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)

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)

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)

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 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.

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.

This article focuses on Bob Zajonc’s views on unconscious emotion, especially in the context of the debates about the independence of affect and cognition. Historically, Bob was always interested in the “mere”—basic, fundamental processes. His empirical demonstrations of precognitive and preconscious emotional processes, combined with his elegant expositions of them, sharply contrasted with cold and complex cognitive models. Interestingly, Bob tended to believe that whereas the causes of emotion can be unconscious, the emotional state itself tends to be conscious. However, (...) he reconsidered this assumption and in his later work showed that subjects in affective priming experiments do not experience conscious affect, but instead act on basic preferences. Today, Bob’s insights continue to inspire research on “unconscious emotion.”. (shrink)

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)

This two-volume collection showcases a wide range of modern approaches to the philosophical study of language. Contributions illustrate how these strands of research are interconnected and show the importance of such a broad outlook. The aim is to throw light upon some of the key questions in language and communication and also to inspire, inform, and integrate a community of researchers in philosophical linguistics. Volume one concentrates on fundamental theoretical topics. This means considering vital questions about what languages are and (...) how they relate to reality, and describing some of the key areas of thought in linguistics and the philosophy of language. Contributors also discuss how philosophy influences related fields such as translation, pragmatics, and argumentation. (shrink)

Teologia już niejednokrotnie składała współczesnej myśli filozoficznej obietnicę poszerzenia horyzontów czy też otwarcia jej na transcendencję. Jednak obietnica ta nie spotkała się – jak dotąd – z przychylnym przyjęciem, gdyż teologia sama nie zdołała przekonać filozofii o swojej racjonalności. Filozofia przestała być bowiem, z jednej strony, wrażliwa na argumentację natury metafizycznej, z drugiej zaś, teologia jest skłonna poddawać się metodologiom nauk szczegółowych, rezygnując przy tym z własnej odrębności. Zarysowana tu hermeneutyka teologii jest w pierwszym rzędzie badaniem teologii jako nauki, próbą (...) uprawomocnienia jej na gruncie jednego z najważniejszych kierunków filozofii współczesnej, mianowicie hermeneutyki filozoficznej. Teologia zostaje tu ujęta w aspekcie immanencji, choć bynajmniej nie zamyka się w jej horyzoncie. Dziejący się świat teologii rozpada się na „kawałki” – miejsca teologiczne – które jednak wchodząc ze sobą w interakcje, uczestniczą w grze teologicznej. Jako pole owej gry zostaje wskazana liturgia, miejsce metodologicznie wyróżnione ze względu na naoczność dokonujących się konsekwencji gry teologicznej. Analiza czerpie z takich źródeł jak teoria miejsc teologicznych Melchiora Cano OP, koncepcja kręgu hermeneutycznego Martina Heideggera oraz projekt hermeneutyki filozoficznej Hansa-Georga Gadamera, ze szczególnym uwzględnieniem roli prze(d)sądów, utożsamianych tutaj z miejscami teologicznymi. Nowość spojrzenia na tradycję katolicką kontekście myśli hermeneutycznej sprawia, że prezentowana książka może trafić nie tylko do teologów i filozofów, ale także do badaczy szeroko rozumianej kultury. (shrink)

This book is a manifesto of reformed Stoicism. It proposes a system of life which is bullet-proof, universal, viable and effective in every cosmic setting. It holds in every possible universe, under any government and within any economic system. We can be reformed Stoics no matter what we believe in. Reformed Stoicism is about enjoying and exercising our agency. In other words, it's about the flow of making autonomous and right decisions, and about celebrating our ability to make them. With (...) no reliance on nature, with the recalibration of metaphysical positions, with skepticism towards grand discourses and universal answers, with an emphasis on the usefulness instead of truthfulness of narratives, with no reference to the vanity argument, with criticism of both conservative and ascetic misinterpretations of Stoicism, with an overall softer and more empathic approach, we can no longer be defined by the generic term "Stoicism". Our time, in short, calls for a fresh interpretation of Stoicism. It is time for a new generation of Stoics. Thus: reformed Stoicism. (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.

Spanning the period between Wittgenstein's return to Cambridge in 1929 and the first version of Philosophical Investigations in 1936, Piotr Dehnel explores the middle stage in Ludwig Wittgenstein's philosophical development and identifies the major issues which engrossed him, including phenomenology, philosophy of mathematics and philosophy of language. Contrary to the dominant perspective, Dehnel argues that this period was intrinsically different from the early and late stages and should not be viewed as a mere transitional phase. The distinctiveness of Wittgenstein's (...) middle work can be seen in his philosophical thinking as it unfolds in a non-linear trajectory: thoughts do not follow upon each other, ideas do not appear sequentially one by one, and insights do not form a straight chain. Dehnel portrays the diffused and multifarious quality of Wittgenstein's middle thinking, enabling readers to form a more comprehensive view of his entire philosophy and acquire a better grasp of his conceptual trajectory, complete with the intricacies and challenges that it poses. (shrink)

Introduction. PHilosophy of Language and Linguistics: The Formal Turn Piotr Stalmaszczyk Gottlob Frege, Philosophy of Language, and Predication Piotr Stalmaszczyk Philosophy, Linguistics and Semantic Interpretation Christian Bassac An Unresolved Issue: Nonsense in Natural Language and Non-Classical Logical and Semantic Systems Elzbieta Chrzanowska-Kluczewska Varieties of Context-Dependence Tadeusz Ciecierski The Logos of Semantic Structure Marie Du í, Bjørn Jespersen and Pavel Materna The Good Samaritan and the Hygienic Cook: A Cautionary Tale About Linguistic Data Chris Fox The Meaning of Multiple (...) Quantified Sentences: Where Formal Semantics Meets Psycholinguistics Justyna Grudzinska The Hybrid Theory of Reference for Proper Names Filip Kawczynski On the Nature of Statistical Language Laws Agnieszka Kulacka Vagueness and Contextualism Joanna Odrowaz-Sypniewska The Myth of Semantic Structure Jaroslav Peregrin Scalar Implicatures, Communication and Language Evolution Salvatore Pistoia Reda Semantics and Contextuality: The Case of Pia s Leaves Stefano Predelli Is Logico-Semantical Analysis of Natural Language Expressions a Translation? Jirí Raclavský Beyond the Fregean Myth: The Value of Logical Values Fabien Schang Modal Calculus of Illocutionary Logic Andrew Schumann Subjectivity in Philosophy and Linguistics Barbara Sonnenhauser Asymmetrical Semantics Mieszko Talasiewicz Truth: An Anti-realist Adequacy Condition Luca Tranchini Belief Reports: Defaults, Intentions and Scorekeeping Giacomo Turbanti On Truth in Time Bartosz Wieckowski Order William J. SUllivan Indexes. (shrink)

In its attempt to come to grips with the nature of the human mind idealism employs such terms as "pure self," "transcendental apperception," "pure con sciousness" and so on.

As the concept of recognition shifts from philosophical theory to other fields of the humanities and social sciences, this volume explores the nature of this border category that exists in the space between sociological and philosophical considerations, related as it is to concepts such as status, prestige, the looking-glass self, respect, and dignity - at times being used interchangeably with these terms. Bringing together work from across academic disciplines, it presents theoretical conceptualizations of recognition, demonstrates its operationalization in historical and (...) literary research, considers recognition as a fundamental problem of sociological theory and examines the concept as a marker of social distances and redistribution. An examination and demonstration of the full potential of recognition as a category, Understanding Recognition: Conceptual and Empirical Studies explores the contemporary meanings and manifestations of recognition and sheds light on its capacity to complement the notions of status, class or prestige. As such, it will appeal to scholars of sociology and social theory, philosophy, history and literary studies. (shrink)

In 2017, Michael Nair-Collins formulated his Transitivity Argument which claimed that brain-dead patients are alive according to a concept that defines death in terms of the loss of moral status. This article challenges Nair-Collins’ view in three steps. First, I elaborate on the concept of moral status, claiming that to understand this notion appropriately, one must grasp the distinction between direct and indirect duties. Second, I argue that his understanding of moral status implicit in the Transitivity Argument is faulty since (...) it is not based on a distinction between direct and indirect duties. Third, I show how this flaw in Nair-Collins’ argument is grounded in the more general problems between preference utilitarianism and desire fulfillment theory. Finally, I present the constructivist theory of moral status and the associated moral concept of death and explain how this concept challenges the Transitivity Argument. According to my view, brain death constitutes a valid criterion of death since brain death is incompatible with the preserved capacity to have affective attitudes and to value anything. (shrink)

Individuation -- Will -- Continuity -- Freedom.

Stoicism is coming back in a big way. Seen as a remedy for the craziness of the times we live in, it is experiencing a great surge in academic and cultural interest. Yet, can one live stoically and be a creative artist at the same time? Delving into its underlying tenets, obscure restrictions and limits of applicability, Stankiewicz critically explores Stoicism and its complex association with artistic creativity. Stoicism and artistic creativity are two great displays of the human spirit. Yet, (...) there are multiple reasons to suspect that they are at odds with each other. Popular culture encapsulates this problem in the figure of the rational, yet emotionally remote Stoic, who achieves serenity through withdrawal, and the contrasting figure of the “cursed poet,” “tormented artist,” or simply a rock star, who lives in a whirl of creative energy, yet falls short of quietude. Is this contrast valid? Other disciplines, including psychology, have explored this problem. But it has never been done philosophically. Pioneering in its philosophical approach, this book discusses how artistic creativity and its problems of identity, expression and self-creation serve as a great testing ground for Stoicism, as well as its theoretical challenges and practical limits. Stankiewicz presents a detailed investigation into the stereotypes of Stoic life that seeks to explain the cause of Stoicism’s modern revival. This book is an essential read for anyone captivated by Stoicism’s complex allure. (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 interdisciplinary study introduces readers to Friedrich Schleiermacher's diverse pathways of reflection and creative practice that are related to the field of translation. By drawing attention to Schleiermacher's various writings on a range of subjects (including philology, criticism, hermeneutics, dialectics, rhetoric and religion), the author makes it clear that the frequently cited lecture Über die verschiedenen Methoden des Übersetzens (On the Different Methods of Translating) represents but a fraction of Schleiermacher's contributions to modern-day insights into translation. The analysis of Schleiermacher's (...) various pathways of reflection on translation presented in this book leads to the conclusion that translation is part of the essence of the world, as it is a fundamental tool of our cognition and a foundation of our existence. In Schleiermacher's works, transfer, translation, mediation, and communication underpin our very existence in the world and our self-awareness. At the same time, they represent fundamental categories for a project that focuses on the consolidation and assimilation - through translation - of that which is foreign, different, diverse. (shrink)

The book is addressed to all readers interested in cognitive science, and especially in research combining a logical analysis with psychological, linguistic and neurobiological approaches. The publication is the result of a collaboration between the Department of Cognitive Science at University of Lodz and the Department of Cognitive Science at Lund University. It is intended to provide a comprehensive presentation of the key research issues undertaken in both Departments, including considerations on meaning, natural language and reasoning, linguistic as well as (...) numerical competence, the theory of decision making, modelling of conceptual representations, cognitive and game theoretic approach to social interactions. (shrink)