Penelope Maddy has recently addressed the set-theoretic multiverseset-theoretic multiverse, and expressed reservations on its status and merits Foundations of mathematics. Essays in honor of W. Hugh Woodin’s 60th birthday. Contemporary mathematics. American Mathematical Society, Providence, pp. 289–322, 2017). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalismset-theoretic naturalism. I first distinguish three main forms of ‘multiversism’multiversism, and then I proceed to analyse MaddyMaddy’s concerns. Among other things, I take into account salient (...) aspects of multiverse-related mathematics, in particular, research programmes in set theory for which the use of the multiverse seems to be crucial, and show how one may provide responses to MaddyMaddy’s concerns based on a careful analysis of ‘multiverse practice’. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.

Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the concept of cardinal number. Neologicism would therefore be threatened by the ‘good company' HP is kept by such alternative assignments. In his review of Mancosu's book, Bob Hale argues, however, that ‘getting different numerosities for different countable infinite collections depends on taking the groups in a certain order – but it is (...) of the essence of cardinal numbers that the cardinal size of a collection does not depend upon how its members are ordered'. This paper's goal is to implement Hale's response to the Good Company problem by producing a Cantorian argument for HP. In Section 2, we present the Heck-Mancosu argument against neologicism. In Section 3, we discuss Hale's defence of Hume's Principle. In Section 4, we discuss Cantor's abstractionist definitions of number. In Section 5, we argue that good abstraction must comply with what we call ‘Gödel’s Minimal Account of Abstraction’ (GMAA). We finally show (Sections 5 and 6) that non-Cantorian theories of cardinality fail to satisfy GMAA. (shrink)

The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...) (H-axioms) in countable transitive models, the collection of which constitutes the `hyperuniverse' (H), has remarkable 1st-order consequences, some of which we review in section 5. (shrink)

L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...) fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e un’analisi di alcune delle questioni filosofiche connesse al Problema del Continuo. (shrink)

The Hyperuniverse Programme, introduced in Arrigoni and Friedman, fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the ‘maximal iterative concept’, and the programme identifies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements in countable (...) transitive models, the collection of which constitutes the ‘hyperuniverse’, has remarkable first-order consequences, some of which we review in Sect. 10.5. (shrink)

In this article, we address fallacious analogical reasoning and the Metaphoric Fallacy to a Deductive Inference (MFDI), recently discussed by B. Lightbody and M. Berman (2010). We claim that the authors’ proposal to introduce a new fallacy is only partly justified. We also argue that, in some relevant cases, fallacious analogical reasoning involving metaphors is only affected by the use of quaternio terminorum.

Although Peano’s negative attitude toward infinitesimals—particularly, geometric infinitesimals—is widely documented, his conception of a single infinite cardinality and, more generally, his views on the infinite are less known. In this article, we reconstruct the evolution of Peano’s ideas on these questions and formulate several hypotheses about their underlying motivations.

Historically, mathematics has often dealt with the ‘expansion’ of previously accepted concepts and notions. In recent years, Buzaglo has provided a formalisation of concept expansion based on forcing. In this paper, I briefly review Buzaglo’s logic of concept expansion and I apply it to Cantor’s ‘creation’ of the transfinite. I argue that, while Buzaglo’s epistemological considerations fit well into Cantor’s conceptions, Buzaglo’s logic of concept expansion might be unsuitable to justify the creation of the transfinite in terms of a logically (...) rigorous derivation of concepts. (shrink)

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how `intuitively plausible' an axiom is, whereas extrinsic justification supports an axiom by identifying certain `desirable' consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor (...) sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the ‘core hypothesis’. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large Cardinals). In (...) the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV’s evidential framework. (shrink)

We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending $\mathsf {ZFC}$ by using his multiverse axioms $\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for $\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of $\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of $\mathsf {MV}_T$ (where (...) T is $\mathsf {ZFC}$ +Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of $\mathsf {ZFC}$. To this end, we take into account several strategies, and assess their prospects in the light of $\mathsf {MV}$ ’s evidential framework. (shrink)

The iterative concept of set is standardly taken to justify ZFC and some of its extensions. In this paper, we show that the maximal iterative concept also lies behind a class of further maximality principles expressing the maximality of the universe of sets V in height and width. These principles have been heavily investigated by the first author and his collaborators within the Hyperuniverse Programme. The programme is based on two essential tools: the hyperuniverse, consisting of all countable transitive models (...) of ZFC, and V -logic, both of which are also fully discussed in the paper. (shrink)

In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows (...) how optimal principles, among those available, may be selected in a justifiable way. (shrink)

The Hyperuniverse Programme, introduced in Arrigoni and Friedman :77–96, 2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the ‘maximal iterative concept’, and the programme identifies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...) in countable transitive models, the collection of which constitutes the ‘hyperuniverse’, has remarkable first-order consequences, some of which we review in Sect. 5. (shrink)

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni-Friedman :77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John (...) Templeton Foundation from January 2013 until September 2015. (shrink)

Extended simple regions have been the focus of recent developments in philosophical logic, metaphysics, and philosophy of physics. However, only a handful of works provides a rigorous characterization of an extended simple region. In particular, a recent paper in this journal defends a definition based on an extrinsic notion of least distance. Call it the Least Distance proposal. This paper provides the first assessment of it. It argues that Least Distance faces difficulties and drawbacks. The paper then goes on to (...) suggest a different proposal, the Diameter proposal that is able to handle such drawbacks and difficulties. (shrink)

The advent of Generative AI, particularly through Large Language Models (LLMs) like ChatGPT and its successors, marks a paradigm shift in the AI landscape. Advanced LLMs exhibit multimodality, handling diverse data formats, thereby broadening their application scope. However, the complexity and emergent autonomy of these models introduce challenges in predictability and legal compliance. This paper analyses the legal and regulatory implications of Generative AI and LLMs in the European Union context, focusing on liability, privacy, intellectual property, and cybersecurity. It examines (...) the adequacy of the existing and proposed EU legislation, including the Artificial Intelligence Act (AIA), in addressing the challenges posed by Generative AI in general and LLMs in particular. The paper identifies potential gaps and shortcomings in the EU legislative framework and proposes recommendations to ensure the safe and compliant deployment of generative models. (shrink)

The landscape of social theory has changed significantly over the three decades since the publication of Anthony Giddens and Jonathan Turner’s seminal Social Theory Today. Sociologists in the twenty-first century desperately need a new agenda centered around central questions of social theory. In Social Theory Now, Claudio E. Benzecry, Monika Krause, and Isaac Ariail Reed set a new course for sociologists, bringing together contributions from the most distinctive sociological traditions in an ambitious survey of where social theory is today (...) and where it might be going. The book provides a strategic window onto social theory based on current research, examining trends in classical traditions and the cutting edge of more recent approaches. From distinctive theoretical positions, contributors address questions about how social order is accomplished; the role of materiality, practice, and meaning; as well as the conditions for the knowledge of the social world. The theoretical traditions presented include cultural sociology, microsociologies, world-system theory and post-colonial theory, gender and feminism, actor network and network theory, systems theory, field theory, rational choice, poststructuralism, pragmatism, and the sociology of conventions. Each chapter introduces a tradition and presents an agenda for further theoretical development. Social Theory Now is an essential tool for sociologists. It will be central to the discussion and teaching of contemporary social theory for years to come. (shrink)

The article argues that AI can enhance the measurement and implementation of democratic processes within political parties, known as Intra-Party Democracy (IPD). It identifies the limitations of traditional methods for measuring IPD, which often rely on formal parameters, self-reported data, and tools like surveys. Such limitations lead to the collection of partial data, rare updates, and significant demands on resources. To address these issues, the article suggests that specific data management and Machine Learning (ML) techniques, such as natural language processing (...) and sentiment analysis, can improve the measurement (ML about) and practice (ML for) of IPD. The article concludes by considering some of the principal risks of ML for IPD, including concerns over data privacy, the potential for manipulation, and the dangers of overreliance on technology. (shrink)

The present article is framed within the biosemiotic glossary project as a way to address common terminology within biosemiotic research. The glossary integrates the view of the members of the biosemiotic community through a standard survey and a literature review. The concept of ‘semiotic threshold’ was first introduced by Umberto Eco, defining it as a boundary between semiotic and non-semiotic areas. We review here the concept of ‘semiotic threshold’, first describing its denotation within semiotics via an examination on the history (...) of the concept, its synonyms, antonyms, etymology, usage in other languages and context in which it is used. Then we present a general overview of the survey among researchers, analyzing the difference in responses for the concept of ‘lower semiotic threshold’ and related concepts. From the answers we also review the difference between the general usage of ‘semiotic threshold’ versus its specific use within biosemiotics, and attempt to make a general synthesis of the concept taking into account what we have learned from the survey and the literature review. (shrink)

Die Bibliotheca Teubneriana, established in 1849, has evolved into the world's most venerable and extensive series of editions of Greek and Latin literature, ranging from classical to Neo-Latin texts. Some 4-5 new editions are published every year. A team of renowned scholars in the field of Classical Philology acts as advisory board: Gian Biagio Conte (Scuola Normale Superiore di Pisa) James Diggle (University of Cambridge) Donald J. Mastronarde (University of California, Berkeley) Franco Montanari (Università di Genova) Heinz-Günther Nesselrath (Georg-August-Universität Göttingen) (...) Dirk Obbink (University of Oxford) Oliver Primavesi (Ludwig-Maximilians Universität München) Michael D. Reeve (University of Cambridge) Richard J. Tarrant (Harvard University) Formerly out-of-print editions are offered as print-on-demand reprints. Furthermore, all new books in the Bibliotheca Teubneriana series are published as eBooks. The older volumes of the series are being successively digitized and made available as eBooks. If you are interested in ordering an out-of-print edition, which hasn't been yet made available as print-on-demand reprint, please contact us:[email protected] All editions of Latin texts published in the Bibliotheca Teubneriana are collected in the online database BTL Online. (shrink)

This volume serves as a reference on the field of cognitive semantics. It offers a systematic and original discussion of the issues at the core of the debate in semiotics and the cognitive sciences. It takes into account the problems of representation, the nature of mind, the structure of perception, beliefs associated with habits, social cognition, autism, intersubjectivity and subjectivity. The chapters in this volume present the foundation of semiotics as a theory of cognition, offer a semiotic model of cognitive (...) integration that combines Enactivism and the Extended Mind Theory, and investigate the role of imagination as the origin of perception. The author develops an account of beliefs that are associated with habits and meaning, grounded in Pragmatism, testing his Narrative Practice Semiotic Hypothesis on persons with autism spectrum disorders. He also integrates his ideas about the formation of the theory of mind with a theory of subjectivity, understood as self-consciousness which derives from semiotic cognitive abilities. This text appeals to students, professors and researchers in the field. (shrink)

Accountability is a cornerstone of the governance of artificial intelligence (AI). However, it is often defined too imprecisely because its multifaceted nature and the sociotechnical structure of AI systems imply a variety of values, practices, and measures to which accountability in AI can refer. We address this lack of clarity by defining accountability in terms of answerability, identifying three conditions of possibility (authority recognition, interrogation, and limitation of power), and an architecture of seven features (context, range, agent, forum, standards, process, (...) and implications). We analyse this architecture through four accountability goals (compliance, report, oversight, and enforcement). We argue that these goals are often complementary and that policy-makers emphasise or prioritise some over others depending on the proactive or reactive use of accountability and the missions of AI governance. (shrink)

Much has been written about recidivist punishments, particularly within the area of criminology. However there is a notorious lack of penal philosophical reflection on this issue. This book attempts to fill that gap by presenting the philosopher’s view on this matter as a way of furthering the debate on recidivist punishments.

Studying the origin of semiosis is a task obscured by terminological and metaphysical issues which create an ambiguous set of definitions for biosemiotics when referring to the concept of emergence. The question is, how emergent can semiosis be? And what are the conditions for semiosis to be an emergent of a certain type? This paper will attempt to briefly deal with the general terminology of emergence from a philosophical point of view and will discuss the characterization of semiosis as an (...) emergent phenomenon based on the distinctions made by Bedau, Kim and Chalmers. Accordingly, we will consider the possibility of strong and weak emergence in an attempt to bring some clarity to what it means for something in biosemiotics to be an emergent and how the philosophical concepts play out when applied to biosemiotic research. In inquiring into the metaphysical status of semiosis, we change our semiotic theories to correspond to the assumptions contained in the elementary objects of our theories. This being the case, the way semiosis–the constitutive element that it is for semiotics–is taken to be with regards to its possible ontology, will conduct to different research objects for the long-term investigation of its origins and necessary conditions. (shrink)

In responding to a question posed by Professor Vincent Carraud regarding a passage in a letter by Etienne Pascal in which he mentions an author who speaks of the mysteries of Pythagorean numbers we have resumed our dossiers on number theory in Mersenne. We have pointed to texts from Mersenne’s correspondence, printed books, and unpublished theological manuscripts between 1638 and 1648 in which the mysteries of the Pythagorean numbers are mentioned with reference to the theory of perfect numbers and with (...) reference to Peter Bongo’s work, which was widespread throughout Europe and particularly in France, De numerorum mysteria (1599 ed. aucta). This has allowed us to provide some indications taken from Mersenne’s unpublished manuscripts in which he addresses the mathematical problem of perfect numbers in exegesis, according to the new dating of his unpublished theological manuscripts that we have established. (shrink)

This commentary draws upon the author's experience in bioethics and as a physician, ordained Rabbi, and certified Mohel (a Jewish professional qualified to perform infant male circumcisions (MC)). People's identity and adherence to a religious belief are frequently cited reasons for deciding whether to circumcise their male children. For Jewish and Muslim males, circumcision is considered essential. In this commentary, the author uses his medical, religious, and bioethical knowledge, expertise, and experience to address common arguments used in opposing nontherapeutic male (...) circumcision of minors. In these narratives, half of the parents agreed with circumcision, and half did not. The parents against circumcision cite human rights violations, security, privacy, and bodily integrity issues and refer to circumcision as genital mutilation, while those who agree argue that circumcision should be done for religious reasons to prevent health issues, promote hygiene and cleanness, avoid cancer, or for later sexual enjoyment. After evaluating the evidence, the author states that parents should be free to either consent or decline MC for a son. Though a reoccurring theme in these narratives was for mothers to leave the circumcision decision up to their male partners, the author concludes that women have considerable power regarding the decision. (shrink)