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The importance of infinity for Spinoza's philosophy can hardly be overstated. Understanding Spinoza means understanding (Spinoza's take on) infinity. In this paper, I present a deflationary account of Spinoza's infinity: Infinities across ontological states (modes, attributes, substance) follow the same general trajectory: From an indeterminate essence, infinitely many things follow. And as a consequence, Spinoza's universe is infinite all the way down. Some think that to Spinoza, infinity is indeterminacy (acosmism). Others say that infinity in substance follows from the essence of substance, while infinity in modes does not. I argue that both views are false. Instead, infinity in modes and infinity in substance emerges from the numerical indeterminacy of essences. I show that this view can be traced back to Descartes' famous wax example and that it is manufactured to avoid a particular contradiction. In order to specify my point, I demonstrate that both ‘absolute infinity’ (God) and ‘in suo genere infinity’ (attributes) must be read through the deflationary account of infinity. I also argue that we do not need an additional qualitative indeterminacy and that causation and conception are not always parallel (such as in reductio proofs). [ABSTRACT FROM AUTHOR]

This work presents a mathematical framework based on uncertain numbers to address the inherent uncertainty in nonlinear systems, a challenge that traditional mathematical frameworks often struggle to fully capture. By establishing five axioms, a formal system of uncertain numbers is developed and embedded within set theory, providing a comprehensive characterization of uncertainty. This framework allows phenomena such as infinity and singularities to be treated as uncertain numbers, offering a mathematically rigorous analytical approach. Subsequently, an algebraic structure for uncertain numbers is constructed, defining fundamental operations such as addition, subtraction, multiplication, and division. The framework is compatible with existing mathematical paradigms, including complex numbers, fuzzy numbers, and probability theory, thereby forming a unified theoretical structure for quantifying and analyzing uncertainty. This advancement not only provides new avenues for research in mathematics and physics but also holds significant practical value, particularly in improving numerical methods to address singularity problems and optimizing nonconvex optimization algorithms. Additionally, the anti-integral-saturation technique, widely applied in control science, is rigorously derived within this framework. These applications highlight the utility and reliability of the uncertain number framework in both theoretical and practical domains. [ABSTRACT FROM AUTHOR]

Previous studies in sports marketing have shown that team love is a key factor influencing loyalty, satisfaction and positive word-of-mouth behaviours. Despite these positive findings, research on team love remains limited, and the concept's theoretical framework is underdeveloped. The study aims to conceptualize football team love using a grounded theory approach and to develop a scale based on this conceptualization. Through a comprehensive literature review, as well as data obtained from focus group and in-depth interviews with football fans, the study delineated the unique conceptual dimensions of team love. Using data obtained from 452 Turkish football fans, exploratory factor analysis (EFA) revealed a three-dimensional (self-team integration, infinity, and passion-driven behaviours), 12-item structure. Confirmatory factor analysis (CFA) further validated and refined, a three-dimensional, nineitem measurement model. This research provides an in-depth, theory-based understanding of the unique construct of football team love, offering valuable insights for both academic and practical applications. The findings lay a foundation for further research and provide strategic guidance for sports marketers to foster stronger connections between football teams and their fans. [ABSTRACT FROM AUTHOR]

In this paper we classify the phase portraits in the Poincaré disk of the Smooth and Discontinuous (SD) Hamiltonian system with the rational Hamiltonian function H (x , y) = y 2 / 2 + P (x) / Q (x , y) , where P (x) = a , ax, a x 2 and Q (x , y) = A x 2 + B y 2 + C . [ABSTRACT FROM AUTHOR]

Modernity is characterised by a specific scientific paradigm based on binary logic. Neither freedom nor creativity have a place in this logical model. On the contrary, triadic logic enables rational consideration of an anthropological and social model compatible with both things: freedom and creativity, and therefore with Christianity. Furthermore, triadic logic provides us with a model of thought compatible with the current physical explanation of reality. [ABSTRACT FROM AUTHOR]

We offer a new angle to explain the comprehension of mathematical infinity. We use Action—Process—Object—Schema (APOS) theory to explore the construction of different aspects of the cognitive Object related to this notion. We focus our attention on different ways of interacting with infinity, and how to act on infinite entities. Our aim is to shed light on the nature of mental constructions that take place when individuals deal with mathematical infinity. An infinite union of finite subsets of N provides the context for our theoretical analysis and the design of interviews conducted with graduate students and instructors. In this study we identify three types of Actions with different complexity levels. When these Actions are performed, they lead to the construction of different facets of the associated Object. We question the construction of an Object conception in infinity-related situations and offer an explanation that might change the way we think about the learning of certain mathematical concepts. We discuss the implications of our research for the advancement of APOS theory. We also offer pedagogical suggestions related to the comprehension of mathematical infinity. [ABSTRACT FROM AUTHOR]

Georg Cantor did detail mathematical work on the theory of transfinite numbers and how the cardinality differences between sets and different levels of infinity can be classified. This study focuses on how Cantor handled the mathematical and logical concepts of infinity and how he revolutionized the world of mathematics by defining infinity as a concrete mathematical concept. This study first examines Cantor's contributions to the theory of infinite sets, especially the mathematical and theological dimensions of the concept of infinity. Cantor defined transfinite numbers as a mathematical reflection of God's absolute infinity, but he argued that these infinities could never fully capture God's absolute nature and provoked a great reaction with these arguments. How his work was received by scientists and theologians of his time and how he gained an important place in modern mathematics thanks to his work is the subject of the rest of this study. Finally, how Cantor associated the mathematical concept of infinity with the absolute nature of God and how this contributed to the debates between theology and mathematics is discussed in this study; the most basic discussions on all these issues are included. [ABSTRACT FROM AUTHOR]

This article contributes to discussions about the problem of evil and Schelling studies by analysing Schelling's conception of the problem in his 1809 Freiheitsschrift essay. I explicate Schelling's critical response to four classic solutions to the problem (embodiment, degree, dualism, and divine forms) and outline his positive solution. My thesis is that Schelling offers a unique theodicy by arguing for a dialectical conception of the infinite omnipotence of God. In contrast to traditional notions of the infinite as the opposite of the finite, Schelling claims that God is only truly infinite if also embodied in the finite, an embodiment enacted through the human freedom to do evil. To explore Schelling's project, I draw parallels between his account of God's omnipotence and Hegel's 'good infinite' and situate Schelling's thesis within Mackie's discussion of the problem of evil in 'Evil and Omnipotence'. [ABSTRACT FROM AUTHOR]

It has long been known that in the context of axiomatic second-order logic (SOL), Hume's Principle (HP) is mutually interpretable with "the universe is Dedekind infinite" (DI). In this paper, we offer a more fine-grained analysis of the logical strength of HP, measured by deductive implications rather than interpretability. Our main result is that HP is not deductively conservative over SOL + DI. That is, SOL + HP proves additional theorems in the language of pure second-order logic that are not provable from SOL + DI alone. Arguably, then, HP is not just a pure axiom of infinity, but rather it carries additional logical content. On the other hand, we show that HP is Π 1 1 conservative over SOL + DI, and that HP is conservative over SOL + DI + "the universe is well ordered" (WO). Next, we show that SOL + HP does not prove any of the simplest and most natural versions of the axiom of choice, including WO and weaker principles. Lastly, we discuss other axioms of infinity. We show that HP does not prove the Splitting or Pairing principles (axioms of infinity stronger than DI). [ABSTRACT FROM AUTHOR]

We address the following questions in this paper: (1) Which set or number existence axioms are needed to prove the theorems of 'ordinary' mathematics? (2) How should Frege's theory of numbers be adapted so that it works in a modal setting, so that the fact that equivalence classes of equinumerous properties vary from world to world won't give rise to different numbers at different worlds? (3) Can one reconstruct Frege's theory of numbers in a non-modal setting without mathematical primitives such as "the number of Fs" ( # F ) or mathematical axioms such as Hume's Principle? Our answer to question (1) is 'None'. Our answer to question (2) begins by defining 'x numbers G' as: x encodes all and only the properties F such that being-actually-F is equinumerous to G with respect to discernible objects. We answer (3) by showing that the mere existence of discernible objects allows one to reconstruct Frege's derivation of the Dedekind-Peano axioms in a non-modal setting. [ABSTRACT FROM AUTHOR]

There is a well-known variety of contact paradoxes which are significantly linked to topology. The aim of this paper is to present a new paradox concerning contact with bodies composed of a denumerable infinity of parts. This paradox establishes the logical necessity, in a Newtonian context, of contact forces (herein called "phantom forces") that violate what is probably our most basic causal intuition, embodied in what I call the Principle of Influence: any force exerted on a body B induces (causes) change of movement of B or (inclusive disjunction) the emergence of internal forces in B. However, the above paradox can be made strictly compatible with a Newtonian framework by introducing phantom forces as ideal elements in the Hilbert sense, though it will be seen that this does not solve all the problems. [ABSTRACT FROM AUTHOR]

Many have the intuition that human persons are both extremely and equally valuable. This seeming extremity and equality of vale is puzzling: if overall value is the sum of one's final value and instrumental value, how could it be that persons share the same extreme value? One way that we can solve the Value Puzzle is by following Andrew Bailey and Josh Rasmussen. Philosophy and Phenomenological Research,103, 264–277 (2020) and accepting that persons have infinite final value. But there are some significant downsides to their way of thinking about values, which relies on the extended real numbers. We offer a different approach: if we model values using the hyperreal numbers, we can capture many of our intuitions about the extremity and relative equality of human value without incurring the substantial theoretical costs of using the extended real numbers. We also examine other ways of modeling the infinite value of persons and evaluate the strengths and weaknesses of these other accounts compared to ours. [ABSTRACT FROM AUTHOR]

Previous studies in sports marketing have shown that team love is a key factor influencing loyalty, satisfaction and positive word-of-mouth behaviours. Despite these positive findings, research on team love remains limited, and the concept’s theoretical framework is underdeveloped. The study aims to conceptualize football team love using a grounded theory approach and to develop a scale based on this conceptualization. Through a comprehensive literature review, as well as data obtained from focus group and in-depth interviews with football fans, the study delineated the unique conceptual dimensions of team love. Using data obtained from 452 Turkish football fans, exploratory factor analysis (EFA) revealed a three-dimensional (self-team integration, infinity, and passion-driven behaviours), 12-item structure. Confirmatory factor analysis (CFA) further validated and refined, a three-dimensional, nine-item measurement model. This research provides an in-depth, theory-based understanding of the unique construct of football team love, offering valuable insights for both academic and practical applications. The findings lay a foundation for further research and provide strategic guidance for sports marketers to foster stronger connections between football teams and their fans.

An examination of Xenophanes’ fragment DK 21 B 34 shows how it to some extent anticipates what is known in contemporary epistemological debate as the “Gettier problem.” According to the argument underlying this problem, it is not enough to have a “justified true belief” in order to be able to say that one has “knowledge.” As Xenophanes’ text has it, even if one were able to say something true, one would not know it yet. This is because, according to Xenophanes, no one is capable of grasping the evident truth of things, since only conjectures about everything are available. But while conjectures cannot rise above the level of pure opinion, they are not all equal: as DK 21 B 18 recognizes, they are perfectible. But what is the basis of this position, which oscillates between scepticism and fallibilism? Since perfectibility never reaches the end (namely the truth), this progress is infinite: the basis of Xenophanes’ epistemology may therefore be his physics of the infinite, which, however, is itself only a conjecture. This entails the risk of self-refutation. To avoid this peril, the doctrine of Parmenides will have to batten down the hatches.

The Infinite Monkeys Theorem has long-established the eventual certainty of the complete works of William Shakespeare being reproduced by a monkey randomly pressing keys on a typewriter. This only considers the infinite limit, with either an infinite number of monkeys and/or an infinite time period of monkey labour. Here, we consider the Finite Monkeys Theorem and look at the probability of a given string being typed by one of a finite number of monkeys within a finite time allocation consistent with estimates for the lifespan of our universe. We also calculate the expected number of keystrokes until a target string would first be produced. Given the expected time until the heat death of the universe, we demonstrate that the widely-accepted conclusion from the Infinite Monkeys Theorem is, in fact, misleading in our finite universe. As such, this places the theorem in a class of probabilistic problems or paradoxes, including the St. Petersburg paradox, Zeno's dichotomy paradox and the Ross–Littlewood paradox wherein the infinite-resource conclusions directly contradict those obtained when considering limited resources, however sizeable.

This paper aims to achieve environmental and economic sustainability in maritime affairs through Green Fishing Ports (GFP). The governance of Fishing Ports with an integrated information system in Indonesia (as a maritime nation) is an intriguing issue for realizing sustainable underwater development, enhancing operational efficiency, improving safety and comfort, and boosting the economic well-being of coastal communities. An effort to achieve that is the management of integrated digital Fishing Ports. This paper employs qualitative research with a comparative case study of the Nizam Zachman Ocean Fishing Port in DKI Jakarta and the Karangantu Archipelago Fishing Port in Banten Province. Data collection utilizes unstructured interviews with six informants knowledgeable in port governance. This paper reveals that the governance of the Nizam Zachman and Karangantu Fishing Ports have operations with an ‘Integrated Fishing Port with Information System for Sustainability’ concept—abbreviated as INFINITY. Through the implementation of INFINITY, the governance of fishing ports can be significantly enhanced. The utilization of INFINITY has led to notable improvements such as increased fish production values, enhanced cooperation within the fisheries sector, and the promotion of marine resource sustainability, thereby contributing to the advancement of a green economy.

The paper deals with Nadija Rebronja’s book 88, which is conceived as 88 piano keys or fragments. The paper recognizes some important links that this book establishes with the mannerist literary tradition, as understood and explained by Gustav René Hocke in his widely acknowledged books Mannerism in Literature and The World as a Labyrinth. Instead of opening up to the totality of all things, 88 resembles a mannerist work that closes itself into its own artistic reality and questions the place that literature and art occupy in today’s world. The paper also deals with the implications that arise from the connection Nadija Rebronja establishes with the tradition of magic(al)realism, particularly regarding the construction of motivational systems and their relation to the phenomenon of magic in this book. By utilising the concepts of simultaneity and catalogue, 88 relates to the tradition of the Russian avant-garde, as understood by Aleksandar Flaker. It is also shown that a reassessment, a significant avant-garde procedure, holds an important place in the 88, since the author re-evaluates the figure of travel and a topos of the town square in relation to classical literature. This reassessment is linked to the tradition of hikaye, instructive stories that have come from the Islamic tradition, as well as a farce - a literary form based on reversal. The final part of this paper deals with the fragmentation in Nadia Rebronja’s88 which can be linked to the atomization within the mannerist literature and the avant-garde understanding of the world. It also explores the ways in which the world created in this book relates to,on the one hand,the “optimal projection” as one of the characteristics of the avant-garde, and the concept of utopia as an alternative to freethinking on the other.

Our mental representation of the infinite has received little research attention in cognitive psychology. In countably infinite sets, the infinity symbol (∞) is presumed to be perceived as larger than any finite natural number. The present study sought to explore if the infinity symbol is processed as "larger than" natural numbers, and, if so, whether it is associated with the special status of "the largest." In a series of four experiments (N = 40, 20, 20, and 40, respectively), participants performed numerical and physical comparisons of the infinity symbol against single- and multidigit numbers. Overall, numerical comparisons yielded slower responses for comparisons between infinity and a number than for comparisons between two numbers. Furthermore, distance-like effects were obtained for comparisons to infinity, suggesting the infinity symbol was treated as larger than all numbers presented. Importantly, however, physical comparisons revealed a normal size congruity effect for comparisons of infinity and single digits, but a reversed effect for comparisons of infinity and multidigit numbers, suggesting that the infinity symbol was automatically processed as smaller than multidigit numbers. These novel findings reveal limitations in abstracting the meanings of infinity from its symbol, indicating that the infinity symbol is not perceived as "the largest" and can be misconceived as a "number" mapped onto the numerical magnitude system. More generally, the results seem to reflect a crude, automatic evaluation of numerical magnitude based on the physical magnitude of the stimuli, namely, their overall length and the number of symbols of which they are comprised. [ABSTRACT FROM AUTHOR]

There are possible worlds in which time is circular and finite in duration, forming a loop of, say, 12,000 years. There are also possible worlds in which time is linear and infinite in both directions and in which history is repetitive, consisting of infinitely many 12,000-year epochs, each two of which are exactly alike with respect to all intrinsic, purely qualitative properties. Could one ever have empirical evidence that one inhabits a world of the first kind rather than a world of the second kind? We argue for the affirmative answer, contra Quine, Newton-Smith, and Bergström. Our argument for that conclusion differs from an argument for the same conclusion due to Weir. Weir's argument is probabilistic and explicitly requires having evidence against determinism. Our argument is a direct appeal to the simplicity of laws, and it involves no probabilistic component. It is modeled on Shoemaker's argument that one could have evidence of time without change. [ABSTRACT FROM AUTHOR]

Leopardi employs the traditional distinction between infinite and indefinite in his works, prompting a question rarely asked in Leopardi studies: what is the role of the true infinite in Leopardi's oeuvre? My contention is that the true infinite should not be dismissed as having no role in Leopardi's production simply because it does not exist in reality. Rather, it is the supreme object of desire, the impossible ideal that his texts pursue. This idea is explored through a comparative reading with the work of Fyodor Dostoevsky, whose belief in the true infinite is not in question. Starting from Leopardi's reflections on the infinite and his 'theory of pleasure', it becomes clear that the infinite cannot be expressed or comprehended by human beings, yet they must have some notion of it. This comparison, therefore, helps to reassess the role of the true infinite in Leopardi, which would otherwise remain hidden. [ABSTRACT FROM AUTHOR]

As often occurs with eminent authors, Emmanuel Levinas has acquired a growing number of followers over time and is now widely regarded as one of the greatest contemporary philosophers. In the modern world, where masks hide the visage of the Other, it is more necessary than ever to rediscover Levinas’ thought and writings in the light of a morality which does not justify egocentric behaviour in times of war, political turmoil or pandemics. A morality which would include us all without exception. The hermeneutical proposal of this research article consists in approaching the author’s thought through a global analysis, including textual, contextual and biographical elements. The unitive understanding of the philosophy of alterity, based on the idea of a transcendental goodness and on a limitless intersubjectivity where humanization unfolds its indestructible summons, is pursued. [ABSTRACT FROM AUTHOR]

In this article, the author take cues from Gilles Deleuze's later works, specifically What Is Philosophy?, reading his and Félix Guattari's concept of infinite speed through a Spinozist lens. The argument is twofold. First, the author demonstrates that the concept of infinite speed serves as an indispensable condition of possibility for Deleuzian epistemology as a whole. He does so by situating the function of infinite speed in Deleuze's work alongside the function of eternity in Spinoza's work, arguing that the former picks up where the latter leaves off—that is, by serving as an epistemological ground. Thus, second, the author argues that despite Deleuze's critique of and departure from key aspects of Spinozist thought—namely, the category of eternity and desirability of substance metaphysics—the notion of infinite speed, alongside other terms, allows one to read Deleuzian philosophy as a continuation of, rather than a significant departure from, Spinozist thought. Thus, by demonstrating the precise role that infinite speed plays in Deleuze's later work, the author both illuminates the metaphysical axiom that grounds Deleuzian epistemology and, via this illumination, shows how that epistemology is consistent with and adherent to specific Spinozist philosophical commitments. [ABSTRACT FROM AUTHOR]

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This paper explores the introduction and teaching of the concept of infinity in mathematics. Since a correct development of the concept of infinity is essential for the successful completion of studies in this field, especially in higher education, this paper collects information known on this topic. This is complemented by a historical background on the subject, related basic concepts, various illustrative examples and exercises, all suitably formulated. All of this is summarised in a methodologically illuminated way, even as a possible resource for teachers. Nevertheless, it should be interesting reading for anyone curious about one of the most important abstract concepts in mathematics, infinity. [ABSTRACT FROM AUTHOR]

It is well-known that the global structure of every space-time model for relativistic cosmology is observationally underdetermined. In order to alleviate the severity of this underdetermination, it has been proposed that we adopt the Cosmological Principle because the Principle restricts our attention to a distinguished class of space-time models (spatially homogeneous and isotropic models). I argue that, even assuming the Cosmological Principle, the topology of space remains observationally underdetermined. Nonetheless, I argue that we can muster reasons to prefer various topological properties over others. In particular, I favor the adoption of multiply connected universe models on grounds of (i) simplicity, (ii) Machian considerations, and (iii) explanatory power. We are able to appeal to such grounds because multiply connected topologies open up the possibility of finite universe models (consistent with our best data), which in turn avoid thorny issues concerning the postulation of an actually infinite universe. [ABSTRACT FROM AUTHOR]

This study delves into the concept of "infinity" within head-mounted display (HMD) virtual reality (VR) exhibitions. In this study, "infinity" is defined as an endless space and a boundless scope of virtual experience and interactions. We investigate how this expansive interpretation creates a unique and special exhibition experience that distinguishes virtual exhibitions from physical exhibitions. To this end, the study examines the effects of the spatial and participatory characteristics of virtual exhibitions on immersion, cybersickness, completion time, and interaction time. The experiment was conducted through surveys, behavioral observations, and interviews. The results showed that infinity of spatial characteristics increased participants' immersion. While participatory characteristics had no direct effect on immersion, there was a significant interaction between spatial and participatory characteristics. The 3D (space) surrounded by walls and create (participation), which allows participants to create their own exhibits, was the only combination that decreased immersion and increased cybersickness. The combination with the highest immersion was "3D infinite (space)" and "change (participation)," indicating that a design that allows users to interact with the exhibition in an infinite space is useful for promoting immersion in virtual exhibitions. Adjusting the proportion of spatial and participatory characteristics according to the purpose of the exhibition is recommended since infinity of spatial characteristics increases interaction time and infinity of participatory characteristics significantly increases completion time. The results of this study have practical implications for organizations developing virtual exhibitions and shed light on the value of infinity-based virtual exhibitions and the specific elements that make them possible. [ABSTRACT FROM AUTHOR]

Divine infinity in the Cartesian corpus can be diffracted into an infinity of distance, proper to the creation of eternal truths (1630), and an infinity of substance, identifiable in Meditatio III (1641). This distinction reveals the structure of Cartesian metaphysics: the conquest by rationality (accomplished by infinity of substance) of the foundation of rationality (manifested by infinity of distance). Hence the duality of infinity: infinity is a Janus bifrons, both founding and founded. To unfold this paradox, we first need to show that infinity fits into the tradition of metaphysics established by Duns Scotus, according to whom being obeys the primacy of concept, intelligibility and therefore univocity. As a result, the Cartesian idea of infinity paradoxically inherits the Scotian tradition, as further demonstrated by its determination of idea maxime vera, against any suspicion of material falsity. Thus, the metaphysical extraterritoriality of infinity may constitute a paradox at the very heart of the idea of infinity itself.

This report delineates the outcomes of an intervention conducted with in-service high school educators, focusing on elucidating three distinct scenarios within geometric and arithmetic domains: the infinitely large, infinitely numerous, and infinitesimally close. Grounded in the theoretical framework of conceptual change, it is posited that when an individual exhibits entrenched conceptions, it signifies a misclassification of the pertinent concept, necessitating a categorical shift to effectuate a transformation in their cognitive schema, particularly concerning the notion of infinity. Thus, the principal objective of this investigation was to ameliorate the entrenched conceptions held by educators pertaining to infinity through a workshop-based intervention. Preceding the workshop, educators predominantly exhibited conceptions aligned with natural and potential infinities. However, after the workshop, a discernible transition was observed, with educators engendering an actual conception of infinity or an omega-epsilon position, exemplified by their acceptance of equivalences such as 0.999…=1 and the parity in the cardinality of sets comprising natural numbers, even numbers, and perfect squares. Nonetheless, notwithstanding this progress, confident educators evinced resistance to embracing the concept of actual infinity, particularly in instances such as the hypothetical scenario depicted in Hilbert's Grand Hotel. Consequently, drawing upon the framework of conceptual change theory, it can be postulated that a complete categorical shift was not universally realized among educators due to their reluctance to revise entrenched beliefs concerning natural or potential infinity.

Leibniz's metaphysics appears to go a long way towards monism: it supports a strong dependence of limited things on the absolute or God and understands this dependence not only as causal dependence but also as a pervasive ontological dependence which involves the communality of nature between absolute and limited. Yet, Leibniz stops short of affirming monism. Why? This paper takes a fresh look at Leibniz's reasons for opposing monism through the lens of a virtually unknown text of 1698 on the metaphysical foundations of the infinite. Against the backdrop of the present-day monistic proposals of Jonathan Schaffer and Michael Della Rocca, the paper identifies and evaluates four different types of monism: 1. Whole-Part Monism; 2. World-Animal Monism; 3. God-Nature Monism and 4. Eleatic Monism. It argues that Leibniz's opposition to Whole-Part Monism, World-Animal Monism, and God-Nature Monism, is due to his conceptions of the infinite and of what it is to be "an absolutely absolute Being". Furthermore, it argues that the only type of monism which could preserve Leibniz's demanding notions of real infinite and absolute is Eleatic Monism. The latter, however, is also rejected by Leibniz due to our first-hand experience of what it is to be a substance. [ABSTRACT FROM AUTHOR]

If the states and goals proper to the mystic are not accounted for in conceptions of moral formation and knowledge, then the object of the mystic's pursuit – the infinitely greater God – inevitably likewise lacks the centrality it deserves. Consequently, our moral wisdom is finitized. To discover afresh a way for moral theology to take the infinitely greater God as its source and end, this paper draws on Gregory of Nyssa's Life of Moses. I argue that Gregory's depiction of moral progress, epektasis, along with his corresponding scriptural hermeneutic, anagogy, present a foundation from which moral and mystical theology can exist in their proper harmony. By contextualizing moral formation within the infinite mystery of God, epektasis establishes an essential continuity between seeing and following God. Likewise, in discovering behind every literal detail of Moses' life a spiritual meaning, Gregory's anagogic hermeneutic opens the virtuous life to the mystic's transcending vision. [ABSTRACT FROM AUTHOR]

Apeiroanxiety is a fear of infinity. Some people may experience anxiety or discomfort when contemplating the vastness or endless nature of certain concepts, such as time or space. Addressing any form of anxiety is important as it can impact an individual's daily functioning, wellbeing, and quality of life. The current study is aimed at developing a reliable and valid measure of apeiroanxiety in the context of the afterlife for the Pakistani Muslim population. The research had three studies: Query Study I generated an item pool of 18 items based on a literature review, semi-structured interviews, and expert opinion. This item pool was administered to a convenient sample of Pakistani Muslim participants (n = 1277). As a result of exploratory factor analysis, a uni-dimensional factor solution with five items was yielded (α = 0.90). Confirmatory factor analysis confirmed the obtained uni-dimensional factor solution in study II of the research with an excellent model fit. In study III, the convergent and discriminant validity of the scale was established on a separate independent sample (n = 485). Results yielded evidence of convergent validity as negative afterlife belief and rumination had a positive correlation with apeiroanxiety. The discriminant validity was established by a non-significant relationship between positive afterlife belief and apeiroanxiety. Moreover, the limitations, suggestions, and implications of the study are discussed. [ABSTRACT FROM AUTHOR]

One of William Lane Craig's key arguments for the finitude of the past is the Successive Addition Argument (SAA). Malpass (2021) has recently developed a novel challenge to the SAA, utilising a thought experiment from the work of Fred Dretske, which is meant to show that it is possible to count to infinity, to argue that there is a counterexample to the SAA's second premise. In this paper, I contend that the Malpass‐Dretske counterexample should not worry advocates of the SAA. First, I argue that one objection Malpass considers—the Potential Infinite Objection—reveals an interesting fact: the SAA's second premise is unnecessarily strong and can be weakened whilst still yielding the same conclusion. Second, I show how another one of the objections considered by Malpass—the Accumulation Objection—is successful, provided some clarification to the SAA's premises. The upshot of both analyses is that we generate two 'new' Successive Addition arguments that not only move the dialectic forward, but shed light on deeper assumptions and motivating intuitions concerning the Kalām. [ABSTRACT FROM AUTHOR]

The way Leibniz applied his philosophy to mathematics has been the subject of longstanding debates. A key piece of evidence is his letter to Masson on bodies. We offer an interpretation of this often misunderstood text, dealing with the status of infinite divisibility in nature , rather than in mathematics. In line with this distinction, we offer a reading of the fictionality of infinitesimals. The letter has been claimed to support a reading of infinitesimals according to which they are logical fictions, contradictory in their definition, and thus absolutely impossible. The advocates of such a reading have lumped infinitesimals with infinite wholes, which are rejected by Leibniz as contradicting the part–whole principle. Far from supporting this reading, the letter is arguably consistent with the view that infinitesimals, as inassignable quantities, are mentis fictiones , i.e., (well-founded) fictions usable in mathematics, but possibly contrary to the Leibnizian principle of the harmony of things and not necessarily idealizing anything in rerum natura. Unlike infinite wholes, infinitesimals—as well as imaginary roots and other well-founded fictions—may involve accidental (as opposed to absolute) impossibilities, in accordance with the Leibnizian theories of knowledge and modality. [ABSTRACT FROM AUTHOR]

Derrida defines différance as the "interruption of Hegelian dialectics." Although scholars have noted that Derrida pursues his critique of Hegel by means of Hegelian concepts, the way that Derrida employs specific Hegelian concepts in his critique, such as non-positionality, self-reference, and contradiction, has not been sufficiently investigated. In this essay, I reconstruct Derrida's critique of Hegel with special focus on the Hegelian concepts of non-positionality, self-reference, and contradiction. [ABSTRACT FROM AUTHOR]

Sixty (35 girls and 25 boys) 9th-grade students’ conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students’ number line descriptions and the explanations coalesced around six features: sequential ordering (i.e., numbers are sequentially represented), positivity-negativity of numbers (i.e., the number line contains positive and negative numbers), non-centrality (i.e., zero does not have to be in the center), infinity, increment flexibility (i.e., number line increments can vary), and continuity (i.e., numbers can be placed anywhere between minus infinity and plus infinity without breaks). The students’ explanations show that these six features emerge in five successive stages in the conceptual understanding of the number line. These stages are (1) no knowledge, (2) sequential ordering and positivity-negativity, (3) infinity and non-centrality, (4) incremental flexibility, and (5) continuity. The last two stages were not found in most descriptions. The results suggest that students’ understanding of the number line is incomplete and may be overestimated by commonly used quantitative assessments of number line knowledge.

The subject of the article is distance, and applications of its analysis, namely: distancial and proximal phenomena and research. The research method is exemplification, explanation of phenomena through characteristic examples, here – distances. Distance has different meanings: distance, detachment, distinction, retreat, distance, alienation, etc., having different natures: physical, figurative, religious, logical, etc. However, if understanding comes late, through distance, then what is the use of it? Depending on the understanding of the infinite, there are two types of distance: 1) a distance leading to an unattainable perfection, an ideal, here infinity turns into emptiness; 2) a distancethat includes the finite and emptiness as ideal moments of movement. The distance is determined through a series of successive approximations, or «near» and «far». In other words, the definition of distance coincides with the definition of truth. For example, K. Levi-Strauss showed that the savage thought as well as modern man, progress occurred not in thinking, but in living conditions. Here people of distant epochs come closer together, through finding a distance between them – this is indicated by the principle of the constancy of distance. An example of a distal study is the definition of the «power distance index», the proximity or remoteness of power from the people. The distances between the past, present and future have ideological significance. Research on this issue notes the reduction of the present time through growth, pressure and acceleration of innovation, alienation from the past and present. A. Einstein, denied the phenomenon of irreversibility, and believed that the division of time into the past, present and future is an illusion. The greatest distance between near and far is created by religion. The article proposes a new trichotomy of truth./false/faith, where faith is the sphere of meaning. Faith forms a distance that transcends human experience – transcendence. Distance and understanding should have a measure – the distance should not be too close or too far, and also go (slip away) into the empty otherworldliness of infinity. Distance correlates with the path, if the path is a necessary condition for existence and understanding, then distance should be a sufficient reason for the path (both understanding and existence). The concept of distance can be attributed to the «dimensional categories» of the synthetic paradigm (the expression of D. V. Pivovarov).

The aim of this article is to reassess Guerric's retraction (i.e. the changes he made to the thesis of the vision of God between Quaestio I and Quaestio III) and to provide some considerations that could, perhaps, contribute to a better understanding of how the condemnations of 1241 forced an author to abruptly contradict his own teaching. We want to shed new light on the difficult repositioning of theologians writing before and after the 1241 moment, who were obliged to openly overturn not only what they had upheld, but also many of the patristic authorities who defended the invisibility of the divine essence.

This paper considers what it would mean to conceive of philosophy as living. In the first part, by way of a discussion on intercultural encounter recently taken up by Cora Diamond, I first illustrate why philosophical conflict cannot be resolved within already given modes of thought or self-contained finite philosophical traditions, but instead transcends those. In the second part, I show why this dynamic plays out not only between cultures but also between the individual and that individual’s own tradition. I do this by drawing from insights of the two major proponents of xinxue 心學 (Learning of the Heart-Mind), Lu Jiuyuan 陸九淵 and Wang Yangming 王陽明. The way xinxue deals with both orthodox and heterodox traditions illuminates how we can understand philosophy as something living. It is neither self-contained and indifferent to its own heritage or to the culturally alien, nor does it consist of the mere accumulation of diverse philosophical contents. The deepening of individuality that xinxue introduces into Chinese philosophy consists in the relation of the individual to what has already been conceived. Ultimately, by grasping this dynamic that happens through the individual, we may better grasp why philosophy is not reducible to given modes of thought nor to self-contained finite philosophical traditions, but instead is infinite.

The paper examines the roles of Crescas and Spinoza in the transition from the medieval to the modern conception of the universe. Crescas is presented as an illustrative example of the tension between Aristotelianism and revealed religion and how the latter brings about the dissolution of the former, thus paving the way for the modern conception of the universe. It is then showed how this modern conception is embodied in Spinoza’s thought, which radicalizes some of its defining traits. This radicalization undermines the traditional conception of the Deus absconditus and leads in Spinoza to the replacement of religion by philosophy as the true divine revelation.

Hasdai Crescas (1340-1411) was a philosopher, rabbi and public person, who lived in a very turbulent period for the Iberian and Provençal Jewish communities of the late Middle Ages. Crescas made a vehement critique of the Aristotelian paradigm received from falsafa, which was used by Maimonides to support and prove the existence, unity and incorporeality of God, conceptualized in the Guide of the Perplexed as the necessary being which is absolutely transcendent in relation to contingent beings, that is, to the world. In Or Hashem, Crescas elaborates an alternative concept of the necessary being, in which the two antithetical notions of divine immanence and transcendence are related to the distinction within the necessary being between its simple essence and its infinite attributes. The simple, one, ineffable essence of the necessary being is expressed in infinite attributes in the eternal and constant act of giving in the univocality of being its good and its actuality to the infinite contingent beings. Crescas advocates that the universe, though ontologically contingent, is infinite in its actuality. God is thus conceived as the eternal and constant first cause, entelechy and Place of the World.

In this article, we discuss the embodiment of infinity as one of fundamental concepts in mathematics. In contrast to the embodiment of many other mathematical concepts, the embodiment of infinity is an endless dynamic process. In embodying +∞, an object moves rightward toward a previously-set limit and passes it. Then, a new limit is set on the right side of the moving object. The moving object continues its movement and passes it as well. The moving object can pass any limit. In other words, there is no impassable limit for it. In embodying -∞, a similar process happens but the movement is leftward. Embodiment of infinitely small quantities has a basic similarity to the embodiment of infinitely large quantities, although it is different in some respects. We call the embodiment of infinity as iterative embodiment. It is iterative because the process of setting a new limit and passing it is repeated endlessly. Finally, it is suggested that in the process of embodying infinitely large and infinitely small quantities, the visual system and the motor system play important roles, as this process involves spatial concepts and movement. [ABSTRACT FROM AUTHOR]

Benardete paradoxes involve a beginningless set each member of which satisfies some predicate just in case no earlier member satisfies it. Such paradoxes have been wielded on behalf of arguments for the impossibility of an infinite past. These arguments often deploy patchwork principles in support of their key linking premise. Here I argue that patchwork principles fail to justify this key premise. [ABSTRACT FROM AUTHOR]

In this article, the Hegelian concepts of bad infinity and true infinity are analyzed in the context of a review of the notion of infinity in the German philosophical tradition with which it engages. Punctually, we will address the formulations of the finitude-infinitude relation in the work of I. Kant, J. G. Fichte, and F. Schlegel before exposing Hegel’s interpretation of how transcendental philosophy and German romanticism approached this issue. Finally, we present the true way in which, according to Hegel, the infinite is to be understood. [ABSTRACT FROM AUTHOR]

We review and compare five ways of assigning totally ordered sizes to subsets of the natural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and a-numerosity. Generalised densities and a-numerosities lack uniqueness, which can be traced to intangibles: objects that can be proven to exist in ZFC while no explicit example of them can be given. As a sixth and final formalism, we consider a recent proposal by Trlifajová (2024), which we call c-numerosity. It is fully constructive and uniquely determined, but assigns merely partially ordered numerosity values. By relating all six formalisms to each other in terms of the underlying limit operations, we get a better sense of the intrinsic limitations in determining the sizes of subsets of N. [ABSTRACT FROM AUTHOR]