Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. After its initial development in the 1970s the notion received considerably less attention than Martin-Löf randomness, but recently interest has increased in a range of randomness concepts. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new (...) characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of "Schnorr reducibility" which allows us to calibrate the Schnorr complexity of reals. We define the class of "Schnorr trivial" reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members. (shrink)

We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

First Published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

In this paper I argue that split-brain syndrome is best understood within an extended mind framework and, therefore, that its very existence provides support for an externalist account of conscious perception. I begin by outlining the experimental aberration model of split-brain syndrome and explain both: why this model provides the best account of split-brain syndrome; and, why it is commonly rejected. Then, I summarise Susan Hurley’s argument that split-brain subjects could unify their conscious perceptual field by using external factors to (...) stand-in for the missing corpus callosum. I next provide an argument that split-brain subjects do unify their perceptual fields via external factors. Finally, I explain why my account provides one with an experimental aberration model which avoids the problems typically levelled at such views, and highlight some empirical predictions made by the account. The nature of split-brain syndrome has long been considered mysterious by proponents of internalist accounts of consciousness. However, in this paper I argue that externalist theories can provide a straightforward explanation of the condition. I therefore conclude that the ability of externalist accounts to explain split-brain syndrome gives us strong reason to prefer them over internalist rivals. (shrink)

The recent first-time publication of works from Edmund Husserl’s later years, especially his Freiburg period, combined with new studies of his method and theories, has stimulated a remarkable shift in perceptions of the scope and significance of Husserl’s transcendental phenomenology. Informed by a deep reading of not just the works published during Husserl’s lifetime but also the countless lectures and manuscripts he wrote in his later years, the essays in The New Husserl provide an alternative approach to Husserl by examining (...) his work and his method as a whole and by probing issues, old and new, that occupied him during this exceptionally productive period. The noted Husserl specialist Klaus Held opens the book with two essays, published here in English for the first time, that provide an insightful and lucid introduction to Husserl’s central texts. Other prominent Husserl scholars treat his most important and lasting contributions to philosophy, such as the concept of intentionality, the theory of types, time-consciousness, consciousness and subjectivity, the phenomenological method, and the problem of generativity. By inviting readers to discover this "new Husserl," the present collection is likely to shape scholarly discussions of Husserl’s thought for some time to come. (shrink)

Can we deploy creative practices to critically address the fatal interlocking of global surveillance technologies, neo-colonial expansionism, environmental degradation and the lethal threat of drone warfare? Throughout the following conversation, Shona Illingworth and Anthony Downey examine these and other questions in relation to the recent publication of Topologies of Air (Sternberg Press and The Power Plant, 2022). Edited by Downey, the book includes discussion and documentation of two major bodies of work by Illingworth, including Topologies of Air (2021) (...) and Lesions in the Landscape (2015), alongside an extended series of essays that analyse the psychological and environmental impact of military, industrial and corporate transformations of airspace and outer space. Employing interdisciplinary research and collaborative processes, Illingworth’s practice, as detailed in the discussion below, uses creative methodologies to visualize and interrogate this proliferating exploitation of airspace. The conversation between Illingworth and Downey also outlines the work of the Airspace Tribunal, an ongoing series of public hearings that brings together diverse disciplines, methodologies, knowledge and lived experiences to propose a new human right that will counter the colonization of the sky and, in time, protect individuals, communities and ecologies from ever-increasing threats from above. (shrink)

Let K denote prefix-free Kolmogorov complexity, and let $K^A$ denote it relative to an oracle A. We show that for any n, $K^{\emptyset ^{(n)}}$ is definable purely in terms of the unrelativized notion K. It was already known that 2-randomness is definable in terms of K (and plain complexity C) as those reals which infinitely often have maximal complexity. We can use our characterization to show that n-randomness is definable purely in terms of K. To do this we extend a (...) certain “limsup” formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $\Delta _2^0$ sets of minimal descriptions. (shrink)

Journal of Political Philosophy, EarlyView.

Concentrating on the major works of Dofine, Marvell, Vaughan and Herbert, Richard Willmott has provided an anthology of metaphysical verse for readers coming to the poetry for the first time. Metaphysical poetry is notorious for its 'difficulty'; in this selection Richard Willmott provides detailed explanatory notes giving in depth information on the period, the poets and 'metaphysical style' and, to ensure a full understanding, line by line exegesis of the poems themselves is given where necessary. The anthology contains about 20-25 (...) poems for each poet represented and a further section of other contemporary poetry which elucidates themes or approaches in the work of the four main poets in the collection and encourages readers to relate an individual poem to a broader context. (shrink)

Loïc Wacquant argues for a radicalization of the habitus concept provided by Pierre Bourdieu, suggesting that habitus is a site and mode for conducting research, not simply an explanatory or theoretical mechanism. Taking seriously this call to examine skills and communities of practice through apprenticeship, however, requires that the theoretical account of habitus be subject to empirical testing. Moreover, enquiry into communities of practice, especially the subtle psychological, behavioural and even neurological consequences of skill acquisition, means that claims about the (...) habitus can be scrutinized by fields such as psychology, neurology and even human biology. Habitus as a truly open path of inquiry will demand, not wholly novel concepts, but a recognition of when claims about practice are simultaneously testable, and falsifiable, by other forms of empirical enquiry, including the human sciences. (shrink)

This book presents new results in computability theory, a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field's connections with disparate areas of mathematical logic and mathematics more generally have grown deeper, and now have a variety of applications in topology, group theory, and other subfields. This monograph establishes new directions in the field, blending classic results with modern research areas such as algorithmic randomness. The significance of the book lies not only (...) in the depth of the results contained therein, but also in the fact that the notions the authors introduce allow them to unify results from several subfields of computability theory. (shrink)

Informal construction -- Formal construction -- Limiting results.

Patrick Downey finds comedy at the heart of the Western philosophical and theological tradition. In Serious Comedy Downey tracks tragedy and comedy from the beginning of Western thought to the twentieth century, beginning with an in-depth examination of Aristotle and three Platonic dialogues: the Republic, the Phaedrus, and the Symposium. In the book's second section Downey argues that the Bible is at heart a comedic narrative and analyzes the philosophical and theological implications of this comedy. In the (...) third section Downey traces comedy and tragedy in classic and modern philosophers such as Machiavelli, Dante, Hegel, Nietzsche, and Kierkegaard. Readers with an interest in classical philosophy and theology, political philosophy, and narrative theory will be particularly interested in this insightful and thoughtful perspective on the Western tradition. (shrink)

Robert I. Soare, Automorphisms of the Lattice of Recursively Enumerable Sets. Part I: Maximal Sets.Manuel Lerman, Robert I. Soare, $d$-Simple Sets, Small Sets, and Degree Classes.Peter Cholak, Automorphisms of the Lattice of Recursively Enumerable Sets.Leo Harrington, Robert I. Soare, The $\Delta^0_3$-Automorphism Method and Noninvariant Classes of Degrees.

The ‘scaling up’ objection says non-representational ecological-enactive accounts will be unable to explain ‘representation hungry’ cognition. Obsessive-compulsive disorder presents a paradigmatic instance of this objection, marked as it is by ‘representation hungry’ obsessive thoughts and compulsive behavior organized around them. In this paper I provide an ecological-enactive account of OCD, thereby demonstrating non-representational frameworks can ‘scale up’ to explain ‘representation hungry’ cognition. First, I outline a non-representational account of mind— a predictive processing operationalization of Sean Kelly’s theory of perception. This (...) account explains the ‘tensions’ and ‘pulls’ which guide and constrain our action-oriented and affect-laden perceptual ‘grip’ upon the world to be underwritten by an imperative to minimize prediction error. I then argue that OCD is best understood as ‘grip gone awry’— malformed predictive models signal inappropriately high error, and this results in extremely strong ‘tensions’ and ‘pulls’ which prescribe actions. Thus, I arrive at the idea that OCD is primarily constituted by ‘not just right’ feelings caused by high error signaling. Finally, I explain that this account provides a causal explanation of the non-representational existential feeling fundamental to OCD. Compulsions are considered manifestations of non-representational grip responsive to this feeling, whilst obsessions are explained to play a meta-role in the condition, being formulated by subjects in a bid to make their ‘not just right’ feeling intelligible. I explain that the meta-role obsessions play makes their representational status largely irrelevant to OCD, and so leave this an open question. Consequently, I provide a non-representational account of OCD, thereby demonstrating that ecological-enactive approaches can respond to the ‘scaling up’ objection. (shrink)

We characterize the class of c.e. degrees that bound a critical triple as those degrees that compute a function that has no ω-c.e. approximation.

We examine the multiplicity of complementation amongst subspaces of V ∞ . A subspace V is a complement of a subspace W if V ∩ W = {0} and (V ∪ W) * = V ∞ . A subspace is called fully co-r.e. if it is generated by a co-r.e. subset of a recursive basis of V ∞ . We observe that every r.e. subspace has a fully co-r.e. complement. Theorem. If S is any fully co-r.e. subspace then S has (...) a decidable complement. We give an analysis of other types of complements S may have. For example, if S is fully co-r.e. and nonrecursive, then S has a (nonrecursive) r.e. nowhere simple complement. We impose the condition of immunity upon our subspaces. Theorem. Suppose V is fully co-r.e. Then V is immune iff there exist M 1 , M 2 ∈ L(V ∞ ), with M 1 supermaximal and M 2 k-thin, such that $M_1 \oplus V = M_2 \oplus V = V_\infty$ . Corollary. Suppose V is any r.e. subspace with a fully co-r.e. immune complement W (e.g., V is maximal or V is h-immune). Then there exist an r.e. supermaximal subspace M and a decidable subspace D such that $V \oplus W = M \oplus W = D \oplus W = V_\infty$ . We indicate how one may obtain many further results of this type. Finally we examine a generalization of the concepts of immunity and soundness. A subspace V of V ∞ is nowhere sound if (i) for all Q ∈ L(V ∞ ) if $Q \supset V$ then Q = V ∞ , (ii) V is immune and (iii) every complement of V is immune. We analyse the existence (and ramifications of the existence) of nowhere sound spaces. (shrink)

We describe new results in parametrized complexity theory. In particular, we prove a number of concrete hardness results for W[P], the top level of the hardness hierarchy introduced by Downey and Fellows in a series of earlier papers. We also study the parametrized complexity of analogues of PSPACE via certain natural problems concerning k-move games. Finally, we examine several aspects of the structural complexity of W [P] and related classes. For instance, we show that W[P] can be characterized in (...) terms of the DTIME ) and NP. (shrink)

Slavoj Zizek is operating from a position of certainty, a position discovered by Jacques Lacan in Seminar XI. In this essay, I examine this position of certainty ("Gewissheit") and the ways this position is distinct from both existential phenomenology and post-structuralism, ultimately arguing that for structuralist psychoanalysis to function requires an intentional forgetting of being.

"With provocations on every page, this book is a philosophical feast. The specialist will find familiar ingredients assembled here in a perspicuous and compelling way, while the nonspecialist will discover a Husserl whose philosophy is made of flesh and blood." —Journal of the History of Philosophy In this thorough study of the full body of his writings, Donn Welton uncovers a Husserl very different from the established view. Arguing against established interpretations, The Other Husserl traces Husserl’s move from static (...) to genetic phenomenology and uses accounts of perception, discourse, subjectivity, and world to elaborate the scope of his systematic phenomenology. This serious reflection on the meaning of phenomenology is the first book in English to outline in full Husserl’s phenomenological method and to argue for its cogency. Welton’s stimulating interpretation highlights Husserl’s relevance for current philosophical debates. (shrink)

The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$ -c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $ -c.a. as shown in [9]. We also show that if cone avoidance is added then (...) there is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$. (shrink)

We investigate the initial segment complexity of random reals. Let K denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K and K. It is well-known that a real α is 1-random iff there is a constant c such that for all n, Kn−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour of K (...) for random α. Following work of Downey, Hirschfeldt and LaForte, we say that αK β iff there is a constant such that for all n, . We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real α so that lim supn K−K=∞ where Ω is Chaitin's random real. One is based upon work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that μ=0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n ≠ m then the initial segment complexities of the natural n- and m-random sets and Ω︀) are different. The techniques introduced in this paper have already found a number of other applications. (shrink)

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory (...) and this Omega operator. But unlike the jump, which is invariant under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. (shrink)

WE CONTINUE TO BE SURPRISED that the only pieces of sustained philosophizing that Husserl published in the 15 years between Ideas I and Internal Time-Consciousness were three articles that appeared in Kaizo, a Japanese periodical, in 1923 and 1924. We also find well-written drafts of two other articles that were to follow in the series but were never completed and submitted. We are even more intrigued when we realize that in these texts Husserl takes up themes never touched in publications (...) before and not touched again until his very late work. While we know that his manuscripts contain continuous and extensive analyses of the essential forms of social and cultural life and of ethical normativity, the Kaizo articles build the only slender bridge actually published on these topics between his attack on "the new Weltanschauung philosophy" in his 1911 Logos article and the 1934-37 texts surrounding the The Crisis of European Sciences and Transcendental Philosophy; slender, indeed, when we think that the articles were published in Japan and that the second and third appeared only in Japanese translation. Yet our intrigue is coupled with anticipation, for we have long wondered what Husserl would say to the Japanese, a society highly developed and yet not Western, a culture that does not have its beginnings in the Greek conception of science and knowledge and which clearly poses the question of cultural diversity. And we have wondered how he could define ethical life in this context, how the analysis would proceed, and how the human and social sciences could play a role in the account that he gives. (shrink)