The Project Gutenberg eBook #41568: An Introduction to ...

1 Project Gutenberg s An Introduction to Mathematics, by Alfred North WhiteheadThis eBook is for the use of anyone anywhere at no cost and withalmost no restrictions whatsoever. You may copy it, give it away orre-use it under the terms of the Project Gutenberg License includedwith this eBook or online at : An Introduction to MathematicsAuthor: Alfred North WhiteheadRelease Date: December 6, 2012 [ eBook #41568]Language: EnglishCharacter set encoding: ISO-8859-1** START OF THIS Project Gutenberg eBook AN Introduction TO MATHEMATICS **Produced by Andrew D. Hwang. (This eBook was produced usingOCR text generously provided by the University ofCalifornia, Santa Barbara, through the Internet Archive.)

2 Transcriber s noteThe camera-quality files for this public-domain ebookmay be eBook was produced using scanned images andOCR text generously provided by the University ofCalifornia, Santa Barbara, through the typographical corrections and presentationalchanges have been made without PDF file is optimized for screen viewing, butmay be recompiled for printing. Please consult thepreamble of the LATEX source file for instructions andother UNIVERSITY LIBRARYOF MODERN KNOWLEDGEAN Introduction TOMATHEMATICSBy A. N. WHITEHEAD, , & NORGATEHENRY HOLT & Co., New YorkCanada: WM. BRIGGS, TorontoIndia: R.

3 & T. WASHBOURNE, THE ABSTRACT NATURE OF MATHEMATICS1II VARIABLES ..7 III METHODS OF APPLICATION ..15IV DYNAMICS ..29V THE SYMBOLISM OF MATHEMATICS ..43VI GENERALIZATIONS OF NUMBER ..54 VII IMAGINARY NUMBERS ..67 VIII IMAGINARY NUMBERS (CONTINUED) ..80IX COORDINATE GEOMETRY ..90X CONIC SECTIONS ..103XI FUNCTIONS ..118 XII PERIODICITY IN NATURE ..134 XIII TRIGONOMETRY ..141 XIV SERIES ..159XV THE DIFFERENTIAL CALCULUS ..179 XVI GEOMETRY ..194 XVII QUANTITY ..202 NOTES ..207 BIBLIOGRAPHY ..209 INDEX ..212AN Introduction TOMATHEMATICSCHAPTER ITHE ABSTRACT NATURE OF MATHEMATICST hestudy of mathematics is apt to commence in disap-pointment.

4 The important applications of the science, thetheoretical interest of its ideas, and the logical rigour of itsmethods, all generate the expectation of a speedy introduc-tion to processes of interest. We are told that by its aid thestars are weighed and the billions of molecules in a drop ofwater are counted. Yet, like the ghost of Hamlet s father,this great science eludes the efforts of our mental weaponsto grasp it Tis here, tis there, tis gone and what wedo see does not suggest the same excuse for illusiveness assufficed for the ghost, that it is too noble for our gross meth-ods.

5 A show of violence, if ever excusable, may surelybe offered to the trivial results which occupy the pages ofsome elementary mathematical reason for this failure of the science to live up to itsreputation is that its fundamental ideas are not explained tothe student disentangled from the technical procedure whichhas been invented to facilitate their exact presentation inparticular instances. Accordingly, the unfortunate learnerfinds himself struggling to acquire a knowledge of a massof details which are not illuminated by any general concep-NATURE OF MATHEMATICS2tion.

6 Without a doubt, technical facility is a first requisitefor valuable mental activity: we shall fail to appreciate therhythm of Milton, or the passion of Shelley, so long as wefind it necessary to spell the words and are not quite certainof the forms of the individual letters. In this sense there is noroyal road to learning. But it is equally an error to confineattention to technical processes, excluding consideration ofgeneral ideas. Here lies the road to object of the following Chapters is not to teach math-ematics, but to enable students from the very beginning oftheir course to know what the science is about, and why itis necessarily the foundation of exact thought as applied tonatural phenomena.

7 All allusion in what follows to detaileddeductions in any part of the science will be inserted merelyfor the purpose of example, and care will be taken to makethe general argument comprehensible, even if here and theresome technical process or symbol which the reader does notunderstand is cited for the purpose of first acquaintance which most people have withmathematics is through arithmetic. That two and two makefour is usually taken as the type of a simple mathematicalproposition which everyone will have heard of. Arithmetic,therefore, will be a good subject to consider in order todiscover, if possible, the most obvious characteristic of thescience.

8 Now, the first noticeable fact about arithmetic isthat it applies to everything, to tastes and to sounds, to ap-ples and to angels, to the ideas of the mind and to the bonesof the body. The nature of the things is perfectly indifferent, Introduction TO MATHEMATICS3of all things it is true that two and two make four. Thuswe write down as the leading characteristic of mathematicsthat it deals with properties and ideas which are applica-ble to things just because they are things, and apart fromany particular feelings, or emotions, or sensations, in anyway connected with them.

9 This is what is meant by callingmathematics an abstract result which we have reached deserves attention. Itis natural to think that an abstract science cannot be ofmuch importance in the affairs of human life, because it hasomitted from its consideration everything of real interest. Itwill be remembered that Swift, in his description of Gul-liver s voyage to Laputa, is of two minds on this point. Hedescribes the mathematicians of that country as silly anduseless dreamers, whose attention has to be awakened byflappers. Also, the mathematical tailor measures his heightby a quadrant, and deduces his other dimensions by a ruleand compasses, producing a suit of very ill-fitting the other hand, the mathematicians of Laputa, by theirmarvellous invention of the magnetic island floating in theair, ruled the country and maintained their ascendency overtheir subjects.

10 Swift, indeed, lived at a time peculiarly un-suited for gibes at contemporary mathematicians. newton sPrincipiahad just been written, one of the great forces whichhave transformed the modern world. Swift might just as wellhave laughed at an a mere list of the achievements of mathematics is anunsatisfactory way of arriving at an idea of its OF MATHEMATICS4It is worth while to spend a little thought in getting at theroot reason why mathematics, because of its very abstract-ness, must always remain one of the most important topicsfor thought. Let us try to make clear to ourselves why ex-planations of the order of events necessarily tend to how all events are interconnected.