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Author: Benjamin Wilck

1 Introduction Euclid’s Elements , c. 300 BCE , 1 is the earliest extant treatise of deductive mathematics in history, and is still regarded as a paradigm of an axiomatic science. 2 By deriving a large number of mathematical theorems from a relatively small number of undemonstrated

Open Access
In: History of Philosophy & Logical Analysis
Author: Holger A. Leuz

is based on axiomatic systems, most notably the system of Peano Arithmetic. A closer look at the ancient number theory contained in Euclid’s Elements reveals that no such rigorous axiomatic foundation for the ontology of numbers and the elementary arithmetical operations is present in that work. It

In: History of Philosophy & Logical Analysis
Author: Benjamin Wilck

geometry. By contrast, Sextus expressly mentions the Epicureans as critics of geometry ( M III .94–107; M I .5, see section 3.3 below). Since variants of definitions that Sextus attacks in M III are found in Euclid’s Elements (= El. ), Heiberg (1888, LXXII ) takes Sextus to implicitly refer to

Open Access
In: History of Philosophy & Logical Analysis
Author: Peter M. Simons

suitable variable, or if there are too few. Take for example Euclid's statement [9] There are infinitely many prime numbers and assurne it is true. On the face-value interpretation this requires the exis- tence of infinitely many prime numbers. On the substitutional interpretation, for it to have the

In: Reflections on Free Logic
Author: Holger A. Leuz

werden kann. 0. Introduction Our contemporary conception of natural numbers and their arithmetic is based on axiomatic systems, most notably the system of Peano Arithmetic. A closer look at the ancient number theory contained in Euclid’s Elements reveals that no such rigorous axiomatic foundation for the

In: Focus: Ancient and Medieval Philosophy/Schwerpunkt: Antike und Mittelalterliche Philosophie

N. Goodman: Fact, Fiction and Forecast , 1965 P. R. Halmos: Naive Mengenlehre, Göttingen 1976 Th. L. Heath: The thirteen books of Euclid’s Elements translated from the text of Heiberg with introduction and commentary, 3 Bde., Cambridge 1925 M. Heidelberger, S. Thiessen: Natur und Erfahrung. Von der

In: Wissenschaftliche Erkenntnis
Author: Martin Lemke

, Cambridge University Press, 1972 Literaturverzeichnis 347 [Pappos 1930] PAPPOS; G. lUNGE, W. T. (Hrsg.): The commentary of Pappos on Book X of Euclid's Elements. Cambridge: Harvard University Press, 1930 [Peano 1889] PEANO, G.: Arithmetices Principa - Nova Methodo Exposita. Rom, Florenz: Fratres Bocca

In: Variationen über ein Thema von Euklid

einer Definition erforderlich ist. Euclid’s Definition von Parallellinien ist von der Art. 1Anmerkung in der Akademie-Ausgabe. 2Bei Textpassagen mit vorangestelltem 𝑔 handelt es sich um Zusätze Kants aus derselben Schriftphase. 3In eckigen Klammern [ ] stehen von Kant ausgestrichene Wörter und

In: Kant zur Philosophie der Mathematik
Author: Peter Janich

Euclid's Elements, eingeleitet und kommentiert von Th. L. Heath, New York 1956 (dt. Euklid. Die Elemente, übersetzt und herausgegeben von C. Thaer, Darm- stadt 1980). 2 D. Hilbert, Grundlagen der Geometrie, Stuttgart 1968. 3 R. Inhetveen, Konstruktive Geometrie. Eine formentheoretische Begründung der

In: Paul Lorenzen und die konstruktive Philosophie

units cannot match the infinitely divisible parts of a geometric magnitude. Aristotle asserts: “It is impossible that something continuous consists of indivisibles, as for instance a line 10 The study of ratio for (geometric) magnitudes in Book V of Euclid’s Elements is entirely separate from the

In: Focus: Ancient and Medieval Philosophy/Schwerpunkt: Antike und Mittelalterliche Philosophie