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Essays on the Theory of Numbers
This volume contains the two most important essays on the logical foundations of the number system by the famous German mathematician J. W. R. Dedekind. 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 been used on an intuitive basis since Greek times. This paper provided a purely arithmetic and perfectly rigorous foundation for the irrational numbers and thereby a rigorous meaning of continuity in analysis.The second essay is an attempt to give a logical basis for transfinite numbers and properties of the natural numbers. 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.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. As a major work of an important mathematician, the book deserves a place in the personal library of every practicing mathematician and every teacher and historian of mathematics. Authorized translations by "Vooster " V. Beman.
79 pages, Kindle Edition
First published January 1, 1901
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August 8, 2020
Mathematics started with counting. Counting is familiar enough for you to know what it is. There is no limit to how high a person can count, given the time to do so. It leads to an infinite set of countable integers. With a bit of manipulation, one can access other types of numbers as well. We have known about the irrational numbers since the Ancient Greeks, but it seems that this author was one of the first mathematicians to apply rigor to the idea.
Julius Wilhelm Richard Dedekind was a German mathematician who wrote both of the essays featured in this book. The first essay covers the irrational numbers, as I mentioned. You can find them using a bit of trigonometry. The square root of two is the hypotenuse of a right triangle with equal sides of one unit length. You can find this number using the Pythagorean Theorem. The essay introduces the Dedekind cut. It is a method to construct a set of real numbers from the rational numbers.
The second essay discusses the transfinite numbers introduced by Georg Cantor. In this essay, Dedekind sets out a series of axioms and uses them to prove his point.
The book is short at 115 pages. It has no problems to solve.
Julius Wilhelm Richard Dedekind was a German mathematician who wrote both of the essays featured in this book. The first essay covers the irrational numbers, as I mentioned. You can find them using a bit of trigonometry. The square root of two is the hypotenuse of a right triangle with equal sides of one unit length. You can find this number using the Pythagorean Theorem. The essay introduces the Dedekind cut. It is a method to construct a set of real numbers from the rational numbers.
The second essay discusses the transfinite numbers introduced by Georg Cantor. In this essay, Dedekind sets out a series of axioms and uses them to prove his point.
The book is short at 115 pages. It has no problems to solve.
July 6, 2023
En este libro se encuentran dos ensayos:
El primero trata sobre los números irracionales, estos surgen a partir de las "Cortaduras de dedekind" completando la recta de los números reales y mostrando qué estos heredan las propiedades de los racionales.
El segundo habla sobre los números transfinitos y su naturaleza, presentando axiomas para estos.
Es un libro bastante interesante para conocer las bases sólidas de la teoria de números.
El primero trata sobre los números irracionales, estos surgen a partir de las "Cortaduras de dedekind" completando la recta de los números reales y mostrando qué estos heredan las propiedades de los racionales.
El segundo habla sobre los números transfinitos y su naturaleza, presentando axiomas para estos.
Es un libro bastante interesante para conocer las bases sólidas de la teoria de números.
November 19, 2016
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?), dove con un anno di anticipo su Peano fornisce una descrizione assiomatica dei numeri naturali. 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. Un'opera utile soprattutto per capire come i concetti matematici non spuntino dal nulla ma siano figli delle diatribe tra matematici, che si leggono benissimo in filigrana.
April 19, 2016
Geometry got BTFO
October 14, 2025
Reconstitution of the Kantian origins of mathematics from the sensibility to the understanding. Fascinating. If mathematics allows us to theorize about certain a priori intuitions because of the fact it stems out of these, then wouldn't those theories only be applicable through empirical tests? Thus a posteriori. Certainly math is a priori, as Kant believes, but as far as how it can be used in application it necessarily depends upon experience with mathematical objects, which exist outside of the Platonic World of the Forms.
November 12, 2021
Excellent book on number theory. Prose, not equations. Explains numbers within the context of numbers. Not about the history of counting but about how numbers relate to each other. Very cool math book. Answered a lot of fundamental questions.
October 7, 2017
Numbers are “things in themselves”
Wittgenstein, I believe, was the one who proposed the human inability to communicate with lions, should we give the regal beasts the ability to speak our language. 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.
And that, in sum, is Dedekind's collection of essays. 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. But this was mathematical theory, and it was fascinating to see a math narrative and vocabulary at work...even though I didn't get it. As Dedekind attempts to "investigate our notions of space and time by bringing them into relation with this number odoain created in our mind, I simply say, do or do not...I have no try and I'm humble enough to admit it. I just wanted to ride along for a second and see how this half lived.
Wittgenstein, I believe, was the one who proposed the human inability to communicate with lions, should we give the regal beasts the ability to speak our language. 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.
And that, in sum, is Dedekind's collection of essays. 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. But this was mathematical theory, and it was fascinating to see a math narrative and vocabulary at work...even though I didn't get it. As Dedekind attempts to "investigate our notions of space and time by bringing them into relation with this number odoain created in our mind, I simply say, do or do not...I have no try and I'm humble enough to admit it. I just wanted to ride along for a second and see how this half lived.
October 19, 2024
TWO ESSAYS BY THE FAMOUS GERMAN MATHEMATICIAN
Julius Wilhelm Richard Dedekind (1831-1916) was a German mathematician who made important contributions to abstract algebra, algebraic number theory and the foundations of the real numbers. This book contains two of his essays: ‘Continuity and Irrational Numbers,’ and ‘The Nature and Meaning of Numbers.’
He begins the first essay, “My attention was first directed toward the considerations which form the subject of this pamphlet in the autumn of 1858… I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic.”
He says of a paper he had just received by Georg Cantor, “the axiom given in Section II… agrees with what I designate in Section III as the essence of continuity. But what advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself.” (Pg. 3)
He explains, “I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else then the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed.” (Pg. 4)
He begins Section III with the statement, “Of the greatest importance, however, is the fact that in the straight line L there are infinitely many points which correspond to no rational number. If the point p corresponds to the rational number a, then, as is well known, the length of o p is commensurable with the invariable unit of measure used in construction, i.e., there exists a third length, a so-called common measure, of which these two lengths are integral multiples… The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number individuals.” (Pg. 8-9)
He asserts, “the way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes---which itself is nowhere carefully defined---and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.” (Pg. 9-10)
He states, “In this property that not all cuts are produced by rational numbers consists the incompleteness of discontinuity of the domain R of all rational numbers. Whenever, then, we have to do with a cut… produced by no rational number, we create a new, an IRRATIONAL number a, which we regard as completely defined by this cut… we shall say that the new number a corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as DIFFERENT or UNEQUAL always and only when they correspond to essentially different cuts.” (Pg. 15)
He wrote in the Preface to the second essay, “In speaking or arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the differences of things. It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind.
"If we scrutinize closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation… must…the whole science of numbers be established.” (Pg. 31-32)
Later, he observes, “If anyone should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and to call attention to the fact that a far advanced, refined scientific training is demanded in order to perceive clearly the essence of continuity and to comprehend that besides rational quantitative relations, also irrational, and besides algebraic, also transcendental quantitative relations are conceivable. All the more beautiful it appears to me that without any notion of measurable quantities and simply by a finite system of simple thought-steps man can advance to the creation of pure continuous number-domain; and only by this means in my own view is it possible for him to render the notion of continuous space clear and definite.” (Pg. 38)
This brief book will be of great interest for anyone studying the foundations and philosophy of mathematics.
Julius Wilhelm Richard Dedekind (1831-1916) was a German mathematician who made important contributions to abstract algebra, algebraic number theory and the foundations of the real numbers. This book contains two of his essays: ‘Continuity and Irrational Numbers,’ and ‘The Nature and Meaning of Numbers.’
He begins the first essay, “My attention was first directed toward the considerations which form the subject of this pamphlet in the autumn of 1858… I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic.”
He says of a paper he had just received by Georg Cantor, “the axiom given in Section II… agrees with what I designate in Section III as the essence of continuity. But what advantage will be gained by even a purely abstract definition of real numbers of a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself.” (Pg. 3)
He explains, “I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else then the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding; the simplest act is the passing from an already-formed individual to the consecutive new one to be formed.” (Pg. 4)
He begins Section III with the statement, “Of the greatest importance, however, is the fact that in the straight line L there are infinitely many points which correspond to no rational number. If the point p corresponds to the rational number a, then, as is well known, the length of o p is commensurable with the invariable unit of measure used in construction, i.e., there exists a third length, a so-called common measure, of which these two lengths are integral multiples… The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number individuals.” (Pg. 8-9)
He asserts, “the way in which the irrational numbers are usually introduced is based directly upon the conception of extensive magnitudes---which itself is nowhere carefully defined---and explains number as the result of measuring such a magnitude by another of the same kind. Instead of this I demand that arithmetic shall be developed out of itself.” (Pg. 9-10)
He states, “In this property that not all cuts are produced by rational numbers consists the incompleteness of discontinuity of the domain R of all rational numbers. Whenever, then, we have to do with a cut… produced by no rational number, we create a new, an IRRATIONAL number a, which we regard as completely defined by this cut… we shall say that the new number a corresponds to this cut, or that it produces this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number, and we regard two numbers as DIFFERENT or UNEQUAL always and only when they correspond to essentially different cuts.” (Pg. 15)
He wrote in the Preface to the second essay, “In speaking or arithmetic (algebra, analysis) as a part of logic I mean to imply that I consider the number-concept entirely independent of the notions or intuitions of space and time, that I consider it an immediate result from the laws of thought. My answer to the problems propounded in the title of this paper is, then, briefly this: numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the differences of things. It is only through the purely logical process of building up the science of numbers and by thus acquiring the continuous number-domain that we are prepared accurately to investigate our notions of space and time by bringing them into relation with this number-domain created in our mind.
"If we scrutinize closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. Upon this unique and therefore absolutely indispensable foundation… must…the whole science of numbers be established.” (Pg. 31-32)
Later, he observes, “If anyone should say that we cannot conceive of space as anything else than continuous, I should venture to doubt it and to call attention to the fact that a far advanced, refined scientific training is demanded in order to perceive clearly the essence of continuity and to comprehend that besides rational quantitative relations, also irrational, and besides algebraic, also transcendental quantitative relations are conceivable. All the more beautiful it appears to me that without any notion of measurable quantities and simply by a finite system of simple thought-steps man can advance to the creation of pure continuous number-domain; and only by this means in my own view is it possible for him to render the notion of continuous space clear and definite.” (Pg. 38)
This brief book will be of great interest for anyone studying the foundations and philosophy of mathematics.
October 9, 2024
Mind blown 🤯
I found myself asking a few questions.
Like, is space a derivative of time?
Do IRR #'s only take place in space? And not time? What is so important about continuity?
Time and space influence each other and the geometry of spacetime can be curved by massive objects. Just think how time can affect the way space is perceived and vice versa. BUT neither space nor time is necessarily a derivative of the other. They are more like two aspects of the same thing with each playing a role in how the universe behaves...like quantum gravity for example💯
I found myself asking a few questions.
Like, is space a derivative of time?
Do IRR #'s only take place in space? And not time? What is so important about continuity?
Time and space influence each other and the geometry of spacetime can be curved by massive objects. Just think how time can affect the way space is perceived and vice versa. BUT neither space nor time is necessarily a derivative of the other. They are more like two aspects of the same thing with each playing a role in how the universe behaves...like quantum gravity for example💯
November 30, 2023
4.5** The essay on Dedekind-cuts was surprisingly easy to follow. Essay on Meaning of Numbers had beautifully simple proofs. Really satisfying to read and be 100% following along for entire chapters. Definitely a master of the craft.
The grammar on this English translation is not perfect and leads to some confusions in certain sentences
The grammar on this English translation is not perfect and leads to some confusions in certain sentences
February 17, 2024
Although the nomenclature may seem dated from our perspective, in Dedekind's historical context almost all of his extrapolation is brilliant. The famous Dedekind "cuts" that shows the necessity for a number line (or system) of Real numbers (irrational + rational) combine perfectly with the proof of the categorical class of the simply infinite system.
December 31, 2018
not all classics are good.
November 8, 2024
Didn't love the proof of Theorem 66 in the second essay...
April 18, 2012
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. His procedure, now called the Dedekind cut, was based upon the right triangle formula in which summing the square of the two legs produces an irrational length hypotenuse. He showed that between any two such rational numbers was an infinite number of irrational numbers. Turning this around he defined an individual irrational number by its bounding pair of rational numbers at its limiting condition (an infinitely small interval).
His second paper was one of the first to attempt an axiomatic set theory for numbers. This sort of approach tends to lose the context of the continuum by treating numbers as “things in themselves”. Apparently because of this he put this statement on the relationship between numbers and the continuum in the paper’s preface:
“Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of building up the science of number and by thus acquiring the continuous number domain that we are preparing accurately to investigate our notions of space and time by bringing them into relation with this number domain created in our mind.”
This contextless approach would lead his contemporary, Georg Cantor to come up with the idea of transfinite numbers (numbers beyond infinity as defined by set theory). 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.
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. His procedure, now called the Dedekind cut, was based upon the right triangle formula in which summing the square of the two legs produces an irrational length hypotenuse. He showed that between any two such rational numbers was an infinite number of irrational numbers. Turning this around he defined an individual irrational number by its bounding pair of rational numbers at its limiting condition (an infinitely small interval).
His second paper was one of the first to attempt an axiomatic set theory for numbers. This sort of approach tends to lose the context of the continuum by treating numbers as “things in themselves”. Apparently because of this he put this statement on the relationship between numbers and the continuum in the paper’s preface:
“Numbers are free creations of the human mind; they serve as a means of apprehending more easily and more sharply the difference of things. It is only through the purely logical process of building up the science of number and by thus acquiring the continuous number domain that we are preparing accurately to investigate our notions of space and time by bringing them into relation with this number domain created in our mind.”
This contextless approach would lead his contemporary, Georg Cantor to come up with the idea of transfinite numbers (numbers beyond infinity as defined by set theory). 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.
February 6, 2008
Two short essays by the Dedekind. 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. It's pretty short, light reading, but illuminating. (It's funny how much is lost hearing about Dedekind cuts from analysis books and logicians, who think it's "old hat" and just "preparation for the interesting stuff".)
The other essay is on the nature of the number concept, and is more philosophical. Being about five times longer than the first, it's also much deeper.
Highly recommended.
The other essay is on the nature of the number concept, and is more philosophical. Being about five times longer than the first, it's also much deeper.
Highly recommended.
February 20, 2014
Remarkable to read, especially having been through algebra and analysis courses in which we took similar and slightly different routes to build the same structures/concepts. Took some time to get used to the notation and terminology, but once I got it, it was a breeze to read.
If you haven't had any exposure to abstract math, this begins easy enough and the difficult parts can serve as a (remarkably lucid) demonstration to you of what abstract math entails.
If you have exposure to advanced math like a real analysis, you'll skip over broad swaths of this as elementary review.
If you haven't had any exposure to abstract math, this begins easy enough and the difficult parts can serve as a (remarkably lucid) demonstration to you of what abstract math entails.
If you have exposure to advanced math like a real analysis, you'll skip over broad swaths of this as elementary review.
June 12, 2007
I wish I'd read this book much earlier in my academic career -- or, at least, that someone would have given me a watered down version of it. Mathematics really is lovely when it's not about drills and flash cards.
July 4, 2010
this is much more a mathematical confession than a textbook. dedekind has a talent (foreign to most mathematicians) for actual, emotional expression. p.s. check out the proof for the existence of infinite sets. I mean, really?!
October 26, 2014
Its been a while ago that I read this book. I only have high school math and wondered if I could read and understand "real" mathematical texts. It turns out I can if I take it slow and take the time to think so can you too. This is an opportunity to see a "real" mathematician at work.
March 31, 2023
Infinitely rereadable.
Displaying 1 - 20 of 20 reviews