The Kolmogorov-Sinai entropy is a fairly exotic mathematical concept which has recently aroused some interest on the philosophers’ part. The most salient trait of this concept is its working as a junction between such diverse ambits as statistical mechanics, information theory and algorithm theory. In this paper I argue that, in order to understand this very special feature of the Kolmogorov-Sinai entropy, is essential to reconstruct its genealogy. Somewhat surprisingly, this story takes us as far back as the (...) beginning of celestial mechanics and through some of the most exciting developments of mathematical physics of the 19th century. (shrink)

In the present paper we discuss: a) how Hilbert’s unification program failed completely, and b) we outline a new electron model based on Helmholtz’s electron vortex and Kolmogorov theory of turbulence. Novelty aspect: we discuss among other things, electron capture event, and von Karman vortex street. We also discuss a new model of origination of charge and matter. This paper is a sequel to a preceding paper on similar theme.

While scientific inquiry crucially relies on the extraction of patterns from data, we still have a far from perfect understanding of the metaphysics of patterns—and, in particular, of what makes a pattern real. In this paper we derive a criterion of real-patternhood from the notion of conditional Kolmogorov complexity. The resulting account belongs to the philosophical tradition, initiated by Dennett :27–51, 1991), that links real-patternhood to data compressibility, but is simpler and formally more perspicuous than other proposals previously defended (...) in the literature. It also successfully enforces a non-redundancy principle, suggested by Ladyman and Ross, that aims to exclude from real-patternhood those patterns that can be ignored without loss of information about the target dataset, and which their own account fails to enforce. (shrink)

Early work on the frequency theory of probability made extensive use of the notion of randomness, conceived of as a property possessed by disorderly collections of outcomes. Growing out of this work, a rich mathematical literature on algorithmic randomness and Kolmogorov complexity developed through the twentieth century, but largely lost contact with the philosophical literature on physical probability. The present chapter begins with a clarification of the notions of randomness and probability, conceiving of the former as a property of (...) a sequence of outcomes, and the latter as a property of the process generating those outcomes. A discussion follows of the nature and limits of the relationship between the two notions, with largely negative verdicts on the prospects for any reduction of one to the other, although the existence of an apparently random sequence of outcomes is good evidence for the involvement of a genuinely chancy process. (shrink)

The framework of Solomonoff prediction assigns prior probability to hypotheses inversely proportional to their Kolmogorov complexity. There are two well-known problems. First, the Solomonoff prior is relative to a choice of Universal Turing machine. Second, the Solomonoff prior is not computable. However, there are responses to both problems. Different Solomonoff priors converge with more and more data. Further, there are computable approximations to the Solomonoff prior. I argue that there is a tension between these two responses. This is because (...) computable approximations to Solomonoff prediction do not always converge. (shrink)

Can an agent's intelligence level be negative? We extend the Legg-Hutter agent-environment framework to include punishments and argue for an affirmative answer to that question. We show that if the background encodings and Universal Turing Machine (UTM) admit certain Kolmogorov complexity symmetries, then the resulting Legg-Hutter intelligence measure is symmetric about the origin. In particular, this implies reward-ignoring agents have Legg-Hutter intelligence 0 according to such UTMs.

The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure (...) of the strength of the theory. I exhibit certain strong counterexamples and establish conclusively that the received view is false. Moreover, I show that the limiting constants provided by the theorem do not in any way reflect the power of formalized theories, but that the values of these constants are actually determined by the chosen coding of Turing machines, and are thus quite accidental. (shrink)

Applying the concepts of Kolmogorov-Chaitin complexity and Turing’s uncomputability from the computability and algorithmic information theories to the irreducible and incomputable randomness of quantum mechanics, a novel argument for the existence of God is presented. Concepts of ‘transintelligence’ and ‘transcausality’ are introduced, and from them, it is posited that our universe must be epistemologically and ontologically an open universe. The proposed idea also proffers a new perspective on the nonlocal nature and the infamous wave-function-collapse problem of quantum mechanics.

There are (at least) three approaches to quantifying information. The first, algorithmic information or Kolmogorov complexity, takes events as strings and, given a universal Turing machine, quantifies the information content of a string as the length of the shortest program producing it [1]. The second, Shannon information, takes events as belonging to ensembles and quantifies the information resulting from observing the given event in terms of the number of alternate events that have been ruled out [2]. The third, statistical (...) learning theory, has introduced measures of capacity that control (in part) the expected risk of classifiers [3]. These capacities quantify the expectations regarding future data that learning algorithms embed into classifiers. Solomonoff and Hutter have applied algorithmic information to prove remarkable results on universal induction. Shannon information provides the mathematical foundation for communication and coding theory. However, both approaches have shortcomings. Algorithmic information is not computable, severely limiting its practical usefulness. Shannon information refers to ensembles rather than actual events: it makes no sense to compute the Shannon information of a single string – or rather, there are many answers to this question depending on how a related ensemble is constructed. Although there are asymptotic results linking algorithmic and Shannon information, it is unsatisfying that there is such a large gap – a difference in kind – between the two measures. This note describes a new method of quantifying information, effective information, that links algorithmic information to Shannon information, and also links both to capacities arising in statistical learning theory [4, 5]. After introducing the measure, we show that it provides a non-universal analog of Kolmogorov complexity. We then apply it to derive basic capacities in statistical learning theory: empirical VC-entropy and empirical Rademacher complexity. A nice byproduct of our approach is an interpretation of the explanatory power of a learning algorithm in terms of the number of hypotheses it falsifies [6], counted in two different ways for the two capacities. We also discuss how effective information relates to information gain, Shannon and mutual information. (shrink)

We generalize the Kolmogorov axioms for probability calculus to obtain conditions defining, for any given logic, a class of probability functions relative to that logic, coinciding with the standard probability functions in the special case of classical logic but allowing consideration of other classes of "essentially Kolmogorovian" probability functions relative to other logics. We take a broad view of the Bayesian approach as dictating inter alia that from the perspective of a given logic, rational degrees of belief are those (...) representable by probability functions from the class appropriate to that logic. Classical Bayesianism, which fixes the logic as classical logic, is only one version of this general approach. Another, which we call Intuitionistic Bayesianism, selects intuitionistic logic as the preferred logic and the associated class of probability functions as the right class of candidate representions of epistemic states (rational allocations of degrees of belief). Various objections to classical Bayesianism are, we argue, best met by passing to intuitionistic Bayesianism—in which the probability functions are taken relative to intuitionistic logic—rather than by adopting a radically non-Kolmogorovian, for example, nonadditive, conception of (or substitute for) probability functions, in spite of the popularity of the latter response among those who have raised these objections. The interest of intuitionistic Bayesianism is further enhanced by the availability of a Dutch Book argument justifying the selection of intuitionistic probability functions as guides to rational betting behavior when due consideration is paid to the fact that bets are settled only when/if the outcome bet on becomes known. (shrink)

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to (...) equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems. (shrink)

I argue for patternism, a new answer to the question of when some objects compose a whole. None of the standard principles of composition comfortably capture our natural judgments, such as that my cat exists and my table exists, but there is nothing wholly composed of them. Patternism holds, very roughly, that some things compose a whole whenever together they form a “real pattern”. Plausibly we are inclined to acknowledge the existence of my cat and my table but not of (...) their fusion, because the first two have a kind of internal organizational coherence that their putative fusion lacks. Kolmogorov complexity theory supplies the needed rigorous sense of “internal organizational coherence”. (shrink)

Seven problems that occur in attempts to measure complexity are pointed out as they occur in four proposed measurement techniques. Each example method is an improvement over the previous examples. It turns out, however, that none are up to the challenge of complexity. Apparently, there is no currently available method that truly gets the measure of complexity. There are two reasons. First, the most natural approach, quantitative analysis, is rendered inadequate by the very nature of complexity. Second, the intrinsic magnitude (...) of complexity is still holding at bay attempts to use both quantitative and qualitative methods combined. Further progress in complexity science and in systems science is required. Any method that simplifies will fail because it ignores what complexity is. Techniques of understanding that do not simplify, but rather provide ways for the mind to grasp and work with complexity are more effective in getting its measure. (shrink)

A principle endorsed by many theories of objective chance, and practically forced on us by the standard interpretation of the Kolmogorov semantics for chance, is the principle that when a proposition P has a chance, any proposition Q that is necessarily equivalent to P will have the same chance as P. Call this principle SUB (for the substitution of necessary equivalents into chance ascriptions). I will present some problems for a theory of chance, and will argue that the best (...) way to resolve these problems is to reject SUB, and similar principles e.g. for the chances of outcomes or the chances of events. Objective chance, it turns out, carves things more finely than necessary equivalence does. (shrink)

Propensities are presented as a generalization of classical determinism. They describe a physical reality intermediary between Laplacian determinism and pure randomness, such as in quantum mechanics. They are characterized by the fact that their values are determined by the collection of all actual properties. It is argued that they do not satisfy Kolmogorov axioms; other axioms are proposed.

In a recent paper by Tranchini (Topoi, 2019), an introduction rule for the paradoxical proposition ρ∗ that can be simultaneously proven and disproven is discussed. This rule is formalized in Martin-Löf’s constructive type theory (CTT) and supplemented with an inferential explanation in the style of Brouwer-Heyting-Kolmogorov semantics. I will, however, argue that the provided formalization is problematic because what is paradoxical about ρ∗ from the viewpoint of CTT is not its provability, but whether it is a proposition at all.

Karl Popper discovered in 1938 that the unconditional probability of a conditional of the form ‘If A, then B’ normally exceeds the conditional probability of B given A, provided that ‘If A, then B’ is taken to mean the same as ‘Not (A and not B)’. So it was clear (but presumably only to him at that time) that the conditional probability of B given A cannot be reduced to the unconditional probability of the material conditional ‘If A, then B’. (...) I describe how this insight was developed in Popper’s writings and I add to this historical study a logical one, in which I compare laws of excess in Kolmogorov probability theory with laws of excess in Popper probability theory. (shrink)

Ontic Structural Realism, as pivotal position in philosophy of science and metaphysics, defends the idea that the world is ultimately constituted of real physical structures. French (2014) regards physical symmetries as the foundational structure of a world without objects. On the other hand, Ladyman and Ross (2007) hold that the world is essentially made of non-redundant informational structure. I argue in this paper that these two positions are by no means incompatible, for instance by interpreting French’s physical symmetries as real (...) structures both encompassing and compressing every piece of information (Kolmogorov complexity) within the world. (shrink)

"Complexity" is a catchword of certain extremely popular and rapidly developing interdisciplinary new sciences, often called accordingly the sciences of complexity. It is often closely associated with another notably popular but ambiguous word, "information"; information, in turn, may be justly called the central new concept in the whole 20th century science. Moreover, the notion of information is regularly coupled with a key concept of thermodynamics, viz. entropy. And like this was not enough it is quite usual to add one more (...) at present extraordinarily popular notion, namely chaos, and wed it with the abovementioned concepts. It is my aim in this paper to critically analyse this conceptual mess from a logical and philosophical point of view concentrating on the concepts of complexity and information and the question concerning the true relation between them. (shrink)

Review of "Exploring Randomness" (200) and "The Unknowable" (1999) by Gregory Chaitin.

Double negations are easily recognised in both the so-called “negative literature” and the original texts of some important scientific theories. Often they are not equivalent to the corresponding affirmative propositions. In the case the law of double negation fails they belong to non-classical logic, as first, intuitionist logic. Through a comparative analysis of the theories including them the main features of a new kind of theoretical organization governed by intuitionist logic are obtained. Its arguing proceeds through doubly negated propositions and (...) ad absurdum arguments. Then, the final doubly negated predicate is translated in the corresponding affirmative one, from which all the consequences are deduced in order to test them against reality. According to this model the true formalization of intuitionist logic is Kolmogorov’s 1932 paper. The above analysis also shows that modal words play an auxiliary or substitutive role to intuitionist propositions. In addition, it is shown that Vasiliev’s paraconsistent logic is represented through doubly negated propositions. By means of these three kinds of logic one can attribute a specific kind of logic to each of the three distinct steps of the process of building a scientific theory from the experimental data to its final theoretical system. It is also shown that the definition of negation is of a structural kind, rather than of an objective kind or a subjective kind: it has to be referred to the organization of the theory to which it belongs. (shrink)

Einstein wrote his famous sentence "God does not play dice with the universe" in a letter to Max Born in 1920. All experiments have confirmed that quantum mechanics is neither wrong nor “incomplete”. One can says that God does play dice with the universe. Let quantum mechanics be granted as the rules generalizing all results of playing some imaginary God’s dice. If that is the case, one can ask how God’s dice should look like. God’s dice turns out to be (...) a qubit and thus having the shape of a unit ball. Any item in the universe as well the universe itself is both infinitely many rolls and a single roll of that dice for it has infinitely many “sides”. Thus both the smooth motion of classical physics and the discrete motion introduced in addition by quantum mechanics can be described uniformly correspondingly as an infinite series converges to some limit and as a quantum jump directly into that limit. The second, imaginary dimension of God’s dice corresponds to energy, i.e. to the velocity of information change between two probabilities in both series and jump. (shrink)

[issue 20211210] Qbism (quantum bayesism) is a philosophical interpretation of quantum mechanics (QM) that places the agent and its expectations at the heart of theory. The QBists advocate a "subjectivist" interpretation of probabilities that allows to understand the quantum laws of Born and to eliminate certain enigmas of interpretation of the QM going "beyond" the interpretation of Copenhagen. The Ontology of Knowledge (OK) is in agreement with the main ideas of the Qbism. For the OdC indeed: -The agent is the (...) focal point of the representation -Representation is specific to the agent -There are no "states of the world" but only "states of knowledge" -There is no probability of evolution of the "state of the world" but only probabilities, for the subject, of evolution of his Knowledge. -These probabilities lead to the future actions and experiences of the agent. The aim of this article is to propose, according to the OdC, ways for an extension of the explanatory power of the QBism. (shrink)

Given functionally equivalent minds, how does the expected quantity of their conscious experience differ across different substrates and how could we calculate this? We argue that a realistic digital brain emulation would be orders of magnitude less conscious than a real biological brain. On the other hand, a mind running on neuromorphic hardware or a quantum computer could in principle be more conscious than than a biological brain.