The notion of potential infinity dominated in mathematical thinking about infinity from Aristotle until Cantor. The coherence and philosophical importance of the notion are defended. Particular attention is paid to the question of whether potential infinity is compatible with classical logic or requires a weaker logic, perhaps intuitionistic.

Online philosophy seminar notes, for virtual conference on the Aristotelian philosophy of mathematics, hosted by University of Geneva (organiser Ryan Miller), June 15, 2023.

Our first goal here is to show how one can use a modal language to explicate potentiality and incomplete or indeterminate domains in mathematics, along the lines of previous work. We then show how potentiality bears on some longstanding items of concern to Mark Steiner: the applicability of mathematics, explanation, and de re propositional attitudes toward mathematical objects.

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, (...) the right modal logic is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Modal logic provides an elegant way to understand the notion of potential infinity. This raises the question of what the right modal logic is for reasoning about potential infinity. In this article I identify a choice point in determining the right modal logic: Can a potentially infinite collection ever be expanded in two mutually incompatible ways? If not, then the possible expansions are convergent; if so, then the possible expansions are branching. When possible expansions are convergent, (...) the right modal logic is S4.2, and a mirroring theorem due to Linnebo allows for a natural potentialist interpretation of mathematical discourse. When the possible expansions are branching, the right modal logic is S4. However, the usual box and diamond do not suffice to express everything the potentialist wants to express. I argue that the potentialist also needs an operator expressing that something will eventually happen in every possible expansion. I prove that the result of adding this operator to S4 makes the set of validities Pi-1-1 hard. This result makes it unlikely that there is any natural translation of ordinary mathematical discourse into the potentialist framework in the context of branching possibilities. (shrink)

Do actual infinities exist or are they impossible? Does mathematical practice require the existence of actual infinities, or are potential infinities enough? Contrasting points of view are examined in depth, concentrating on Aristotle’s ancient arguments against actual infinities. In the long 19th century, we consider Cantor’s successful rehabilitation of the actual infinite within his set theory, his views on the continuum, Zeno's paradoxes, and the domain principle, criticisms by Frege, and the axiomatisation of set theory by Zermelo, as well (...) as Zermelo’s assertion of the primacy of potential infinity in mathematics. (shrink)

Against any obscurantist stand, denying the interest of natural sciences for the comprehension of human meaning and language, but also against any reductionist hypothesis, frustrating the specificity of the semiotic point of view on nature, the paper argues that the deepest dynamic at the basis of meaning consists in its being a mechanism of ‘potentiality navigation’ within a universe generally characterized by motility. On the one hand, such a hypothesis widens the sphere of meaning to all beings somehow endowed with (...) the capacity of moving and/or perceiving movement. On the other hand, through a new evolutionist interpretation of the concept of generativity, such a hypothesis preserves the peculiarity of human meaning, meant as essentially founded on a certain intuition of infinity. Two corollaries stem from this hypothesis: first, religiosity can be considered as a matrix of grammars of infinity, aiming at regimenting its flight of potentialities. Second, non-genetic transmission of cultural information exerts determinant influence also at the level of that very deep mechanism of the human predicament that is the cognitive navigation of motor potentialities. A re-reading of the structuralist epistemology, scientific literature on the nervous cells of jellyfish, and some recent experiments on the mirror neurons of dancers, as well as certain intuitions of Teilhard de Chardin, are the main arguments of the paper. (shrink)

Two fundamental categories of any ontology are the category of objects and the category of universals. We discuss the question whether either of these categories can be infinite or not. In the category of objects, the subcategory of physical objects is examined within the context of different cosmological theories regarding the different kinds of fundamental objects in the universe. Abstract objects are discussed in terms of sets and the intensional objects of conceptual realism. The category of universals is discussed in (...) terms of the three major theories of universals: nominalism, realism, and conceptualism. The finitude of mind pertains only to conceptualism. We consider the question of whether or not this finitude precludes impredicative concept formation. An explication of potential infinity, especially as applied to concepts and expressions, is given. We also briefly discuss a logic of plural objects, or groups of single objects (individuals), which is based on Bertrand Russell’s (1903, The principles of mathematics, 2nd edn. (1938). Norton & Co, NY) notion of a class as many. The universal class as many does not exist in this logic if there are two or more single objects; but the issue is undecided if there is just one individual. We note that adding plural objects (groups) to an ontology with a countable infinity of individuals (single objects) does not generate an uncountable infinity of classes as many. (shrink)

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm (...) of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. (shrink)

This article discusses the history of the concepts of potential infinity and actual infinity in the context of Christian theology, mathematical thinking and metaphysical reasoning. It shows that the structure of Ancient Greek rationality could not go beyond the concept of potential infinity, which is highlighted in Aristotle's metaphysics. The limitations of the metaphysical mind of ancient Greece were overcome through Christian theology and its concept of the infinite God, as formulated in Gregory of Nyssa's (...) theology. That is how the concept of actual infinity emerged. However, Gregory of Nyssa's understanding of human rationality went still further. He said that access to the infinity of God was to be found only in an asymptotic ascension, as expressed in apophatic theology, and this view endangered the rationality of theology. Deeply influenced by the apophatic tradition, Nicholas of Cusa avoided this danger by showing that infinity could be accessed by symbolic representation and asymptotic mathematical reasoning. Thus, he made an immense contribution in making infinity rationally accessible. This endeavor was finally realized by Georg Cantor. This rational accessibility revealed discernible structures of infinity, such as the continuum hypothesis, and the cardinal numbers. However, the probable logical inconsistency of Georg Cantor's all-encompassing Absolute Infinity points to an intuitive understanding of infinity that goes beyond its rational structures – as in apophatic theology. (shrink)

Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other (...) hand, Aristotle says that infinity "exists in actuality as a process that is now occurring" (234). Bowin makes clear that Aristotle doesn't explicitly solve this problem, so we are left to work out the best reading we can. His proposed solution is that "infinity must be...a per se accident...of number and magnitude" (250). (Bryn Mawr Classical Review 2008.07.47). (shrink)

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. We 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. We 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. (shrink)

Puzzles can arise in value theory and deontic (permissibility) theory when infinity is involved. These puzzles can arise for ethics, for prudence, or for any normative perspective. For the sake of simplicity, we focus on the ethical versions of these problems. We start by addressing problems that can arise in determining what is permissible, either in a given choice situation when there are an infinite number of options or in infinite sequence of choice situations, each with only finitely many (...) options. There is a common theme here: standard and plausible decision rules, such as the dominance principle or maximising principles, run into conflict with other plausible principles. We then address problems that can arise in determining whether one option is more valuable than another, when the options have infinite, or undefined, value. This includes cases where a single bearer of value has infinite or undefined value, as well as cases where each bearer of value is finite but the total for the collection is infinite or undefined totals. Our focus is on introducing the puzzles, rather than canvassing potential solutions. (shrink)

Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to (...) avoid introduction of potential and actual infinities. If decidability and consistency are desired, keep formal systems finite. Infinity is a useful heuristic concept, but has no place in proof theory. Implications: We attempt to debunk many of the mysticisms and uncritical adulations of Gödelian arguments and to ground mathematical foundations in intersubjectively verifiable operations of limited observers. We hope that these insights will be useful to anyone trying to make sense of claims about the nature of formal systems. If we return to the notion of formal systems as concrete, finite systems, then we can be clear about the nature of computations that can be physically realized. In practical terms, the answer is not to proscribe notions of the infinite, but to recognize that these concepts have a different status with respect to their verifiability. We need to demarcate clearly the realm of free creation and imagination, where platonic entities are useful heuristic devices, and the realm of verification, testing, and proof, where infinities introduce ill-defined entities that create ambiguities and undecidable, ill-posed sets of propositions. Constructivist content: The paper attempts to extend the scope of radical constructivist perspective to mathematical systems, and to discuss the relationships between radical constructivism and other allied, yet distinct perspectives in the debate over the foundations of mathematics, such as psychological constructivism and mathematical constructivism. (shrink)

In the Reality we know, we cannot say if something is infinite whether we are doing Physics, Biology, Sociology or Economics. This means we have to be careful using this concept. Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of ω? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on another worldly absolute truth. Many mathematicians (...) believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that our knowledge of the physical world is imperfect and falls short of what we can apprehend with mathematical thinking. The objective of this paper is to present an epistemological rather than an historical vision of the mathematical concept of infinity that examines the dialectic between the actual and potential infinity. (shrink)

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. (shrink)

In the late 1940s and early 1950s Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as the precursor to the more well-known dialogical logic and one could assumed that the same philosophical motivations were present in both works. However we want to show that this is not always the case. In particular, we claim, that Lorenzen’s well-known rejection of the actual infinite as stated in Lorenzen (1957) was not a major motivation (...) for operative logic and mathematics. In this article, we claim that this is in fact not the case. Rather, we argue for a shift that happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operativism is concerned with a critique of the Cantorian notion of set and related questions about the notion of countability and uncountability; only later, his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

Modal logic has been used to analyze potential infinity and potentialism more generally. However, the standard analysis breaks down in cases of divergent possibilities, where there are two or more possibilities that can be individually realized but which are jointly incompatible. This paper has three aims. First, using the intuitionistic theory of choice sequences, we motivate the need for a modal analysis of divergent potentialism and explain the challenges this involves. Then, using Beth–Kripke semantics for intuitionistic logic, we (...) overcome those challenges. Finally, we apply our modal analysis of divergent potentialism to make choice sequences comprehensible in classical terms. (shrink)

The article analyses the explication of the infinity in the philosophical courses taught at Kyiv-Mohyla Academy at the 17th and 18th centuries. It examines 12 philosophical courses – since 1645 (the course by Inokentii Gizel) until 1751 (the course by Georgii Konyskyi). It shows how the infinity was defined and in which kinds it was divided in different courses. In general, all the professors, as well as other scholastic philosophers, agree that categorematic infinity exists only in God, (...) but syncategorematic is present in the created world. Regarding the question whether God, being omnipotent, can create a categorematic infinity in the world, the Mogilyans are divided into several camps: (1) Inokentii Gizel, Stefan Yavorskyi, Inokentii Popovskyi, Sylvestr Pinovskyi, Platon Malynovskyi gave a positive answer to the question; (2) Yosyf Volchanskyi, Ilarion Levytskyi, Amvrosii Dubnevych, Sylvestr Kulabka believed that this kind of infinity is in principle impossible, hence God cannot create it; (3) Teofan Prokopovych and Georgii Konyskyi took a sceptical stand and consider that the human mind as such could not solve this problem. The article analyses which arguments were offered by each camp, and gives suggestions what influenced the position of certain Mohylian professors. Most probably, Mohylian philosophers who supported the possibility of actual infinity in the created world were influenced by nominalistically oriented Jesuit philosophers, like Pedro Hurtando de Mendoza and Rodrigo Arriaga. The other Mohylians backed a more traditional idea, supported by Thomas Aquinas, that only one actual infinity can exist and it is God. (shrink)

The aim of this paper is not only to deal with the concept of infinity, but also to develop some considerations about the epistemological status of cosmology. These problems are connected because from an epistemological point of view, cosmology, meant as the study of the universe as a whole, is not merely a physical (or empirical) science. On the contrary it has an unavoidable metaphysical character which can be found in questions like “why is there this universe (or a (...) universe at all)?”. As a consequence, questions concerning the infinity of the universe in space and time can correctly arise only taking into account this metaphysical character of cosmology. Accordingly, in the following paper it will be shown that two different concepts of physical infinity of the universe (the relativistic one and the inflationary one) rely on two different ways of solution of a metaphysical problem. The difference between these concepts cannot be analysed using the classical distinctions between actual/potential infinity or numerable/continuum infinity, but the introduction of a new “modal” distinction will be necessary. Finally, it will be illustrated the role of a philosophical concept of infinity of the universe. (shrink)

Aristotle was the first not only to distinguish between potential and actual infinity but also to insist that potential infinity alone is enough for mathematics ...

In the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the better-known dialogical logic, and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s well-known rejection of the actual infinite, as stated in Lorenzen, was not a major motivation for operative (...) logic and mathematics. Rather, we argue that a shift happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operationism is concerned with a critique of the Cantorian notion of set and with related questions about the notions of countability and uncountability; it is only later that his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity. (shrink)

This dissertation shows that Proclus provides a consistent reading of Plato's late dialogues, and develops a three level ontology which stands on its own. By augmenting the reserve of Platonist philosophy with Post Platonic developments of Greek mathematics and astronomy and physics, at points where Platonism ceased to provide operating principles, Proclus, reached for formulations which went beyond Plato. His own metaphysics, though sometimes obscured by theurgic allusions, grounds Being in an infinite One. ;One of the problems that Proclus attempts (...) to solve through the posit of three infinities, is that an iterative potential infinity, seen in the interval structure of the physical world, could divide to oblivion if uncolonized by Principle. The 'One Being', which brings Limit and noetic sameness, is itself entrapped in Self Reflexivity. An infinity of a third kind, one beyond all Limit and Unlimit is posed to solve this problem. This is Proclus' NeoPlatonic solution to the fact that Being alone, albeit under the rule of intellect, cannot guarantee itself. ;Platonism assumes a "universe" i.e. One grounding Many, which predetermines the place of all structure. $\rm A\nu\lambda o\gamma\iota\alpha$ is a principle of substitution which governs the visible cosmos and makes it comprehensible by mathematical formulations. Plato's emphasis on the self same as the epitome of Being, with circularity as the highest form for thought, helps Proclus provide a consistent account of: Creation ; Infinity as a Simultaneous Whole ; Soul and Nous, Dialectic is valorized over other forms of thought as most like $$ the eternal truth and a means to assimilation. Procession and reversion forms a circle which demonstrates a metaphysical successor of cosmic revolutions found to exist as the heavenly bodies carry out their noetic patterns. ;Finally Eternity encapsulates Time; but again its infinite circularity is only recouped by a One beyond all Limit which can carry out the Providence of a final cause. (shrink)

A reflection upon Husserl's notion of an "Idea in a Kantian sense" calls for an inquiry into the relationship between experience and infinity. This question is first considered in Kant's doctrine of antinomies. It is shown that, in the Critique of Pure Reason, infinity is held to be a mere idea, which, however, has an indispensable regulative function in experience. It is at this point that Kant is compared with Husserl, who, drawing upon the notion of regulative principle (...) elaborated in the Critique of Pure Reason, conceives of a thing in its particular reality as an Idea in a Kantian sense. A majordifference between the two thinkers is particularly emphasized: Kant uses his analysis of the antinomies for justifying his distinction between the ' thing in itself and 'appearance'; Husserl, on the contrary, tries to overcome this opposition. It is argued forthat this difference between the two philosophers arises from two different notions of infinity: whereas Kant has a potential infinity in view, Husserl, who is familiar with Cantors mathematical and philosophical thoughts, relies upon a scientifically established form of actual, but nevertheless open infinity. (shrink)

Throughout the long centuries of western metaphysics the problem of the infinite has kept surfacing in different but important ways. It had confronted Greek philosophical speculation from earliest times. It appeared in the definition of the divine attributed to Thales in Diogenes Laertius (I, 36) under the description "that which has neither beginning nor end. " It was presented on the scroll of Anaximander with enough precision to allow doxographers to transmit it in the technical terminology of the unlimited (apeiron) (...) and the indeterminate (aoriston). The respective quanti tative and qualitative implications of these terms could hardly avoid causing trouble. The formation of the words, moreover, was clearly negative or privative in bearing. Yet in the philosophical framework the notion in its earliest use meant something highly positive, signifying fruitful content for the first principle of all the things that have positive status in the universe. These tensions could not help but make themselves felt through the course of later Greek thought. In one extreme the notion of the infinite was refined in a way that left it appropriated to the Aristotelian category of quantity. In Aristotle (Phys. III 6-8) it came to appear as essentially re quiring imperfection and lack. It meant the capacity for never-ending increase. It was always potential, never completely actualized. (shrink)

We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.

What is music? Modern society has come to view music largely as entertainment and commodity. In response, We Are Music: An Existential Journey Toward Infinity provides the reader with a holistic starting point. Music has unlimited potential to transform and enlighten, and is only impeded when bound by materialism, physicalism, and reductionism. We Are Music is an attempt to bring music back to the core of humanity as an agent of positive empowerment, self-actualization, and beyond. Embracing interconnectivity, music (...) is more deeply experienced alongside the arts, science, social sciences, math, philosophy, history, and, above all, spirituality. An endless spiral, self-perception and identity can be vastly expanded, if not questioned and transcended. Music is an infinite field and any attempt to define and describe it is problematically finite and consequently limited. Herein lies the impossibility of the book which both excites and disturbs; the paradox of being and not being. A roadmap for music lovers toward self-realization, We Are Music is for those who desire to delve deeper into the power and illusion of self through music. (shrink)

Recent work on Kant’s conception of space has largely put to rest the view that Kant is hostile to actual infinity. Far from limiting our cognition to quantities that are finite or merely potentially infinite, Kant characterizes the ground of all spatial representation as an actually infinite magnitude. I advance this reevaluation a step further by arguing that Kant judges some actual infinities to be greater than others: he claims, for instance, that an infinity of miles is strictly (...) smaller than an infinity of earth-diameters. This inequality follows from Kant’s mereological conception of magnitudes (quanta): the part is (analytically) less than the whole, and an infinity of miles is equal to only a part of an infinity of earth-diameters. This inequality does not, however, imply that Kant’s infinities have transfinite and unequal sizes (quantitates). Because Kant’s conception of size (quantitas) is based on the Eudoxian theory of proportions, infinite magnitudes (quanta) cannot be assigned exact sizes. Infinite magnitudes are immeasurable, but some are greater than others. (shrink)

This chapter contains sections titled: * The Fundamental Question * 1 Whatever Begins to Exist Has a Cause * 2 The Universe Began To Exist * 3 The Cause of the Universe * Notes.

The medieval distinction between categorematic and syncategorematic words is usually given as the distinction between words which have signification or meaning in isolation from other words and those which have signification only when combined with other words . Some words, however, are classified as both categorematic and syncategorematic. One such word is Latin infinita ‘infinite’. Because infinita can be either categorematic or syncategorematic, it is possible to form sophisms using infinita whose solutions turn on the distinction between categorematic and syncategorematic (...) uses of infinita. As a result, medieval logicians were interested in identifying correct logical rules governing the categorematic and syncategorematic uses of the term. In this paper, we look at 13th–15th-century logical discussions of infinita used syncategorematically and categorematically. We also relate the distinction to other medieval distinctions with which it has often been conflated in modern times, and show how and where these conflations go wrong. (shrink)

Discussions about naturalness, artificiality and unnaturalness in this article are motivated by the field of Human Cognitive Enhancement because of its potential for altering human personality and identity. This article at first proposes a concept of human naturalness as interaction between physis and logos. Then it presents an intuitive understanding of naturalness in terms of the inherent inability of language to fully describe all attributes of an object that is natural. The analytical core of the article proposes a formal (...) model of naturalness utilising Vopenka's phenomenologically grounded Alternative Set Theory, comprising and formalising the concept of natural infinity. A brief introduction to AST is presented as well. Naturalness and artificiality are modelled as two structurally different naturally infinite semisets within AST. This key structural difference is then analysed and applied back to the domain of HCE. (shrink)

Human language has the characteristic of being open and in some cases polysemic. The word “infinite” is used often in common speech and more frequently in literary language, but rarely with its precise meaning. In this way the concepts can be used in a vague way but an argument can still be structured so that the central idea is understood and is shared with to the partners. At the same time no precise definition is given to the concepts used and (...) each partner makes his own reading of the text based on previous experience and cultural background. In a language dictionary the first meaning of “infinite” agrees with the etymology: what has no end. We apply the word infinite most often and incorrectly as a synonym for “very large” or something that we do not perceive its completion. In this context, the infinite mentioned in dictionaries refers to the idea or notion of the “immeasurably large” although this is open to what the individual’s means by “immeasurably great.” Based on this linguistic imprecision, the authors present a non Cantorian theory of the potential and actual infinite. For this we have introduced a new concept: the homogon that is the whole set that does not fall within the definition of sets established by Cantor. (shrink)

This is an article whose intended scope is to deal with the question of infinity in formal mathematics, mainly in the context of the theory of large cardinals as it has developed over time since Cantor’s introduction of the theory of transfinite numbers in the late nineteenth century. A special focus has been given to this theory’s interrelation with the forcing theory, introduced by P. Cohen in his lectures of 1963 and further extended and deepened since then, which leads (...) to a development and further refinement of the theory of large cardinals ultimately touching, especially in view of the discussion in the last section, on the metatheoretical nature of infinity. The whole undertaking, which takes into account major stages of the research in large cardinals theory, tries to present a defensible argumentation against an ontology of infinity actually rooted in the notion of subjectivity within the world. This means that rather than talking of a general ontology of infinity in the ideal platonic or in the aristotelian sense of potentiality, even in the alternative sense of an ontology of the event in A. Badiou’s sense, one can argue from a subjective point of view about the impossibility of defining cardinalities greater than the first uncountable one \ that would correspond to a distinct existence in real world terms or would be supported by a mathematical intuition in terms of reciprocity with experience. The argumentation from the particular standpoint includes also certain comments on the delimitative character of Gödel’s constructive universe L and the influence of the constructive approach in narrowing the breadth of an ‘ontology’ of infinity. (shrink)

In this paper we attempt to clear out the ground concerning the Aristotelian notion of density. Aristotle himself appears to confuse mathematical density with that of mathematical continuity. In order to enlighten the situation we discuss the Aristotelian notions of infinity and continuity. At the beginning, we deal with Aristotle’s views on the infinite with respect to addition as well as to division. In the sequel, we focus our attention to points and discuss their status with respect to the (...) actuality–potentiality distinction. Then we focus our attention to the nature of continuity, which Aristotle tends to consider as leading to infinite divisibility and potential density of the extended. (shrink)

It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle (...) has good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logical’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed. (shrink)

Free choice sequences play a key role in the intuitionistic theory of the continuum and especially in the theorems of intuitionistic analysis that conflict with classical analysis, leading many classical mathematicians to reject the concept of a free choice sequence. By treating free choice sequences as potentially infinite objects, however, they can be comfortably situated alongside classical analysis, allowing a rapprochement of these two mathematical traditions. Building on recent work on the modal analysis of potential infinity, I formulate (...) a modal theory of the free choice sequences known as lawless sequences. Intrinsically well-motivated axioms for lawless sequences are added to a background theory of classical second-order arithmetic, leading to a theory I call $MC_{LS}$. This theory interprets the standard intuitionistic theory of lawless sequences and is conservative over the classical background theory. (shrink)

Linnebo and Shapiro have recently given an analysis of potential infinity using modal logic. A key technical component of their account is to show that under a suitable translation ◊ of nonmodal language into modal language, nonmodal sentences ϕ 1, …, ϕ n entail ψ just in case ϕ 1 ◊, …, ϕ n ◊ entail ψ ◊ in the modal logic S4.2. Linnebo and Shapiro establish this result in nonfree logic. In this note I argue that their (...) analysis of potential infinity should be carried out in a free logic. I then extend their key theorems to the setting of negative free logic. (shrink)

The topic of this article is the problem of World and Infinity. The concepts “Weltanschauung,” (I. Kant) and “Umgreifende” (K. Jaspers) are introduced. A transcendental analysis of the concepts "World" and "Infinite" as ideas of reason is carried out. The theory of the world by L. Tengely is analyzed. Metaphysical and mathematical interpretations of the infinite are compared (actual and potential infinity, openness, horizon).

The purpose of this special issue and the conference that inspired it was to address the issue of conceptual integration in a science of consciousness. We felt this to be important, for while current efforts to scientifically investigate consciousness are taking place in an interdisciplinary context, it often seems as though the very terms being used to sustain a sense of interdisciplinary cooperation are working against it. This is because it is this very array of common concepts that generates a (...) sense of unity among consciousness researchers, despite the fact the concepts mean different things in different disciplines. These Concepts of Consciousness include the following: realism, representation, intentionality, information, control, memory and self. Given this list, we believed we could best approach the issue of potential conceptual integration by addressing each concept from different perspectives and asking the following: how do uses of the concept differ, must these meanings be synthesized in order for there to be a unified science of consciousness, is a unified conceptual scheme necessary to establish an independent science of consciousness, is a unified conceptual scheme possible, if it is not possible, why not, and if it is possible, what might it look like? To this end we invited, for each concept, two scholars who made extensive use of the identified concept in their work. The papers entailed in this special issue constitute the outcome of this effort, and in what follows we offer a brief examination of possible forms of integration the papers seem to collectively suggest. (shrink)

The five participants in this dialogue critically discuss Zeno of Elea's paradox of Achilles and the tortoise. They consider a number of solutions to and restatements of the paradox, together with their philosophical implications. Among the issues investigated include the appearance-reality distinction, Aristotle's distinction between actual and potential infinity, the concept of a continuum, Cantor's continuum hypothesis and theory of transfinite ordinals, and, as a solution to Zeno's puzzle, the distinction between infinite and indeterminate or inexhaustible divisibility.