Consistency and Representations: The Case of Actual Infinity

1999 ◽  
Vol 30 (2) ◽  
pp. 213 ◽  
Author(s):  
Pessia Tsamir ◽  
Dina Tirosh
Keyword(s):  
Actual Infinity

Kant’s Mereological Account of Greater and Lesser Actual Infinities

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniel Smyth
Keyword(s):  
Recent Work ◽  
Spatial Representation ◽  
Actual Infinity

Abstract 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.

scholarly journals TWO TRADITIONS OF THE ACTUAL INFINITY INTERPRETATION

Metaphysics ◽  
2021 ◽  
pp. 47-62
Author(s):  
V. N Katasonov
Keyword(s):  
Rene Descartes ◽  
The Other ◽  
Theory Of Knowledge ◽  
Actual Infinity ◽  
Nicholas Of Cusa ◽  
René Descartes

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.

Pavel Florensky: At the Boundary of Immanence and Transcendence

2020 ◽  
pp. 292-308
Author(s):  
Christoph Schneider
Keyword(s):  
Visual Arts ◽  
Philosophy Of Language ◽  
Religious Philosophy ◽  
Actual Infinity ◽  
Diachronic Change ◽  
Pavel Florensky ◽  
Postmodern Thought ◽  
Platonic Tradition

This chapter discusses four themes in the religious philosophy of Pavel Florensky (1882–1937): Georg Cantor’s mathematics, truth, philosophy of language, and the visual arts. Apart from Church doctrines, the key ideas that emerge in his work are ‘antinomy’, ‘discontinuity’, ‘actual infinity’ and ‘realism’. Deeply rooted in the Christian-Platonic tradition, Florensky is critical of rationalism, empiricism, Kantianism, and positivism. He anticipates postmodern insights in the sense that his worldview allows for synchronic difference as well as time and diachronic change. But unlike postmodern thought, which tends to interpret synchronic difference and the flux of time in terms of relativistic perspectivism and historicism, Florensky provides difference and change with a realist underpinning. And despite his emphasis on antinomicity and discontinuity in his conception of truth, he affirms the grandeur of reason and rejects irrationalism and fideism.

Analysis without actual infinity

1981 ◽  
Vol 46 (3) ◽  
pp. 625-633 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Natural Science ◽  
Order Theory ◽  
Finite Part ◽  
Consistency Proof ◽  
Actual Infinity ◽  
First Order ◽  
Finite Models ◽  
First Order Theory ◽  
Potential Applications

AbstractWe 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.

scholarly journals Objects as Limits of Experience and the Notion of Horizon in Mathematical Theories

Phainomenon ◽  
2012 ◽  
Vol 25 (1) ◽  
pp. 131-153
Author(s):  
Stathis Livadas
Keyword(s):  
Set Theory ◽  
The Other ◽  
Actual Infinity ◽  
Human Perceptions ◽  
Subjective Origin

Abstract The present work is an attempt to bring attention to the application of several key ideas of Husserl ‘s Krisis in the construction of certain mathematical theories that claim to be altemative nonstandard versions of the standard Zermelo-Fraenkel set theory. In general, these theories refute, at least semantically, the platonistic context of the Cantorian system and to one or the other degree are motivated by the notions of the lifeworld as the pregiven holistic field of experience and that of horizon as the boundary of human perceptions and the de facto constraint in reaching limit-idealizations. Moreover, 1 try to give convincing reasons for the existence of an ultimate constitutional ‘vacuum’ of a subjective origin that is formally refiected in the application of a notion of actual infinity in dealing generally with the mathematical infinite.

scholarly journals Mathematics and God’s Point of View

2016 ◽  
Vol 44 (1) ◽  
pp. 81-96
Author(s):  
Zbigniew Król
Keyword(s):  
Mathematical Reasoning ◽  
Point Of View ◽  
Historical Process ◽  
Actual Infinity ◽  
Main Factors ◽  
Subject Of Cognition

Abstract In this paper the final stages of the historical process of the emergence of actual infinity in mathematics are considered. The application of God’s point of view – i.e. the possibility to create mathematics from a divine perspective, i.e. from the point of view of an eternal, timeless, omniscience and unlimited subject of cognition – is one of the main factors in this process. Nicole Oresme is the first man who systematically used actual infinity in mathematical reasoning, constructions and proofs in geometry.

scholarly journals An Analysis of Zeno’s Paradox and Non-measurable Sets Based on Dialectical Infinity

2021 ◽  
Vol 4 (4) ◽  
pp. p30
Author(s):  
Zhang Hong ◽  
Zhou Hong Qiang
Keyword(s):  
Line Segment ◽  
Basic Problem ◽  
Clear Understanding ◽  
Actual Infinity ◽  
Mathematical Definition ◽  
Measurable Set ◽  
Long Time ◽  
And Mathematics ◽  
Definition Of ◽  
Zeno's Paradox

The Problem of Continuity and Discreteness is the basic problem of philosophy and mathematics. For a long time, there is no clear understanding of this problem, which leads to the stagnation of the problem, because the essence of the problem is a problem of finity and infinity. The essence of the philosophical thought on which the mathematical definition of “line segment is composed of dots” is the idea of actual infinity, and geometric dot is equivalent to algebraic zero in terms of measure properties. In view of the above contradictions, this paper presents two solutions satisfying both the philosophical and mathematical circles based on the view of dialectical infinity, and the authors make a deep analysis of Zeno’s paradox and the non-measurable set based on both solutions.

scholarly journals L’infini entre deux bouts. Dualités, univers algébriques, esquisses, diagrammes

2020 ◽  
Vol 41 (2) ◽  
Author(s):  
René Guitart
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Mathematical Thinking ◽  
Mathematical Work ◽  
Mathematical Activity ◽  
Actual Infinity ◽  
Zeno's Paradoxes ◽  
Mathematical Problems ◽  
Technical Details ◽  
The One

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises.

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