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Paradoxes of the Infinite and Ontological Dilemmas Between Ancient Philosophy and Modern Mathematical Solutions

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Book cover Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11973))

Abstract

The concept of infinity had, in ancient times, an indistinguishable development between mathematics and philosophy. We could also say that his real birth and development was in Magna Graecia, the ancient South of Italy, and it is surprising that we find, in that time, a notable convergence not only of the mathematical and philosophical point of view, but also of what resembles the first “computational approach” to “infinitely” or very large numbers by Archimedes. On the other hand, since the birth of philosophy in ancient Greece, the concept of infinite has been closely linked with that of contradiction and, more precisely, with the intellectual effort to overcome contradictions present in an account of Totality as fully grounded. The present work illustrates the ontological and epistemological nature of the paradoxes of the infinite, focusing on the theoretical framework of Aristotle, Kant and Hegel, and connecting the epistemological issues about the infinite to concepts such as the continuum in mathematics.

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Notes

  1. 1.

    If there are n new guests the simplest choice is to use the function , instead, in case of an infinite countable number, the function (see [22, 39]).

  2. 2.

    The unimaginable numbers are numbers extremely large so that they cannot be written through the common scientific notation (also using towers of exponents) and are behind every power of imagination. To write them some special notations have been developed, the most known of them is Knuth’s up-arrow notation (see [30]). A brief introduction to these numbers can be found in [10], while more information is contained in [7, 11, 25].

  3. 3.

    See Physics, 204a8–204a16, in [4, Vol. I].

  4. 4.

    See Metaphysics, IX.6, 1048a-b, in [4, Vol. I].

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Aknowledgments

This work is partially supported by the research projects “IoT&B, Internet of Things and Blockchain”, CUP J48C17000230006, POR Calabria FESR-FSE 2014–2020.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Università della Calabria, Cubo 31/A, 87036, Arcavacata di Rende, CS, Italy

    Fabio Caldarola & Gianfranco d’Atri

  2. Via degli Orti, 12, 89861, Tropea, VV, Italy

    Domenico Cortese

  3. Department of Environmental and Chemical Engineering (DIATIC), Università della Calabria, Cubo 42/B, 87036, Arcavacata di Rende, CS, Italy

    Mario Maiolo

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  1. Fabio Caldarola
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Correspondence to Fabio Caldarola .

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Editors and Affiliations

  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Caldarola, F., Cortese, D., d’Atri, G., Maiolo, M. (2020). Paradoxes of the Infinite and Ontological Dilemmas Between Ancient Philosophy and Modern Mathematical Solutions. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11973. Springer, Cham. https://doi.org/10.1007/978-3-030-39081-5_31

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