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)

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 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)

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 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.

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)

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.

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)

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)

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)

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)

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)

The aim of this paper is to analyze the implications flowing from adopting methodological naturalism in science, with special emphasis on the relation between science and religion. Methodological naturalism, denying supernatural and teleological explanations, influences the content of scientific theories, and in practice leads to vision of science as compatible with ontological naturalism and in opposition to theism. Ontological naturalism in turn justifies the acceptance of methodological naturalism as the best method to know the reality. If we accept realistic interpretation (...) of scientific theories, then methodological naturalism conflicts science with religion. Theistic evolution does not seem to be a proper way to reconcile Darwinism and methodological naturalism with theism. Many of such propositions are boiled down to deism. Although evolution can be interpreted theistically, it is not the way in which majority of modern scientists and respectable scientific institutions understand it. (shrink)

Choice reaction times to visual stimuli may be influenced by preceding subliminal stimuli . Some authors reported a straight priming effect i.e., responses were faster when primes and targets called for the same response than when they called for different responses. Others found the reversed pattern of results. Eimer and Schlaghecken [Eimer, M. & Schlaghecken, F. . Links between conscious awareness and response inhibition: evidence from masked priming. Psychonomic Bulletin & Review, 9, 514–520.] showed recently that straight priming occurs whenever (...) a prime is not efficiently masked thereby the information provided by the prime is accessible for consciousness. In the present study, a hypothesis is tested that straight priming is due to mediation of consciousness. To test this hypothesis, prime validity was manipulated. We showed that even when no mask was used so that participants could fully and consciously perceive the prime and participants were informed that primes were mostly invalid, for the short prime–target ISI interval straight priming occurred. The priming was inverse when the ISI was 800 ms. This indicates that participants were able to use the information provided by the prime to prepare the response opposite to that cued by the prime but only if the time between the prime and the target was long enough. (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 paper considers whether virtue ethics should be regarded as excluding duty ethics or any of its essential elements. The argument suggested here consists of two steps: (1) an argument that there are two different versions of virtue ethics (moderate and strong) and that moderate virtue ethics does not exclude the duty ethics; (2) an analysis of various difficulties with the strong version of virtue ethics, which shows that moderate virtue ethics is more plausible because of its capacity to avoid (...) these difficulties. This capacity makes moderate virtue ethics more attractive as an ethical theory because it covers the entire range of moral phenomena described both by strong versions of virtue ethics and by duty ethics without the attendant difficulties. (shrink)

Esej ten kreśli nową strategię uniwersalizacji historii, która wyłania się z analizy żydowskiejpraktyki filozofowania w erze nowożytnej. Nazywam ją „strategią marańską”, budując analogięmiędzy sytuacją conversos, którzy zostali zmuszeni do przyjęcia chrześcijaństwa, przechowującprzy tym judaizm „utajony”, a filozoficzną interwencją nowoczesnych myślicieli żydowskich,którzy wkroczyli w idiom zachodniej filozofii, jednocześnie nasycając go motywami wywiedzionymiz ich „partykularnego” kontekstu: nie po to jednak, by podważyć perspektywęuniwersalistyczną, lecz po to, by ją przekształcić. Dla Waltera Benjamina i Jacques’a Derridywłaściwy uniwersalizm okazuje się równoznaczny z „zadaniem tłumacza”: projektem (...) sklejaniarozbitej całości od środka, horyzontalnie, przez wysiłek wielojęzyczności, czyli bez przyjmowaniawyniosłej pozycji ogólnego metajęzyka, który mógłby uzurpować pozycję hegemona. (shrink)

This paper deals with two fragments from Diodorus’ Bibliothêkê that are unanimously considered to belong to the narrative of the First Slave Revolt in Sicily. It is the main concern of this paper to demonstrate that they most likely did not, but instead originate from an unknown preface to Book 34. The article begins with a brief introduction into Diodorus’ prefaces and discusses the Byzantine transmission of both fragments. Against this backdrop, three main steps are consecutively applied to prove the (...) hypothesis. First, the narrative order of both fragments within the Byzantine collections is re-examined. Furthermore, the paper establishes a thematic and argumentative relationship between the two fragments. In the last step, the structure and the style of both fragments are analysed. (shrink)

This article argues that if contemporary philosophical counseling wishes to perpetuate the spirit of ancient philosophy, it must also incorporate metaphysics. This kind of incorporation means, among other things, regarding as essential the implementation of the meditation practices of certain historical schools of philosophy. Meditation was an inherent feature not only of Eastern schools of thought, but also within Western philosophy. These practices lead to a psychophysical state, known as ataraxia, which causes the perception of reality to undergo a radical (...) change, including occasional “altered states of consciousness,” as they are known by psychology. The author calls such practices “live metaphysics.” He also maintains that, without the inclusion of such metaphysical practices, philosophical counselling and coaching lose their eudemonic character and become a means of hedonistic gratification in the service of the liberated ego of the contemporary man. (shrink)

Internal audit provides an independent and objective opinion to the Board on risk management, control and governance, by measuring and evaluating their effectiveness in achieving an organisation’s agreed objectives. It also provides an independent and objective consultancy service to help line management improve the organisation’s processes. Internal auditors should respect in their work some ethical rules, especially these included in the Code of Ethics formed by the IIA. This code is intended to clarify the standard of conduct expected from all (...) members of the internal audit unit. The main four principles that should be observed are as follows: integrity, objectivity, competency and confidentiality Being ethical in internal audit is necessary in order to win the trust of the environment in which auditors work. However, declaring one’s readiness to respect the Code of Ethics is not enough. Much more important is to implement and comply with given rules in carrying out audit duties. (shrink)

_The Life of Reason in an Age of Terrorism_ brings together seventeen original essays that discuss George Santayana’s social and political thought within the context of contemporary considerations, especially terrorism.

This is an expository paper on the problem of independent axiomatization of any set of sentences. This subject was investigated in 50's and 60's, and was abandoned later on, though not all fundamental questions were settled then. Besides, some papers written at that time are hardly available today and there are mistakes and misunderstandings there. We would like to get back to that unfinished business to clarify the subject matter, correct mistakes and answer questions left open by others. We shall (...) deal with results of many authors. However, they will be exposed in a different manner, with complete proofs and, often, with refinements and supplements. Some questions will be brought up to date and related to other questions in logic. New results and questions will also be added. Stress will be laid on constructive aspects; that is, we will examine the problem of the possibility of independent axiomatization as well as algebraic means by use of which independent sets of axioms can be given. (shrink)

This article is a critical discussion of the book Setting Health-Care Priorities by Torbjörn Tännsjö. This critique targets mainly Tännsjö’s method, but also several unjustified conclusions and some implicit assumptions.

Values, Valuations, and Axiological Norms in Richard Rorty's Neopragmatism sympathetically discusses Richard Rorty's neopragmatist philosophy. This book brings together a range of interpretations and possibilities on a variety of humanistic topics, including philosophy, literature, culture, film, economics, social issues, politics, and more. Skowroński involves the work of philosophers such as Kant, Dewey, Santayana, and Kołakowski as he delves into various philosophical problems using the lens of Rorty’s neopragmatist thought.

There is an ongoing investigation by scholars of ethics and economics into whether human decision making and the resultant acts should be guided by rules and procedures or by judgment and discretion. Although each of these modes offers advantages and disadvantages to decision makers, they are by no means neutral in their effect on professional development. The paper presents an in-depth view of discretionary decisions using an Aristotelian-Thomistic framework. This is the first of the series of papers which focus on (...) the application of realistic philosophical principles to ethical professional behaviour and decision making in daily practice. Results indicate that classical philosophical theories which use virtues and truth as indispensable components still may give rise to desirable and moral conduct among individuals who apply them. (shrink)

We present a dynamical model of interaction between recognition memory and affect, focusing on the phenomenon of “warm glow of familiarity.” In our model, both familiarity and affect reflect quick monitoring of coherence in an attractor neural network. This model parsimoniously explains a variety of empirical phenomena, including mere-exposure and beauty-in-averages effects, and the speed of familiarity and affect judgments.

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational (...) fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules. (shrink)

We present several naturally defined σ-ideals which have Borel bases but, unlike for the classical examples, these ideals are not of bounded Borel complexity. We investigate set-theoretic properties of such σ-ideals.