Essays on the Theory of Numbers by Richard Dedekind

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Dedekind, Richard, 1831-1916 Dedekind, Richard, 1831-1916
English
Ever wondered what numbers really are? Not just counting things, but the deep, philosophical foundation of mathematics itself? That's exactly what Richard Dedekind tackles in this slim but mighty book. Forget dry textbooks—this is a series of two powerful essays where a brilliant 19th-century mathematician asks the simple, terrifying question: 'What is a number?' He wasn't satisfied with just using numbers; he wanted to build them from scratch using pure logic. The result is a mind-bending journey into the bedrock of our reality. It's like watching someone lay the first brick of a skyscraper you've lived in your whole life without ever seeing the foundation.
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almost every music teacher has hitherto undertaken to teach Singing. This book will not, I am sure, add to the mystery. A careful perusal of its contents should clear away many misconceptions, and place the student on the right road to that end which he or she has in view. PREFACE TO THE FOURTH EDITION. That another large edition of this little Manual should be called for in so comparatively short a time is sufficiently encouraging testimony of the worth of the book and the favour it has found at the hands of students of singing and others--a result the more gratifying to the publishers since hitherto it has been issued with an anonymous title-page. Often has the authorship of the little volume been industriously defended and disputed--not by myself--both in this country and America; but, on the whole, the identity of its originator has been well maintained. For my part matters might have remained so, especially as I am not insensible to the fact that there is much "preaching" herein--as indeed there must be in such a work, and some of the advice is of such a nature that its giver runs the risk of being placed upon an exceeding high pinnacle of moral excellence, or of being accounted the personification of all the virtues--both of which distinctions might scarcely be merited. The appearance of my name upon the title-sheet is the result of no wish of mine, and I have consented to it only out of deference to the pressing request of the publishers. A chapter on the Physiological Surroundings of the Voice has been added to this edition. FREDERICK J. CROWEST. PREFACE TO THE SEVENTH EDITION. Once again my publishers have informed me of the necessity for reprinting this Manual, which has been so successful both here, in America, and in the Colonies. I have nothing to alter in the work, but I must express my thanks for the marked support that has been given to this straightforward advice. Did space permit, much might be said relative to that growing complaint, "Teachers' (or Board School) Laryngitis" as it is called. For some time past I have been authorized to make observations at the chief Throat Hospitals in connection with this constantly increasing mischief, arising from an injurious use of the voice in Teaching, and which only proper Voice Production will remedy. While I cannot give advice here, I shall be happy to answer any communications of sufferers from this complaint. FREDERICK J. CROWEST. 24 AMPTHILL SQUARE, LONDON, N.W. CONTENTS. _Page_ PRELIMINARY REMARKS 9 ON HABITS, DIET, &c. 13 Early Rising--Cleanliness--The Hair--The Teeth--Exercise--Dress-- Diet--Smoking--Late Hours--General Morality. PRONUNCIATION AND STUDY OF WORDS 21 Nervousness--Pronunciation--Vowel-Sounds--Consonants--English-- Emphasis--Position of Lips--Study of "Words"--General Education--Dramatic Study. VOICES AND THEIR VARIOUS QUALITIES 28 Voices and their Names--Compass--Soprano--Mezzo-Soprano-- Mezzo-Contralto--Contralto--Alto--Tenore-Leggiero--Tenore-Robusto-- Barytone and Basso-Cantante--Bass--Buffo--Qualities of Voices. INSTRUCTION, BOOKS AND MASTERS 47 Schools of Singing--Masters--Institutions--A First Opinion--Bad Lessons--Bad Teachers--Books of Exercises. PRACTICE 53 Individuality--Perseverance--Facial Expression--Self-Accompaniment-- Position of Body, Arms, Hands, Throat, Tongue, Larynx-- Head-Singing--Throatiness--Clearing the Throat--High Notes--The Scale--Forte, Mezzo-Voce, and Piano--Tone--Chest, Falsetto, and Head--Scale Practice--First Exercises (_with Examples_)--Duration of Practice--Singing in Tune--How to Begin--Variety--Chorus Singing--Humming--Studying Songs--Imitation--General Musical Study. ON STYLE AND EXPRESSION 71 Traditional Styles--Modern German _Lieder_--"The Cathedral School"--Oratorio (_with Examples_)--Opera--Ballads--Recitative (_with Examples_)--Slurring--Sentiment--Decision--Imitation--Public Singing--"Holding" an Audience--Mistakes in Public. ON TIME IN SINGING 87 The Metronome--Accent--Exercises: four beats to the bar; three beats to the bar; two beats to the bar. ON THE CHOICE OF MUSIC 103 Music to Suit the Voice--"Original Keys"--Execution--Fashion-- Forming a "Repertoire." PHYSIOLOGICAL SURROUNDINGS 109 The Larynx--The Thyroid--The Pharynx--The Voice; A Wind, Reed, or String Instrument. EXERCISES 115 Exercises 1, 2, for Uniting Notes--Exercise 3, for...

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This isn't a story with characters and a plot in the usual sense. The 'story' here is the story of an idea. In the first essay, Dedekind wrestles with the concept of infinity. He gives us a clear, logical way to understand what it means for one set of things to be 'the same size' as another, even if both are endless. The second essay is the real showstopper. Here, Dedekind sets out to define what numbers actually are. He doesn't start with 1, 2, 3. Instead, he starts with the idea of a 'cut' in the line of rational numbers, and from these 'cuts,' the real numbers (including irrationals like √2) emerge naturally. He builds the continuum from the ground up.

Why You Should Read It

You should read this to feel the thrill of pure thought. Dedekind's writing is surprisingly clear and patient. You can feel his desire to make this profound idea accessible. There's a beautiful elegance to his construction—it feels inevitable once you see it. It shows that the most abstract mathematics isn't just cold calculation; it's a creative act of defining the world logically. It changed how all future mathematicians thought about their subject.

Final Verdict

This book is perfect for the curious reader who enjoys philosophy or history of science, or anyone who's ever looked at a number and thought, 'But what are you?' It's not a light read—you'll need to go slow and think—but it's incredibly rewarding. If you liked 'Gödel, Escher, Bach' or wonder about the foundations of knowledge, give Dedekind a few hours of your time. He'll reshape how you see the logic of the world.



📢 Public Domain Notice

This historical work is free of copyright protections. It is now common property for all to enjoy.

Jackson Sanchez
5 months ago

Clear and concise.

5
5 out of 5 (1 User reviews )

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