Torricelli's Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century

Mancosu, Paolo; Vailati, Ezio

doi:10.1086/355637

Cited by 21 publications

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“…While paradoxes of the infinite have often popped up in geometrical contexts-witness the long debates on the one-to-one correspondence between the points of two segments of different lengths (see Mancosu, 1996, chap. 5) or Torricelli's determination that there is a solid of infinite length with finite volume and no center of gravity (see Mancosu & Vailati, 1991;Mancosu, 1996, chap. 5)-in this paper I will only focus on the paradoxes concerning the determination of sizes of infinite collections.…”

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Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?

Mancosu

2009

The Review of Symbolic Logic

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Abstract. Cantor's theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same 'size' in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the 'size' of A should be less than the 'size' of B (part-whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part-whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part-whole principle, envisaged by Bolzano (Kitcher). §1. Introduction. Two central issues seem to have determined the reflection on mathematical infinity in Western thought. The first concerns its existence. The second whether it can be measured. In this paper, I will only deal with the second aspect of the issue although, of course, the two issues cannot always be separated. The structure of the paper is as follows. First, I will retrace some of the major historical positions that were taken with respect to the paradoxical properties displayed by infinite sets of natural numbers with emphasis on whether there could be an arithmetic of infinite sets. In the second part, I will describe recent mathematical developments that offer a way to measure the size of infinite sets of natural numbers while preserving the part-whole principle. In the conclusion, I will offer some philosophical reflections as to how these recent mathematical developments impact various historical and philosophical claims found in the literature, including Gödel's claim, which I contest, as to the inevitability of Cantor's definition of infinite number. This is the notion of inevitability referred to in the title of this paper, namely the claim that if one wants to generalize the notion of number from the finite to the infinite there is only one possible way to go and that is the Cantorian notion of cardinal number.

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confidence: 99%

Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?

Mancosu

2009

The Review of Symbolic Logic

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“…1 A cylindrical indivisible of Gabriel's horn at point P and the cylinder Torricelli constructed fundamental principle of the theory of indivisibles, the volumes of the two figures are equal. So the volume of this infinitely long solid is 2π×OA (Carroll et al 2013;Mancosu and Vailati 1991). Mancosu and Vailati (1991) note that Torricelli took for granted the equality of the indivisibles when P=O, that is, when the lateral surface degenerates into a straight line.…”

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“…So the volume of this infinitely long solid is 2π×OA (Carroll et al 2013;Mancosu and Vailati 1991). Mancosu and Vailati (1991) note that Torricelli took for granted the equality of the indivisibles when P=O, that is, when the lateral surface degenerates into a straight line. Mancosu and Vailati (1991) point out that the above theorem brings out the infinitistic nature of Torricelli's result.…”

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“…Mancosu and Vailati (1991) note that Torricelli took for granted the equality of the indivisibles when P=O, that is, when the lateral surface degenerates into a straight line. Mancosu and Vailati (1991) point out that the above theorem brings out the infinitistic nature of Torricelli's result. The notion of actually infinite length is present in the statement of the theorem.…”

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“…But, for Wallis, Torricelli's infinitely long solid did not create an issue as long as it was considered as a mathematical object. He shared Leibniz' opinion that it was just as spectacular as, for instance, the fact that the infinite series 1 2 þ 1 4 þ 1 8 þ 1 16 þ 1 32 þ … is equal to 1 (Bråting 2012;Mancosu and Vailati 1991).…”

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On Painter’s Paradox: Contextual and Mathematical Approaches to Infinity

Wijeratne

Zazkis

2015

Int. J. Res. Undergrad. Math. Ed.

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In mathematics education research paradoxes of infinity have been used in the investigation of students' conceptions of infinity. We analyze one such paradoxthe Painter's Paradox -and examine the struggles of a group of Calculus students in an attempt to resolve it. The Painter's Paradox is based on the fact that Gabriel's horn has infinite surface area and finite volume and the paradox emerges when finite contextual interpretations of area and volume are attributed to the intangible object of Gabriel's horn. Mathematically, this paradox is a result of generalized area and volume concepts using integral calculus, as the Gabriel's horn has a convergent series associated with volume and a divergent series associated with surface area. This study shows that contextual considerations hinder students' ability to resolve the paradox mathematically. We suggest that the conventional approach to introducing area and volume concepts in Calculus presents a didactical obstacle. A possible alternative is considered.

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Mathematik und Religion in der frühen Neuzeit

Breger

1995

Ber Wissenschaftsgesch

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Some protestant Mathematicians had a strong preoccupation with the Day of Judgement. Stifel, Faulhaber, Napier and Newton made calculations in order to determine the date of the end of the world. Craig gave mathematical rules for a decline in the reliability of Christian tradition; in order to prevent a reliability of nearly zero, the Day of Judgement must come before. Furthermore, some conflicts between theology and mathematics are discussed. The Council of Konstanz condemned Wyclif's theory of the continuum. It seems that the relation between part and whole was a crucial point between the Aristotelians and the "moderns".

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Torricelli's Infinitely Long Solid and Its Philosophical Reception in the Seventeenth Century

Cited by 21 publications

References 0 publications

Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?

Measuring the Size of Infinite Collections of Natural Numbers: Was Cantor’s Theory of Infinite Number Inevitable?

On Painter’s Paradox: Contextual and Mathematical Approaches to Infinity

Mathematik und Religion in der frühen Neuzeit

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