The paper continues my earlier Chat with OpenAI’s ChatGPT with a Focused LLM Experiment (FLEX). The idea is to conduct Large Language Model (LLM) based explorations of certain areas or concepts. The approach is based on crafting initial guiding prompts and then follow up with user prompts based on the LLMs’ responses. The goals include improving understanding of LLM capabilities and their limitations culminating in optimized prompts. The specific subjects explored as research subject matter include a) diagonalization techniques as practiced (...) by Cantor, Turing, Gödel, and the advances, such as the Forcing techniques introduced by Paul Cohen and later investigators, b) Knowledge Hierarchies & Mapping Exercises, c) Discussions of IJ Good’s Speculations Concerning the First Ultraintelligent Machine, AGI, and Super- intelligence. Results suggest variability between major models like ChatGPT-4, Llama-3, Cohere, Sonnet and Opus. Results also point to strong dependence on users’ preexisting knowledge and skill bases. The paper should be viewed as ’raw data’ rather than a polished authoritative reference. (shrink)

Von Zeit zu Zeit hat man Preisaufgaben uber die Frage gestellt, welche Fortschritte die Philosophie in einem bestimmten Zeitraume gemacht habe. Der Zeitabschnitt pflegte auf der einen Seite durch den Namen eines grosen Denkers, auf der andern durch die "Gegenwart"? abgegrenzt zu werden. Man schien also vorauszusetzen, das uber die philosophischen Fortschritte der Menschheit bis zu jenem Denker hin einigermasen Klarheit herrsche, das es aber von da ab zweifelhaft sei, welche neuen Errungenschaften die letzte Zeit hinzugefugt habe.

We argue that the need for commentary in commonly used linear calculi of natural deduction is connected to the “deletion” of illocutionary expressions that express the role of propositions as reasons, assumptions, or inferred propositions. We first analyze the formalization of an informal proof in some common calculi which do not formalize natural language illocutionary expressions, and show that in these calculi the formalizations of the example proof rely on commentary devices that have no counterpart in the original proof. We (...) then present a linear natural deduction calculus that makes use of formal illocutionary expressions in such a way that unique readability for derivations is guaranteed – thus showing that formalizing illocutionary expressions can eliminate the need for commentary. (shrink)

Ichbin überzeugt, daß die Hauptwiderstände gegen unsere Auffassung daher rühren, daß der Unterschied zwischen der Falschheit und der Sinn|losigkeit eines Satzes nicht beachtet wird. Der Satz: ”Das Reden von einer metaphysischen Außenwelt ist sinnleer“ sagt nicht: ”Es gibt keine metaphysische Außenwelt“, sondern etwas toto coelo anderes. Der Empirist sagt dem Metaphysiker nicht: ”Deine Worte behaupten etwas Falsches“, sondern ”Deine Worte behaupten überhaupt nichts!“ Er widerspricht ihm nicht, sondern er sagt: ”Ich verstehe dich nicht“.

A surprising feature of the White Rose anti-Nazi resistance pamphlets is their appeal to a foundational classical Chinese text, the Laozi (otherwise known as the Daodejing), to buttress their critique of fascism and authoritarianism. I argue that from the perspective of a 1942 educated readership, the act of quoting the Laozi functioned as a subtle and pointed nod to anti-fascist intellectuals in pre-war Germany, many of whom had interpreted the Laozi as an anti-authoritarian and pacifist text. To a sympathetic reader, (...) the Laozi therefore constituted an apt reference point for critiquing National Socialism. I then introduce a complication for this wartime reading of the Laozi from the perspective of its ambiguous reception in ancient Chinese political thought. I more specifically discuss an ancient line of interpretation of the Laozi that points in the direction of authoritarianism — in stark tension with the White Rose message of passive resistance and popular revolt. (shrink)

Problems in normative ethics paradigmatically concern what it is obligatory or permissible for an individual to do. Yet sometimes, each of us ought to do something individually in virtue of what we ought to do together. Unfortunately, traversing these two different levels at which a moral obligation can arise – individual and collective – is fraught with difficulties that easily lure us into conclusions muddying our understanding of collective obligations. This paper seeks to clearly lay out a systematic problem central (...) to the relation between collective and individual duties in view of alleviating some such confusion and facilitating more concerted attempts at resolving it: Collective duties require individuals to act in order for the collective duty to be fulfilled. Yet typically, a collective duty does not entail any one set of individual duties that would prescribe such contributory actions: The work we need to do in order to fulfil a duty can be allocated differently to the individuals collectively bearing it. Much like in matters of distributive justice, then, deriving individual duties from a collective one requires us to employ a distributive scheme, which raises separate normative concerns that have rarely taken centre stage in the debate so far. (shrink)

What is Understanding? This is the first of a series of Chats with OpenAI’s ChatGPT (Chat). The main goal is to obtain Chat’s response to a series of questions about the concept of ’understand- ing’. The approach is a conversational approach where the author (labeled as user) asks (prompts) Chat, obtains a response, and then uses the response to formulate followup questions. David Deutsch’s assertion of the primality of the process / capability of understanding is used as the starting point. (...) Understanding is posited as a major step in the Big Picture description of emergence in the Universe. Understanding follows the emergence of Brains and Minds, culminating in Power Over the Universe. My evaluation suggests that Chat does indeed offer utility, but also has limitations manifested by overly general responses that do not focus tightly on the questions asked. Chat does yield some gems that could be easily missed. The approach used here includes creating Metaprompts, wherein Chat is asked to create more efficient prompts based on initial promptsThis approach appears to hold significant promise, and will likely be of benefit to those exploring various facets of Artificial Intelligence (AI), Large Language Models (LLMs) and designers of Advanced General Intelligence (AGI). Other findings include the need for creating new metrics and ways of discussing intelligence, and various adjacent areas and subprocesses associated with intelligence. Discussions of future directions is also included. (shrink)

This article provides a re-evaluation of David Hume's intensive reading of the classics at an important moment of his literary and intellectual career. It sets out to reconstruct the extent and depth of this reading as well as the uses – scholarly, philosophical and polemical – to which Hume put the information he had gathered in the course of it. The article contends that Hume read the classics against the grain to collect data on a wide range of cultural information (...) which he could utilise for a number of literary and philosophical projects he was engaged in during the early 1750s. This reading soon came to pervade almost all aspects of Hume's literary activities of that period and resulted in what is here described as a fragmentary history of classical antiquity. As a result Hume's reading of the classics emerges from this article as both more extensive and more significant than has so far been acknowledged. (shrink)

The traditional use of the expression 'pseudoproblem' is analysed in order to clarify the talk of pseudoproblems and related phenomena. The goal is to produce a philosophically serviceable terminology that stays true to its historical roots. This explicative study is inspired by and makes use of the method of logical reconstruction. Since pseudoproblems are usually expressed by pseudoquestions a formal language of questions is presented as a possible reconstruction language for alleged pseudoproblems. The study yields an informal theory of pseudoproblems (...) whose presuppositions are critically investigated right away. At least one result remains: Claims of pseudoproblemship and their refutations must not be voiced casually - they are to be relativized and need substantial interpretive effort. (shrink)

Building on the work of Peter Hinst and Geo Siegwart, we develop a pragmatised natural deduction calculus, i.e. a natural deduction calculus that incorporates illocutionary operators at the formal level, and prove its adequacy. In contrast to other linear calculi of natural deduction, derivations in this calculus are sequences of object-language sentences which do not require graphical or other means of commentary in order to keep track of assumptions or to indicate subproofs. (Translation of our German paper "Ein Redehandlungskalkül. Ein (...) pragmatisierter Kalkül des natürlichen Schließens nebst Metatheorie"; online available at http://philpapers.org/rec/CORERE.). (shrink)

Climate change, pollution, and deforestation have a negative impact on global mental health. There is an environmental justice dimension to this challenge as wealthy people and high-income countries are major contributors to climate change and pollution, while poor people and low-income countries are heavily affected by the consequences. Using state-of-the art data mining, we analyzed and visualized the global research landscape on mental health, climate change, pollution and deforestation over a 15-year period. Metadata of papers were exported from PubMed®, and (...) both relevance and relatedness of terms in different time frames were computed using VOSviewer. Co-occurrence graphs were used to visualize results. The development of exemplary terms over time was plotted separately. The number of research papers on mental health and environmental challenges is growing in a linear fashion. Major topics are climate change, chemical pollution, including psychiatric medication in wastewater, and neurobiological effects. Research on specific psychiatric syndromes and diseases, particularly on their ethical and social aspects is less prominent. There is a growing body of research literature on links between mental health, climate change, pollution, and deforestation. This research provides a graphic overview to mental healthcare professionals and political stakeholders. Social and ethical aspects of the climate change-mental health link have been neglected, and more research is needed. (shrink)

This essay reconsiders David Hume’s thinking on the fate of the British Empire and the future of established religion. It provides a detailed reconstruction of the development of Hume’s views on Britain’s successive attempts to impose or regain its authority over its North American colonies and compares these views with the stance taken during the American Crisis by Adam Smith and Josiah Tucker. Fresh light is shed on this area of Hume’s later political thought by a new letter, appended to (...) the essay, which at the same time provides an illuminating glimpse of his abiding preoccupation with the future of established religion. It is argued that this evidence of Hume’s privately held views belies the notion that his thinking on political and religious matters was fundamentally opposed to that of his friends among the philosophes. It is consequently misleading to regard Hume as an opponent of the more radical wing of the Enlightenment. (shrink)

Memetic Science is the name of a new field that deals with the quantitativeanalysis of cultural transfer.The units of cultural transfer are entities called "memes". In a nutshell, memes are to cultural and mental constructs as genesare to biological organisms. Examplesof memesare ideas,tunes, fashions, and virtuallyany culturaland behavioral unit that gets copiedwitha certaindegree of fidelity. It is arguedthat the under standing of memes is of similar importance and consequence as the understanding of processes involving DNA and RNA in molecular biology.Thispaperpresentsa (...) rigorousfoundation fordiscussion ofmemes and approaches to quantifying relevantaspects of memegenesis, inter action, mutation, growth,deathand spreadingprocesses. It is also argued inthispaper that recombinant memetics is possible incomplete analogy to recombinant DNA/ genetic engineering. Special attention is paid to memes in written modern English. (shrink)

It is popular to hold that emotions are evaluative. On the standard account, the evaluative character of emotion is understood in epistemic terms: emotions apprehend or make us aware of value properties. As this account is commonly elaborated, emotions are experiences with evaluative intentional content. In this paper, I am concerned with a recent alternative proposal on how emotions afford awareness of value. This proposal does not ascribe evaluative content to emotions, but instead conceives of them as evaluative at the (...) level of intentional mode or attitude. I first argue that this proposal fails to make emotions intelligible as value apprehensions. There are reasons to suppose that emotions do not apprehend value to begin with, but are related to values in a different, non-epistemic sense. I then go on to show that the notion of an evaluative intentional mode can still help elucidate the evaluative character of emotion. I argue that there is a plausible non-epistemic understanding of the view that emotions are evaluative modes. On this account, emotions are not ways of apprehending values, but ways of acknowledging values. (shrink)

A prominent version of emotional cognitivism is the view that emotions are preceded by awareness of value. In a recent paper, Jonathan Mitchell (2019) has attacked this view (which he calls the content-priority view). According to him, extant suggestions for the relevant type of pre-emotional evaluative awareness are all problematic. Unless these problems can be overcome, he argues, the view does not represent a plausible competitor to rivaling cognitivist views. As Mitchell supposes, the view is not mandatory since its core (...) motivations can be accommodated by competing views, too. I argue that Mitchell's case against the content-priority view is unconvincing as it misconceives the principal motivation for the view. As I show, properly reconstructed, this motivation provides a strong case for the indispensability of the view to any adequate cognitivist treatment of emotion. Moreover, Mitchell's survey of candidates for pre-emotional value awareness can be seen to rest on problematic phenomenological assumptions. (shrink)

It is a popular thought that emotions play an important epistemic role. Thus, a considerable number of philosophers find it compelling to suppose that emotions apprehend the value of objects and events in our surroundings. I refer to this view as the Epistemic View of emotion. In this paper, my concern is with a rivaling picture of emotion, which has so far received much less attention. On this account, emotions do not constitute a form of epistemic access to specific axiological (...) aspects of their objects. Instead it proposes that they are ways of taking a stand or position on the world. I refer to this as the Position-Taking View of emotion. Whilst some authors seem sympathetic to this view, this it has so far not been systematically motivated and elaborated. In this paper, I fill this gap and propose a more adequate account of our emotional engagement with the world than the predominant epistemic paradigm. I start by highlighting the specific way in which emotions are directed at something, which I contrast with the intentionality of perception and other forms of apprehension. I then go on to offer a specific account of the valence of emotion and show how this account and the directedness of emotions makes them intelligible as a way of taking a position on something. (shrink)

It is popular to hold that our primary epistemic access to specific response-dependent properties like the fearsome or admirable (or so-called ‘affective properties’) is constituted by the corresponding emotion. I argue that this view is incompatible with a widely held meta-ethical view, according to which affective properties have deontic force. More specifically, I argue that this view cannot accommodate for the requirement that deontic entities provide guidance. If affective properties are to guide the formation of the corresponding emotion, our primary (...) access to them cannot be provided by that same emotion. (shrink)

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between expressive power (...) and meta-theoretic properties when comparing first-order and second-order logic. (shrink)

It is widely assumed that emotions are evaluative. Moreover, many authors suppose that emotions are important or valuable as evaluations. According to the currently dominant version of cognitivism, emotions are evaluative insofar as they make us aware of value properties of their intentional objects. In attributing to emotions an epistemic role, this view conceives of them as epistemically valuable. In this paper, I argue that proponents of this account mischaracterize the evaluative character of emotions and, a fortiori, their value. Moreover, (...) I propose an alternative view of emotional evaluation, according to which emotions are practically rather than epistemically important. As I argue, emotions are ways of acknowledging their intentional objects as (dis)valuable. As such, they do not apprehend values but make them count. I elaborate this idea by drawing an analogy with legal and political sanctions. The resulting view has it that emotions are practically important in that they affirm the cares and concerns which serve as standards of emotional evaluation. (shrink)

There is a wide range of things we do out of emotion. For example, we smile with pleasure, our voices drop when we are sad, we recoil in shock or jump for joy, we apologize to others out of remorse. It is uncontroversial that some of these behaviors are actions. Clearly, apologizing is an action if anything is. Things seem less clear in the case of other emotional behaviors. Intuitively, the drop in a sad person’s voice is something that happens (...) to her, rather than something she actively performs. Perhaps more interestingly, even jumping for joy can seem a problematic case: although its execution involves the active performance of certain movements, it has been argued to contrast, e.g., with an act of apology, in that it is not performed in order to achieve some end, such as repairing a relationship. This can make this behavior seem considerably different from paradigm actions. Our central concern in this paper is with which emotional behaviors should be classed as actions and why... (shrink)

According to a well-known view in the philosophy of mind, intentional attitudes by their very nature satisfy requirements of rationality (e.g. Davidson 1980; Dennett 1987; Millar 2004). This view (which I shall call Constitutivism) features prominently as the ‘principle of minimal rationality’ in de Sousa’s monograph The Rationality of Emotion (1987). By explicating this principle in terms of the notion of the formal object of an attitude, de Sousa articulates an interesting and original version of Constitutivism, which differs in important (...) respects from other extant versions. In this paper, I explore this version of Constitutivism against the background of recent developments in the theory of rationality and make explicit its ramifications for the long-standing dispute over whether the mind is essentially normative. My focus will be on how to conceive of the form of the rationality requirements that attitudes as such must satisfy according to this principle. I argue that, although de Sousa seems officially to endorse a structuralist conception of rationality, according to which these requirements are requirements of coherence, his considerations on formal objects suggest that they are more aptly conceived in terms of a reasons-responsive conception of rationality. I further argue that which of these two readings we choose makes a significant difference to the prospect of vindicating the essential normativity of mind by invoking the principle of minimal rationality. (shrink)

It is sometimes alleged that the study of emotion and the study of value are currently pursued as relatively autonomous disciplines. As Kevin Mulligan notes, “the philosophy and psychology of emotions pays little attention to the philosophy of value and the latter pays only a little more attention to the former.” (2010b, 475). Arguably, the last decade has seen more of a rapprochement between these two domains than used to be the norm (cf. e.g. Roeser & Todd 2014). But there (...) still seems to be considerable potential for exchange and dialogue if the situation is compared with their intimate relationship in central strands of early realist phenomenology. The philosopher perhaps most representative of this ecumenical approach is Husserl’s early student Dietrich von Hildebrand (1889-1977). From the very early stages of his philosophical career, Hildebrand has developed one of the most original, comprehensive and nuanced accounts of emotions at whose core is a detailed examination of their connection to value. While his central concern with the ethical significance of our affective life is in many ways continuous with Scheler’s work and draws crucially on Reinach’s philosophy of mind, Hildebrand’s own reflections considerably expand on and substantially modify the picture of the ontology and normative role of emotions defended by these authors. In this article, I reconstruct Hildebrand’s view of emotions with a particular focus on those aspects which represent his most distinctive contribution to this subject. (shrink)

This paper explores a currently popular view in the philosophy of emotion, according to which emotions constitute a specific form of evaluative aspect-perception (cf. esp. Roberts 2003, Döring 2004, Slaby 2008). On this view, adequate or fitting emotions play an important epistemic roe vis à vis evaluative knowledge. The paper specifically asks how to conceive of the adequacy or fittingness conditions of emotion. Considering the specific, relational nature of the evaluative properties disclosed by emotions, it is argued that a suitable (...) standard of fittingness has both a world-involving and person-involving aspect: Fitting emotions require both a fundamentum in re as well as a fundamentum in persona. (shrink)

Dieser Forschungsbericht gibt einen Überblick über die aktuelle Debatte über motivierende Gründe in der Handlungs- und Erkenntnistheorie. Folgende drei Fragen werden schwerpunktmäßig behandelt: a) Was für eine Art von Entität sind motivierende Gründe? b) Welche Beziehung besteht zwischen einer Handlung oder Einstellung und ihren motivierenden Gründen? c) Welche kognitiven Bedingungen gelten für die Zuschreibung motivierender Gründe?

Emotions, Values and Agency. By Tappolet Christine.

Aim Excellent organization of the internal processes is essential for a well-functioning health care system. Particularly, the process of treatment and interaction with the patients is important because the perception of treatment and interaction has a direct relation to the patients’ satisfaction. The latter has a strong influence on the recovery of the patients. The impact of the maturity level of the treatment and interaction process on patient orientation is investigated at the example of an extensive survey of hospital managers (...) in Ukraine. -/- Method The analysis is based on an empirical data collection conducted with hospital managers and their deputies in a large Ukrainian city, which is acknowledged as a ‘model city’ with regard to a comprehensive health care reform in the country. By means of the Business Process Maturity Model (BPMM) we have tested our hypothesis that a high level of process maturity leads to a high level of patient orientation. For this purpose, we have developed a scale for measuring the level of process maturity and have checked the method on validity and reliability. A total of 79 hospital managers and deputies, who represent all public health care facilities of the city (hospitals, primary care clinics, etc.), participated in the study. -/- Results The study reveals a clear relationship between the level of maturity of the treatment and interaction process and the level of patient orientation of the respective health care facility. The variance, which is explained by the independent variable process maturity level, is 16.3 %. All formative dimensions have a strong significant influence on the construct “patient orientation”. Likewise the maturity level of the treatment and interaction process shows a strong significant influence on patient orientation. -/- Conclusion For health care facilities it is of great importance to achieve a high maturity level of the treatment and interaction process to live up to the claim of being patient oriented. The maturity of the process can be tested using the questionnaire developed in the course of our study. (shrink)

We are in a time of rapid change on multiple levels. Change can be seen as positive by one group and negative by another. As a result, different perspectives on any given change can draw completely different conclusions. In these proceedings we want to address different approaches to change from all kinds of perspectives within the realm of citizen science and participatory research. We discuss both active, transformative change, and the observation of change monitored by citizen science in all kinds (...) of disciplines. We highlight the potential of citizen science to be a change maker in research and society, and as a tool to manage the change happening around us. The proceedings "Change - The transformative power of citizen science" showcase a selection of topics that have been presented and discussed at the ECSA/ACSC 2024 double conference in Vienna and highlight the transformative power, citizen science can have. (shrink)

Cantor’s proof that the powerset of the set of all natural numbers is uncountable yields a version of Richard’s paradox when restricted to the full definable universe, that is, to the universe containing all objects that can be defined not just in one formal language but by means of the full expressive power of natural language: this universe seems to be countable on one account and uncountable on another. We argue that the claim that definitional contexts impose restrictions on (...) the scope of quantifiers reveals a natural way out. (shrink)

In 1924, the Munich-school phenomenologist Moritz Geiger argued that there are dynamic essences. His two examples are the tragic, and being human, his main ideas are that what it takes to be tragic varies over time historically and that what makes an organism human varies across different stages of its ontogenetic development. He hence points to two ways in which essences may be dynamic, that is, subject to change. The current paper takes Geiger’s view seriously and assumes that it (...) poses an explanatory challenge for the, then and now, standard view that essences are ‘Platonic’, i.e. cannot be subject to change. In the first part of the paper, I introduce Geiger’s view and a bit of its historical context. In the second, I first formulate the challenge it poses to the standard ‘platonic’ view of essence, then discuss how this challenge can be met by a contemporary view of essence discussing three potential responses. The first relies on a notion of relativized essence, the second on the distinction between determinables and determinates, the third and last one on multidimensional properties. Finally, I argue that the last of these three proposals may be preferable to the other two. (shrink)

I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into (...) a set that is bigger than it—that can arise from Cantor’s account if the Axiom of Choice is false, illuminates the debate over whether the Axiom of Choice is true, is a mathematically fruitful alternative to Cantor’s account, and sheds philosophical light on one of the oldest unsolved problems in set theory. (shrink)

Moritz Geiger (1880–1937) in Phänomenologische Ästhetik paper postulates aesthetics to become an autonomous science. The new science is intended to analyze aesthetic values and to discover the rules of their regulations. It tends to be separated from aesthetics as the sub-discipline of philosophy (especially under the influence of metaphysics) and aesthetics as a field of applying other sciences (mainly psychology). It may be achieved by the usage of a phenomenological method.

Yleensä Wienin piirin loogista empirismiä kannattaneet filosofit mielletään antirealisteiksi. Tässä artikkelissa kuitenkin argumentoidaan, että piirin johtohahmo Moritz Schlick oli eräänlainen realisti myös nykystandardien valossa. Näin ollen – niin yllättävältä kuin se kuulostaakin – positivismi ja realismi ovat yhteensovitettavissa. Schlick tosin erotti kannattamansa empiirisen realismin jyrkästi metafyysisestä realismista, jota hän piti merkityksettömänä. Schlickin realismin esittelyn lisäksi artikkelissa tarkastellaan yleisellä tasolla hänen epistemologiaansa, kielifilosofiaansa ja metafysiikan kritiikkiään. Tekstissä pohditaan myös Schlickin suosiman realismin asemaa ja relevanssia nykyisessä realismikeskustelussa.

This analysis shows Cantor's diagonal definition in his 1891 paper was not compatible with his horizontal enumeration of the infinite set M. The diagonal sequence was a counterfeit which he used to produce an apparent exclusion of a single sequence to prove the cardinality of M is greater than the cardinality of the set of integers N.

The mathematician Georg Cantor strongly believed in the existence of actually infinite numbers and sets. Cantor’s “actualism” went against the Aristotelian tradition in metaphysics and mathematics. Under the pressures to defend his theory, his metaphysics changed from Spinozistic monism to Leibnizian voluntarist dualism. The factor motivating this change was two-fold: the desire to avoid antinomies associated with the notion of a universal collection and the desire to avoid the heresy of necessitarian pantheism. We document the changes in (...) class='Hi'>Cantor’s thought with reference to his main philosophical-mathematical treatise, the Grundlagen (1883) as well as with reference to his article, “Über die verschiedenen Standpunkte in bezug auf das aktuelle Unendliche” (“Concerning Various Perspectives on the Actual Infinite”) (1885). (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add (...) Zermelo's choice function. This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (shrink)

Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to (...) the recent debate on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)

Sequel to Part I. In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addresses Russell’s own various attempts to solve these (...) paradoxes, including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ‘logical fictions’ or ‘logical constructions’ instead. (shrink)

In this paper, I introduce an iterative method aimed at exploring numbers within the interval [0, 1]. Beginning with a foundational set, S0, a series of algorithms are employed to expand and refine this set. Each algorithm has its designated role, from incorporating irrational numbers to navigating non-deterministic properties. With each successive iteration, our set grows, and after infinite iterations, its cardinality is proposed to align with that of the real numbers. This work is an initial exploration into this approach, (...) and further investigation is encouraged to understand its full implications and potential. (shrink)

In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions, and equivalence classes of coextensional properties. Part I focuses on Cantor’s theorem, its proof, how it can be used (...) to manufacture paradoxes, Frege’s diagnosis of the core difficulty, and several broad categories of strategies for offering solutions to these paradoxes. (shrink)

Moritz Schlicks Plädoyer für ein empirisches Fundament unserer Erkenntnis enthält weder reduktionistische noch phänomenalistische Extrempositionen. Seine Beispiele für Fundamentalsätze haben allesamt die Form: Hier jetzt so und so; aber nicht alle diese Sätze sind Fundamentalsätze. Was muss man für die letzten drei Wörter dieses Schemas einsetzen, um wirklich beim Fundament anzukommen? Ich schlage vor, die Frage durch Rückgriff auf das interpretationstheoretische Prinzip des Wohlwollens zu beantworten. Demzufolge sind diejenigen Sätze (der Form Hier jetzt so und so) Fundamentalsätze, über die (...) kein inhaltlicher Streit aufkommen kann; wer bei solchen Sätzen – unter denselben Umständen – zu einem anderen Urteil kommt als wir, der bezeugt dadurch keine Meinungsverschiedenheit in der Sache, sondern nur einen anderen Gebrauch sprachlicher Ausdrücke. Bei welchen Ausdrücken funktioniert dieses Kriterium besonders gut? Einerseits bei Ausdrücken, die mit Gesichtsfeldern zu tun haben; noch besser bei Ausdrücken aus dem eindeutig phänomenalistischen Bereich unserer Sprache. (shrink)

For any new internet-based product, service, or technology to succeed, it must satisfy the criterion of providing access to or creating a network of possible users, products, and services. This is the Conceptual Access-Network (CANTOR) Thesis proposed. In addition to the main issues of success and how and why internet technology evolves, the principle can also meet the objective of explaining what underlies a range of traditional and nontraditional technologies beyond the internet. Through qualitative exploration, the tenets of the (...) access-network thesis are applied to natural and synthetic forms of endocrine substances including insulin and highways. A discussion of technologically generated knowledge is included, as well as how phenomenotechnique can be used to establish a communication path between technology and what it produces. Dr. Stanley Milgram's intimate stranger phenomenon which was first explored in 1967, before the internet existed, will ultimately be distorted by the future of internet-based technologies. As previously discovered or established concepts are combined or permuted, the future will display numerous manifestations, combinations, and permutations. We conclude that successful internet-based products, services, or technologies simply would not work without an access network. From this article, we have gained a better understanding of the current development stage of internet-based products and technology, thereby enabling society to better anticipate the future of internet-based products and technology, highlight significant ethical considerations, and avoid unwanted outcomes. (shrink)

In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument. According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of (...) f(a). The definiton is incomplete. The definition of "f(a)=1+f(a)" can not decide anything about the value of f(a) too. The definiton is incomplete.<br><br>According to Wittgenstein, the contradiction, in Cantor's proof, originates from the hidden presumption that the definition of Cantor’s number is complete. The contradiction shows that the definition of Cantor’s number is incomplete. <br><br>According to Wittgenstein’s analysis, Cantor’s diagonal argument is invalid. But different with Intuitionistic analysis, Wittgenstein did not reject other parts of classical mathematics. Wittgenstein did not reject definitions using self-reference, but showed that this kind of definitions is incomplete.<br><br>Based on Thomson’s diagonal lemma, there is a close relation between a majority of paradoxes and Cantor’s diagonal argument. Therefore, Wittgenstein’s analysis on Cantor’s diagonal argument can be applied to provide a unified solution to paradoxes. (shrink)

Ortega y Gasset is known for his philosophy of life and his effort to propose an alternative to both realism and idealism. The goal of this article is to focus on an unfamiliar aspect of his thought. The focus will be given to Ortega’s interpretation of the advancements in modern mathematics in general and Cantor’s theory of transfinite numbers in particular. The main argument is that Ortega acknowledged the historical importance of the Cantor’s Set Theory, analyzed it and (...) articulated a response to it. In his writings he referred many times to the advancements in modern mathematics and argued that mathematics should be based on the intuition of counting. In response to Cantor’s mathematics Ortega presented what he defined as an ‘absolute positivism’. In this theory he did not mean to naturalize cognition or to follow the guidelines of the Comte’s positivism, on the contrary. His aim was to present an alternative to Cantor’s mathematics by claiming that mathematicians are allowed to deal only with objects that are immediately present and observable to intuition. Ortega argued that the infinite set cannot be present to the intuition and therefore there is no use to differentiate between cardinals of different infinite sets. (shrink)

The purpose of this paper is to examine in detail a particularly interesting pair of first-order theories. In addition to clarifying the overall geography of notions of equivalence between theories, this simple example yields two surprising conclusions about the relationships that theories might bear to one another. In brief, we see that theories lack both the Cantor-Bernstein and co-Cantor-Bernstein properties.

Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. This paper considers how the doctrine of symbolism and the philosophers' different commitments to the laws of logic, especially the Law of Non-Contradiction (LNC), enables these thinkers to articulate a transcendental apophatic approach to divinity. (shrink)

I argue that Composition as Identity blocks the plural version of Cantor's Theorem, and that therefore the plural version of Cantor's Theorem can no longer be uncritically appealed to. As an example, I show how this result blocks a recent argument by Hawthorne and Uzquiano.