involves some sort of selective processes. Finally, I show that a second criterion, concerning novel explanatory regularities following the emergence of a pattern, captures the robustness of emergence displayed by some cases of emergence (according to the first criterion). Emergent processes fulfilling both criteria are therefore exemplified in evolutionary biology by some so-called “innovations”, and mostly by the new units of fitness or new kinds of adaptations (like sexual reproduction, multicellular organisms, cells, societies) sometimes called “major transitions in evolution”, that recent research programs (Maynard-Smith and Szathmary 1995 ; Michod 1999 ) aims at explaining. (shrink)

Top-down synthetic biology makes partly synthetic cells by redesigning simple natural forms of life, and bottom-up synthetic biology aims to make fully synthetic cells using only entirely nonliving components. Within synthetic biology the notions of complexity and emergence are quite controversial, but the imprecision of key notions makes the discussion inconclusive. I employ a precise notion of weak emergent property, which is a robust characteristic of the behavior of complex bottom-up causal webs, where a complex causal web is one that (...) is incompressible and its behavior cannot be derived except by crawling through all of the gory details of the interactions in the web. The central thesis of this article is that synthetic biology centrally is the activity of engineering the desired weak emergent properties of synthetic cells. Synthetic biology has many different ways to engineer desired weak emergent properties of synthetic cells, including Edisonian trial and error, standardized parts, refactoring, and reprogramming synthetic genomes. The article ends by noting two philosophical consequences of engineering weak emergence. One is epistemological: synthesis is crucial for discovering weak emergent properties. The other is metaphysical: simple life forms are nothing but complex chemical mechanisms. (shrink)

James McAllister’s 2003 article, “Algorithmic randomness in empirical data” claims that empirical data sets are algorithmically random, and hence incompressible. We show that this claim is mistaken. We present theoretical arguments and empirical evidence for compressibility, and discuss the matter in the framework of Minimum Message Length (MML) inference, which shows that the theory which best compresses the data is the one with highest posterior probability, and the best explanation of the data.

This article distinguishes two different senses of information-theoretic approaches to statistical mechanics that are often conflated in the literature: those relating to the thermodynamic cost of computational processes and those that offer an interpretation of statistical mechanics where the probabilities are treated as epistemic. This distinction is then investigated through Earman and Norton’s ([1999]) ‘sound’ and ‘profound’ dilemma for information-theoretic exorcisms of Maxwell’s demon. It is argued that Earman and Norton fail to countenance a ‘sound’ information-theoretic interpretation and this paper (...) describes how the latter inferential interpretations can escape the criticisms of Earman and Norton ([1999]) and Norton ([2005]) by adopting this ‘sound’ horn. This article considers a standard model of Maxwell’s demon to illustrate how one might adopt an information-theoretic approach to statistical mechanics without a reliance on Landauer’s principle, where the incompressibility of the probability distribution due to Liouville’s theorem is taken as the central feature of such an interpretation. (shrink)

It is demonstrated that Maxwellian electrodynamics was created as a result of the old pre-Maxwellian programmes’s reconciliation: the electrodynamics of Ampère–Weber, the wave theory of Young–Fresnel and Faraday’s programme. Maxwell’s programme finally superseded the Ampère–Weber one because it assimilated the ideas of the Ampère–Weber programme, as well as the presuppositions of the programmes of Young–Fresnel and Faraday. Maxwell’s victory became possible because the core of Maxwell’s unification strategy was formed by Kantian epistemology. Maxwell put forward as a basic synthetic principle (...) an idea that radically differed from that of rival approaches by its open, flexible and contra-ontological character. “Action at a distance”, “incompressible fluid”, “molecular vortices” were contrived analogies for Maxwell, capable only of directing the researcher to the “right” mathematical relations Kantian epistemology subsequently enabled Helmholtz and Hertz to arrive at a version of Maxwell’s theory that served as a heuristical basis for the discovery of radio waves. Finally, though neither of Einstein’s relativistic ideas came directly from Kant, they were made possible by the Kantian worldview that had permeated Einstein’s thinking. The Maxwellian revolution can be described in terms of Habermas’s communicative rationality encouraging the establishment of mutual understanding between the various scientific communities. Maxwell’s programme constituted a progressive step in respect to its rivals because it constituted a basis of communication and interpenetration between the main paradigms of 19th century physics. (shrink)

The algorithmic theory of laws claims that the laws of nature are the algorithms in the best possible compression of all empirical data. This position assumes that the universe is compressible and that data received from observing it is easily reproducible using a simple set of rules. However, there are three sources of evidence that suggest that the universe as a whole is incompressible. The first comes from the practice of science. The other two come from the nature of the (...) universe itself: the presence of chaotic behavior and the nature of quantum systems also suggests that the universe is incompressible. This paper evaluates these sources and argues that none provides a convincing case to reject the algorithmic theory of laws. (shrink)

It is discerned what light can bring the recent historical reconstructions of maxwellian optics and electromagnetism unification on the following philosophical/methodological questions. I. Why should one believe that Nature is ultimately simple and that unified theories are more likely to be true? II. What does it mean to say that a theory is unified? III. Why theory unification should be an epistemic virtue? To answer the questions posed genesis and development of Maxwellian electrodynamics are elucidated. It is enunciated that the (...) Maxwellian Revolution is a far more complicated phenomenon than it may be seen in the light of Kuhnian and Lakatosian epistemological models. Correspondingly it is maintained that maxwellian electrodynamics was elaborated in the course of the old pre-maxwellian programmes’ reconciliation: the electrodynamics of Ampére-Weber, the wave theory of Young-Fresnel and Faraday’s programme. To compare the different theoretical schemes springing from the different language games James Maxwell had constructed a peculiar neutral language. Initially it had encompassed the incompressible fluid models; eventually – the vortices ones. The three programmes’ encounter engendered the construction of the hybrid theory at first with an irregular set of theoretical schemes. However, step by step, on revealing and gradual eliminating the contradictions between the programmes involved, the hybrid set is “put into order” (Maxwell’s term). A hierarchy of theoretical schemes starting from ingenious crossbreeds (the displacement current) and up to usual hybrids is set up. After the displacement current construction the interpenetration of the pre-maxwellian programmes begins that marks the commencement of theoretical schemes of optics, electricity and magnetism real unification. Maxwell’s programme surpassed that of Ampére-Weber because it did absorb the ideas of the Ampére-Weber programme, as well as the presuppositions of the programmes of Young-Fresnel and Faraday properly co-ordinating them with each other. But the opposite statement is not true. The Ampére-Weber programme did not assimilate the propositions of the Maxwellian programme. Maxwell’s victory over his rivals became possible because the gist of Maxwell’s unification strategy was formed by Kantian epistemology looked in the light of William Whewell and such representatives of Scottish Enlightenment as Thomas Reid and Sir William Hamilton. Maxwell did put forward as basic synthetic principles the ideas that radically differed from that of Ampére-Weber approach by their open, flexible and contra-ontological, genuinly epistemological, Kantian character. For Maxwell, ether was not the ultimate building block of physical reality, from which all the charges and fields should be constructed. “Action at a distance”, “incompressible fluid”, “molecular vortices”, etc. were contrived analogies for Maxwell, capable only to direct the researcher at the “right” mathematical relations. Key words: J.C. Maxwell, unification of optics and electromagnetism, I. Kant, T. Reid, W. Hamilton . (shrink)

This essay presents arguments for the claim that in the best of all possible worlds (Leibniz) there are sources of unpredictability and creativity for us humans, even given a pancomputational stance. A suggested answer to Chaitin’s questions: “Where do new mathematical and biological ideas come from? How do they emerge?” is that they come from the world and emerge from basic physical (computational) laws. For humans as a tiny subset of the universe, a part of the new ideas comes as (...) the result of the re-configuration and reshaping of already existing elements and another part comes from the outside as a consequence of openness and interactivity of the system. For the universe at large it is randomness that is the source of unpredictability on the fundamental level. In order to be able to completely predict the Universe-computer we would need the Universe-computer itself to compute its next state; as Chaitin already demonstrated there are incompressible truths which means truths that cannot be computed by any other computer but the universe itself. (shrink)

This discussion note responds to objections by Twardy, Gardner, and Dowe to my earlier claim that empirical data sets are algorithmically incompressible. Twardy, Gardner, and Dowe hold that many empirical data sets are compressible by Minimum Message Length technique and offer this as evidence that these data sets are algorithmically compressible. I reply that the compression achieved by Minimum Message Length technique is different from algorithmic compression. I conclude that Twardy, Gardner, and Dowe fail to establish that empirical data sets (...) are algorithmically compressible.Keywords: Algorithmic compression; Algorithmic randomness; Empirical data; Huffman compression; Minimum Message Length technique. (shrink)

The present article proposes an Informational-Theoretic interpretation of logical analysis applied to natural language in Tractatus Logico-Philosophicus. Natural language is characterized by descriptive definitions in order to compress information according to empirical regularities. However, notations fitted to empirical patterns do not explicitly reflect the logical structure of language that enables it to represent those very patterns. I argue that logical analysis is the process of obtaining incompressible and uniformly distributed codes, best fitted to express the possible combinations of facts instead (...) of eventual empirical regularities. That implies mutual translatability between different representations of the same facts, which justifies the possibility to decompose natural language and to rebuild it in mathematized ideographical languages, such as physics. Logical analysis decomposition process, then, results in a reductionism that sets the basis for conceptions of logic and language not only in Tractatus, but also in Some Remarks on Logical Form, which, in turn, is shown to be unattainable in Philosophical Remarks. That posterior book introduces a fragmentation of linguistic systems and analytical methods that can no longer be interpreted under computation theory alone. (shrink)

This essay presents arguments for the claim that in the best of all possible worlds (Leibniz) there are sources of unpredictability and creativity for us humans, even given a pancomputational stance. A suggested answer to Chaitin’s questions: “Where do new mathematical and biological ideas come from? How do they emerge?” is that they come from the world and emerge from basic physical (computational) laws. For humans as a tiny subset of the universe, a part of the new ideas comes as (...) the result of the re-configuration and reshaping of already existing elements and another part comes from the outside world as a consequence of openness and interactivity of biological and cognitive systems. For the universe at large it is randomness that is the source of unpredictability on the fundamental level. In order to be able to completely predict the Universe-computer we would need the Universe-computer itself to compute its next state. As Chaitin demonstrated there are incompressible truths, which means truths that cannot be computed by any other computer but the universe itself. (shrink)

Analysis of three kinds of flow of viscous, incompressible, electrically conducting fluids in high-aspect-ration rectangular channels subjected to transverse magnetic fields: turbulent flow in the presence of a d-c magnetic field, and both laminar and turbulent induction-driven flows.

Key words: rationality, communication, maxwellian revolution, Ampere-Weber research programme, synthesis, Kantian epistemology.. Why did Maxwell’s programme supersede the Ampere-Weber one? – To answer the question one has to consider the intertheoretic context of maxwellian electrodynamics genesis and development. It is demonstrated that maxwellian electrodynamics was created as a result of the old pre-maxwellian programmes reconciliation: the electrodynamics of Ampere-Weber, the wave theory of Young-Fresnel and Faraday’s programme. The programmes’ meeting led to construction of the hybrid theory at first with an (...) irregular set of theoretical schemes. However, step by step, on revealing and gradual eliminating the contradictions between the programmes involved, the hybrid set is “put into order”.A hierarchy of theoretical schemes starting from the crossbreeds and up to usual hybrids is set up. And after the displacement current construction the interpenetration of the pre-maxwellian programmes begins that markes the commencement of theoretical schemes of optics and electromagnetism real unification. Maxwell’s programme did supersede the Ampere-Weber one because it did assimilate the ideas of the Ampere-Weber programme, as well as the presuppositions of the programmes of Young-Fresnel and Faraday properly co-ordinating them with each other. But the opposite proposition is not true. Ampere-Weber programme did not assimilate the propositions of the Maxwellian programme. Maxwell’s victory became possible because the core of Maxwell’s unification strategy was formed by Kantian epistemology looked through the prism of William Whewell and such representatives of Scottish Enlightenment as Thomas Reid and William Hamilton. Maxwell did put forward as a basic synthetic principle the idea that radically differed from that of Ampere-Weber approach by its open, flexible and contra-ontological, strictly epistemological, Kantian character. For Maxwell, ether was not the last building block of physical reality, from which all the charges and fields should be constructed. “Action at a distance”, “incompressible fluid”, “molecular vortices” were contrived analogies for Maxwell, capable only to direct the researcher at the “right” mathematical relations. Namely the application of Kantian epistemology enabled Hermann von Helmholtz and his pupil Heinrich Hertz to arrive at such a version of Maxwell’s theory that served a heuristical basis for the radio waves discovery. (shrink)

Maxwellian electrodynamics genesis is considered in the light of the author’s theory change model previously tried on the Copernican and the Einstein revolutions. It is shown that in the case considered a genuine new theory is constructed as a result of the old pre-maxwellian programmes reconciliation: the electrodynamics of Ampere-Weber, the wave theory of Fresnel and Young and Faraday’s programme. The “neutral language” constructed for the comparison of the consequences of the theories from these programmes consisted in the language of (...) hydrodynamics with its rich content of analogous models ranging from the uncompressible fluid up to molecular vortices. The programmes’ meeting led to construction of the whole hierarchy of crossbred objects beginning from the displacement current and up to common hybrids. After that the interpenetration of the pre-maxwellian programmes began that marked the beginning of theoretical schemes of optics and electromagnetism unification. Maxwell’s programme did assimilate some ideas of the Ampere-Weber programme, as well as the presuppositions of the programmes of Fresnel and Faraday; and the significance of this fact for further methodology of scientific research programmes development is discussed. It is argued that the core of Maxwell’s unification strategy was formed by Kantian epistemology looked through the prism of William Whewell and such representatives of Scottish Enlightenment as Thomas Reid and William Hamilton. All these enabled Maxwell to start to unify not only optics and electromagnetism, but British and continental research traditions as well. Maxwell’s programme did supersede the Ampere-Weber one because Maxwell did put forward as a synthetic principle the idea, that differed from that of Ampere-Weber by its flexible and contra-ontological, strictly epistemological, Kantian character. For Maxwell, ether was not the last building block of physical reality, from which fields and charges should be constructed . “Action at a distance”, “incompressible fluid”, “molecular vortices” were only analogies for Maxwell, capable to direct the researcher on the “right” mathematical relations. From the “representational” point of view all this hydrodynamical models were doomed to failure efforts to describe what can not be described in principle – things in themselves, the “nature” of electrical and magnetic phenomena. On the contrary, Maxwell aimed his programme to find empirically meaningful mathematical relations between the electrodynamics basic objects, i.e. the creation of inter - coordinated electromagnetic field equations system. Namely the application of this epistemology enabled Hermann von Helmholtz and his pupil Heinrich Hertz to arrive at such a version of Maxwell’s theory that served a heuristical basis for the radiowaves discovery. (shrink)

A theoretical model is presented for pattern formation in an epithelium. The epithelial model consists of a thin, incompressible, viscoelastic membrane on an elastic foundation (substrate), with the component cells assumed to have active contractile properties similar to those of smooth muscle. The analysis includes the effects of large strains and material nonlinearity, and the governing equations were solved using finite differences. Deformation patterns form when the cells activate while lying on the descending limb of their total (active + passive) (...) stress-stretch curve. Various one-dimensional and two-dimensional simulations illustrate the effects of spatial and temporal variations in passive stiffness, as well as the effects of foundation stiffness and stretch activation. The model can be used to examine the mechanical aspects of pattern formation in morphogenetic processes such as angiogenesis and myocardial trabeculation. (shrink)

A phenomenon resulting from a computationally irreducible (or computationally incompressible) process is supposedly unpredictable except via simulation. This notion of unpredictability has been deployed to formulate recent accounts of computational emergence. Via a technical analysis, I show that computational irreducibility can establish the impossibility of prediction only with respect to maximum standards of precision. By articulating the graded nature of prediction, I show that unpredictability to maximum standards is not equivalent to being unpredictable in general. I conclude that computational irreducibility (...) fails to fulfill its assigned philosophical roles in theories of computational emergence. (shrink)

Qu’est-ce donc que la musique des aliens ou pour les aliens? Des cinq notes de la mélodie de Rencontres du troisième type aux explorations sonores de Sun Ra ou des intelligents dansants de l’électro, elle n’est peut-être que l’expression la plus accomplie d’une tendance très humaine à projeter sur l’extraterrestre certains reflets plus ou moins déformés de soi, surjouant quelque insaisissable mystère, révélation ou écart incompressible venu de l’ailleurs pour retrouver finalement, tout là-bas, une bribe d’ici.

I challenge the usual approach of defining emergence in terms of properties of wholes “emerging” upon properties of parts. This approach indeed fails to meet the requirement of nontriviality, since it renders a bunch of ordinary properties emergent; however, by defining emergence as the incompressibility of a simulation process, we have an objective meaning of emergence because the difference between the processes satisfying the incompressibility criterion and the other processes does not depend on our cognitive abilities. Finally, this (...) definition fulfills the nontriviality and the scientific‐adequacy requirements better than the combinatorial approach, emergence here being a predicate of processes rather than of properties. †To contact the author, please write to: Institut d'Histoire et de Philosophie des Sciences et des Techniques (CNRS/Université Paris I Sorbonne), 13 rue du Four 75006, Paris; e‐mail: [email protected]. (shrink)

Among many properties distinguishing emergence, such as novelty, irreducibility and unpredictability, computational accounts of emergence in terms of computational incompressibility aim first at making sense of such unpredictability. Those accounts prove to be more objective than usual accounts in terms of levels of mereology, which often face objections of being too epistemic. The present paper defends computational accounts against some objections, and develops what such notions bring to the usual idea of unpredictability. I distinguish the objective unpredictability, compatible with (...) determinism and entailed by emergence, and various possibilities of predictability at emergent levels. This makes sense of practices common in complex systems studies that forge qualitative predictions on the basis of comparisons of simulations with multiple values of parameters. I consider robustness analysis as a way to ensure the ontological character of computational emergence. Finally, I focus on the property of novelty, as it is displayed by biological evolution, and ask whether computer simulations of evolution can produce the same kind of emergence as the open-ended evolution attested in Phanerozoic records. (shrink)

According to a traditional view, scientific laws and theories constitute algorithmic compressions of empirical data sets collected from observations and measurements. This article defends the thesis that, to the contrary, empirical data sets are algorithmically incompressible. The reason is that individual data points are determined partly by perturbations, or causal factors that cannot be reduced to any pattern. If empirical data sets are incompressible, then they exhibit maximal algorithmic complexity, maximal entropy and zero redundancy. They are therefore maximally efficient carriers (...) of information about the world. Since, on algorithmic information theory, a string is algorithmically random just if it is incompressible, the thesis entails that empirical data sets consist of algorithmically random strings of digits. Rather than constituting compressions of empirical data, scientific laws and theories pick out patterns that data sets exhibit with a certain noise. (shrink)

Synthesizing important research traditions in information theory, structuralist semiotics, and generative linguistics, at least three main types of semiotic indeterminacy must be distinguished: Kolmogorov’s notion of randomness defined as sequential incompressibility, de Saussure’s principle of contingency of sign which ensures the possibility of translation between different sign systems, and Chomsky’s idea of indefiniteness in generative mechanisms as a requirement for the explanation of semiotic creativity. These types of semiotic indeterminacy form an abstract system useful for the description of concrete (...) sign processes in their syntactic, semantic, and pragmatic dimension. In his philosophical reflections on modern biology, Jacques Monod used the conceptual opposition chance versus necessity to analyse several phenomena of indeterminacy (especially in molecular biology). The biosemiotic approach to life permits to apply the suggestedsystem of semiotic indeterminacy on these phenomena. (shrink)

We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable randoms is equivalent (...) to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal. (shrink)

We critically analyse the point of view for which laws of nature are just a mean to compress data. Discussing some basic notions of dynamical systems and information theory, we show that the idea that the analysis of large amount of data by means of an algorithm of compression is equivalent to the knowledge one can have from scientific laws, is rather naive. In particular we discuss the subtle conceptual topic of the initial conditions of phenomena which are generally incompressible. (...) Starting from this point, we argue that laws of nature represent more than a pure compression of data, and that the availability of large amount of data, in general, is not particularly useful to understand the behaviour of complex phenomena. (shrink)

We study reals with infinitely many incompressible prefixes. Call $A \in 2^{\omega}$ Kolmogorot random if $(\exists^{\infty}n) C(A \upharpoonright n) \textgreater n - \mathcal{O}(1)$ , where C denotes plain Kolmogorov complexity. This property was suggested by Loveland and studied by $Martin-L\ddot{0}f$ , Schnorr and Solovay. We prove that 2-random reals are Kolmogorov random. Together with the converse-proved by Nies. Stephan and Terwijn [11]-this provides a natural characterization of 2-randomness in terms of plain complexity. We finish with a related characterization of 2-randomness.

Euler’s interpretation of Newton’s gravity (NG) as Archimedes’ thrust in a fluid ether is presented in some detail. Then a semi-heuristic mechanism for gravity, close to Euler’s, is recalled and compared with the latter. None of these two ‘‘gravitational ethers’’ can obey classical mechanics. This is logical since the ether defines the very reference frame, in which mechanics is defined. This concept is used to build a scalar theory of gravity: NG corresponds to an incompressible ether, a compressible ether leads (...) to gravitational waves. In the Lorentz–Poincaré version, special relativity is compatible with the ether, but, with the heterogeneous ether of gravity, it applies only locally. A correspondence between metrical effects of uniform motion and gravitation is assumed, yet in two possible versions (one is new). Dynamics is based on a (non-trivial) extension of Newton’s second law. The observational status for the theory with the older version of the correspondence is summarized. (shrink)

A hierarchy of functions with respect to their role as bounds in the Turing reducibility of functions is introduced and studied. This hierarchy leads to a certain notion of incompressibility of sets which is also investigated.

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Löf random relative to ∅. We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C ≥ |x|-c. The ‘only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity. Next we prove some results (...) on lowness. Among other things, we characterize the 2-random sets as those 1-random sets that are low for Chaitin's Ω. Also, 2-random sets form minimal pairs with 2-generic sets. The r.e. low for Ω sets coincide with the r.e. K-trivial ones. Finally we show that the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness can be separated in every high degree while the same notions coincide in every non-high degree. We make some remarks about hyperimmune-free and PA-complete degrees. (shrink)

In this work we study the relativistic mechanics of continuous media on a fundamental level using a manifestly covariant proper time procedure. We formulate equations of motion and continuity (and constitutive equations) that are the starting point for any calculations regarding continuous media. In the force free limit, the standard relativistic equations are regained, so that these equations can be regarded as a generalization of the standard procedure. In the case of an inviscid fluid we derive an analogue of the (...) Bernoulli equation. For irrotational flow we prove that the velocity field can be derived from a potential. If in addition, the fluid is incompressible, the potential must obey the d'Alembert equation, and thus the problem is reduced to solving the d'Alembert equation with specific boundary conditions (in both space and time). The solutions indicate the existence of light velocity sound waves in an incompressible fluid (a result known in previous literature (19) ). Relaxing the constraints and allowing the fluid to become linearly compressible one can derive a wave equation, from which the sound velocity can again be computed. For a stationary background flow, it has been demonstrated that the sound velocity attains its correct values for the incompressible and nonrelativistic limits. Finally viscosity is introduced, bulk and shear viscosity constants are defined, and we formulate equations for the motion of a viscous fluid. (shrink)