Introduction to Mathematical Philosophy

Bertrand Russell

Description

Whilst the greatest effort has been made to ensure the quality of this text, due to the historical nature of this content, in some rare cases there may be minor issues with legibility. Groundbreaking in its day the aim of this fantastic, if dated book by renowned philosopher Bertrand Russell is nothing less than to demonstrate that all of mathematics is describable by a single system of logic. It does so in a way which, while not simplistic, is still accessible to anyone willing to put in some thought. A foundation in logic and philosophy would certainly be a help when reading this book however.<br><br>This approach was later demonstrated to be impossible, with Kurt Godel's incompleteness theorems proving that for any consistent logical system there are some problems which cannot be solved within that system. Godel's work would not have been the same without these earlier attempts however, so they provide an essential context for understanding these later developments.<br><br>This does not mean that Russell's work is devoid of insight or use for its own sake however, as most of the individual sections as well as the history of mathematical philosophy are very valuable. Those whose interest is piqued by this book may wish to move on to Russell and Whitehead's Principia Mathematica for a more difficult and in depth work.

Sujets

• Philosophie et théologie

Informations

• Publié par: Forgotten Books
• Date de parution: 27 novembre 2019
• Nombre de lectures: 0
• EAN13: 9780243648719
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

T H andis inten ed essentially as an Introdu tion does aim at giving an e haustive discussion the problems with which it deals It seemed desirable set forth certain results hitherto only available to those who have mastered logical symbolism in a form the minimum of dificulty to the beginner The utmost endeavour has been made to avoid dogmatism on such questions as are still open serio s doubt and this endeavour has some e tent dominated the choice of topics considered The beginnings of mathematical logic are less defibut arenitely known than its later po tions at lea t equal philosophical interest uch of what is set forth in the following chapters is not prop rly to be called philosophy though the matters concerned were included in philosophy so long as satisfactory science of them e isted The nature of infifor e ample nity and continuity belonged in former days to philosophy but belongs now to mathematics athematical in the strict sense cannot perhaps be held include such definite s ientific results as have been obtained in this region the philosophy of mathematics will naturally deal with questions on the frontier of knowledge as which comparative certainty is not yet attained But spe ulation such questions is hardly likely fruitful unless the more scientifithe principles mathematicsc parts are known A thereforebook dealing with those parts may claim be an mathematical philosophy though i t hardly claim e cept where it steps outside its province be actually dealing with a part philosophy It does deal

d however with a body of knowledge which to those who a ept it appears to invalidate much traditional philosophy and even a good deal what is current in the present day In this way as well as by its bearing on sti unsolved problems mathemati al logi is relevant to ph lo ophy th s reason as well as a ount the in insic importance the subj e t some purpo e may be served by a suc inct account the m in re u t mathemati l lo ic in a form requ ring ne ther a knowledge mathematics nor an a titude for mathemati al symbolism Here however as e sewhere the method is more important than the results from the point of v ew of further r earch and the method annot well be e lained with n the framework of su h a book as the following It is to be hoped that some readers may be interested to advance to a study of method by whi h ma hematical logic can made helpful in investigating the traditional roblems philosophy But that is a topic with whi h the following pages have attempted to deal BERT AND R U SS E LL

T H O E athemati aldistinction between relying the Philosophy and the Philosophy of athematics think that this book is out place in the present Library may be referred to what the author himself says this head in the Preface It is not necessary agree with what he there suggests as to the readj ustment of thefield of philosophy by the transference from it mathem atics such problems as those of lass continuity infinity in order perceive the bearing of the definitions and discussions that follow on the work of traditional philosophy If philosophers cannot consent to relegate the criticism of these categories to any of the special sciences it is essential at any rate that they s ould know the precise meaning that the science mathematics in which these concepts play lar a part ssigns to them If on the other hand there be mathematicians to whom thes e definitions and discussions seem to be an elabora tion and compli ation the simple it may be well to remind them from the side of philosophy that here as elsewhere apparent sim li ity may conceal a comple ity which it is the business of somebody whether philosopher mathematician like author of this volume both in one to unravel

R EFA E EDIT O NR S O TE T H E SERIES O F NAT U RAL N UM ERS DEFI N ITI O N O F N UM ER FI N IT U DDE AN M AT H E M ATI C AL I N D U C TI O N T H E DEFI N ITI O N O F O RDER I ND S O F RELATI O N S SI M ILARIT O F RELATI O N S RA I O DREAL AN N AL CO M PLE N UM ERS I NFI N ITE C ARD IN AL N UM ERS I N F N ITE SERIES AND O R I N ALS LI M ITS AN D CO NTI N U IT LI M ITS AN D CO NTI N U IT O F F U N C TI O N S SELE C TI O AN D TN S H E MU LTIPLI C A IATI E O M T H E A I O M O NFI N IT F I LAN D O I C AL T PES I N CO M D TLIT AN PATI I H E T H E O R O F DED U C TI O N P O P O ITI O N AL F U N C TI O N S D ES C RIPTI O N S C LASSES AT H E M ATI C D LS AN O I C I N DE

R I ATU R AL N UM ERS AT H E M ATI C S we start from its mostis a study which when famili ar portions may be pursued in either of two op osite rections The more familiar direction is constructive towards gradually increasing comple ty from in egers to fract ons real umbers comple numbers from ad ition and multi pl cation to and integration and to gher mathematics The other direction which is l ss familiar proceeds by analysing greater and greater abstractness and lo cal simplicity instead Of asking what can be defined and deduced from what is assumed to begin with we ask instead what more general ide s and principles can be found in terms which what was our starting point can be defined or deduced It is the fact this opposite direction that characterises mathematical philosophy as Opposed to ordinary mathematics But it should be understood that the stinction is not in the subj ect matter but in the state of mind of the investigator E arly Greek geometers passing from the empirical ules of Egyptian land surveying to the general propositions by w h t ose rules were found to be j ustifisable and thence Euclid a ioms and postulates were engaged in mathematical p los according to the above defithewhen once tion but a ioms and postulates had been reached their dedu tive employ ment asfind it in Euclid belonged to mathematics in the

d mathem atics and or inary sense The distinction between mathemati al p losophy is which depends upon the interest inspiring the research and upon the stage which the res earch has reached not upon the pro ositions with whi h the resear h is on erned We may state the same stinction in another way The most obvious and easy t ngs in ma hematics are not thos e that come logi a ly at the beginning they are t ngs that from the point of view lo ical deduction come somewhere in the middle Just as the easiest bod es see are those that are neither very near very far neither very sm ll nor very great so the easiest conce tions to grasp are thos e that are neither ve y comple nor very simpl e using simpl e in a sense And as we need two sorts Of inst uments the teles ope and the microscope the enlargement of visual powers so we need two sorts instruments the enlargement of logical powers one to take us forward to the higher mathematics the other to take us backward to the logi al foundations of the t ngs that we are inclined take for granted in mathemati s We shallfind that by analysing ordinary mathemati al notions we acquire fresh insight new powers and the means reaching whole mathemati al subj ects by adopting fre h lines Of advance fter our backward j ourney It i s the purpose of this book to e plain mathemati philos simply and untechnically without enl arging upon thos e portions wh ch are so doubtful d cult that an elementary treatme t is s ar ely possible A full treatment will be found in the treatment in the pres ent vo ume intended merely as an introduction is To the average educated person of the resent day the Ob ious starting point mathematics would b e the series of whole numbers

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N N Probably only person with some mat em ti al knowledge wo d think begin ing with instead with I llbut we w pr sume this degree knowledge we il take as starting point th e series

and it is this series th t we shall mean when we speak of the series natural numbers I t is only at a gh stage civilis ation that we ould take th s seri s as starting point I t must have required many ag s scover that a brace pheasants and a couple days were bo h instan e number the degree abstracti on involved is far from easy And the dis overy that I is a number must have been difiult it is a very recent addition the Greeks and Rom ans had no such digit If we had been embarking upon mathematical philosophy in earlier days should have had start with something less abstra t than the s eries natural numbers whi h we should reach as a stage b ack ard j ourney When the logical foundations mathe have grown more fam li ar we shall be able start fur her back at what a late stage in analysis But for the moment the natural num ers s eem to represent what is easiest and most familiar in mathematics But though familiar they not understoo Very people are prepared with a definition what is meant by number or It is not very di ult see that starting from any other of the natural numbers be reached by rep eated additions Of but shall have define what we mean by ad ing and what we mean by repeated The e questions are by means easy I t was believed until recen ly that some at le t thesefirst no ions ari hmetic must be a epted as simple and primi tive to be defined Since all terms h t are defined are defined by means of other terms it is lear that human knowledge must always be content to accept some terms as intelligibl e without definition in order