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)

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)

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.

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)

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.

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)

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)

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)

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)

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)

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 comment is analysing the last section of a paper by Piotr Blaszczyk, Mikhail G. Katz, and David Sherry on alleged misconceptions committed by historians of mathematics regarding the history of analysis, published in this journal in the first issue of 2013. Since this section abounds of wrong attributions and denouncing statements regarding my research and a key publication, the comment serves to rectify them and to recall some minimal methodological requirements for historical research.

During its century-long unfolding, spreading in numerous directions, Husserlian phenomenology while loosening inner articulations, has nevertheless maintained a somewhat consistent profile. As we see in this collection, the numerous conceptions and theories advanced in the various phases of reinterpretations have remained identifiable with phenomenology. What conveys this consistency in virtue of which innumerable types of inquiry-scientific, social, artistic, literary – may consider themselves phenomenological? Is it not the quintessence of the phenomenological quest, namely our seeking to reach the very foundations (...) of reality at all its constitutive levels by pursuing its logos? Inquiring into the logos of the phenomenological quest we discover, indeed, all the main constitutive spheres of reality and of the human subject involved in it, and concurrently, the logos itself comes to light in the radiation of its force (Tymieniecka). Papers by: Kristana Arp, Gary Backhaus, Mafalda Blanc, Piotr Blaszczyk, Manuel Bremer, Jonathan Lahey Dronsfield, Peter Abumhenre Egbe, Jesus Adrian Escudero, Wayne Froman, Jorge Garcia-Gomez, David Grunberg, Dasuke Kamei, Arion Kelkel, Filip Kolen, Tze-wan Kwan, Leonard Lawlor, Grahame Lock, Nancy Mardas, Nikolay Milkov, Cezary J. Olbromski, Helena De Preester, Anna-Teresa Tymieniecka, Gertrudis Van de Vijver, Luca Vanzago, Anatoly Zotov. (shrink)

In recent years there has been growing interest in adapting virtue ethics to the ethics of technology. However, it has most typically been invoked to address some particular issue of moral importance, and there is only a limited range of works dealing with the methodological question of how virtue ethics may contribute to this field. My approach in this paper is threefold. I start with a brief discussion of Aristotelian virtue ethics, with a view to constructing a framework in which (...) to then address at least some of the aspects of the growing technicization of life that call for scrutiny. I subsequently proceed further by outlining three possible models of virtue-theoretical analysis within the ethics of technology, giving special attention to those changes in human agency where technicization can be seen to exert a significant influence. Finally, I focus on the third such possibility, the extended virtue model, as a basis for calling for a more inclusive account of moral agency. (shrink)

The measure of trust that people vest in their fellow citizens or institutions depends on three factors: the `reflected trustworthiness' of the target as estimated by themselves in a more or less rational manner, the attitude of `basic trustfulness' deriving from socialization, and the `culture of trust' pervading their society and normatively encouraging the trusting orientation. The author presents a model of a structural context conducive for the emergence of the culture of trust, and then argues that democratic organization contributes (...) to trust-generating conditions, such as normative certainty, transparency, stability, and accountability. This influence is found to be doubly paradoxical. First, democracy breeds the culture of trust by institutionalizing distrust at many levels of democratic organization. And second, the strongest influence of democracy on the culture of trust may be expected when the institutionalized distrust remains a resource used sparingly and only when there appear significant breaches of trust. (shrink)

How should ethical standards be maintained within engineering and engineering education? The present paper addresses this question with relation to the dominant models of engineering ethics (EE) to show that their limits might be overcome by incorporating the vocabulary of neo-Aristotelian virtue ethics. On the basis of the MacIntyrean concept of practice, the secondary role of engineering is highlighted which echoes similar debates concerning education. This similarity is picked up to argue that the role of the engineer in relation to (...) the end-users of their projects should be understood as on a par with the teacher-student relationship. This enables us not only to redefine the internal good of engineering and the ground for EE, but also to indicate the key virtues and educational models for engineering that they should be key parts of. (shrink)

The aim of this paper is to analyze the basis for the moral obligation to remember. As the moral relation to the past is primarily a matter of shared identity, the kind of obligation in question splits into two related issues, namely, that of political, state-oriented and state-organized memory on which the political identity rests and that of memory labour grounded in social identities based in shared, time-extended projects. Drawing upon tensions between these two, I discuss time control and the (...) accumulation of identity as central to memory labour and, referring to John Zerzan’s critique of symbolical roots of power, pinpoint the moral basis of such an accumulation. On the basis of this, I argue for nesting the duty to remember in acknowledging the agent’s recognition of the relatedness and dependency of their agency and possibilities of flourishing which can be obtained thanks to adjusting the field of the virtue of practical wisdom so that it includes members of the time-extended community. (shrink)

In the paper, several data reduction techniques for machine learning from big datasets are discussed and evaluated. The discussed approach focuses on combining several techniques including stacking, rotation, and data reduction aimed at improving the performance of the machine classification. Stacking is seen as the technique allowing to take advantage of the multiple classification models. The rotation-based techniques are used to increase the heterogeneity of the stacking ensembles. Data reduction makes it possible to classify instances belonging to big datasets. We (...) propose to use an agent-based population learning algorithm for data reduction in the feature and instance dimensions. For diversification of the classifier ensembles within the rotation also, alternatively, principal component analysis and independent component analysis are used. The research question addressed in the paper is formulated as follows: does the performance of a classifier using the reduced dataset be improved by integrating the data reduction mechanism with the rotation-based technique and the stacking? (shrink)

Basing itself on Edith Stein's research into philosophy of psychology and the humanities, this article focuses particularly on discussions of the theory of association. Stein's approach, rooted in Husserlian phenomenological perspective, seems to represent a significant contribution to the establish of an intellectual framework for the exploration of the philosophy of consciousness, and also seems helpful for inquiries into the issue pf practical know-how pertaining to remembering things, as well as the use of schemata in intellectual activity generally.

The expression “moral experience”, along with the concomitant notion of experience itself, seems to have been understood in divergent ways. Taking as a background three views currently operative in our culture - emotivism, the ethics of duty, and the notion of an ethics “beyond good and evil” - a conception of ethical experience will be presented based on the findings of Edith Stein as elaborated in her work "Philosophy of Psychology and the Humanities".

This paper offers a critical review of the notion of a “singular cognitive role”, which is central to some recent theories of singular thought. According to those theories, whether a thought is singular depends on the role it plays in the subject’s cognitive activity. We compare the two most developed accounts of this type: Crane’s :21–43 2011, The Objects of Thought2013) and Jeshion’s. Both theories aim to capture the notion of a singular cognitive role in terms of mental files. We (...) argue that Jeshion’s theory is much more promising as it provides a more detailed and non-circular characterization of the mental files responsible for singular thinking. We examine the prospects for enhancing that account in light of three major concerns about it present in the literature. First, we discuss whether the weakest aspect of the whole proposal, i.e. the condition for the initiation of mental files, can be replaced without debilitating the rest of the theory. Second, we examine Goodman’s : 437–461 2014, The Philosophical Quarterly2015) argument against mental file theories of singular thought and show that it does not affect Jeshion’s account. Third, we argue against Sawyer :264–283 2012) that Jeshion’s notion of a singular cognitive role is in strong opposition to acquaintance-based theories of singular thought and cannot be incorporated into any of them. Thus, the paper concludes that Jeshion’s proposal, despite its shortcomings, is currently the most promising account of a singular cognitive role. (shrink)

In this paper, I propose a specific version of theism which I would call apophatic theism. In the first part of the paper, I argue that this in the only tenableversion of theism. Due to the fact that it may seem indistinguishable from a very strong form of agnosticism (or atheism understood in the etymological sense of the word: as a-theism where ‘a’ means ‘without’), in the second part of my paper, I try to distinguish apophatic theism from agnosticism (or (...) a-theism), and from so called “Wittgensteinian” view of religion, which also may seem similar to the position I propose. (shrink)

The article investigates the canonical plays of William Shakespeare—Hamlet, Macbeth, A Midsummer Night’s Dream and The Tempest—in an attempt to determine the nature of Shakespeare’s position on the early modern tendency to demonize fairy belief and to view fairies as merely a form of demonic manifestation. Fairy belief left its mark on all four plays, to a greater or lesser extent, and intertwined with the religious concerns of the period, it provides an important perspective on the problem of religion in (...) Shakespeare’s works. The article will attempt to establish whether Shakespeare subscribed to the tendency of viewing fairies as demonic agents, as epitomized by the Daemonologie of King James, or opposed it. Special emphasis will also be put on the conflation of fairies and Catholicism that one finds best exemplified in Thomas Hobbes’ Leviathan. The article draws on a wealth of recent scholarship on early modern fairies, bringing together historical reflection on the changing perception of the fairy figure, research into Shakespeare’s attitude towards Catholicism and analyses of the many facets of anti-Catholic polemic emerging from early modern Protestant discourse. (shrink)

The articles in this collection focus on philosophical approaches to proper names. The issues discussed include abstract names, empty names, naming and name-using practices, definite descriptions, individuals, reference, designation, sense and semantics. The contributions show the importance and lasting influence of theories proposed by John Stuart Mill, Gottlob Frege, Bertrand Russell, Donald Davidson, and Saul Kripke. Individual chapters assess traditional analyses and modern controversies, and contribute to the debate on proper names in contemporary philosophy of language.

Lay perceptions of lying are argued to consist of a lie prototype. The latter was found to entail the intention to deceive, belief in falsity and falsity. We proposed and found that the perceptions of the benefits of others are also an important factor that influences the extent, to which an act of intentional misleading someone to foster a false belief is labeled as a lie. Drawing from the intuitionist model of moral judgments we assumed that moral judgment of the (...) behaviour would mediate the relationship. In Study 1 we analyzed data coming from a crosscultural project and found that perceived intention to benefit others was negatively related to lie labeling and that this relationship was mediated by the moral judgment of that act. In Study 2 we found that manipulating the benefits of others influenced the extent, to which an act of intentional misleading in order to foster a false belief is labeled as a lie and that, again, this relationship is mediated by the moral judgment of that act. (shrink)

The uniqueness of Edith Stein’s approach to lived experience emerges only in light of intentionality as reasonableness. The “personal touch” or authentic affectivity means in this context one’s own “living body” in regard to a threefold dimension of the human experiencing: the personal, the humanistic, and the spiritual, and seems to echo those of Immanuel Kant’s, i.e., the soul, the world and God. Consequently, not whatever kind of own’s commitment is at stake. Moreover, no less important is the role of (...) community and its various types. For sure, Stein’s genuine account is found in dialogue with the phenomenologists of her time. It paves the way toward a community of life and life itself. Therefore, it seems to be possible to some extend to accord Stein’s account with contemporary discussions of the meaning of life and “a fundamental transformation of human existence.”. (shrink)