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Product Details
| ISBN-13: | 9781566633574 |
|---|---|
| Publisher: | Dee, Ivan R. Publisher |
| Publication date: | 04/17/2001 |
| Series: | Philosophers in 90 Minutes Series |
| Pages: | 96 |
| Product dimensions: | 4.98(w) x 7.98(h) x 0.29(d) |
About the Author
Read an Excerpt
Chapter One
Russell's Life and Works
Bertrand Russell was born in 1872 into one ofthe most distinguished aristocratic families inEngland. This was the height of the Victorianera, when the British Empire was approachingits apogee. Hypocrisy was the order of the day,amidst widespread social and psychological repression.However, both Russell's parents heldenlightened liberal views—his father lost his seatin Parliament for espousing the cause of birthcontrol.
Young Bertrand's childhood was overshadowedby death. Both his parents, and his sister,died by the time he was five. His parents had instructedthat their two sons be placed under theguardianship of an atheist friend, but this wascontested in the courts by young Bertrand's powerfulgrandfather, Lord Russell, who had twicebeen prime minister. The court overruled theparents' will, so Bertrand and his older brotherwere taken to live with Lord and Lady Russell atPembroke Lodge in Richmond Park, on the outskirtsof London. Queen Victoria herself wroteto congratulate Lady Russell, adding, "I trustthat your grandsons will grow up all that youcould wish." (As Bertrand Russell wryly commentedmany years later, this wish "was deniedher.") Within a year Lord Russell himself wasdead. Young Bertrand lay in bed dreading themoment when Lady Russell too would die, anevent that he childishly assumed was bound tohappen soon. He focused his mind on thethought of his beloved parents, a fading pictureof certainty, sweetness, and light.
Life at Pembroke Lodge was very different.Lady Russell was a willful puritan,though paradoxicallyshe retained the liberal political viewsof her husband. Her "angel child," as she calledBertrand, was brought up under a regime of coldbaths before breakfast and blinkered morality.Such matters as sex and trade were simply notmentioned. Lady Russell decided that her angelshould remain uncontaminated by contact withother children. He was educated at home by tutors,with occasional lessons from his amiableelder brother Frank, who was seven yearsBertrand's senior and evidently considered a lostcause, for he was sent away to school.
It was Frank who introduced Bertrand to thesubject that would transform his life. Russell describeshow at the age of eleven he began studyinggeometry under his brother's tuition. Theystarted working their way through Euclid's Elements,and, in Russell's telling words, "I had notimagined there was anything so delicious in theworld." Even when they came to Euclid's difficultfifth proposition, Russell found no difficulty,prompting a surprised comment fromFrank. Again in Russell's own words: "This wasthe first time it had dawned upon me that Imight have some intelligence." In his isolation hehad simply had no one with whom to comparehimself. But for the adolescent Bertrand this wassomething more than the rapturous discovery ofsome hitherto undreamed-of wonder. The waythat Russell regarded mathematics was characteristicallyoriginal from the start. Frank explainedto Bertrand that Euclid had establishedthe whole of geometry by proof, thus making itstheorems utterly certain and incontestable. ButBertrand was disappointed to discover that Euclidhad in fact based his geometry upon a seriesof basic axioms. What about proofs for these?Frank replied that there weren't any. Bertrandobstinately refused to go on until Frank producedsome. Frank explained to Bertrand that hejust had to accept these axioms or they wouldnot be able to progress further. Because Bertrandwas dying to learn more of this wonderful geometry,he reluctantly accepted. This love of the formalbeauty and certainty of mathematics, as wellas the pressing desire for these to be based uponsome bedrock of unquestionable truth, wouldkeep Russell alive for the next thirty years.
This is no fanciful exaggeration. Russell's lifeat Pembroke Lodge remained unhealthily solitary,his feelings for his fellow human beings almostentirely sublimated. He relates how hewould frequently go into the garden to lookdown over Richmond Park and the far vista ofthe Thames valley. Here he would gaze at thesunset and think about committing suicide. Theonly thing that prevented him from taking hislife was the wish to discover more about the "delicious"abstract beauty of mathematics. He explainshow he was searching "for somethingbeyond what the world contains, somethingtransfigured and infinite ... it's like a passionatelove for a ghost.... I have always desired tofind some justification for the emotions inspiredby certain things that seemed to stand outsidehuman life and to deserve feelings of awe."
The psychology of these words would seemtransparent. But Russell's unconscious desire tobe reunited with his parents does not explainaway his passionate involvement in mathematics.From his earliest years he exhibited an exceptionalclarity of thought which was ideallysuited to mathematics. Yet this clarity oftenmasked near-impenetrable complexities, and notonly in mathematics. Russell would always feelthe need to give clear and candid expression tohis thoughts, yet things would seldom be asclear-cut as he wished them to appear. His solitarycogitations soon led him to reject any woollynotion of God, especially the personal God sobeloved by his grandmother. Throughout his lifeRussell would profess, with rational and persuasiveclarity, his atheistic belief—his "vain searchfor God"—yet at the same time retain an attitudetoward mathematics that he expressed interms of mystic religiosity. He believed in the abstractworld of mathematics and was driven tosearch in it for the certainty that had, during hisearly childhood, vanished from his life.
At sixteen Russell was sent to a Londoncrammer, where he was a boarder for almost twoyears. The pupils were mainly being crammedfor the army exams, and Russell found them adecidedly coarse and ignorant lot. This accurateassessment of most potential army officerswould unfortunately color Russell's entire viewof humanity to the end of his days. Despite hismuch-avowed concern for the plight of his fellowbeings, Russell would always find it difficultto disguise a certain aristocratic aloofness. Thiswould intensify into disdain when he found himselfconfronted by those who chose to devotetheir lives to less noble pursuits, such as soldiers,statesmen, and authorities of any sort.
In 1890, at the age of eighteen, Russell won ascholarship to Trinity College, Cambridge,where Isaac Newton had studied and taught. ForRussell's first three years he studied mathematics,which proved a bitter disappointment. Forthe most part, British mathematics had languishedin the 150 years since Newton's day, andnowhere was this more apparent than at hisalma mater. The celebrated "wrangler" examinations,designed to discover the finest mathematiciansin Cambridge, required little morethan formidable rote learning and ever more ingeniousmathematical conjuring tricks. This wasa mockery of the abstract beauty which had soinspired Russell, and in his fourth year he turnedin disgust to philosophy.
Here he discovered the abstract world to endall abstract worlds, in the form of the all-embracingmetaphysical system first conceivedby the early-nineteenth-century German philosopherHegel. A modern variant of Hegel's AbsoluteIdealism was taught at Cambridge by J.M. E. McTaggart. According to this, both timeand matter were unreal. Only the AbsoluteSpirit, which contained everything, had reality.This ultimate reality was a whole, whose partswere all interrelated. Russell would liken thiswhole to a jelly: the moment you touched onepart of it, the whole quivered. Yet unlike a jelly,this whole could not be cut up into separateparts. According to McTaggart, although this ultimatereality existed in an idealistic world aboveand beyond the so-called reality we experienced,it was still possible to deduce its nature. Thiscould be done by starting from certain self-evidenttruths and just two empirical premises—namely,that something exists, and that it hasparts. As is evident, this Absolute Idealism notonly uncannily resembled the world of mathematics,it also went beyond it, subsuming themerely mathematical into the greater scheme ofthings in an overall philosophy. Russell was entranced.Here was a philosophy that fulfilled histwin needs—for the certainty of geometry andthe mystical sublime.
But Russell also found that he possessedmerely human needs. Even before he went toCambridge he had met and fallen in love with anAmerican Quaker called Alys Pearsall Smith. Hewas only seventeen, she was twenty-two—a five-yearage difference which represented a yawninggap in their development. Russell did not declarehis calf love, letting it mature in secret. Alys heldadvanced social views but remained strictly religiousand devoted part of her time to deliveringspeeches at temperance rallies. Only four yearslater did Russell reveal his feelings, whereuponhe was pleasantly surprised to find them reciprocated.In such an era of suppressed emotion,when so few had any experience of dealing withtheir feelings, even platonic love could quicklybecome a consuming passion. Within monthsBertie and Alys were planning to be married.Lady Russell's reaction was predictable. Outragedthat this American gold digger had corruptedher angelic Bertie, she did everything inher power to put an end to the romance. Bertiestood his ground—amidst the tears, railings, accusationsof ingratitude, and threats—promisingthat he would marry Alys as soon as he wastwenty-one. Then he would be legally free tomake his own decisions and would also inheritan income sufficient to support them both. In1894 they were married.
Russell had graduated with a first-class degreein moral science (philosophy) and was nowelected a fellow of Trinity College. This appointmententailed no binding duties other than research.Mr. and Mrs. Russell set off on a tour ofEurope, where they settled for an extended periodin Germany. Here Russell developed an interestin politics and even wrote a book calledGerman Social Democracy, which became hisfirst published work.
When Russell finally returned to Cambridgehe was introduced to G. E. Moore, who waswidely regarded as the new young intellectualstar of the university. Moore's attitude to philosophywas obstinately robust. He rejected McTaggart'sidealism on the grounds that it simplydefied common sense. Moore insisted on believingin the physical world he experienced. Russell'sthinking had been undergoing a sea change,and he quickly established a rapport withMoore. Russell began to realize that the Hegelianworld of Absolute Idealism bore no relation tothe actualities of physical experience. But scienceand material reality simply could not be ignored.Russell found himself adopting an empirical materialistview of the world. Experience is what isreal, and what we experience is the materialworld. Yet he found himself unable to relinquishhis mystical belief in mathematics. "The greatestmen who have been philosophers have felt theneed of both science and mysticism." The needto reconcile these apparent disparities made philosophy"a greater thing than either science orreligion." Russell now attempted to do just this,by embarking upon an investigation into theprinciples of mathematics. His thinking hadcome full circle. The twenty-six-year-old manwas tackling the question raised by the eleven-year-oldboy on his first encounter with Euclid.How could one discover the ultimate principlesupon which mathematics was based? As Russellput it at the time: "Although the work is almostwholly mathematical, its interest is almostwholly philosophical." His search was for ultimatecertainty.
Euclid had begun with axioms; these werethe basis of geometry. But what was the basis ofthese axioms? They weren't just random—surelythey had to conform to something? Russell concludedthat this ultimate something could onlybe logic. The basic axioms of geometry, and likewisethe fundamental concepts of mathematicsas a whole, had to be logical. So what was thelogical basis from which mathematics was derived?
In July 1900 Russell attended the InternationalCongress of Philosophy in Paris. Here hemet the Italian mathematical logician GiuseppePeano, who had been working for several yearsupon the foundations of number. Peano's aimwas to reach beyond the idea of number as simpleintuition, and instead establish a logicalmethod on which the concept of number couldbe founded, and from which the numbers themselvescould be generated. In the course of this hehad developed a series of fundamental logicalsymbols which enabled concepts and propositionsto be analyzed into their ultimate constituentparts. For instance, he introducedseparate symbols for "a class which has onemember" and "the member of this class." Thissubtlety enabled him to overcome the previouslogical confusion between the concepts "is amember of," "is contained in," and "is equalto." Russell was deeply impressed: he had neverbefore encountered such precise logical rigor. Hehad been having extreme difficulty in his attemptto unravel the basic principles of mathematics.But now: "My sensations resembled those onehas after climbing a mountain in a mist, when,on reaching the summit, the mist suddenlyclears, and the country becomes visible for fortymiles in every direction."
Previously Russell had seen the universe as abowl of jelly; now he likened it to a bucket ofshot. The whole had given way to a myriad ofdiscrete parts. This required a diametrically differentapproach. Instead of synthesis, it was nownecessary to apply analysis—which derives fromthe Greek word to unravel. The bucket of shotconsisted of separate parts, each in contact onlywith those around it. Any understanding of thisnew discrete universe required an analysis of therelations between the separate parts.
The emphasis was now on the atomic natureof the universe, which was amenable to logicalanalysis. Here Russell was not so much referringto physical atoms as to the ancient Greek ideathat gave rise to the notion of atoms. Accordingto the fifth-century B.C. philosopher Democritus,if one went on dividing matter one must in theend come to something indivisible. This wouldbe uncuttable—in Greek a-tomos, hence theword atom. Democritus had not reached this notionby experiment but purely by logical reasoning.This was now Russell's aim too. He wishedto reach the indivisible atoms of logic on whichmathematics was based. Initially he had arrivedat the basic concept of "number," "order," and"whole and part." But Peano had shown himhow to go beyond the immediate intuition ofnumber, by demonstrating it could be generatedfrom certain even more fundamental postulates:
1. 0 is a number.
2. The successor of any number is a number.
3. No two numbers have the same successor.
4. 0 is not the successor of any number.
Russell was not entirely persuaded by thislogic, but he recognized at once that its methodheld the key. Instead of his previous concept of"whole and parts," he decided to use the notionof "class" (as in "the class of all apples," "theclass of all unsolved problems," etc.). Class is alogical distinction: it is based upon the fundamentallogical law of identity. A thing cannot atthe same time be both itself and not itself. (Theworld is divided into "apples" and "things thatare not apples.") Russell was able to show thatthe notion of class is prior to that of number. Forexample, we can conceive of the class of appleswithout collecting all apples and placing themtogether. Without counting the number of applesin this class, we can still say some very definitethings about it. That is: This class will include nopears; its members will all be fruit; and so on.From this it could be seen that the notion of classis logically prior to that of number. In otherwords, the logical notion of class was more fundamentalthan that of number.
Russell then proceeded to use the notion ofclass to generate the concept of number, andthen all the individual numbers. A simplified versionof Russell's method goes as follows:
—The class of all objects that are not identicalto themselves has 0 members.
—But all empty classes have the same members,so they are not equal to one another; theyare in fact identical. They are the same class.
—There is thus just one empty class. Hencefrom 0 we have generated the notion of 1.
—The class of empty classes thus containsone member. So the class of empty classes and itsmember make 2. And the class of the class ofempty classes and its member lead us to generatethe number 3, and so forth.
—The whole of mathematics could be generatedout of the logical notion of classes, whichderived from the fundamental logical notion ofidentity.
With this method Russell had establishedtwo vital points. He had shown that the truths ofmathematics could be translated into the truthsof logic. And this showed that mathematics infact had no distinct subject matter of its own,such as numbers. All mathematical truths werethus in theory ultimately reducible to logicalform. This meant that they could be proved bylogic. (The wish of the eleven-year-old child hadcome true: even the axioms of geometry could beproved!)
In 1903, Russell published The Principles ofMathematics. This work established him as amajor philosophical thinker, especially in Europewhere this topic had become a matter of intensespeculation. With the new century, philosophywas moving away from the grandiose speculationsof metaphysics, epitomized by Hegel, andhad begun to concentrate on the more preciseproblem of human knowledge. What was thebasis of our knowledge, and how could weknow if it was true? The first step had been toapply this question to the most certain and infallibleknowledge available, namely mathematics.And the answer appeared to lie in logical analysis.It was generally agreed that Russell had notanswered the question completely; there was stillthe problem of less rigid forms of knowledgesuch as science. But thinkers recognized thatRussell had made a significant step toward answeringone of the problems that had troubledphilosophers since ancient Greek times.
Since Russell's point was essentially philosophical,The Principles of Mathematics hadbeen written in plain English (or as near to thatmythical entity as philosophers can come). Butas Peano had shown, language can frequentlygloss over crucial logical distinctions. Russellnow intended to write a second volume whichwould set down his argument in the more preciseform of logical symbols, thus overcoming possiblemisinterpretations. The immense difficultiesarising from this project led him to collaboratewith the Cambridge mathematician AlfredNorth Whitehead, who had taught Russell duringhis undergraduate days. Whitehead was theonly mathematician at Cambridge whom Russelladmired; Whitehead also had a thorough knowledgeof philosophy and logic. This would be anequal partnership. Together the two of them setabout developing a symbolic logic which extendedPeano's original conception. This was thebeginning of Principia Mathematica, a collaborationthat would eventually take Russell andWhitehead no less than ten years. In Russell'swords, they would show that "logic is the youthof mathematics and mathematics is the manhoodof logic." They would start with an irreducibleminimum of logical concepts, represented inclear symbolic form. They would then advance,step by logical step, to show how the whole oflogic, and then mathematics, could be derivedfrom these basic concepts alone. This would bean immense project, often requiring fiendish ingenuity,involving many hundreds of pages coveredwith logical symbols. But it would be worthit. What it would establish would be absoluteand irrefutable: the status of human knowledgewould be transformed forever. This would be thegreatest advance in philosophic certainty sinceAristotle's initial discovery of logic more thantwo millennia earlier.
Three years into this project, disaster struck.Russell discovered a flaw which ran to the heartof their logical argument. This was a paradoxwhich seemed to render the very notion ofclasses self-contradictory. Today it is known asRussell's Paradox.
Imagine a library, which besides its shelves ofbooks also includes two catalogs. The first cataloglists all books that refer to themselves—forexample, "as mentioned previously in Chapter2." The second catalog lists all books in the librarythat do not refer to themselves. In whichcatalog is the second catalog listed? If it is listedin itself, it immediately becomes a book thatdoes refer to itself. But it cannot be listed in thefirst catalog because it does not refer to itself.The paradox appears irreducible.
But what has this got to do with classes? InRussell's form, the argument runs as follows: Insteadof the two catalogs, we have two classes.First there is the class of all classes that are membersof themselves. For instance, the class of allclasses is a member of itself, because it is itself aclass. Second, we have the class of all classes thatare not members of themselves. Among these isthe class of all numbers, which is not itself a
(Continues...)
Excerpted from Bertrand Russell in 90 Minutes by Paul Strathern. Copyright © 2001 by Paul Strathern. Excerpted by permission. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Table of Contents
| Introduction | 7 |
| Russell's Life and Works | 13 |
| Criticisms and Comments | 73 |
| Chronology of Significant Philosophical Dates | 77 |
| Chronology of Russell's Life | 83 |
| Recommended Reading | 89 |
| Index | 91 |