Deep learning developers typically rely on deep learning software frameworks (DLSFs)—simply described as pre-packaged libraries of programming components that provide high-level access to deep learning functionality. New DLSFs progressively encapsulate mathematical, statistical and computational complexity. Such higher levels of abstraction subsequently make it easier for deep learning methodology to spread through mimesis (i.e., imitation of models perceived as successful). In this study, we quantify this increase in abstraction and discuss its implications. Analyzing publicly available code from Github, we found that (...) the introduction of DLSFs correlates both with significant increases in the number of deep learning projects and substantial reductions in the number of lines of code used. We subsequently discuss and argue the importance of abstraction in deep learning with respect to ephemeralization, technological advancement, democratization, adopting timely levels of abstraction, the emergence of mimetic deadlocks, issues related to the use of black box methods including privacy and fairness, and the concentration of technological power. Finally, we also discuss abstraction as a symptom of an ongoing technological metatransition. (shrink)

Can there be grounding without necessitation? Can a fact obtain wholly in virtue of metaphysically more fundamental facts, even though there are possible worlds at which the latter facts obtain but not the former? It is an orthodoxy in recent literature about the nature of grounding, and in first-order philosophical disputes about what grounds what, that the answer is no. I will argue that the correct answer is yes. I present two novel arguments against grounding necessitarianism, and show that grounding (...) contingentism is fully compatible with the various explanatory roles that grounding is widely thought to play. (shrink)

Whether or not quantum physics can account for molecular structure is a matter of considerable controversy. Three of the problems raised in this regard are the problems of molecular structure. We argue that these problems are just special cases of the measurement problem of quantum mechanics: insofar as the measurement problem is solved, the problems of molecular structure are resolved as well. In addition, we explore one consequence of our argument: that claims about the reduction or emergence of molecular structure (...) cannot be settled independently of the choice of a particular resolution to the measurement problem. Specifically, we consider how three standard putative solutions to the measurement problem inform our under- standing of a molecule in isolation, as well as of chemistry’s relation to quantum physics. (shrink)

While it might seem prima facie plausible that divine foreknowledge is all that is needed for prophecy, this seems incorrect. To issue a prophecy, God hasto know not just how someone will act, but how someone would act were the prophecy issued. This makes some think that Middle Knowledge is required.I argue that Thomas Flint’s two Middle Knowledge based accounts of prophecy are unsatisfactory, but one of them can be repaired. However the resources needed for repair also yield a sketch (...) of a foreknowledge-only account of prophecy. (shrink)

La philosophie de David Hume entend rompre avec une théorie classique des facultés, redessinant la place que l'imagination joue dans notre vie psychique et déployant ainsi une géographie mentale inédite. Instrument privilégié de son scepticisme philosophique, cette théorie de l'imagination pose cependant autant de questions qu'elle n'en résout, du fait même qu'associant imagination et conception, elle se voit chargée de rendre compte de tendances contradictoires au sein de l'esprit humain. Une clarification terminologique et conceptuelle permet alors, faute de résoudre ces (...) tensions, d'éclairer les nouveaux enjeux proposés par une telle approche. (shrink)

Discussion of the relevance of grounding to substantiveness, theory-choice, and “location problems” in metaphysics.

Achieving space domain awareness requires the identification, characterization, and tracking of space objects. Storing and leveraging associated space object data for purposes such as hostile threat assessment, object identification, and collision prediction and avoidance present further challenges. Space objects are characterized according to a variety of parameters including their identifiers, design specifications, components, subsystems, capabilities, vulnerabilities, origins, missions, orbital elements, patterns of life, processes, operational statuses, and associated persons, organizations, or nations. The Space Object Ontology provides a consensus-based realist framework (...) for formulating such characterizations in a computable fashion. Space object data are aligned with classes and relations in the Space Object Ontology and stored in a dynamically updated Resource Description Framework triple store, which can be queried to support space domain awareness and the needs of spacecraft operators. This paper presents the core of the Space Object Ontology, discusses its advantages over other approaches to space object classification, and demonstrates its ability to combine diverse sets of data from multiple sources within an expandable framework. Finally, we show how the ontology provides benefits for enhancing and maintaining longterm space domain awareness. (shrink)

Some combinations of attitudes--of beliefs, credences, intentions, preferences, hopes, fears, and so on--do not fit together right: they are incoherent. A natural idea is that there are requirements of "structural rationality" that forbid us from being in these incoherent states. Yet a number of surprisingly difficult challenges arise for this idea. These challenges have recently led many philosophers to attempt to minimize or eliminate structural rationality, arguing that it is just a "shadow" of "substantive rationality"--that is, correctly responding to one's (...) reasons. -/- In *Fitting Things Together*, Alex Worsnip pushes back against this trend--defending the view that structural rationality is a genuine kind of rationality, distinct from and irreducible to substantive rationality, and tackling the most important challenges for this view. In so doing, he gives an original positive theory of the nature of coherence and structural rationality that explains how the diverse range of instances of incoherence can be unified under a general account, and how facts about coherence are normatively significant. He also shows how a failure to focus on coherence requirements as a distinctive phenomenon and distinguish them adequately from requirements of substantive rationality has led to confusion and mistakes in several substantive debates in epistemology and ethics. -/- Taken as a whole, Fitting Things Together provides the first sustained defense of the view that structural rationality is a genuine, autonomous, unified, and normatively significant phenomenon. (shrink)

Professional philosophers and advanced students working in metaphysics and the philosophy of science will find this book both provocative and stimulating.

The present study investigated whether individual differences between psychologists in thinking styles are associated with accuracy in diagnostic classification. We asked novice and experienced clinicians to classify two clinical cases of clients with two co-occurring psychological disorders. No significant difference in diagnostic accuracy was found between the two groups, but when combining the data from novices and experienced psychologists accuracy was found to be negatively associated with certain decision making strategies and with a higher self-assessed ability and preference for a (...) rational thinking style. Our results underscore the idea that it might be fruitful to look for explanations of differences in the accuracy of diagnostic judgments in individual differences between psychologists (such as in thinking styles or decision making strategies used), rather than in experience level. (shrink)

A counterpossible is a counterfactual with an impossible antecedent. Counterpossibles present a puzzle for standard theories of counterfactuals, which predict that all counterpossibles are semantically vacuous. Moreover, counterpossibles play an important role in many debates within metaphysics and epistemology, including debates over grounding, causation, modality, mathematics, science, and even God. In this article, we will explore various positions on counterpossibles as well as their potential philosophical consequences.

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators. It starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantic and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist (...) can use the book as a source of references. Results and methods of many directions in propositional modal logic, from completeness and duality to algorithmic problems, are collected and systematically presented in one volume. (shrink)

Otto Neurath’s empiricist methodology of economics and his contributions to politi- cal economy have gained increasing attention in recent years. We connect this research with contemporary debates regarding the epistemological status of thought experiments by reconstructing Neurath’s utopias as linchpins of thought experiments. In our three reconstructed examples of different uses of utopias/dystopias in thought experiments we employ a reformulation of Häggqvist’s model for thought experiments and we argue that: (1) Our reformulation of Häggqvist’s model more adequately complies with many (...) uses of thought experiments, especially with the open-ended discussions of utopias and dys- topias in thought experiments. (2) As a strict logical empiricist, Neurath is committed to a strictly empiricist account of thought experiments. John Norton’s empiricist argument view can indeed account for the justifications of empirical beliefs and genuine discoveries targeted by scientific utopianism in three distinct (yet connected) ways, all of which Neur- ath already contemplated: (2.I) Dealing with utopias and thought experiments on a regular basis increases creativity and inventiveness. (2.II) Particular ways of presenting knowl- edge facilitate scientific discovery and social progress. (2.III) The use of utopias in thought experiments can prompt conceptual change and allow access to new phenomena. We con- clude by highlighting that, even though thought experiments support a positive attitude for exploring new social possibilities, Neurath points out that active decisions are unavoidable. The exploration of alternatives and the awareness of a need for decisions in policy discus- sion avert a technocratic outlook in social science. (shrink)

We have begun work on two separate but related ontologies for the study of neurological diseases. The first, the Neurological Disease Ontology (ND), is intended to provide a set of controlled, logically connected classes to describe the range of neurological diseases and their associated signs and symptoms, assessments, diagnoses, and interventions that are encountered in the course of clinical practice. ND is built as an extension of the Ontology for General Medical Sciences — a high-level candidate OBO Foundry ontology that (...) provides a set of general classes that can be used to describe general aspects of medical science. ND is being built with classes utilizing both textual and axiomatized definitions that describe and formalize the relations between instances of other classes within the ontology itself as well as to external ontologies such as the Gene Ontology, Cell Ontology, Protein Ontology, and Chemical Entities of Biological Interest. In addition, references to similar or associated terms in external ontologies, vocabularies and terminologies are included when possible. Initial work on ND is focused on the areas of Alzheimer’s and other diseases associated with dementia, multiple sclerosis, and stroke and cerebrovascular disease. Extensions to additional groups of neurological diseases are planned. The second ontology, the Neuro-Psychological Testing Ontology (NPT), is intended to provide a set of classes for the annotation of neuropsychological testing data. The intention of this ontology is to allow for the integration of results from a variety of neuropsychological tests that assay similar measures of cognitive functioning. Neuro-psychological testing is an important component in developing the clinical picture used in the diagnosis of patients with a range of neurological diseases, such as Alzheimer’s disease and multiple sclerosis, and following stroke or traumatic brain injury. NPT is being developed as an extension to the Ontology for Biomedical Investigations. (shrink)

In this paper, I explore a fundamental but under-appreciated distinction between two ways of understanding the desire-satisfaction theory of well-being. According to proactive desire satisfactionism, a person is benefited by the acquisition of new satisfied desires. According to reactive desire satisfactionism, a person can be benefited only by the satisfaction of their existing desires. I first offer an overview of this distinction. I then canvass several ways of developing a general formulation of desire satisfactionism that would capture the reactive view, (...) and argue that all come with significant costs. (shrink)

Could it be true that even though we as a group ought to do something, you as an individual ought not to do your part? And under what conditions, in particular, could this happen? In this article, I discuss how a certain kind of case, introduced by David Copp, illustrates the possibility that you ought not to do your part even when you would be playing a crucial causal role in the group action. This is because you may have special (...) agent-relative reasons against participating that are not shared by the group as a whole. I defend the claim that these are indeed cases in which you ought not to do your part in what the group ought to do. I then argue that we can expect these cases to produce a troubling kind of rational conflict. (shrink)

Thomas Samuel Kuhn (1922–1996) is one of the most influential philosophers of science of the twentieth century, perhaps the most influential. His 1962 book The Structure of Scientific Revolutions is one of the most cited academic books of all time. Kuhn’s contribution to the philosophy of science marked not only a break with several key positivist doctrines, but also inaugurated a new style of philosophy of science that brought it closer to the history of science. His account of the development (...) of science held that science enjoys periods of stable growth punctuated by revisionary revolutions. To this thesis, Kuhn added the controversial ‘incommensurability thesis’, that theories from differing periods suffer from certain deep kinds of failure of comparability. (shrink)

This Is A New Release Of The Original 1868 Edition.

Philosophical discussions often involve appeals to verdicts about particular cases, sometimes actual, more often hypothetical, and usually with little or no substantive argument in their defense. Philosophers — on both sides of debates over the standing of this practice — have often called the basis for such appeals ‘intuitions’. But, what might such ‘intuitions’ be, such that they could legitimately serve these purposes? Answers vary, ranging from ‘thin’ conceptions that identify intuitions as merely instances of some fairly generic and epistemologically (...) uncontroversial category of mental states or episodes to ‘thick’ conceptions that add to this thin base certain semantic, phenomenological, etiological, or methodological conditions. As this chapter discusses, thick conceptions turn out to have their own methodological problems; some may even leave philosophers in the methodologically untenable position of being unable to determine when anyone is doing philosophy correctly. (shrink)

This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

Consider the following three-step dialectics. (1) Even if God (consistently) commanded torture of the innocent, it would still be wrong. Therefore Divine Command Metaethics (DCM) is false. (2) No: for it is impossible for God to command torture of the innocent. (3) Even if it is impossible, there is a non-trivially true per impossibile counterfactual that even if God (consistently) com­manded torture of the innocent, it would still be wrong, and this counterfac­tual is incompatible with DCM. I shall argue that (...) the last step of this dialectics is flawed because it would rule out every substantive metaethical theory. (shrink)

This volume contains the Arabic translations of a lost treatise by Alexander of Aphrodisias "On the Principles of the Universe" with English translation, introduction and commentary. It also includes an Arabic and Syriac glossary. The introduction and commentary deal in detail with the manuscripts, the translators and the exegetical tendencies of the text, as well as with its reception in Arabic philosophy. The main theme of the work is the motion of the heavenly bodies and their influence on the (...) physical world. (shrink)

Thomas Kuhn transformed the philosophy of science. His seminal 1962 work "The Structure of Scientific Revolutions" introduced the term 'paradigm shift' into the vernacular and remains a fundamental text in the study of the history and philosophy of science. This introduction to Kuhn's ideas covers the breadth of his philosophical work, situating "The Structure of Scientific Revolutions" within Kuhn's wider thought and drawing attention to the development of his ideas over time. Kuhn's work is assessed within the context of other (...) philosophies of science notably logical empiricism and recent developments in naturalized epistemology. The author argues that Kuhn's thinking betrays a residual commitment to many theses characteristic of the empiricists he set out to challenge. Kuhn's influence on the history and philosophy of science is assessed and where the field may be heading in the wake of Kuhn's ideas is explored. (shrink)

This essay examines Gloria Anzaldúa’s critical appropriation of two Mexican philosophers in the writing of Borderlands/La Frontera: Samuel Ramos and Octavio Paz. We argue that although neither of these authors is cited in her seminal work, Anzaldúa had them both in mind through the writing process and that their ideas are present in the text itself. Through a genealogical reading of Borderlands/La Frontera, and aided by archival research, we demonstrate how Anzaldúa’s philosophical vision of the “new mestiza” is a critical (...) continuation of the broader tradition known as la filosofía de lo mexicano, which flourished during a golden age of Mexican philosophy (1910–1960). Our aim is to open new directions in Latinx and Latin American philosophy by presenting Anzaldúa’s Borderlands/La Frontera as a profound scholarly encounter with two classic works of Mexican philosophy, Ramos’ Profile of Man and Culture in Mexico and Paz’s The Labyrinth of Solitude. (shrink)

Common errors on the mind.--Errors of suppressed correlatives.--The civil service examinations.--The classical controversy.--Metaphysics and debating societies.--The university ideal, past and present.--The art of study.--Religious tests and subscriptions.--Procedure of deliberative bodies.

' But, although such a being (a purely intellectual being) might perhaps be conceived to exist, and although, in studying our internal frame, ...

This article examines Gloria Anzaldúa’s critical appropriation of Mexican philosophical sources, especially in the writing of Borderlands/La Frontera. We argue that Anzaldúa effectively contributed to la filosofía de lo mexicano by developing an Inter-American Philosophy of Mexicanness. More specifically, we recover “La Mexicana en la Chicana” by paying careful attention to Anzaldúa’s Mexican sources, both those she explicitly cites and those we have discovered while conducting archival research using the Gloria Evangelina Anzaldúa Papers at the Benson Latin American Collection at (...) the University of Texas at Austin. The eight Mexican philosophical sources we examine and discuss here are: José Vasconcelos (1882-1959), Miguel León-Portilla (1926-2019), Juana Armanda Alegría (1938- ), Octavio Paz (1914-1998), Samuel Ramos (1897-1959), Rosario Castellanos (1925-1974), Sor Juana Inés de La Cruz (1648-1695), and Jorge Carrión (1913-2005). (shrink)

The hard problem of induction is to argue without begging the question that inductive inference, applied properly in the proper circumstances, is conducive to truth. A recent theorem seems to show that the hard problem has a deductive solution. The theorem, provable in ZFC, states that a predictive function M exists with the following property: whatever world we live in, M ncorrectly predicts the world’s present state given its previous states at all times apart from a well-ordered subset. On the (...) usual model of time a well-ordered subset is small relative to the set of all times. M’s existence therefore seems to provide a solution to the hard problem. My paper argues for two conclusions. First, the theorem does not solve the hard problem of induction. More positively though, it solves a version of the problem in which the structure of time is given modulo our choice of set theory. (shrink)

Achieving space situational awareness requires, at a minimum, the identification, characterization, and tracking of space objects. Leveraging the resultant space object data for purposes such as hostile threat assessment, object identification, and conjunction assessment presents major challenges. This is in part because in characterizing space objects we reference a variety of identifiers, components, subsystems, capabilities, vulnerabilities, origins, missions, orbital elements, patterns of life, operational processes, operational statuses, and so forth, which tend to be defined in highly heterogeneous and sometimes inconsistent (...) ways. The Space Domain Ontologies are designed to provide a consensus-based realist framework for formulating such characterizations in a way that is both consistent and computable. Space object data are aligned with classes and relations in a suite of ontologies built around the existing Space Object Ontology. They are stored in a dynamically updated Resource Description Framework triple store, which can be queried to support space situational awareness and the needs of spacecraft operators and analysts. This paper provides an overview of the Space Domain Ontologies and their development and use. It presents the motivation for and advantages of the Space Domain Ontologies, including the benefits they provide for enhancing and maintaining long-term space situational awareness. (shrink)

Unlike some other reproductions of classic texts (1) We have not used OCR(Optical Character Recognition), as this leads to bad quality books with introduced typos. (2) In books where there are images such as portraits, maps, sketches etc We have endeavoured to keep the quality of these images, so they represent accurately the original artefact. Although occasionally there may be certain imperfections with these old texts, we feel they deserve to be made available for future generations to enjoy.

If the total state of the universe is encodable by a real number, Hardin and Taylor have proved that there is a solution to one version of the problem of induction, or at least a solution to a closely related epistemological problem. Is this philosophical application of the Hardin-Taylor result modest enough? The paper advances grounds for doubt. [A longer and more detailed sequel to this paper, 'Proving Induction', was published in the Australasian Journal of Logic in 2011.].

Hate speech law can be found throughout the world. But it is also the subject of numerous principled arguments, both for and against. These principles invoke a host of morally relevant features and practical considerations . The book develops and then critically examines these various principled arguments. It also attempts to de-homogenize hate speech law into different clusters of laws/regulations/codes that constrain uses of hate speech, so as to facilitate a more nuanced examination of the principled arguments. Finally, it argues (...) that it is morally fitting for judicial and legislative judgments about the overall warrant of hate speech law to reflect principled compromise. Principled compromise is characterized not merely by compromise over matters of principled concern but also by compromise which is itself governed by ideals of moral duty or civic virtue. (shrink)

George Berkeley (1685-1753) is the superstar of Irish Philosophy. He entered Trinity College, Dublin, in 1700 and became a fellow in 1707. In 1724 he resigned his Fellowship to become Dean of Derry, and in 1734 he was made Bishop of Cloyne. He settled in Oxford in 1752 and died the following year. The work of George Berkeley is marked by its diversity and range. His writings take in such topics as mathematics, psychology, politics, health, economics, deism and education, as (...) well as that with which he is most associated - philosophy. Whatever topic he dealt with, his grasp of the subject matter was always impressive and his criticisms of his contemporaries often acute. Among the most noteable of the British Empiricists, he took his starting point from Locke's new 'way of ideas', but he rejected abstract ideas and the possibility of real existence outside perception. It is for this doctrine that he is most known - that there is no such thing as matter, and that things are collections of ideas which can exist only in our minds and only for so long as they are perceived. This philosophy, summed up in his phrase 'to be is to be perceived', he termed 'immaterialism'. A.C. Fraser's collection includes a series of what were previously unpublished notes by Berkeley on all the main topics of his philosophy. This, the 1901 edition, was the first complete edition of his works. (shrink)

In the late Middle Ages the philosopher and mystic Meister Eckhart preached that to know the truth you must be the truth. But how to be the truth? Eckhart's answer comes in the form of an imperative: release yourself, let be. Only then will you be able to understand that the deepest meaning of being is releasement. Only then will you become who you truly are. This book interprets Eckhart's Latin and Middle High German writings under the banner of an (...) imperative of releasement, and then shows how the twentieth-century thinker Martin Heidegger creatively appropriates this idea at several stages of his career. Heidegger had a lifelong fascination with Eckhart, referring to him as "the old master of letters and life." Drawing on archival material and Heidegger's marginalia in his personal copies of Eckhart's writings, Moore argues that Eckhart was one of the most important figures in Heidegger's philosophy. This book also contains previously unpublished documents by Heidegger on Eckhart, as well as the first English translation of Nishitani Keiji's essay "Nietzsche's Zarathustra and Meister Eckhart," which he initially gave as a presentation in one of Heidegger's classes in 1938. (shrink)

Many introductions to this field start with the problem of justifying scientific knowledge but Alexander Bird begins by examining the subject matter, or metaphysics, of science. Using topical scientific debates he vividly elucidates what it is for the world to be governed by laws of nature. This idea provides the basis for explanations and causes and leads to a discussion of natural kinds and theoretical entities. With this foundation in place he goes on to consider the epistemological issues of (...) how science arrives at knowledge, favouring a treatment of scientific reasoning based on inference to the best explanation. Drawing on contemporary work in epistemology, Bird argues that scepticism about induction should not be a problem for science and examines the consequences of this position for controversies surrounding the ideas of scientific progress and scientific revolution. Bird's insightful treatment makes Philosophy of Science an ideal text for undergraduate courses. The guides to further reading provided in each chapter help the reader pursue interesting topics and facilitate the use of the book in conjunction with primary sources. (shrink)

Scotland has played an immense role in European high culture through the centuries, and among its cultural links none have been greater than those with France. This book shows that the links with France stretch back deep into the Middle Ages, and continue without a break into the 18th century, the Age of Enlightenment.

Any serious student attempting to better understand the nature, methods, and justification of science will value Alex Rosenberg's and Lee McIntyre's updated and substantially revised Fourth Edition of Philosophy of Science: A Contemporary Introduction. Weaving lucid explanations with clear analyses, the volume is as a much-used, thematically-oriented introduction to the field.

This essay fills in some historical, conceptual, and pedagogical gaps that appear in the most visible and recent professional efforts to “revive” Philosophy as a Way of Life (PWOL). I present “American Philosophy and Self-Culture” as an advanced undergraduate seminar that broadens who counts in and what counts as philosophy by immersing us in the lives, writings, and practices of seven representative U.S.-American philosophers of self-culture, community-building, and world-changing: Benjamin Franklin (1706–1790), Frederick Douglass (1818–1895), William Ellery Channing (1780–1842), Henry David (...) Thoreau (1817–1862), Margaret Fuller (1810-1850), Henry Bugbee (1915–1999), and Gloria Anzaldúa (1942–2004). Students enter the class with preconceptions about who philosophers are, what they do, how they write, and the languages in which they write. Students walk out with new senses of self, place, and language that emerge through new ways of seeing, doing, and writing philosophy. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)