This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of (...) contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all properties of the numbers, achieving a key (...) neofregean advantage, while preserving the key structuralist advantage, which objects play the number role does not matter. (shrink)

Frege's definition of the real numbers, as envisaged in the second volume of Grundgesetze der Arithmetik, is fatally flawed by the inconsistency of Frege's ill-fated Basic Law V. We restate Frege's definition in a consistent logical framework and investigate whether it can provide a logical foundation of real analysis. Our conclusion will deem it doubtful that such a foundation along the lines of Frege's own indications is possible at all.

PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is radically (...) alternative to Frege’s and which is grounded on the existence of individuals rather than on the existence of concepts. (shrink)

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this paper, (...) we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction. (shrink)

PG (Plural Grundgesetze) is a predicative monadic second-order system which is aimed to derive second-order Peano arithmetic. It exploits the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. In this paper, a model-theoretical consistency proof for the system PG is provided.

PG (Plural Grundgesetze) is a predicative monadic second-order system which exploits the notion of plural quantification and a few Fregean devices, among which a formulation of the infamous Basic Law V. It is shown that second-order Peano arithmetic can be derived in PG. I also investigate the philosophical issue of predicativism connected to PG. In particular, as predicativism about concepts seems rather un-Fregean, I analyse whether there is a way to make predicativism compatible with Frege’s logicism.

PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence of (...) individuals rather than on the existence of concepts. (shrink)

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system (...) will be compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted. (shrink)

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system (...) will be compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted. (shrink)

Several natural languages such as English contain prima facie different kinds of referential and quantificational expressions. In particular, natural languages.

This book offers a plurality of perspectives on the historical origins of logicism and on contemporary developments of logicist insights in philosophy of mathematics. It uniquely provides up-to-date research and novel interpretations on a variety of intertwined themes and historical figures related to different versions of logicism. The essays, written by prominent scholars, are divided into three thematic sections. The first section focuses on major authors like Frege, Dedekind, and Russell, providing a historical and theoretical exploration of such figures in (...) the philosophical and mathematical milieu in which logicist views were first expounded. Section II sheds new light on the interconnections between these founding figures and a number of influential other traditions, represented by authors like Hilbert, Husserl, and Peano, as well as on the reconsideration of logicism by Carnap and the logical empiricists. Finally, the third section assesses the legacy of such authors and of logicist themes for contemporary philosophy of mathematics, offering new perspectives on highly debated topics--neo-logicism and its extension to accounts of ordinal numbers and set-theory, the comparison between neo-Fregean and neo-Dedekindian varieties of logicism, and the relation between logicist foundational issues and empirical research on numerical cognition--which define the prospects of logicism in the years to come. This book represents a comprehensive account of the development of logicism and its contemporary relevance for the logico-philosophical foundations of mathematics. It will be of interest to graduate students and researchers working in philosophy of mathematics, philosophy of logic, and the history of analytic philosophy. (shrink)

Farmers’ markets (FMs) represent a crucial player in urban food systems, being the interconnection of local agricultural production and consumption, and serving as spaces for both economic exchange and community building. Despite their transformative potential, there is a scarcity of research that comprehensively investigates the dynamics of FMs network structure and the influence of the actors shaping FMs. The present article delves into the network of relationships within FMs in the Italian city of Bologna. This study adopts the Social Network (...) Analysis (SNA) methodology applied with the Net-Map tool. The research objectives are to visualise the underlying network structure, map the dynamics, and identify the key actors who play pivotal roles in Bologna’s FMs and their level of influence. The research carried out interviews with ten FMs stakeholders, revealing the network of relationships between 54 actors, divided into three categories—civil society, food economy and public administration. Actors were linked by a total of 428 relationships across three types of networks: support, economic, and hindering. Findings indicate a strong support network structure characterised by varying degrees of centrality among different actors. Farmers emerge as a central node due to their pivotal role in providing fresh, local produce. Additionally, local institutions contribute significantly to FMs resilience and growth. Our research demonstrates the importance of recognising the embeddedness of FMs within the local context. By understanding the network structure and influential actors in FMs, policymakers can devise more effective policies for promoting local agriculture, and enhancing the sustainability of urban–rural exchanges. In conclusion, the present study offers valuable insights into the network dynamics of FMs, highlighting their crucial role in the sustainable development of urban and local food systems. (shrink)

In health communication, metaphor can be considered as a reasoning device to let people understand an abstract concept in terms of a concrete one (Lakoff and Johnson 1980; Bowdle and Gentner 2005). Both the positive and negative communicative effects of metaphors have been largely pointed out in a variety of medical fields, from oncology (Semino et al. 2016, 2018) to mental health (Frezza and Zoccolotti 2019). The use of metaphors in vaccine communication has been less considered, though it might be (...) crucial to let people understand vaccination as an important collective health phenomenon. A previous study (Scherer et al. 2015) focused on the communicative effects of conventional metaphors for vaccination, but not specifically on novel metaphors deliberately used to explain the concept of population immunity in vaccination. We hypothesized that metaphors used to explain vaccination as a collective phenomenon could improve the communicative aims of pro-vaccination texts in many respects, ranging from understandability to trust in expertise. We designed a study to test the hypothesis: two groups of participants were presented with a text about vaccination as a collective phenomenon, described in either literal or metaphorical terms. They were asked to evaluate the text on different communicative aspects from both an individual and a collective point of view. The results confirmed the hypothesis that metaphor is an effective reasoning device and that it also enhances the overall communicative effects of the message, in terms of understandability, persuasion, feeling of control over the health situation, trust in expertise, and uptake of experts’ advice. However, the results show that this effect is significantly nuanced in the individual vs. collective response type. (shrink)

This paper studies internal (or intra-)mathematical explanations, namely those proofs of mathematical theorems that seem to explain the theorem they prove. The goal of the paper is a rigorous analysis of these explanations. This will be done in two steps. First, we will show how to move from informal proofs of mathematical theorems to a formal presentation that involves proof trees, together with a decomposition of their elements; secondly we will show that those mathematical proofs that are regarded as having (...) explanatory power all display an increase of conceptual complexity from the assumptions to the conclusion. (shrink)

This volume provides an extensive overview of the Ethics of Artificial Intelligence for the Sustainable Development Goals. The authors are experts contributing with perspectives from different fields. The comprehensive collection of chapters illustrates the pressing governance problems related to using AI for the SDGs, and case studies describing how AI is advancing and can advance the achievement of the Goals. Students, scholars, and practitioners working on AI for SDGs, the ethical governance of AI, sustainability, and the fourth revolution can find (...) this book a helpful reference. (shrink)

This annual edited volume presents an overview of cutting-edge research areas within digital ethics as defined by the Digital Governance Research Group of the University of Oxford. It identifies new challenges and opportunities of influence in setting the research agenda in the field. The 2022 edition of the Yearbook presents research on the following topics: autonomous weapons, cyber weapons, digital sovereignty, smart cities, artificial intelligence for the Sustainable Development Goals, vaccine passports, and sociotechnical pragmatism as an approach to technology. This (...) text appeals to students, researchers, and professionals in the field. (shrink)

Criminal liability for acts committed by AI systems has recently become a hot legal topic. This paper includes three different contributions. The first contribution is an analysis of the extent to which an AI system can satisfy the requirements for criminal liability: accomplishing an actus reus, having the corresponding mens rea, possessing the cognitive capacities needed for responsibility. The second contribution is a discussion of criminal activity accomplished by an AI entity, with reference to a recent case involving an online (...) bot, the Random Darknet Shopper. This discussion will provide the context for the analysis of commonalities and differences between criminal activities by humans and by artificial systems. The third contribution concerns the evaluation of different ways of addressing criminal activities by AI systems in a regulatory perspective. (shrink)

Developmental System Theory is a theoretical reinterpretation of biological phenomena challenging the conventional gene-centered account of development and evolution. In this paper, I focus on Griffiths and Gray’s version of Developmental Systems Theory and I particularly analyze their reconceptualization of inheritance. First, I present their concept of expanded and diffused inheritance; then, I examine and criticize their refusal of the multiple inheritance system model; finally, I present and contrast Griffiths and Gray’s extension of what they call the “causal parity thesis” (...) from development to evolution. I argue that their proposal is an interesting and programmatic philosophical perspective on biological phenomena but, because of their commitment to holism, fails to provide both more heuristic tools for empirical investigation in biology and a more realistic representation of the biological world. (shrink)

Metaphor has been considered as a cognitive process, independent of the verbal versus visual mode, through which an unknown conceptual domain is understood in terms of another known conceptual domain. Metaphor might instead be viewed as a cognitive process, dependent on the mode, which leads to genuinely new knowledge via ignorance. First, I argue that there are two main senses of ignorance at stake when we understand a metaphor: we ignore some existing properties of the known domain in the sense (...) that we disregard or neglect them; we ignore some “non-existing” properties of the known domain in the sense that they are not a piece of information belonging to the known domain, but emerge in metaphor interpretation. Secondly, I consider a metaphor as a reasoning device, guiding the interpreters along a path of inferences to a conclusion, which attributes to the target some properties of the source. In this path, interpreters might discover the ignored existing properties of the known domain and/or recover the “non-existing” properties, inferring or imagining the missing piece of information. Finally, I argue that, especially in visual metaphors, this process is guided by a “sentiment of rationality”, tracking a disruption of existing familiar conceptualisations of objects and/or actions and a recovery of ignored properties. (shrink)

In recent years, bioethical discourse around the topic of ‘genetic enhancement’ has become increasingly politicized. We fear there is too much focus on the semantic question of whether we should call particular practices and emerging bio-technologies such as CRISPR ‘eugenics’, rather than the more important question of how we should view them from the perspective of ethics and policy. Here, we address the question of whether ‘eugenics’ can be defended and how proponents and critics of enhancement should engage with each (...) other. (shrink)