### Related Papers

Syrianus on the Platonic Tradition of the Separate Existence of Numbers

Melina G . Mouzala

Aristotle in books M and N of his Metaphysics, attacks the Form-numbers or the so called eidetic numbers and differentiates them from the monadic or unitary, i.e. the mathematical, numbers, namely the ordinary numbers which are addible to one another (sumblētoi) and composed of undifferentiated units (monades). Syrianus in his Commentary on Aristot-le's Metaphysics defends the existence of eidetic numbers and justifies their necessity, by following a line of argument which puts forward the main characteristics of their divine and immaterial nature. In parallel he analyzes and explains the ontological status of mathematical numbers in a way which reveals that they have a kind of separate existence. This paper attempts to bring to the fore the most salient aspects of this argumentation and sets out to show how a neoplatonic Platonist such as Syrianus, understood the nature not only of the Form-numbers, but also of the mathematical numbers, through the transmission of the relevant Platonic tradition, especially with regard to Plato's unwritten doctrines.

One, Two, Three… A Discussion on The Generation of Numbers in Plato’s Parmenides

George Florin Calian

In M. Sialoros (ed.), Revolution and Continuity in Greek Mathematics, De Gruyter, Berlin, pp. 295-318

Substantiae sunt sicut numeri. Aristotle on the structure of numbers.

Aristotle's contribution to the metaphysics of numbers is often described in terms of a critical response to the Platonist paradigm. Plato, we are told, conceives of numbers as abstract entities entirely distinct from the physical objects around us, while Aristotle takes the more mundane view that numbers are pluralities of physical objects considered in a particular way, a way relevant to mathematics. Without rejecting altogether this familiar picture, this paper aims to show that Aristotle has another major contribution to offer to the history of philosophy of mathematics. In the Metaphysics, he claims that numbers too can be analysed in terms of matter and form (hylomorphism). On the hylomorphic model, a number has both a material component (the units in the number) and a formal one (the structure that keeps the units together). The paper fully explores the motivations behind Aristotle's hylomorphic conceptions of numbers as well as its most significant implications.

Technical Transactions. Kraków. – 2014. – 1-NP. – p. 211-223.

Concepts of a number of C. Méray, E. Heine, G. Cantor, R. Dedekind, and K. Weierstrass.

2014 •

Galina Sinkevich

The article is devoted to the evolution of conception of number in XVIII-XIX c. The Ch. Méray`, H. Heine`, R. Dedekind`, G. Cantor` and K. Weierstrass` constructions of a number considered.

Numbers, Ontologically Speaking: Plato on Numerosity

George Florin Calian

in Numbers and Numeracy in Classical Greece, Brill (Mnemosyne Supplements Series)

Historical Changes in the Concepts of Number, Mathematics and Number Theory

Nicky Gregory

In Defense of Plato on Mathematical Ideas: A Commentary on Aristotle's Metaphysics XIII 6-8

Ryan Haecker

Where Plato had robustly conceived of numbers as Mathematical Ideas generated by the supreme Principles and multiply instantiated in numerically distinct sensible objects, Aristotle rejects Mathematical Ideas and thinly re-conceives of numbers as no more than abstract concepts generalized by the intellect from quantities of numerical distinct sensible substances. Aristotle’s many criticisms of Plato’s theory of Mathematical Ideas are, however, an ignorant argument (ignorationes elenchi) that, not only disregards the eidetic generation of numbers from the supreme Principles, but may only plausibly succeed against his own forced re-conception of eidetic numbers as mathematical numbers. The many absurdities that Aristotle purports to derive from Plato's theory of Mathematical Ideas are thus the consequence of his own, rather than Plato's, conception of mathematical objects. The following commentary will (§I) describe how Aristotle re-conceives of Plato's Mathematical Ideas of eidetic numbers; (§II) defend Plato's theory of Mathematical Ideas against Aristotle's criticisms in Metaphysics XIII 6-8; and (§III) prosecute the case for Plato's transcendental argument for eidetic numbers against Aristotle's abstraction theory for mathematical numbers.

Sociological Focus, vol.6, no. 2, pp. 107-116

The Evolution of Number

1983 •

Richard Startup

How is the historical development of mathematics to be understood sociologically? The writings of Mannheim and Wittgenstein suggest two contrasting approaches which may be evaluated through a detailed analysis of the development of the number concept. Five main number systems can be distinguished of which the most basic is the system of whole numbers and, given that importance attaches to generalizing power, there can be shown to be a tendency for the number concept to be progressively augmented. This process may be understood by reference to the basic pre-Darwinian evolutionary idea of "unfolding." Complementing this intrinsic element, in actual historical instances a variety of extrinsic factors may be seen to be operative, facilitating or retarding development. To explain historical change fully an elaborate normative structure must be analyzed incorporating a macro-institutional level, the "rules of the mathematical game", notational considerations, and relations internal to mathematics itself (e.g., between number and geometry).

Additional Notes on Numbers

Additional Notes on Numbers

2023 •

Jay Sklar

Jay Sklar's commentary on Numbers for The Story of God series is guided by the question, "What is most necessary to know to teach or preach well on this passage?" Any of its technical or in-depth discussions, or interactions with the secondary literature, remain focused on those aspects of the passage most central to its meaning. This book is guided by the question, "What else might the reader want to know when interacting with the secondary literature, or when wanting even further detail on various aspects of the passage?" It thus complements the commentary so that readers wanting to go even further in their study of Numbers are equipped to do so. Keeping the preacher and teacher in mind, it also has appendices that provide tips on narrative preaching and a full range of possible sermon series.

THE INVENTION OF NUMBER

Olivier Keller