Richard Dedekind (1831 - 1916) was one of the pioneers of number theory and this book contains the English translations of his two most important papers: “Continuity and Irrational Numbers” from 1872 and “The Nature and Meaning of Numbers” from 1887.
The first paper shows how he came up with a purely number based procedure that defined the inexact irrational numbers like square root of 2.
The translation is rather painfully literal, and does not convey much idea of the graceful style of the original; but it is, on the whole, correct.
The reference is to the motto on the title-page of the German edition, which was coined by the author in imitation of the Platonic dictum, αει ο θεος γεωμετρίζει.
The first presents Dedekind's theory of the irrational number-the Dedekind cut idea-perhaps the most famous of several such theories created in the 19th century to give a precise meaning to irrational numbers, which had be This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. The second essay is an attempt to give a logical basis for transfinite numbers and properties of the natural numbers.
This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis.This has everything to do with context, perspective, understandings, and connections.The word "color" would mean a slew of different things to the now English-speaking creature and its usage of, say, "speed" would mean something completely different than a rate of movement.Curiosità: per lui l'induzione matematica è un teorema, perché usa come assioma la possibilità di avere catene infinite di insiemi.A mio parere il primo articolo è molto più chiaro del secondo, che mette insieme un approccio puramente deduttivo con alcune considerazioni che appaiono poste più o meno a caso: la traduzione pedissequa non aiuta certo.99.9% of this book went beyond my level of comprehension; it's the lion's language...still English, but totally incomprehensible, which is exactly why I trekked through it.Numbers and mathematics have only been presented to me through formulas, problems, equations, theorems, and proofs.This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. The contents of these essays belong to the foundations of mathematics and will be welcomed by those who are prepared to look into the somewhat subtle meanings of the elements of our number system. Questo libretto della Dover contiene la traduzione inglese di due articoli fondamentali scritti dal matematico tedesco: Stetigkeit und irrationale Zahlen (Continuità e numeri irrazionali), nel quale definisce i numeri irrazionali mediante il procedimento che poi verrà detto taglio di Dedekind, e Was sind und was sollen die Zahlen (Cosa sono e cosa dovrebbero essere i numeri? It examines the notion of natural numbers, the distinction between finite and transfinite (infinite) whole numbers, and the logical validity of the type of proof called mathematical or complete induction.This sort of break from reality eventually lead to the whole approach being discredited by Kurt Godel’s famous incompleteness theorems of 1931.Since then number theorists / logicians have tried to come up with context based logics to axiomatize number theory, apparently without success. One contains the original presentation of the famous "Dedekind cut" formulation of the real numbers, plus developments up to the well-known "sup" property of bounded sets of reals.