The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum (...) with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

Foundations of Science recently published a rebuttal to a portion of our essay it published 2 years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his 2005 book, Conflicts between generalization, rigor, and intuition. He further argues that our attempt to show that Cauchy is part of a long infinitesimalist tradition confuses text with context and thereby misunderstands the significance of Cauchy’s use of infinitesimals. Here we defend our original analysis of various misconceptions and misinterpretations concerning (...) the history of infinitesimals and, in particular, the role of infinitesimals in Cauchy’s mathematics. We show that Schubring misinterprets Proclus, Leibniz, and Klein on non-Archimedean issues, ignores the Jesuit context of Moigno’s flawed critique of infinitesimals, and misrepresents, to the point of caricature, the pioneering Cauchy scholarship of D. Laugwitz. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

In his 1887's Mitteilungen zur Lehre von Transfiniten, Cantor seeks to prove inconsistency of infinitesimals. We provide a detailed analysis of his argument from both historical and mathematical perspective. We show that while his historical analysis are questionable, the mathematical part of the argument is false.

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to (...) teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms. (shrink)

In this paper, we present a reinterpretation of Roman Ingarden's theory of intentional objects. There are four types of intentional objects in Ingarden's ontology, we offer a detailed analyses of an intentional object that is a correlate of a text. Such an object is characterised by Ingarden as a two-sided and schematised formation. We focus on the notion of schematism. We classify different interpretations of schematism and propose our own definition of schematism of a purely intentional object.

All through the XXth century it has been repeated that "there is an exact correspondence, almost coincidence between Euclid's definition of equal ratios and the modern theory of irrational numbers due to Dedekind". Since the idea was presented as early as in 1908 in Thomas Heath's translation of Euclid's Elements as a comment to Book V, def. 5, we call it in the paper Heath's thesis. Heath's thesis finds different justifications so it is accepted yet in different versions. In the (...) paper its historical and mathematical version is reconstructed. We next reconstruct Eudoxos' theory of proportions in an axiomatic fashion. Finally, we show that Heath's thesis both in the historical and mathematical version is false. To this end a counterexample is given; it is based upon a specific interpretation of the uniform distribution theorem. (shrink)

Galileo's paradox of infinity involves comparing the set of natural numbers, N, and the set of squares, {n2 : n ∈ N}. Galileo sets up a one-to-one correspondence between these sets; on this basis, the number of the elements of N is considered to be equal to the number of the elements of {n2 : n ∈ N}. It also characterizes the set of squares as smaller than the set of natural numbers, since ``there are many more numbers than squares". (...) As a result, it concludes that infinities cannot be compared in terms of greater--lesser and the law of trichotomy does not apply to them. Cantor's cardinal numbers provide a measure for sets. Cantor gives a definition of the relation greater–lesser between cardinal numbers and establishes the law of trichotomy for these numbers. Yet, when Cantor's theory is applied to subsets of N, it gives that any set can be either finite or of the power ℵ0. Thus, although the set of squares is the subset of N, they are of the same cardinality. Benci, Di Nasso introduces specific numbers to measure sets called numerosities. With numerosities, the following claim is true: numerosity of A < numerosity of B, whenever A ⊈ B. In this paper, we present a simplified version of the theory of numerosities that applies to subsets of N. This theory complies with Galileo's presupposition that when A ⊈ B, then the number of elements in A is smaller than the number of elements in B. Specifically, we show that as the numerosity of N is the number α, the numerosity of the set of squares is the integer part of the number √α, that is ⌊√α⌋, and the inequality ⌊√α⌋ < α holds. (shrink)

In this paper we show that the field of the real numbers is an intentional object in the sense specified by Roman Ingarden in his Das literarische Kunstwer and Der Streit um die Existenz der Welt. An ontological characteristics of a classic example of an intentional object, i.e. a literary character, is developed. There are three principal elements of such an object: the author, the text and the entity in which the literary character forms the content. In the case of (...) the real numbers the triad consists of Richard Dedekind, his work Steitigkeit und irratinale Zahlen, and the intentional object determined by this work. Showing that we are indeed faced with the inten-tional object we analyse three moments: two-sidedness of the formal structure of the intentional object, the moment of its existentional derivation, and schematism of the intentional object. Since there are many constructions of the real numbers known in mathematics, we show what relation obtains between Dedekind's and Cantor's constructions of the real numbers as these are taken as intentional objects. Moreover, we show what relation obtains between Dedekind's construction and the axiomatic characteristics of the real numbers. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Did Leibniz exploit infinitesimals and infinities à la rigueur or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Hidé Ishiguro defends the latter position, which she reformulates in terms of Russellian logical fictions. Ishiguro does not explain how to reconcile this interpretation with Leibniz’s repeated assertions that infinitesimals violate the Archimedean property (i.e., Euclid’s Elements, V.4). We present textual evidence from Leibniz, as well as historical evidence from the early decades of the calculus, to undermine Ishiguro’s (...) interpretation. Leibniz frequently writes that his infinitesimals are useful fictions, and we agree, but we show that it is best not to understand them as logical fictions; instead, they are better understood as pure fictions. (shrink)

In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under scrutiny. Following Ian Hacking, we point out that in the history of mathematics the amount of contingency is larger than is usually thought. As a case study, we analyze the historians’ approach to interpreting James Gregory’s expression ultimate terms in his paper attempting to prove the irrationality of \. Here (...) Gregory referred to the last or ultimate terms of a series. More broadly, we analyze the following questions: which modern framework is more appropriate for interpreting the procedures at work in texts from the early history of infinitesimal analysis? As well as the related question: what is a logical theory that is close to something early modern mathematicians could have used when studying infinite series and quadrature problems? We argue that what has been routinely viewed from the viewpoint of classical analysis as an example of an “unrigorous” practice, in fact finds close procedural proxies in modern infinitesimal theories. We analyze a mix of social and religious reasons that had led to the suppression of both the religious order of Gregory’s teacher degli Angeli, and Gregory’s books at Venice, in the late 1660s. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

Topic modeling—a text‐mining technique often used to uncover thematic structures in large collections of texts—has been increasingly frequently used in the context of the analysis of scholarly output. In this study, we construct a corpus of 19,488 texts published since 1971 in seven leading journals in the field of bioethics and philosophy of medicine, and we use a machine learning algorithm to identify almost 100 topics representing distinct themes of interest in the field. On the basis of intertopic correlations, we (...) group the content‐based topics into eight clusters, thus providing a novel, fine‐grained intellectual map of bioethics and philosophy of medicine. Moreover, we conduct a number of diachronic analyses, examining how the “prominence” of different topics has changed across time. In this way, we are able to observe the distinct patterns in which bioethics and philosophy of medicine have evolved and changed their focus over the past half a century. (shrink)

It is proved that the following conditions are equivalent: (a) there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, (b) there exists a nowhere dense ultrafilter on ω. Therefore, the existence of such algebras is undecidable in ZFC. In "forcing language" condition (a) says that there exists a non-trivial σ-centered forcing not adding Cohen reals.

Predictive processing (PP) has been repeatedly presented as a unificatory account of perception, action, and cognition. In this paper, we argue that this is premature: As a unifying theory, PP fails to deliver general, simple, homogeneous, and systematic explanations. By examining its current trajectory of development, we conclude that PP remains only loosely connected both to its computational framework and to its hypothetical biological underpinnings, which makes its fundamentals unclear. Instead of offering explanations that refer to the same set of (...) principles, we observe systematic equivocations in PP‐based models, or outright contradictions with its avowed principles. To make matters worse, PP‐based models are seldom empirically validated, and they are frequently offered as mere just‐so stories. The large number of PP‐based models is thus not evidence of theoretical progress in unifying perception, action, and cognition. On the contrary, we maintain that the gap between theory and its biological and computational bases contributes to the arrested development of PP as a unificatory theory. Thus, we urge the defenders of PP to focus on its critical problems instead of offering mere re‐descriptions of known phenomena, and to validate their models against possible alternative explanations that stem from different theoretical assumptions. Otherwise, PP will ultimately fail as a unified theory of cognition. (shrink)

Samuel J. Wheeler defends Wittgenstein's criticism of Cantor's set theory against the objections raised by Hilary Putnam. Putnam claims that Wittgenstein's dismissal of the basic tenets of this set theory concerning the noncountability of the set of real numbers was unfounded and ill‐conceived. In Wheeler's view, Putnam's charges result from his failure to grasp Wittgenstein's intention and, in particular, to consider the difference between empirical and logical impossibility. In my paper, I argue that Wheeler's defence is unsuccessful and, at the (...) same time, that Putnam's objections go too far. (shrink)

Citizen identification systems (known also as ‘ID card systems’, or ‘national identity management systems’, even though those definitions are not identical) are receiving a mixed acceptance, with their privacy, security and usability being criticised, specifically in the UK. This paper investigates whether it is possible to improve social acceptance of such systems in cases where they are incompatible with the perceived value of privacy, but without significantly changing their original architecture. The paper analyses requirements using four different scenarios that address (...) long-term privacy issues. Relatively small alterations to such systems are suggested that may significantly improve their adoption. (shrink)

The article presents a thought experiment aimed at indicating a possibility for thinking education beyond the logic of progress. In its first part, the argument reconstructs the entanglement of the modern idea of progress (as found in Francis Bacon and Comenius) and education, while tracking down the specific coupling of obedience and conquest at work. Through such an analysis a link between the ideas of progress and of emancipation is determined, which leads to the acknowledgement of the difficulty of the (...) task of imagining education outside of the logic of progress. In the second part an attempt is made to match this task by suggesting that when studying with a teacher, the logic of progress is deactivated. A phenomenological analysis of the practices involved in studying with a teacher points to a specific way of living together that such a collective study involves. This way of living together is formed by continual exercises in the humble equality realized through shared attention to something worthy of study: this stems from attentiveness to what exists, and it develops an attitude of care and respect for being as such. (shrink)

This paper discusses differences between two major schools in philosophy of criminal law, retributivism and consequentialism, with regard to the risk of punishing the innocent. As it is argued, the main point of departure between these two camps in this respect lies in their attitude towards the high evidentiary threshold in a criminal trial: while retributivism seems to strongly support setting this standard high, consequentialists may find it desirable to relax it in some cases. This discussion is set in the (...) context of proxy criminalization, i.e. a situation, in which some suspicious behaviour is criminalized. Since proxy criminalization may be understood as an effective lowering of the evidentiary threshold, its employment is justifiable from the consequentialist perspective, while being highly problematic for the retributivists. (shrink)

This theoretical paper is offered in the spirit of advancing the debate on the socioemotional wealth construct and its impact on how family firms conceptualize and practise corporate social responsibility. The study builds on Kellermanns et al.’s :1175–1182, 2012) claim that the SEW dimensions can be positively and negatively valenced as well as makes a distinction between the selective and instrumental approach to CSR and the holistic and normative one. Drawing on these considerations, it provides a theoretical underpinning in favour (...) of the view that SEW has ambivalent nature and therefore can produce detrimental outcomes for stakeholders of family companies. In this way, the study challenges the implicit assumption prevalent in the literature that SEW is “a prosocial and positive stimulus”. Crucially, it expands on the SEW construct by arguing that, given its ambivalent nature, SEW, as such, is at odds with the “strategic, whole-business view of responsibility”. Consequently, it posits that family firms—because of their concern with SEW—may be more likely to adopt the instrumental and selective rather than strategic and normative approach. Hence, it also makes the case for regarding the latter as a reference point to investigate the family company’s attitude towards social responsibility. It concludes by summarising the argument and offering future research avenues. (shrink)

The paper discusses a relatively underexamined element of Wittgenstein’s philosophy of mathematics associated with his critique of set theory. I outline Wittgenstein’s objections to the theories of Dedekind and Cantor, including the confounding of extension and intension, the faulty definition of the infinite set as infinite extension and the critique of Cantor’s diagonal proof. One of Wittgenstein’s major objections to set theory was that the concept of the size of infinite sets, which Cantor expressed by means of symbols אₒ and (...) c, had no application, i.e. that there was no grammatical technique that could show how such expressions were to be used. Notions of set theory are, so to speak, exterior – they find themselves outdoors, outside of what we usually do. They form a discourse that takes us beyond the horizon of everydayness and commonality. They are like an engine idling of the language of mathematics. (shrink)

The paper presents an analysis of the anumāna chapter of Jayarāśi’s Tattvôpaplava-siṁha and the nature of his criticism levelled against the anumāna model. The results of the analysis force us to revise our understanding of Jayarāśi Bhaṭṭa as a sceptic. Instead, he emerges as a highly critical philosopher. In addition, the nature of Jayarāśi’s criticism of the anumāna model allow us to conclude that anumāna should not be equated with inference, but rather is its limited subset, and may at best (...) be rendered as ‘disputational inference’, ‘debational inference’ or even ‘dialogical inference’. Jayarāśi applies a range of logical laws which clearly represent patterns of what can be classified as a priori reasoning and analytical justifications for knowledge, which were traditionally not reckoned sound. Against the backdrop of Jayarāśi’s criticism of anumāna, the paper also attempts to provide an explanation to why Indian philosophy and logic did not develop any concept of proper symbols and variables. (shrink)

The paper examines the Tattvôpaplava-siṁha of Jayarāśi Bhaṭṭa, and presents an analysis of his positive arguments that can be traced in the work. Despite the widely held opinion that Jayarāśi was a sceptic or held no positive opinions, the author concludes that, first, Jayarāśi does not fit a standard description of a sceptic. What may appear as an approach to philosophical problems, typical of a sceptic, turns out to be Jayarāśi’s particular method of critical examination on the part of a (...) rationalist. Second, a number of positive views Jayarāśi entertains can be identified in his work, and most of them overlap with much of the doctrine of the Cārvākas and Lokāyatas and materialist tradition recorded as early as the Sāmañña-phala-sutta. Therefore, Jayarāśi should be classified as a representative of the Cārvāka/Lokāyata tradition. (shrink)

Locke’s theory of property is irreconcilable with intellectual property. Property-like titles in ideal objects cannot be introduced within the framework of the natural law, because they could constrain others from acts necessary for their survival. Nevertheless, followers of Locke’s theory of politics choose to belittle this conclusion and even Locke himself supported early copyright legislation. The inconsistency is important, for it depicts the problem of legitimization of intellectual property as political and demonstrates liberal reification of various aspects of social life.

The article is an introduction to the issue of Polish literature written in Germany after 1989. The first part of the work focuses around the current literary discussions on defining and developing the terminology and methods enabling research into this new literature – written abroad after the fall of the Iron Curtain. Post­-emigration literature, migration literature, minority literature, multi­-, inter­-, trans­-cultural and diasporic literature – these are only a few of the suggested terms. The second part of the article refers (...) to the works of selected representatives of Polish literature written after 1989 and the issues they discuss in their works. The purpose of the article is to briefly present the complexity of an important problem in the views of Polish, German and Slavic philologists. At the same time, it aims to show that study of the phenomenon in question is still insufficient. (shrink)

Artykuł stanowi krytyczne omówienie książki Kevina Duttona zatytułowanej Mądrość psychopatów. Lekcja życia pobrana od świętych, szpiegów i seryjnych morderców. Celem pracy jest przybliżenie zasadniczych tez stawianych przez autora oraz zaproszenie do namysłu nad psychopatią – zarówno jako fenomenem naukowym, jak i egzystencjalnym.

Blumenthal-Barby and her colleagues (2022) situate their discussion of philosophy and bioethics in the context of (reportedly) widely held assumption that, when compared to the early days of bioethics, the role of philosophy is now diminished across the field – the assumption we call the Disconnection Thesis. This assumption can be summarized, to use the authors’ own words, by the phrase “philosophy’s glory days in bioethics are over“. While in no place of the article they explicitly endorse the Disconnection Thesis, (...) at least some of the authors had previously endorsed it in print (Savulescu 2015). Such expressions of collective expert wisdom might be a valuable source of information on the discipline's history, but they should not be accepted uncritically. Given the explosion in the size and scope of bioethical research in recent decades, any scholar’s familiarity with the area is necessarily based on selective reading and might be biased. Hence, in this commentary, we examine what kind of more rigorous evidence could corroborate the Disconnection Thesis. In other words, if the role of philosophy in bioethics has been indeed diminishing, what kind of observable patterns should we expect to see? Drawing on our previous research (Bystranowski, Dranseika, Żuradzki 2022), here we focus on two useful perspectives: citation analysis and topic modeling. While the first approach allows us to indirectly measure the level of engagement of bioethicists with philosophical literature (by measuring the proportion of references from articles published in journals in bioethics that cite philosophy journals), the latter provides a window into the content and argumentative style of bioethical texts. (shrink)