We show how to extract effective bounds Φ for $\bigwedge u^1 \bigwedge v \leq_\gamma tu \bigvee w^\eta G_0$ -sentences which depend on u only (i.e. $\bigwedge u \bigwedge v \leq_\gamma tu \bigvee w \leq_\eta \Phi uG_0$ ) from arithmetical proofs which use analytical assumptions of the form \begin{equation*}\tag{*}\bigwedge x^\delta\bigvee y \leq_\rho sx \bigwedge z^\tau F_0\end{equation*} (γ, δ, ρ, and τ are arbitrary finite types, η ≤ 2, G0 and F0 are quantifier-free, and s and t are closed terms). If τ (...) ≤ 2, (*) can be weakened to $\bigwedge x^\delta, z^\tau\bigvee y \leq_\rho sx \bigwedge \tilde{z} \leq_\tau z F_0$ . This is used to establish new conservation results about weak König's lemma. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers. (shrink)

Kohlenbach, U., Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation, Annals of Pure and Applied Logic 64 27–94.We consider uniqueness theorems in classical analysis having the form u ε U, v1, v2 ε Vu = 0 = G→v 1 = v2), where U, V are complete separable metric spaces, Vu is compact in V and G:U x V → is a constructive function.If is proved by arithmetical means from analytical assumptions x (...) εXy ε Yx z ε Z = 0) only , then we can extract from the proof of → an effective modulus of uniqueness, which depends on u, k but not on v1,v2, i.e., u ε U, v1, v2 ε Vu, k εG, G 2-φuk→dv2-itk). Such a modulus φ can e.g. be used to give a finite algorithm which computes the zero of G on Vu with prescribed precision if it exists classically.The extraction of φ uses a proof-theoretic combination of functional interpretation and pointwise majorization. If the proof of → uses only simple instances of induction, then φ is a simple mathematical operation. (shrink)

The discourse on climate politics so far is an expert and elitist discourse in which peoples, societies, citizens, workers, voters and their interests, views and voices are very much neglected. So, in order to turn climate change politics from its head onto its feet you have to take sociology into account. There is an important background assumption which shares in the general ignorance concerning environmental issues and, paradoxically, this is in corporated in the specialism of environmental sociology itself — this (...) is the category of ‘the environment’. If ‘the environment’ only includes everything which is not human, not social, then the concept is sociologically empty. If the concept includes human action and society, then it is scientifically mistaken and politically suicidal. (shrink)

A pointwise version of the Howard-Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel'sT as well as Kleene/Feferman's ) is applied to systems of intuitionistic and classical arithmetic (H andH c) in all finite types with full induction as well as to the corresponding systems with restricted inductionĤ↾ andĤ↾c.H and Ĥ↾ are closed under a generalized (...) fan-rule. For a restricted class of formulae this also holds forH c andĤ↾c.We give a new and very perspicuous proof that for each one can construct a functional such that $\tilde \Phi \alpha $ is a modulus of uniform continuity for Φ on {β1∣∧n(βn≦αn)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for Φ.The type structure ℳ of all pointwise majorizable set-theoretical functionals of finite type is used to give a short proof that quantifier-free “choice” with uniqueness (AC!)1,0-qf. is not provable within classical arithmetic in all finite types plus comprehension [given by the schema (C)ϱ:∨y 0ϱ∧x ϱ(yx=0⇔A(x)) for arbitraryA], dependent ω-choice and bounded choice. Furthermore ℳ separates several μ-operators. (shrink)

The intended title was “Universe Oriented Ontology” or “Multiverse Oriented Ontology”, or “Universe or Multiverse Metaphysics”. I mention this as it gives an idea about the meaning and intention of the title and the work as well as the titles I considered and why I moved away from them to the present one. The sub-title provides a further hint towards the intentions of the work, namely: ” Beyond Earth- and Human-centricity’. I opted for ‘transcending’ rather than beyond, as I am (...) still in the process of describing the process of transcending and many of the ideas I am obliged to use from our conceptual system and practices are still earth- and anthropo-centered, And, they have not yet gone to a state beyond those two -isms. I say something about universe-centeredness, then about planet earth as point and frame of reference and anthropo-centricity. The socio-cultural practice of philosophy and the doing of philosophy is merely one of many human, social and cultural practices. I post different notions of philosophy by a number of writers, some of them philosophers. These notions are about their perceptions of what philosophy is. Few of them go into detail about the subject-matter of philosophy. No one really deal with the aims, purposes and objectives of the discipline and none deal with the nature of the doing of philosophy or philosophizing. Activities that I suggest resemble certain features of the processes of theorizing. I end with explorations of possible characteristics of original- and creative thinkers. I do this by mentioning a number of themes of meta-philosophy listed and described by Peter Suber. I make a number of comments in them and highlight aspects relevant to these type of thinkers. One find them of course in all disciplines and socio-cultural [racticesand disciplines be they the arts,humanities, sciences, etc. I am of course specifically concerned with original andcreative thinkers inthe Western tradition of philosophy. As I am a radical and absolute sceptic, I end with an article on this theme. It is said by everyone, for example Hume, Russell and Pascal, that such a radical position is psychologically impossible and that it cannot be lived 24/7. Pascal for example opted for believe or faith and he and a number of writers suggest that that is the only way out and the final position of radical sceptics. I disagree with him. (shrink)

Meta-philosophical investigations of philosophy, the discourse, methods, syllabuses, the nature of philosophy, ontology, meta-ontology.

Human beings employ concepts not merely to re-constitute their worlds, realities, including their selves, minds, consciousness, lives and loves but to fabricate and constitute these things. .Concepts, conceptual practices, usage and meanings are loaded and associated with predetermined -isms, presuppositions, assumptions, attitudes, beliefs, restrictions, perspectives, frames of reference, and other phenomena that will determine how they are used, their effects, results, consequences, etc. Concepts, conceptual practices, usage and meanings are loaded and associated with pre-determined -isms, pre-suppositions, assumptions, attitudes, beliefs, restrictions, (...) perspectives, frames of reference, and other phenomena that will determine how they are used, their effects, results, consequences, etc. -/- The above is earth- and anthropo-centered and restricted. The origins, nature, past, present and future is explored. This is suggested as point of reference and not the minute and irrelevant planet earth. Changes, modifications even the destruction of this planet will have little effect on and consequences for our galaxy and the universe. Against or in this universal context the nature, the functions, aims, objectives, methods, techniques, relevance, meaning and possibility of philosophy and philo- sophizing is explored. -/- Reductionistic humans are obsessed with and drawn to minimalist and generalized patterns or sets and systems of ideas as explanations and underlying foundations of complex realities and phenomena. (shrink)

In this paper a new method, elimination of Skolem functions for monotone formulas, is developed which makes it possible to determine precisely the arithmetical strength of instances of various non-constructive function existence principles. This is achieved by reducing the use of such instances in a given proof to instances of certain arithmetical principles. Our framework are systems ${\cal T}^{\omega} :={\rm G}_n{\rm A}^{\omega} +{\rm AC}$ -qf $+\Delta$ , where (G $_n$ A $^{\omega})_{n \in {\Bbb N}}$ is a hierarchy of (weak) subsystems (...) of arithmetic in all finite types (introduced in [14]), AC-qf is the schema of quantifier-free choice in all types and $\Delta$ is a set of certain analytical principles which e.g. includes the binary König's lemma. We apply this method to show that the arithmetical closures of single instances of $\Pi^0_1$ -comprehension and -choice contribute to the growth of extractable bounds from proofs relative to ${\cal T}^\omega$ only by a primitive recursive functional in the sense of Kleene. In subsequent papers these results are widely generalized and the method is used to determine the arithmetical content of single sequences of instances of the Bolzano-Weierstraß principle for bounded sequences in ${\Bbb R}^d$ , the Ascoli-lemma and others. (shrink)

In [15], [16] G. Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals Φ A of order type 0 which realize the Herbrand normal form A H of A. Subsequently more perspicuous proofs of this fact via functional interpretation (combined with normalization) and cut-elimination were found. These proofs however do (...) not carry out the no-counterexample interpretation as a local proof interpretation and don't respect the modus ponens on the level of the no-counterexample interpretation of formulas A and A → B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (δ) and--at least not constructively-- (γ) which are part of the definition of an 'interpretation of a formal system' as formulated in [15]. In this paper we determine the complexity of the no-counterexample interpretation of the modus ponens rule for (i) PA-provable sentences, (ii) for arbitrary sentences A, B ∈ L(PA) uniformly in functionals satisfying the no-counterexample interpretation of (prenex normal forms of) A and A → B, and (iii) for arbitrary A, B ∈ L(PA) pointwise in given α( 0 ) -recursive functionals satisfying the no-counterexample interpretation of A and A → B. This yields in particular perspicuous proofs of new uniform versions of the conditions (γ), (δ). Finally we discuss a variant of the concept of an interpretation presented in [17] and show that it is incomparable with the concept studied in [15], [16]. In particular we show that the no-counterexample interpretation of PA n by α( n (ω))-recursive functionals (n ≥ 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (δ). (shrink)

In this paper, we present another case study in the general project of proof mining which means the logical analysis of prima facie non-effective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation developed in Kohlenbach , Oxford University Press, Oxford, 1996, pp. 225–260) to analyze Cheney's simplification 189) of Jackson's original proof 320) of the uniqueness of the best L1-approximation of continuous functions fC[0,1] by polynomials pPn of degree n. Cheney's (...) proof is non-effective in the sense that it is based on classical logic and on the non-computational principle WKL . The result of our analysis provides the first effective uniform modulus of uniqueness . Moreover, the extracted modulus has the optimal ε-dependency as follows from Kroó 331). The paper also describes how the uniform modulus of uniqueness can be used to compute the best L1-approximations of a fixed fC[0,1] with arbitrary precision. The second author uses this result to give a complexity upper bound on the computation of the best L1-approximation in Oliva 66–77). (shrink)

The so-called weak König's lemma WKL asserts the existence of an infinite path b in any infinite binary tree . Based on this principle one can formulate subsystems of higher-order arithmetic which allow to carry out very substantial parts of classical mathematics but are Π 2 0 -conservative over primitive recursive arithmetic PRA . In Kohlenbach 1239–1273) we established such conservation results relative to finite type extensions PRA ω of PRA . In this setting one can consider also a uniform (...) version UWKL of WKL which asserts the existence of a functional Φ which selects uniformly in a given infinite binary tree f an infinite path Φf of that tree. This uniform version of WKL is of interest in the context of explicit mathematics as developed by S. Feferman. The elimination process in Kohlenbach [10] actually can be used to eliminate even this uniform weak König's lemma provided that PRA ω only has a quantifier-free rule of extensionality QF-ER instead of the full axioms of extensionality for all finite types. In this paper we show that in the presence of , UWKL is much stronger than WKL: whereas WKL remains to be Π 2 0 -conservative over PRA, PRA ω ++ UWKL contains full Peano arithmetic PA. We also investigate the proof–theoretic as well as the computational strength of UWKL relative to the intuitionistic variant of PRA ω both with and without the Markov principle. (shrink)

It is well known by now that large parts of mathematical reasoning can be carried out in systems which are conservative over primitive recursive arithmetic PRA . On the other hand there are principles S of elementary analysis which are known to be equivalent to arithmetical comprehension and therefore go far beyond the strength of PRA . In this paper we determine precisely the arithmetical and computational strength of weaker function parameter-free schematic versions S− of S, thereby exhibiting different levels (...) of strength between these principles as well as a sharp borderline between fragments of analysis which are still conservative over PRA and extensions which just go beyond the strength of PRA. (shrink)

When it comes to understanding the ontology of individual existence-that is the everyday behaviors that we all perform and hardly ever think about-the voice has a particularly complicated status. Together with writing, voice is the medium expressing ideas that, broadly speaking, we have previously formed in our minds. At the same time, voices trigger vague images and associations that do not have determinate forms. They remain in our memory, and we may even "hear" them as if immediately present, but we (...) are unable to detach the propositional contents they articulate from the centrifugal effects that they have on our lives. Writing in both a personal and philosophical register, Hans Ulrich Gumbrecht explores the complexity of the voice as an understudied philosophical, social, and existential phenomenon. He starts out with a focus on its core intellectual problem as "the knot of the voice" - referring to the inseparable proximity between meanings, images, and the physical perceptions on which they depend. In conversation with Nietzsche, Husserl, Heidegger, Derrida, Lyotard, Luhmann, and above all Roland Bartes, Gumbrecht addresses topics that range from the social functions of the voice to its status in different historical contexts, and to the ways in which the perception of voices animates imagination. Throughout, incisive analyses of moments such as Julius Caesar's purportedly high-pitched voice, the surprisingly fragile authority of God's voice in the Torah and in the Gospel, and Gumbrecht's own personal attachment to the voices of popular singers such as Edith Piaf, Elvis Presley, and Adele, create a portrait of the voice that is both philosophically challenging and entertaining to read. (shrink)

This article investigates the changing justifications of one of the hallmarks of orthodox psychoanalytic practice, the neutral and abstinent stance of the psychoanalyst, during the middle decades of the 20th century. To call attention to the shifting rationales behind a supposedly cold, detached style of treatment still today associated with psychoanalysis, explanations of the clinical utility of neutrality and abstinence by ‘classical’ psychoanalysts in the United States are contrasted with how intellectuals and cultural critics understood the significance of psychoanalytic abstinence. (...) As early as the 1930s, members of the Frankfurt School discussed the cultural and social implications of psychoanalytic practices. Only in the 1960s and 1970s, however, did psychoanalytic abstinence become a topic within broader intellectual debates about American social character and the burgeoning ‘therapy culture’ in the USA. The shift from professional and epistemological concerns to cultural and political ones is indicative of the changing appreciation of psychoanalysis as a clinical discipline: for psychoanalysts as well as cultural critics, I argue, changing social mores and the professional decline of psychoanalysis infused the image of the abstinent psychoanalyst with nostalgic longing, making it a symbol of resistance against a culture seen to be in decline. (shrink)

In this note we show that the so-called weakly extensional arithmetic in all finite types, which is based on a quantifier-free rule of extensionality due to C. Spector and which is of significance in the context of Gödel"s functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for Π0 1-axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept fixed) was known.

LetA H be the Herbrand normal form ofA andA H,D a Herbrand realization ofA H. We showThere is an example of an (open) theory ℐ+ with function parameters such that for someA not containing function parameters Similar for first order theories ℐ+ if the index functions used in definingA H are permitted to occur in instances of non-logical axiom schemata of ℐ, i.e. for suitable ℐ,A In fact, in (1) we can take for ℐ+ the fragment (Σ 1 0 -IA)+ (...) of second order arithmetic with induction restricted toΣ 1 0 -formulas, and in (2) we can take for ℐ the fragment (Σ 1 0,b -IA) of first order arithmetic with induction restricted to formulas VxA(x) whereA contains only bounded quantifiers.On the other hand, $$PA^2 \vdash A^H \Rightarrow PA \vdash A,$$ wherePA 2 is the extension of first order arithmeticPA obtained by adding quantifiers for functions andA∈ℒ(PA). This generalizes to extensional arithmetic in the language of all finite types but not to sentencesA with positively occurring existential quantifiers for functions. (shrink)

Kant setzt sich im Vorwort seiner Schrift "Metaphysische Anfangsgründe der Naturwissenschaft" mit dem Vorwurf auseinander, die Deduktion der Verstandeskategorien in der "Kritik der reinen Vernunft" würde dem Gedanken einer prästabilierten Harmonie Vorschub leisten. Gleichsam ins kritische Mark getroffen weist er diesen Gedanken der dogmatischen Metaphysik zurück. Nicht zuletzt durch diesen Vorwurf angeregt überarbeitet er für die zweite Auflage seiner Kritik insbesondere die Deduktion, die sich nun in zwei Schritten vollzieht. Die Überlegungen sollen verdeutlichen, daß sich diese zwei Schritte zum einen (...) als ein analytisches und zum anderen als ein synthetisches Argument verstehen lassen. 1790 deutet Kant in seiner sogenannten "Streitschrift gegen Eberhard" allerdings in der Tat den Gedanken einer prästabilisierten Harmonie an. Wenngleich er damit eine Argumentationslücke in seinem transzendentalphilosophischen System offenbart, läßt sich doch zeigen, daß er auch hier nicht in dogmatische Metaphysik zurückfällt, sondern vielmehr aus empirischer Sicht die "Zusammenstimmung" der Erkenntnisvermögen Sinnlichkeit und Verstand erörtert. (shrink)

The concept of the national is often perceived, both in public and academic discourse as the central obstacle for the realization of cosmopolitan orientations. Consequently, debates about the nation tend to revolve around its persistence or its demise. We depart from this either-or perspective by investigating the formation of the ‘cosmopolitan nation’ as a facet of world risk society. Modern collectivities are increasingly preoccupied with debating, preventing and managing risks. However, unlike earlier manifestations of risk characterized by daring actions or (...) predictability models, global risks can no longer be calculated or forecast. Accordingly, more influence accrues to the perception of risk, largely constructed by media representations. Cosmopolitanized risk collectivities are engendered through the anticipation of endangered futures which are, for the most part, communicated through an increasingly global media scape. While global media events produce shared exposure, risk conceptions retain distinctive political-cultural features as their respective meanings are prefigured by path-dependent pasts. Nevertheless, the promulgation of risk societies, we argue, results in a reimagination of nationhood which takes place in the context of: global norms (e.g. human rights); globalized markets; transnational migrations; global generations and their embeddedness in civil society movements; and the local interpenetration of world religions to name but a few of the global backdrops shaping new associational intersections. We develop our argument in four interrelated steps. Contrary to essentialized notions of nationalism or universal versions of cosmopolitanism, we address the cosmopolitan reconfiguration of nationhood by differentiating between presumptions of thick belonging and the actual proliferation of cosmopolitan affiliations. In a second step we overcome the territorial fixation of the social sciences by shifting our attention to temporal dimensions, with a particular focus on competing conceptions of the future. In a third step we demonstrate how these cosmopolitan transformations of nationhood are taking place in the context of a world risk society regime that marshals a set of cosmopolitan imperatives situating the global other in our midst. In a fourth step we illustrate these developments by exploring how the mediatization of risk, and concomitant notions of the future, contribute to the reimagination of cosmopolitan risk collectivities. (shrink)

This paper discusses what kind of quantitative information one can extract under which circumstances from proofs of convergence statements in analysis. We show that from proofs using only a limited amount of the law-of-excluded-middle, one can extract functionals , where L is a learning procedure for a rate of convergence which succeeds after at most B-many mind changes. This -learnability provides quantitative information strictly in between a full rate of convergence and a rate of metastability in the sense of Tao (...) . In fact, it corresponds to rates of metastability of a particular simple form. Moreover, if a certain gap condition is satisfied, then B and L yield a bound on the number of possible fluctuations. We explain recent applications of proof mining to ergodic theory in terms of these results. (shrink)

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in a finite types together with various forms of the axiom of choice and a numerical omniscience schema which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a non-constructive μ-operator and his well-known results on the strength of this operator from the 70's. In this note we consider a weaker form LNOS of NOS which (...) suffices to derive the strong form of binary König's lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semi-classical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. (shrink)

In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems Tnω in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive functions of low growth. We reduce the use of instances of these principles in Tnω-proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical (...) content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper.As corollaries we obtain new conservation results for fragments of analysis over fragments of arithmetic which strengthen known purely first-order conservation results.We also characterize the provably recursive functions of type 2 of the extensions of Tnω based on these fragments of arithmetical comprehension, choice and uniform boundedness. (shrink)

This chapter summarises the autors' work in embodied robotics, emphasising the need for scientific tools to measure chaos and sensitivity to intial conditions, the role of novelty and development, and the relevance of human behaviour in natural environments.

Meta-philosophical investigation of the nature, aims, methods of philosophy. Original, creative-thinking vs derivative , secondary thinking. Exploration of the limits of the philosophical discourse by subject-matter or fields shared with other disciplines eg logic, thinking, cognition, arguments, etc. Inter-disciplinary topics and issues can lead philosophers to become involved in other disciplines and thereby destroy the philosophical discourse and philosophizing.

The last few years have seen a marked increase of interest in the ethics of psychotherapy. As the field of mental health has recently taken on a new level of prominence, renewed commitment to the ethical analysis of psychiatric and psychological treatment is clearly required. In this review essay, we survey recent work on psychotherapy ethics, taking a critical yet generally sympathetic view of the new literature. There are important considerations that remain neglected or overlooked entirely, in our view, and (...) several substantive and methodological trends are in need of diagnosis and critical response. We structure the discussion around several themes, including the inherently moral nature of psychotherapy, the heterogeneity of such practices, the significance of social, cultural, and historical context, and, finally, questions concerning prevailing methodological assumptions. (shrink)

Goodman's theorem states that intuitionistic arithmetic in all finite types plus full choice, HA + AC, is conservative over first-order intuitionistic arithmetic HA. We show that this result does not extend to various subsystems of HA, HA with restricted induction.

ABSTRACT Theodor Herzl once contemptuously remarked that he regards Freiland as a joke. This statement surprises if one compares his novel Altneuland to Theodor Hertzka's Freiland. To say the least, both utopias share many themes and narratives structures. While Altneuland became the world-renowned manifesto of Zionism, Freiland cherished popularity only at the time of its publication. Both novels are products of Vienna's fin-de-siècle modernism. Herzl's utopia is set in Palestine, Hertzka places Freiland in the empty space of East Africa. His (...) vision of a new civilization in Africa coincided with European colonialism, nationalism and the surge of anti-Semitism in Vienna. In the following, the essay investigates if Hertzka merely criticizes the culture of fin-de-siècle Vienna, or, if he produces a unique alternative? (shrink)

Large Groups are not constituted simply by adding further members to small groups. There is a qualitative difference between the social actions which take place in small communities and those in large ones. Large communities are irreducibly characterized by anonymity, i.e., the members of large groups don’t know of most of the other members as individual. Therefore, social action in large groups is based on a sign process: each member of a large group is understood as a representative of the (...) other anonymous members of the group as well as a sign for the group as a whole. (shrink)

Weber's analysis of capitalism evolved in arguing against Marx's theory. It claims no less than giving a better account of modern capitalism, historically, as weil as categorially, than Marx himself. As a sociological theory of economy, based on an analytical theory ofhuman action, it also offers an approach to a critical ethics of the capitalist economy, which, after the self-induced end of socialism, has become an issue of greater urgency than ever before.