Actual Infinity Research Articles

Durand of Saint Pourçain and the actual infinite. A reflection on divine omnipotence (Super Sent., I, 43, 2)

In his Commentary on the first Book of Sentences (d. 43, q. 2, red. B), Durand of Saint Pourçain (1270-­1334) poses the question regarding the divine possibility of producing infinite actual things, contributing to the debate about the divine power of creating what is considered to be contradictory (actual infinity). Taking into account the philosophical and theological sources of Durand's Commentary on the issue, the article focuses on his accurate solution to the problem of God's production of actual infinites by analysing some arguments such as that of the production of the individuals of a species, like the souls, as well as the argument of the divisibility of the continuum and the original assumption of the commensurability of infinite quantities. The article shows that for Durand God cannot create an actual infinite, but only a potential one.

Генезис науки XVII века: опыты по «приручению» бесконечности

The author of the article considers the problem of actual infinity to be the meta-basis of the genesis of New European science. The author shows the advantages of this approach to identify the internal logic of the genesis of science of the XVII century, as well as to clarify the prospects of modern science, experiencing a crisis of its foundations. The author shows that Nikolai Kuzansky’s ontological project reveals the conditions necessary for the mathematization of nature for the transformation of the finite into the actually infinite, which overcomes the hopelessness of Zeno’s similar attempts. Zeno showed that there is no transition from the finite to the actually infinite. To transform the fi nite into the actually infi nite, the latter must already be given. This kind of presence is revealed by Kuzansky in the apophatic idea of the divine minimum coinciding with the maximum. But at the same time, Kuzansky speaks only about the imaginary, and not the real possibility of transforming nature on the basis of actual infinity. And in order to turn ‘scientific ignorance’ into ‘scientific knowledge’, it was necessary to present, outside the theological context, the possibility of transforming nature on the basis of actual infinity, discovered by Kuzansky. This task was set by the founders of the New European science. The author of the article analyzes and compares the attempts of Galileo and Descartes to ‘tame’ the actual infinity. The author shows that the difference in their positions on the question of the achievability of reliable knowledge, which laid important milestones in the history of the genesis of New European science (the transition from the scholastic tradition to probabilistic epistemology), is due to the awareness of the complexity of the problem of actual infinity. As a result, the author shows that neither the theoretical speculations of the founders of science, nor the attraction of modern additional resources unknown to antiquity or the Middle Ages (experiment, pinhole camera) were unable to change anything in the ratio of the finite and the infinite.

On the immanentization of the religious ideal in the culture (according to "Diaries" of Protopresbyter Alexander Schmemann)

The problem of the immanentization of the religious ideal is explicated in the context of cultural history as a synchronous "movement" from idea to ideal and from ideal to idea. In the process of realizing the idea of the transcendent within the framework of the Christian tradition, its immanentization takes place – reduction to ultimate forms that can be considered based on the intuition of potential infinity (ancient tradition) and based on the intuition of actual infinity (Christian philosophy). The intuition of actual infinity is actualized in the eschatological relation to the Other, and the intuition of potential infinity in the principle of hierarchy, universal for the Christian tradition. The synthesis of these intuitions, with the predominance of one of them, is a tragedy for the Christian tradition, which determines the peculiarities of the spiritual searches of thinkers. The reflection of this tragedy in the works of Alexander Schmemann suggests a return to the "attitude" characteristic of early Christianity for the future, correlating with the preserved tradition based on hierarchy. At the same time, the philosopher's spiritual quest reflects the process of devaluation of the hierarchical principle in Christian culture, stimulating the inversion of the hierarchy, when transcendence can be realized through low-ranking humor and foolishness. The work uses the method of historical and philosophical reconstruction, which allows to identify the features of the interaction of the transcendent and immanent in certain philosophical and religious concepts. The synchronic method explicates the mutual correlations between philosophical and religious concepts belonging to the same historical period. The diachronic method makes it possible to actualize the features of the immanentization of the religious ideal in the work of thinkers belonging to different historical periods. The scientific novelty of the research lies in the combination of reflection on the relationship between the transcendent and the immanent, carried out on a rational basis by European philosophy and the study of the experience of transcendence described by specific Christian writers and thinkers (in particular St. Augustine). Thanks to this combination, the study realized the possibility of considering the relationship of the transcendent and immanent in the history of European philosophy using the logic of the symbol by A.F. Losev, when the structure of the interaction of the symbolized and the symbolizing is extrapolated to the correlation of the transcendent and immanent, and the scope of the concept of "transcendent-immanent" becomes equivalent to its content. The process of immanentization of the transcendent is considered simultaneously from the outside (from the point of view of the dialectic of concepts) and from the inside from the point of view of changing the paradigm of religious life in culture, which allows us to take into account its features.

Turing Machine Halting Problem, Russell’s Paradox and Gödel Incompleteness Theorem

The Turing Machine Halting Problem is a major problem in computer theory, Russell’s Paradox is the root of the Third Mathematical Crisis, and the Gödel Incompleteness Theorem is a major discovery in modern logic. The three have had a profound impact on the development of science and have attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned; in particular, Wittgenstein regards it as a certain logical paradox, and Russell’s Paradox has not yet been settled. This paper makes a detailed analysis of the three based on the view of dialectical infinity. The author notes that the Principle of Comprehension based on the view of actual infinity is the root of Russell’s Paradox. The Turing Machine Halting Problem shows that it is impossible to make an actual-infinite ultimate judgment of the constantly generated infinite world, but the philosophical significance of the Gödel Incompleteness Theorem is that our understanding of the world is essentially potentially infinite. At the end of the article, the author raises several questions about the proof of the Gödel Incompleteness Theorem, finds out the specific paradox form in the proof, points out the high consistency of its proof method and Russell’s Paradox, which strongly supports Wittgenstein’s view. The author points out that the philosophical basis of the proof of the Gödel Incompleteness Theorem is the idea of actual infinity, the proof of the theorem is based on a logically invalid circular formula, the contradiction of the proof originates from the Gödel formula itself, and cannot be attributed to the incompleteness of the system, so the proof is wrong. Therefore, the conclusion of this paper is that the world is constantly developing and changing, and our human understanding of the world is essentially a potential infinite, that is, the world is Aristotelian, not Platonic.

An Explanation to the Concept of Actual Infinity and Potential Infinity through Set Theory and Calculus

The concept of infinity refers to either an unending process or a limitless quantity. Aristotle introduced two types of infinity: potential infinity and actual infinity. Potential infinity refers to a never-ending process, and actual infinity refers to a collection containing infinitely many elements. This paper presents a descriptive study of the concept of infinity and discusses its properties through set theory and calculus. Infinity plays a central role in the formation and development of mathematics, specifically in limit, derivative, and integral. Moreover, the similarities and differences between potential infinity and actual infinity are explained with the help of set theory and integral differential calculus. The relationship between mathematics and infinity is a vital one. Since infinity is an elusive and contradictory idea without mathematical tools, it is hard to understand it, and there is no other knowledge to explain and make it understandable. By the way, in the absence of infinity, mathematics will never survive. This paper provided some examples to show that without employing mathematics, solving problems involving infinity based on human intuitions or weak induction may provide inaccurate results or lead to contradictions. Therefore, this paper suggested that using mathematical tools is essential for solving problems involving infinity.

Infinity and the Possibility of Metaphysics. The Debate on Kant's Antinomies in France (1850s-­1920s)

The article focuses on a neglected chapter in the French reception of Kant: the debate on antinomies. Initially understood, by Victor Cousin's eclecticism, as symbolizing the abstractness of the Kantian system, antinomies were the focus of cogent interpretations in the second half of the nineteenth century. The debate that arose following Charles Renouvier's solution, based on the rejection of the mathematical notion of actual infinity, was intense, involving major philosophers of the time, such as François Evellin, Louis Couturat, Henri Bergson and Léon Brunschvicg. The stakes were decisive: the nature of reason, the relationship between philosophy and science, and the possibility of a positive metaphysics. The Author reconstructs and analyzes the various interpretations, showing how the debate on antinomies played a decisive role in the formation of French philosophy between the nineteenth and twentieth centuries.

THE MAGIC OF METAPHYSICS: SUPER-RECONSTRUCT OF THE SUBCONSCIOUS

Algebraic unknown x is known virtually ( unknown-known ), ω actual infinity is virtually finite ( infinite-finite ), the transcendental number e is determined by the limit in which the variable is virtually constant ( variable-constant ). The excitement of the subconscious prepares the neutrality of the consciousness. Based on the two-level coordinates of the Cartesian square, a metaphysical clock is constructed.

Kant's Turn: Between Actual and Potential Infinity

Introduction. The purpose of the article is to show that Kant's turn is the answer to the problem of actual infinity, in the light of which it becomes clear how Immanuel Kant contributed to the turn of Western epistemology to non-classics. The relevance of the research is due to the need to make modern epistemology more “sighted” in relation to its own prospects and opportunities.Methodology and sources. The author of the article reconstructs and critically analyzes Kant's approach to neutralizing the problem of actual infinity on the basis of the involvement of both primary sources and historical-philosophical and historical-scientific works, modern research in the field of the theory of cognition. Results and discussion. Kant's separation of “in itself” and “for us” separates the finite and dependent world from the infinite and independent. The latter is emptied of its theological content (inherited from Nicholas of Cusa) and, instead of a central meaning, receives a peripheral one, is declared an unknowable thing in itself. And the former is endowed with some properties of the latter and is declared a self-sufficient education. So the transcendent becomes transcendental, and the problem of the connection between the finite and the infinite, discussed by Kant's predecessors (Galileo, Descartes, etc.), is neutralized and passes into the category of antinomies. But the discovery of the inconsistency of Kant's separation “in itself” and “for us” makes the problem of the ratio of the finite and the infinite relevant again. However, now the former becomes completely blind to the latter, so that everything that is beyond “for us” can be attributed to “in itself”. Thus, in place of the actual infinity as transcendent, a potential infinity arises that does not know the limits of its immanence. The author of the article shows that such substitution, while remaining unidentified, can disorient the modern theory of cognition even more.Conclusion. Revealing how Kant's reinterpretation of the relation of the finite and the infinite contributed to the substitution of the actual infinity by the potential infinity makes it clear that such a substitution does not neutralize the original problem of the actual infinity, but that such a substitution is itself a real problem, the sharpening of which promises great changes in the theory of knowledge.

Many infinities, or only one?

In this short note we first account for Aristotle's views on infinity, by clarifying the way his notion of potential infinity should be understood, in the light of his notion of entelechy. We then present four distinct ways in which mathematicians attempted to tame the notion of actual infinity, and we ask the question of whether they are indeed four different ways, or whether they ultimately are variations on the same concept. We suggest that what mathematicians are doing is indeed to find a way to construct a form of actual infinity that subsumes, into itself, the potential infinity, and observe how the presence of the former brings to a different view about potential infinity than Aristotle's.

The Concept of Infinity under German Idealism and Modern Set Theory

The purpose of this paper is to compare and analyse the concepts of infinity in German idealism and Modern set theory. The fisrt part of this paper analyses Kant and Hegels views on mathematical infinity, where Kant suggests a notion of potential infinity constructed by means of intuition, while Hegel that of actual infinity. The second part illustrates the characteristics of Cantorian transfinite sets and both the progress it has made and the limitations. Following the two main parts, not only a clear contrast between the notions of infinity under German idealism and Cantorian set theory can be made but their close link can also be shown. By analysing perspectives of Freges logicism, Hilberts formalism, Brouwers intuitionism, and Badious comments, it is clear German idealism has lost its mainstream position in understanding the mathematical concept of infinity. However, the concept of infinity in German idealism can be supplied as a powerful facilitator for philosophers and mathematicians to understand this concept in the realm of mathematics.

The Problem of the Actual Infinity and the Path of the New European Classical Epistemology: from Genesis to the Modern Crisis

Introduction. The purpose of the paper is to show the prospects of the problem of actual infinity for understanding the ways of New European classical epistemology. The relevance of the work is due to the need to fill a gap in the history of New European epistemology, which today makes it difficult to understand the essence of its crisis. Such a lacuna was formed by the lack of grounds for the transition from the probabilistic concept of natural science, which takes into account apophaticism associated with the idea of actual infinity, to positive science, which presupposes a positive understanding of actual infinity.Methodology and sources. The author of the paper reconstructs and critically analyzes the attempts of the founders of New European science (Galileo and Descartes) to reform the ontological project of Nikolai Kuzansky due to the need for a positive understanding of the actual infinity. The author distinguishes between the reception of the problem of actual infinity in the New European philosophy and its objective state. Based on a comparative analysis, the author of the article establishes the identity of the logic of the development of the problem of actual infinity in the studies of Zeno of Elea and in New European epistemology, thereby testifying to its enduring significance.Results and discussion. The author shows that in the development of the problem of actual infinity, an objective regularity manifests itself, consisting in the fact that an unjustified assumption of the possibility of changing physical nature on the basis of the actual infinite really leads to a change in nature, but only the basis for this change is not actual, but potential infinity. This logic was reproduced in the history of New European epistemology. Thereby, mathematical natural science really turned out to be possible, but not on what its creators put in the basis (actual infinity), but on a derivative of it (potential infinity), and as a result it strengthens the position of supporters of non-classics.Conclusion. The problem of actual infinity makes it possible to understand the logic of the development of New European epistemology from its genesis to the modern crisis, and shows a deep interconnection between these two points.

수직선 및 수체계, 실수의 성질 의미와 관계에 대한 중학생의 오류와 그 오류 사이의 관계

Objectives This study explores middle school students' errors in explaining the concept of the number line, number system, and completeness among the properties of real numbers. Furthermore, it aims to explore the relationship between each error.
 Methods To conduct this study, 12 middle school students were recruited and interviewed. The students' responses were analyzed and compared with textbooks and tutorials. Through this analysis, the connection between errors in the number line and number system, and the meaning and relationship of the properties of real numbers was examined.
 Results As a result, errors in number representation and number systems were found to be linked to improper use of definitions or theorems in explaining completeness. Additionally, errors in the method of mapping irrational numbers on the number line, and the belief that it is impossible to map irrational numbers on the number line were linked to logically improper reasoning in explaining completeness.
 Conclusions To address these errors, the study suggests three implications for the direction of textbook narrative and teachers' pedagogical methods. First, it is necessary to present a link that shows that the number line can be enlarged. Second, the distinction between fractions, decimals, and number systems should be emphasized. Third, it is necessary to present various endpoints in the section on the number line, and a method to freely change the perspective of potential and actual infinity. Based on these results, the study derives implications for the direction of textbook narrative and teachers' pedagogical methods.

On Hegel’s View of Dialectical Infinity

It is well known that the problem of finity and infinity is the basic problem of mathematics, and it is also the basic problem of Philosophy. From the perspective of philosophy and mathematics, this paper comprehensively reviews and analyzes Hegel’s view of dialectical infinity, introduces Engels’discussion on infinity, deeply analyzes the characteristics of the thought of actual infinity, and points out: Hegel’s thought of real infinity is completely different from the thought of actual infinity, the Being of infinity (objective infinity) is not equal to the completed infinity (subjective infinity), the mathematical limit is a real infinity, and real infinity is the inner law of infinite things and truth; the view of actual infinity views the objective material world from the viewpoint of static rather than motion, denying the contradiction between finity and infinity, so it is actually a downright idealist. In this paper, the author puts forward the Infinite Exchange Paradox, which strongly questions the idea of actual infinity in Hilbert Hotel Problem, and points out the internal irreconcilable contradiction in the idea of actual infinity. At the same time, we made a detailed comparison of Hegel’s view of infinity and the view of mathematical infinity, and on this basis, the author gives a complete definition of the view of dialectical infinity: abandoning the wrong aspects of the potential infinity and actual infinity, and actively absorbing correct aspects of both, that is, not only to recognize the existence and knowability of infinite objectivity, but also to admit the imcompletion of infinite process. The reexcavation of Hegel’s view of dialectical infinity and the criticism of the actual infinity thought aim to find possible philosophical solutions for Russell’s Paradox and the problem of Continuum Hypothesis.

Actual Infinity: Spinoza’s Substance Monism as a Reply to Aristotle’s Physics

I conceive of Spinoza’s substance monism as a response to Aristotle’s prohibition against actual infinity for one key reason: nature, being all things, is necessarily infi nite. Spinoza encapsulates his substance monism with the phrase, “Deus sive Natura,” implying that there is only one infinite substance, which also possesses an infi nity of attributes, of which we are but modes. These logical delineations of substance never actually break up God’s reality. Aristotle’s well-known argument against the reality of an actual infinity in his Physics prohibits the existence of an actually infinite bodily substance because it would necessarily “destroy” (Physics 204b26-27) all other elements or bodies. On Aristotle’s view, there is a fundamental and concrete distinction between things: each substance is primarily a this (Categories 3b10). I maintain that Spinoza’s rationalism and radicalization of the principle of sufficient reason lends him greater explanatory potential than Aristotle to justify the (non) existence of actual infinity.

Actual Infinity, Freedom and Modern Theology

Actual Infinity, Freedom and Modern Theology

Kalam Cosmological Argument and the actual infinity

The kalam cosmological argument has received a renewed attention in philosophical literature after several presentations and defenses of its versions offered by William Lane Craig. The crux of the debate about Craig?s a priori arguments in its favor has had mainly to do with the analysis of the concept of actual infinity and its implications for the possibility that the universe has an infinite past. While Craig?s arguments point to the counter-intuitiveness of the implications of the existence such a past, his critics offer reasons to think that his analysis is flawed in different respects, mostly due to inappropriate application of mathematical concept of an infinite set. I aim to show that the criticisms on offer have flaws of their own and, hence, fail to show that Craig?s reasoning is insufficient to offer serious reasons to doubt the coherence of the notion of the infinite past. I argue that a more serious threat to Craig?s arguments lies in certain types of symmetry between the past and the future.

Something Valid This Way Comes: A Study of Neologicism and Proof-Theoretic Validity

AbstractThe interplay of philosophical ambitions and technical reality have given birth to rich and interesting approaches to explain the oft-claimed special character of mathematical and logical knowledge. Two projects stand out both for their audacity and their innovativeness. These are logicism and proof-theoretic semantics. This dissertation contains three chapters exploring the limits of these two projects. In both cases I find the formal results offer a mixed blessing to the philosophical projects.Chapter 1. Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. I re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. I conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.Chapter 2. There have been several recent results bringing into focus the super-intuitionistic nature of most notions of proof-theoretic validity. But there has been very little work evaluating the consequences of these results. In this chapter, I explore the question of whether these results undermine the claim that proof-theoretic validity shows us which inferences follow from the meaning of the connectives when defined by their introduction rules. It is argued that the super-intuitionistic inferences are valid due to the correspondence between the treatment of the atomic formulas and more complex formulas. And so the goals of proof-theoretic validity are not undermined.Chapter 3. Prawitz (1971) conjectured that proof-theoretic validity offers a semantics for intuitionistic logic. This conjecture has recently been proven false by Piecha and Schroeder-Heister (2019). This chapter resolves one of the questions left open by this recent result by showing the extensional alignment of proof-theoretic validity and general inquisitive logic. General inquisitive logic is a generalisation of inquisitive semantics, a uniform semantics for questions and assertions. The chapter further defines a notion of quasi-proof-theoretic validity by restricting proof-theoretic validity to allow double negation elimination for atomic formulas and proves the extensional alignment of quasi-proof-theoretic validity and inquisitive logic.Abstract prepared by Will Stafford extracted partially from the dissertation.E-mail: stafford@flu.cas.czURL: https://escholarship.org/uc/item/33c6h00c

Актуальная бесконечность: псевдопроблема или метаоснование западноевропейской философии и науки?

The author of the article reveals the theological context of the origin of the concept of actual infinity and clarifies the problem of actual infinity. The author shows that this problem is not a paradoxical category of thinking, but a problem of the unity of two realities (eternal, unchanging and infinite, and temporary, changeable and finite), which has been misunderstood. The author raises the question of the relevance of the problem of actual infinity brought by Christianity for modern secularized science and philosophy. The author shows that the problem of the unity of the two realities was declared much earlier than Christianity. This problem was already dealt with by the ancient Eleans. They initiated the one-sided view and incorrect understanding of this problem, which opened the main path of development of the entire Western European philosophy. With the advent of Christianity, all the dangers identified by the Eleans (and above all by Zeno) and then still unclear on this path received a new sharpness and now real force. The author of the article shows that the regularity of the relation of the finite, the actually infinite, and the potentially infinite, revealed by Zeno, was the basis for changing the classical rationality to the non-classical one. In turn, the fact of the collapse of the classics has become evidence of modernity that the problem of actual infinity is not a mental paradox, but contains the real possibility of changing the finite nature. But this change is not carried out in the direction suggested by the recognition of actual infinity itself, but in another direction, opposite to it, but closely connected with it. The disclosure of the essence of this connection will be the disclosure of the problem of actual infinity.

FLORENSKY’S PHILOSOPHICAL IDEAS AND THE THEORY OF SETS OF CANTOR

The article considers the role of the idea of actual infinity in the works of Florensky. The introduction briefly traces the history of ideas about the actual infinity in European culture to the works of George Cantor. The reaction of European scientists and religious figures to the emergence of the “naïve” theory of Cantor sets is characterized. A detailed analysis of the connection between Florensky and George Cantor’s ideas is given. Many quotations from the 1904 work on the symbols of Infinity are given, which illustrate the influence of Cantor’s works on Florensky. The presentation of Florensky’s religious and philosophical ideas of Cantor about the actual infinity is given. Emphasized understanding Florensky transfinite numbers Cantor as symbols.

TWO TRADITIONS OF THE ACTUAL INFINITY INTERPRETATION

The article considers two traditions in the interpretation of the actual infinity. One is associated with the name of Nicholas of Cusa, the other with the name of Rene Descartes. It is shown how Nicholas of Cusa within the framework of his idea of the coincidentia oppositorum overcomes the traditional Aristotelian norms of philosophizing, while Descartes puts the finitist ideology at the foundation of both his theology and the theory of knowledge.