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.

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)

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)

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.

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)

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)

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)

The aim of this paper is to investigate Nishida Kitarō’s way of philosophizing in the light of the concept of “transition” in order to deepen our understanding of both Nishida’s philosophy and our thinking about and in transitions, using the concept of “boundary” or “border” (Grenze) as a catalyst. For that purpose, we focus on Nishida’s essay “Place” (「場所」), passing through different parts of the text as if through successive gates on a path of transition between one place and the (...) next, until we reach the final place of “absolute nothingness.” Dwelling on this place, we turn our attention to its internal structure and try to depict it along the outlines of a boundary, following the movements taking place in Nishida’s essay. The second part proposes an interpretation of the place of nothingness as an interminable practice of boundary-crossing that doesn’t come to a halt in a final, all-encompassing place, but dynamically situates itself on countless intersecting planes. After a more or less abstract analysis of the concept of “boundary,” we will apply and concretize this approach by using the example of the skin. To this end, we expose five main features of the skin as boundary: permeability, enclosure, excessiveness, interstitiality and reciprocal self-formation. (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)

Despite the remarkable advances in behavioral and brain sciences over the last decades, the mind/body (brain) problem is still an open debate and one of the most intriguing questions for both cognitive neuroscience and philosophy of mind. Traditional approaches have conceived this problem in terms of a contrast between physicalist monism and Cartesian dualism. However, since the late sixties, the landscape of philosophical views on the problem has become more varied and complex. The Multiple Realization Thesis (MRT) claims that mental (...) properties can be (or are) realized, and mental processes can be (or are) implemented by neural correlates of different kinds. Thus, MRT challenges the psychoneural type-identity theory and the corresponding reductionism. Many philosophers have acknowledged the a priori plausibility of MRT. However, the existence of empirical evidence in favor of intraspecific, human multiple realizations of mental processes and properties is still controversial. Here we illustrate some cases that provide empirical evidence in support of MRT. Recently, it has been proposed that foveal agnosic vision, like peripheral vision, can be restored by increasing object parts’ spacing (Strappini et al., 2017a; Crutch and Warrington, 2007). Agnosic fovea and normal periphery are both limited by crowding, that impairs object recognition, and provides the signature of visual integration. Here we define a psychological property of restored object identification and we cross-reference the data of visually-impaired patients with different etiologies. In particular, we compare the data of two stroke patients, two patients with posterior cortical atrophy (PCA), six cases of strabismic amblyopia, and one case with restored sight. We also compare these patients with unimpaired subjects tested in the periphery. We show that integration (i.e. restored recognition) seems to describe quite accurately the visual performance in all these cases. While the patients have different etiologies and different neural correlates, the unimpaired subjects have no neural damage. Thus similarity in the psychological property given the differences in the neural substrate can be interpreted in relation to MRT and provide evidence in its support. Finally, we will frame our contribution within the current debate concerning MRT providing new and compelling empirical evidence. (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)

What is a gene? Does it represent a natural kind, or is it just a tool for genomics? A clear answer to these questions has been challenged by postgenomic discoveries. In response, I will argue that the gene can be deemed a natural kind as it satisfies some requirements for genuine kindhood. Specifically, natural kinds are projectible categories in our best scientific theories, and they represent nodes in the causal network of the world (as in Khalidi. Natural Categories and Human (...) Kinds: Classification in the Natural and Social Sciences. Cambridge University Press, Cambridge, 2013; Khalidi. Synthese 195: 1379–1396, 2018; Khalidi. Are Sexes Natural Kinds, In: Dasgupta S, Weslake B (eds) Current Controversies in Philoso-phy of Science. Routledge, New York, 2020; Khalidi. Philos Sci 88:1–21, 2021). In Sect. 2, I will present a brief history of the gene and the controversy over its status. In Sect. 3, I will introduce the account of natural kinds considered in this paper. In Sect. 4, I will first present the relevant definition of genes and how they can be classified. Then, I will argue that the gene can be considered a natural kind as it satisfies the criteria for natural kindhood. Section 5 concludes. (shrink)

Grammar as a discipline devoted to the study of language was greatly advanced by the Alexandrian philologists, and especially by Aristarchus, as demonstrated by Stephanos Matthaios. In order to edit Homer and other literary authors, whose texts were often written in archaic Greek and presented many linguistic problems, the Alexandrians had to recognize linguistic grammatical categories and declensional patterns. In particular, to determine the correct orthography or accentuation of debated morphological forms they often employed analogy, which is generally defined as (...) the doctrine that grammatical forms must follow strict rules of declension. Modern scholars have often opposed the Alexandrian doctrine of analogy to the Pergamene doctrine of ‘anomaly’, which favoured spoken usage to determine debated forms. Detlev Fehling and David Blank, however, have shown that this strong opposition never really existed and it is mostly due to Varro. More correctly, ancient grammarians identified inflectional rules as well as forms derived from spoken usage or otherwise aberrant forms—however, respect for spoken usage in the latter case was not labelled ‘anomaly’, which was never a technical term of ancient grammar. Rather, and especially in the Roman period, grammarians used the term ‘pathology’ to account for and explain irregular forms. (shrink)

Sociopolitical attitudes are often the root cause of conflicts between individuals, groups, and even nations, but little is known about the origin of individual differences in sociopolitical orientation. We test a combination of economic and evolutionary ideas about the degree to which the mating market, sex, age, and income affect sociopolitical orientation. We collected data online through Amazon’s Mechanical Turk from 1108 US participants who were between 18 and 60, fluent in English, and single. While ostensibly testing a new online (...) dating website, participants created an online dating profile and described people they would like to date. We manipulated the participants’ popularity in the mating market and the size of the market and then measured participants’ sociopolitical attitudes. The sociopolitical attitudes were reduced to five dimensions via Principal Components Analysis. Both manipulations affected attitudes toward wealth redistribution but were largely not significant predictors of the other dimensions. Men reported more unrestricted sociosexual attitudes, and more support for benevolent sexism and traditional family values, than women did, and women supported wealth redistribution more than men did. There was no sex difference in accepting nonconforming behaviors. Younger people and people with lower incomes were more liberal than older people and people with higher incomes, respectively, regardless of sex. Overall, effects were largely not interactive, suggesting that individual differences in sociopolitical orientation may reflect strategic self-interest and be more straightforward than previously predicted. (shrink)

The global spread of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) or coronavirus disease 19 (COVID-19) has led to the imposition of severely restrictive measures by governments in the Western hemisphere. We feel a contrast between these measures and our freedom. This contrast, we argue, is a false perception. It only appears to us because we look at the issue through our contemporary moral philosophy of utilitarianism and an understanding of freedom as absence of constraints. Both these views can be (...) substituted with more sophisticated alternatives, namely an ethics of virtue and a notion of freedom of the will. These offer a fuller picture of morality and enable us to cooperate with the current restrictions by consciously choosing to adhere to them instead of perceiving them as draconian and immoral. We ask whether we should collaborate with the restrictions and argue that considerations of virtue will lead to an affirmative answer. More broadly, virtue ethics permits to deal with the practical concerns about how an individual should behave during this pandemic, given the current lockdown measures or lack thereof. -/- In section 1, we present how utilitarianism and a notion of freedom as negative liberty support the opposition to restrictive measures. In section 2, we outline an alternative based on an ethics of virtue and a more elaborated notion of free will. In the concluding section 3, we argue that considerations of virtue should guide the individual and public response to the emergency. (shrink)

The purpose of this randomized controlled study was to investigate the impact of a 12-week physical education program on the self-efficacy of overweight schoolgirls. We randomly assigned 60 overweight schoolgirls to either an experimental moderate to vigorous aerobic exercise group or a control group that received non-specific regular PE lessons with activities chosen by the curricular teacher mainly focused on team games and sports skills that aimed to achieve general psycho-physical wellness. To assess the starting level of students and significant (...) changes reached, at baseline and after training, a battery of standardized assessment motor tests and a psychometric scale were administered. At the end of the intervention, the experimental group reported a considerable decrease in body mass index and a large improvement in self-efficacy. No significant changes were found in the CG. The results suggested that the 12-week moderate to a vigorous aerobic exercise program is an effective weight loss intervention and a vehicle to promote a range of outcomes important to the qualitative growth of adolescents. In fact, it could provide a positive and significant impact on the self-efficacy of overweight schoolgirls. (shrink)

This paper is a critical response to Andreas Bartels’ sophisticated defense of a structural account of scientific representation. We show that, contrary to Bartels’ claim, homomorphism fails to account for the phenomenon of misrepresentation. Bartels claims that homomorphism is adequate in two respects. First, it is conceptually adequate, in the sense that it shows how representation differs from misrepresentation and non-representation. Second, if properly weakened, homomorphism is formally adequate to accommodate misrepresentation. We question both claims. First, we show that homomorphism (...) is not the right condition to distinguish representation from misrepresentation and non-representation: a “representational mechanism” actually does all the work, and it is independent of homomorphism – as of any structural condition. Second, we test the claim of formal adequacy against three typical kinds of inaccurate representation in science which, by reference to a discussion of the notorious billiard ball model, we define as abstraction, pretence, and simulation. We first point out that Bartels equivocates between homomorphism and the stronger condition of epimorphism, and that the weakened form of homomorphism that Bartels puts forward is not a morphism at all. After providing a formal setting for abstraction, pretence and simulation, we show that for each morphism there is at least one form of inaccurate representation which is not accommodated. We conclude that Bartels’ theory – while logically laying down the weakest structural requirements – is nonetheless formally inadequate in its own terms. This should shed serious doubts on the plausibility of any structural account of representation more generally. (shrink)

Cryonics—also known as cryopreservation or cryosuspension—is the preservation of legally dead individuals at ultra-low temperatures. Those who undergo this procedure hope that future technology will not only succeed in reviving them, but also cure them of the condition that led to their demise. In this sense, some hope that cryopreservation will allow people to continue living indefinitely. This book discusses the moral concerns of cryonics, both as a medical procedure and as an intermediate step toward life extension. In particular, Minerva (...) analyses the moral issues surrounding cryonics-related techniques by focusing on how they might impact the individuals who undergo cryosuspension, as well as society at large. (shrink)

After preliminarily pointing to the undeniable differences between Jonas’ philosophical biology and Schelling’s philosophy of nature, I contend that, besides their divergencies, the two philosophers agree on several important points. I then show to what extent, based on these elements of convergence, their two approaches could even be taken as complementary. In the core of my paper I lay emphasis on what I believe to be the main ground for the complementarity of the two philosophical inquiries, that is to say, (...) their common radicalization and superseding of Kant’s principle of self-organization. In this respect – all differences considered – they both can be said to delineate a philosophy of the organic that can fruitfully contribute, even better than Kant’s input, to the currently widely discussed topic of biological autonomy. (shrink)