The fundamental ideas concerning computation and recursion naturally find their place at the interface between logic and theoretical computer science. The contributions in this book, by leaders in the field, provide a picture of current ideas and methods in the ongoing investigations into the pure mathematical foundations of computability theory. The topics range over computable functions, enumerable sets, degree structures, complexity, subrecursiveness, domains and inductive inference. A number of the articles contain introductory and background material which it is hoped will (...) make this volume an invaluable resource for specialists and non-specialists alike. (shrink)

Subrecursive hierarchy classifications are used to compare the complexities of recursive functions according to their derivations in a version of Kleene's equation calculus, and their computations by term-rewriting. In each case ordinal bounds are assigned, and it turns out that the respective complexity measures are given by a version of the Fast Growing Hierarchy, and the Slow Growing Hierarchy. Known comparisons between the two hierarchies then provide ordinal trade-offs between derivation and computation. Characteristics of some well-known subrecursive classes are also (...) read off. (shrink)

The class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from "within". On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been "coded" at previous levels. The question is: how can a (...) recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The "accessible" recursive functions thus generated turn out to be those provably recursive in $(\prod_{1}^{1}-CA)_{0}$. (shrink)

Girard’s maxim, that Peano Arithmetic is a theory of one inductive definition, is re-examined in the light of a weak theory EA formalising basic principles of Nelson’s predicative Arithmetic.

This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.

Wilfred Sieg and Stanley S. Wainer. Program Transformation and Proof Transformation.

There is a very simple way in which the safe/normal variable discipline of Bellantoni–Cook recursion [S. Bellantoni, S. Cook, A new recursion theoretic characterization of the polytime functions, Computational Complexity 2 97–110] can be imposed on arithmetical theories like PA: quantify over safes and induct on normals. This weakens the theory severely, so that the provably recursive functions become more realistically computable . Earlier results of D. Leivant [Intrinsic theories and computational complexity, in: D. Leivant , Logic and Computational Complexity, (...) Lecture Notes in Computer Science, vol. 960, Springer-Verlag, 1995, pp. 177–194] are re-worked and extended in this new context, giving proof-theoretic characterizations of complexity classes between Grzegorczyk’s E2 and E3. (shrink)

Following our [6], though with somewhat different methods here, further variants of Goodstein sequences are introduced in terms of parameterized Ackermann–Péter functions. Each of the sequences is shown to terminate, and the proof-theoretic strengths of these facts are calibrated by means of ordinal assignments, yielding independence results for a range of theories: PRA, PA,$\Sigma ^1_1$-DC$_0$, ATR$_0$, up to ID$_1$. The key is the so-called “Hardy hierarchy” of proof-theoretic bounding finctions, providing a uniform method for associating Goodstein-type sequences with parameterized normal (...) form representations of positive integers. (shrink)

The philosophical interest in experimental practice in neuroscience has brought renewed attention to the study of the development and use of techniques and tools for data production. John Bickle has argued that the construction and progression of theories in neuroscience are entirely dependent on the development and ingenious use of research tools. In Bickle's account, theory plays a tertiary role, as it depends on what the tools allow researchers to manipulate, and the tools, in turn, are developed not in order (...) to test theories but as solutions to engineering problems. However, Bickle's account is not entirely precise in explaining what informs researchers' decision-making in their atheoretical laboratory tinkering. Identifying the sources that guide researchers in tool development and use is crucial if one wishes to contribute to the philosophical or meta-scientific understanding of experimental practice in neuroscience. In the following paper, I claim that decision-making in tools' development and use in neuroscience is doubly guided. Pre-existing theory and concepts determine information's relevance, whereas tools' functioning in controlled situations determines information's reliability. Accordingly, experimenters' decision-making is situated both in the context of analysing, modelling or interpreting information and in the context of producing information. I study the case of the tungsten microelectrode developed by David Hubel during the 1950s. First, I show that pre-existing theory and concepts determine in advance what information would be relevant to obtain from the microelectrode. Second, I show that Hubel's tinkering follows the guidelines derived from the very structure of what we recognise as reliable experimentally produced information. Finally, I suggest that data-production processes allow experimenters to assess what to expect from an experimental system in terms of concept- and theory-generation and confirmation, thereby endorsing Bickle's tenet on the tertiary role of theory in neuroscience. (shrink)

There is increasing interest in the EU about the central place of eIdentity (eID) in people’s lives. eID is increasingly seen as a bridge between the commercial viability of models based on large-scale provision of e-services and users’ need for privacy and security in online transactions. This paper examines technological, social and legal developments in the field of eID and asks whether there is the need for a new regulatory framework that both preserves users’ identity and enables the provision of (...) advanced services. Firstly, the paper interprets recent market moves in the eID field as a response to a rising regulatory tide. Secondly, it examines some of the challenges arising from Web2.0, and four emerging socio-legal issues associated with eID—behavioural profiling, social engineering, redlining and other unsocial practices. Thirdly, the paper examines the capacity of the current regulatory framework to absorb this turbulence, and finds it wanting. Finally, it advances a novel model of eID regulation that may help regulators create an identity-preserving, transaction-friendly eID environment. (shrink)

In his entertaining and informative book Graphic Discovery, Howard Wainer unlocked the power of graphical display to make complex problems clear. Now he's back with Picturing the Uncertain World, a book that explores how graphs can serve as maps to guide us when the information we have is ambiguous or incomplete. Using a visually diverse sampling of graphical display, from heartrending autobiographical displays of genocide in the Kovno ghetto to the "Pie Chart of Mystery" in a New Yorker cartoon, (...) Wainer illustrates the many ways graphs can be used--and misused--as we try to make sense of an uncertain world. Picturing the Uncertain World takes readers on an extraordinary graphical adventure, revealing how the visual communication of data offers answers to vexing questions yet also highlights the measure of uncertainty in almost everything we do. Are cancer rates higher or lower in rural communities? How can you know how much money to sock away for retirement when you don't know when you'll die? And where exactly did nineteenth-century novelists get their ideas? These are some of the fascinating questions Wainer invites readers to consider. Along the way he traces the origins and development of graphical display, from William Playfair, who pioneered the use of graphs in the eighteenth century, to instances today where the public has been misled through poorly designed graphs. We live in a world full of uncertainty, yet it is within our grasp to take its measure. Read Picturing the Uncertain World and learn how. (shrink)

The general objective of this essay is to systematize Jean-Luc Nancy’s post- deconstructive reflections on the concept of evidence. A general claim of this paper is that the post-deconstructive concept of evidence is genuinely an epistemic concept of evidence insofar as it refers to structures involved in verification processes. Evidence is the presentation of a state of affairs that relates the presentation not only to what we claim about this state of affairs but also to the singular circumstances of its (...) production. Verification (the production of a claim’s truth) results directly from the singularities involved in the production or presentation of evidence. This means that evidence is never exhausted in the truth it produces or the knowledge it validates but remains a priori exposed or available to produce and validate unknown knowledge about unknown states of affairs. (shrink)

Data production in experimental sciences depends on localised experimental systems, but the epistemic properties of data transcend the contingencies of the processes that produce them. Philosophers often believe that experimental systems instantiate but do not produce the epistemic properties of data. In this paper, we argue that experimental systems' local functioning entails intrinsic capacities to produce the epistemic properties of data. We develop this idea by applying Derrida's model of arche-writing to study a case of theory-oriented experimental practice. Derrida's model (...) relativises or dissolves the conceptual distinction between the moment of data production and a subsequent moment of data dissemination. It thus has consequences for understanding both data production (despite being intrinsically local, data production a priori generates transferrable and modellable information) and data dissemination (when modelling information, researchers needs to refer this information to the context of its production). We study a case of data production in a non-exploratory experimental system designed to test a pre-existing hypothesis in visual neuroscience. A case of theory-oriented experimental practice should allow us to identify the autonomous functioning of experimental systems in data production more clearly, insofar as it allows us to study the limits of pre-existing theory in the activities of these systems. We suggest that pre-existing concepts, hypotheses and theories condition the relevance but not the production of experimental data. (shrink)

This article presents evidence regarding aspects of the gendered nature of care women with gynaecological cancer receive from their (usually) male surgeons and oncologists in Australia. We argue that despite women’s general preference for female gynaecologists, those with a gynaecological cancer develop a strong therapeutic relationship with their male medical specialist, not extended to their (usually) female nurses and other allied health professionals. Given the highly sensitive and sexualized nature of gynaecological cancer, this requires explanation. These findings can be partly (...) explained by examining the division of labour between nurses and doctors, specifically issues of control over this process and the development of specializations. The findings also bring into stark relief the way in which power and status differences can be used by medicine to create a positive therapeutic relationship with patients while simultaneously de-eroticizing the intimate procedures necessary in assisting women throughout their cancer treatment. Importantly, this relationship also has relevance for policy makers, particularly those concerned with the highly gendered division of labour of the medical specialty workforce in Australia. (shrink)

We show the following results on Wainer's notation for a minimal subrecursive inaccessible ordinal τ: First, we give a constructive proof of the collapsing theorem. Secondly, we prove that the slow-growing hierarchy and the fast-growing hierarchy up to τ have elementary properties on increase and domination, which completes Wainer's proof that τ is a minimal subrecursive inaccessible. Our results are obtained by showing a strong normalization theorem for the term structure of the notation. MSC: 03D20, 03F15.

The article reviews various theories of laughter on the basis of their conventionally accepted classification. Their substantive intersections, especially significant for A. F. Losev and L. V. Karasev, are revealed. This leads us to the conclusion that there can be as many emotions and meanings expressed by laughter as there are hypotheses about what laughter points to. This conclusion drew our attention to the essential qualities of laughter, regardless of its objects, forms, and purposes: virtuality and eventuality. The theories of (...) M. Bakhtin, M. T. Ryumina, and G. Deleuze are considered in connection with these concepts. Specific attention is paid to the concept of laughter by J.-L. Nancy, which, while not excluding the ideas of the previously listed authors, as well as J. Derrida's and M. Heidegger's, is original in that it presents laughter as a presence, an autonomous event or affect, which is capable of being expressed in text and speech. (shrink)

In this book Sergey Horujy undertakes a novel comparative analysis of Foucault s theory of practices of the self and the Eastern Orthodox ascetical tradition of Hesychasm, revealing great affinity between these two radical subject-less approaches to anthropology. As he facilitates the dialogue between the two, he offers both an original treatment of ascetical and mystical practices and an up-to-date interpretation of Foucault that goes against the grain of mainstream scholarship. In the second half of the book Horujy transitions from (...) the dialogue with Foucault to his own work of Christian philosophy, rooted in -- but not limited to -- the Eastern Christian philosophical and theological tradition. Horujy s thinking exemplifies the postsecular nature of our contemporary period and serves as a powerful invitation to think beyond religious-secular divides in philosophy and Eastern-Western divides in intellectual history. (shrink)