Infinity and the savage mind (original) (raw)
Related papers
Chasing a tortoise: The notion of infinity in mathematics throughout history
Philhist 15: Proceedings of Interactions in the History of Philosophy Conference, 2015
Infinity, which is by nature a highly elusive concept, and in a way, the most elusive concept of all, has been a matter for merely mythological and cosmological consideration. It is due to Anaximander, the Pythagoreans and Zeno of Elea that the notion was gradually put into a more rational and “mathematical” perspective, and ultimately gained its first systematic analysis with Aristotle. Here, by “mathematical”, I mean the Aristotelian account of infinity which was implemented in the work of the Greek mathematicians. The Aristotelian analysis of the notion predominated the Western thought until the set-theoretic approach emerged. Although, infinity is now purified from its once-accepted cosmological connotations, it still seems that the discovery of the true nature of the notion can never be attained from a purely mechanical and materialistic viewpoint. In this paper, I investigate the views on infinity from the Greek period up until today, highlighting certain mathematicians and thinkers who made the most remarkable and intriguing contributions to the matter.
IS INFINITY PURELY ARITHMETICAL IN NATURE
In this article we highlight some of the main contours of the urge towards the infinite in order to focus on the twofold role of infinity in mathematics. Our brief discussion of the discovery of the whole-parts relation explains the switch from infinity as endlessness to infinity turned 'inwards', evinced in the infinite divisibility of (spatial) continuity. The traditional Aristotelian distinction between the potential infinite and the actual infinite constitutes the background of our subsequent analysis which touches upon Zeno's paradoxes and Aristotle's objections to the actual infinite. Since Descartes mathematicians increasingly reverted the relation between the potential infinite and the actual infinite by considering the latter as the basis of the former. The historical dominance of the potential infinite was eventually challenged by Cantor in his transfinite arithmetic. Weyl even portrayed mathematics as the science of the infinite. However, this view prompts us to analyse in more detail what the difference between the potential infinite and the actual infinite really is. Cantor's own definition of these two kinds of infinity serves as the starting-point of our ensuing analysis. It is argued that set theory (while employing the actual infinite) crucially depends upon 'borrowing' (imitating) key features from space, namely the just-mentioned whole-parts relation and the spatial (time) order of simultaneity (at once). A spatially deepened account of the nature of real numbers has to consider them as being present at once. Attention is also given to the objections raised by Paul Bernays, the co-worker of David Hilbert, regarding the assumed arithmetization of modern mathematics. Bernays argues that it is the totality character of continuity (which is originally a geometrical notion) resisting a complete airthmetization of mathematics. It is striking that the spatial feature of wholeness receives opposing interpretations in the thought of Bernays and Brouwer. The former explores the totality character of the continuum whereas the latter focuses on the whole-parts relation. Ultimately the impossibility to articulate the nature of the at once infinite without (implicitly or explicitly) exploring key elements of space therefore uproots the claims of arithmeticism. Although the potential infinite is purely arithmetical in nature, the actual infinite is not, because no single account of it succeeded in avoiding the above-mentioned key spatial characteristics. Lorenzen aptly points out that arithmetic provides no motif for introducing the at once infinite. Therefore the question posed in the title of this article, namely: "Is infinity purely arithmetical in nature?" should be answered in a twofold way: (i) The potential infinite (successive infinite) is a purely arithmetical concept, whereas (ii) the actual infinite (at once infinite) is not purely numerical in nature. Some of the key elements of the argument is captured in the Figure inserted in paragraph 22.
Boll. Accademia Gioenia, 2023
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.
Educational Studies in Mathematics, 2005
This paper applies APOS Theory to suggest a new explanation of how people might think about the concept of infinity. We propose cognitive explanations, and in some cases resolutions, of various dichotomies, paradoxes, and mathematical problems involving the concept of infinity. These explanations are expressed in terms of the mental mechanisms of interiorization and encapsulation. Our purpose for providing a cognitive perspective is that issues involving the infinite have been and continue to be a source of interest, of controversy, and of student difficulty. We provide a cognitive analysis of these issues as a contribution to the discussion. In this paper, Part 1, we focus on dichotomies and paradoxes and, in Part 2, we will discuss the notion of an infinite process and certain mathematical issues related to the concept of infinity. KEY WORDS: actual and potential infinity, APOS Theory, classical paradoxes of the infinite, encapsulation, history of mathematics, human conceptions of the infinite, large finite sets
The article is about the theological and philosophical history of the mathematical concept of infinity
A cultural overview on the concept of infinity
International academic semi-annual “Journal of Education, Culture and Society” - ISSN: 2081-1640, 2019
The human being meets the idea of infinity very early, still child, when realizes that he can go on with natural numbers until it is possible to count. The idea of infinity attracts and rejects, sometimes becomes object of desire and sometimes of study and systematic research. Infinity is the testimony that intellect, even starting from experience, can overcome limits and boundaries. Our own limited experience on the Earth suggests the existence of something beyond it. Scientists, artists, philosophers, musicians, writers, mathematicians have often assumed towards this concept positions and opinions sensitively different. Infinity would be, by its etymology and nature, what escapes all possible classification and measurement, while mathematics tends to classify and measure every object it examines, and it has been able to measure it. Infinity follows us from primary school until university, but often remains a misunderstood concept in the mathematical sense.