Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the recent debate (...) on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)

The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a case (...) study on extremal proofs. Based on this case study, we compare our analysis of proof understanding to Avigad’s ability-based analysis of proof understanding. (shrink)

Gottlob Frege has exerted an enormous influence on the evolution of twentieth-century philosophy, yet the real significance of that influence is still very much a matter of debate. This book provides a completely new and systematic account of Frege's philosophy by focusing on its cornerstone: the theory of sense and reference. Two features distinguish this study from other books on Frege. First, sense and reference are placed absolutely at the core of Frege's work; the author shows that no adequate account (...) of the theory can avoid analysing the notion of thought that underpins it, or explaining how it has clarified our concept of judgement. Second, the theory is situated within the development of Frege's thought; the author reveals how the theory caused Frege to alter many of his fundamental views. In doing so the author presents a clearer picture of the problems the theory was intended to solve, and delineates more sharply the characteristic features of Frege's philosophy. (shrink)

This introductory chapter provides an overview of the recent debate about responsibility and distributive justice. It traces the recent philosophical focus on distributive justice to John Rawls and examines two arguments in his work which might be taken to contain the seeds of the focus on responsibility in later theories of distributive justice. It examines Ronald Dworkin's ‘equality of resources’, the ‘luck egalitarianism’ of Richard Arneson and G. A. Cohen, as well as the criticisms of their work put forward by (...) Elizabeth Anderson, Marc Fleurbaey, Susan Hurley, and Jonathan Wolff. Key concepts such as responsibility (individual and collective), luck (thin and thick; brute and option), control, desert, and equality of opportunity are delineated, and the implementation of responsibility-sensitive accounts of justice is considered. The chapters of this book are positioned in relation to the wider literature on responsibility and distributive justice, and a brief outline of the chapters is provided. (shrink)

Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics, including set theory and model theory, are covered.

The principle of proportionate causality is often cited as a cause for concern that Thomistic metaphysics may be irreconcilable with a theory of biological evolution. St. Thomas does hold that for the generation of what he calls perfect animals, a generator of the same species is required. This study clarifies what the proportionate causes of generated organisms are for Thomas, examining his views about spontaneous generation, reproductive generation, and hybridization, while also articulating the roles of both the heavenly bodies and (...) their separate movers as universal causes of generation. This study establishes that Thomas’s assertion of the need for a univocal generator for perfect animals is grounded not in the principle of proportionate causality, but rather in physical and biological doctrines received from Aristotle and in a causal principle that seems reconcilable with biological evolution, namely, that a remote universal cause requires more mediating causes to produce more powerful effects. (shrink)

Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is set (...) to the lim inf of previous register contents if that limit is finite; otherwise the register is reset to 0. The theory of these machines has several similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. The machines can decide all ${\Pi^1_1}$ sets, yet are strictly weaker than ITTMs. As in the ITTM situation, we introduce a notion of ITRM-clockable ordinals corresponding to the running times of computations. These form a transitive initial segment of the ordinals. Furthermore we prove a Lost Melody theorem: there is a real r such that there is a program P that halts on the empty input for all oracle contents and outputs 1 iff the oracle number is r, but no program can decide for every natural number n whether or not ${n \in r}$ with the empty oracle. In an earlier paper, the third author considered another type of machines where registers were not reset at infinite lim inf’s and he called them infinite time register machines. Because the resetting machines correspond much better to ITTMs we hold that in future the resetting register machines should be called ITRMs. (shrink)

The interest contemporary philosophy takes in Kant's notion of apperception is restricted to his criticism of the Cartesian Ego and to his refutation of scepticism, but there is a profound lack of concern for the notion itself and for the act of spontaneity in particular which is connected with the use of the word T. Starting from a comparison of Wittgenstein's account of this use with Kant's considerations it is argued that the latter aims at a theory of formal conditions (...) of knowledge which includes the availability of the notion of the I. It is clarified what the determination of apperception as an 'act of spontaniety' amounts to (B: 132). Kant's scattered remarks on the ability of having the representation of the I, of using the word 'I', are considered in order to show that what he called 'the logical I' has something to do with the capacity of performing an act of judgment. It is argued that such an account is not to be found in contemporary discussions of 'essential indexicals', 'first-person view' and mental self-ascriptions. (shrink)

What are the relationships between philosophy and the history of philosophy, the history of science and the philosophy of science? This selection of essays by Lorenz Krüger (1932-1994) presents exemplary studies on the philosophy of John Locke and Immanuel Kant, on the history of physics and on the scope and limitations of scientific explanation, and a realistic understanding of science and truth. In his treatment of leading currents in 20th century philosophy, Krüger presents new and original arguments for a deeper (...) understanding of the continuity and dynamics of the development of scientific theory. These result in significant consequences for the claim of the sciences that they understand reality in a rational manner. The case studies are complemented by fundamental thoughts on the relationship between philosophy, science, and their common history. (shrink)

The authors survey some debates about the nature and structure of physical theories and about the connections between our physical theories and naturalized metaphysics. The discussion is organized around an “ideal view” of physical theories and criticisms that can be raised against it. This view includes controversial commitments regarding the best analysis of physical modalities and intertheory relations. The authors consider the case in favor of taking laws as the primary modal notion, discussing objections related to alleged violations of the (...) laws, the apparent need to appeal to causality, and the status of probability. The “ideal view” includes a commitment that fundamental physical theories are explanatorily sufficient. The authors discuss several challenges to recovering the manifest image from fundamental physics, along with a distinct challenge to reductionism based on acknowledging the contributions of less fundamental theories in physical explanations. (shrink)

In this section, we apply the notions obtained above to a famous historical example of a false proof. Our goal is to demonstrate that this proof shows a sufficient degree of distinctiveness for a formalization in a Naproche-like system and hence that automatic checking could indeed have contributed in this case to the development of mathematics. This example further demonstrates that even incomplete distinctivication can be sufficient for automatic checking and that actual mistakes may occur already in the margin between (...) the degree of distinctiveness necessary for formalization and complete distinctiveness. (shrink)

We consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of ZFC for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting and (...) resetting infinite-time register machines, and α-Turing machines for countable admissible ordinals α. (shrink)

Infinite Time Register Machines are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. , and . We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in [6] and applied to ITRM 's in [3]. A real x is ITRM -recognizable iff there is an ITRM -program P such that PyPy stops with output 1 iff y=xy=x, and otherwise stops with output 0. In [3], it is (...) shown that the recognizable reals are not contained in the ITRM -computable reals. Here, we investigate in detail how the ITRM -recognizable reals are distributed along the canonical well-ordering shrink)

According to the PubMed resource from the U.S. National Library of Medicine, over 750,000 scientific articles have been published in the ~5000 biomedical journals worldwide in the year 2007 alone. The vast majority of these publications include results from hypothesis-driven experimentation in overlapping biomedical research domains. Unfortunately, the sheer volume of information being generated by the biomedical research enterprise has made it virtually impossible for investigators to stay aware of the latest findings in their domain of interest, let alone to (...) be able to assimilate and mine data from related investigations for purposes of meta-analysis. While computers have the potential for assisting investigators in the extraction, management and analysis of these data, information contained in the traditional journal publication is still largely unstructured, free-text descriptions of study design, experimental application and results interpretation, making it difficult for computers to gain access to the content of what is being conveyed without significant manual intervention. In order to circumvent these roadblocks and make the most of the output from the biomedical research enterprise, a variety of related standards in knowledge representation are being developed, proposed and adopted in the biomedical community. In this chapter, we will explore the current status of efforts to develop minimum information standards for the representation of a biomedical experiment, ontologies composed of shared vocabularies assembled into subsumption hierarchical structures, and extensible relational data models that link the information components together in a machine-readable and human-useable framework for data mining purposes. (shrink)

The ArgumentL. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notoriousGrundlagenstreitcentered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which (...) last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a non-standard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy.Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework — not from Kant's philosophy of mathematics but from his general metaphysics — which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy. (shrink)

The physical activity-related health competence model assumes that individuals require movement competence, control competence, and self-regulation competence to lead a healthy, physically active lifestyle. Although previous research has already established some measurement factors of the three dimensions, no attempts have so far been made to statistically aggregate them on the sub-competence level. Therefore, the goal of the present study was to test two additional factors for PAHCO and subsequently model the second-order structure with two samples from the fields of rehabilitation (...) and prevention. We conducted two questionnaire surveys with persons with multiple sclerosis and teaching students undergoing a basic qualification course in physical education. After performing exploratory items analysis, we used second-order confirmatory factor analysis and multidimensional scaling to investigate whether the scales could be bundled in accordance with the PAHCO model. The CFAs with 10 factors demonstrated a good model fit. In contrast, the second-order analysis with a simple loading structure on the three sub-competencies revealed an unacceptable model fit. Instead, a second-order model variant was preferred [comparative fit index = 0.926, root mean square error of approximation = 0.048, standardized root mean square residual = 0.065] in which body awareness and self-efficacy had theory-conform cross-loadings. The results of multidimensional scaling were in line with the extracted second-order structure. The present results suggested that the extension of the measurement instrument to 10 first-order factors was psychometrically justified for the two populations. The results from the second-order analyses provided the basis for the creation of sum scores, representing manifest indicators of movement competence, control competence, and self-regulation competence. Future studies are needed that cross-validate the extended measurement model with other populations and that relate the sub-competencies of PAHCO to indicators of health-enhancing physical activity. (shrink)

We introduce and study some variants of a notion of canonical set theoretical truth. By this, we mean truth in a transitive proper class model M of ZFC that is uniquely characterized by some $$\in$$ ∈ -formula. We show that there are interesting statements that hold in all such models, but do not follow from ZFC, such as the ground model axiom and the nonexistence of measurable cardinals. We also study a related concept in which we only require M to (...) be fixed up to elementary equivalence. We show that this theory-canonicity also goes beyond provability in ZFC, but it does not rule out measurable cardinals and it does not fix the size of the continuum. (shrink)

The abstract of this chapter was initially published with error. The chapter has been updated with the corrected abstract as given below.