BERTRAND RUSSELL
The Spirit of Solitude, 1872-1921
By Ray Monk
Illustrated. 695 pages. Free Press. $35.

Bertrand Russell might possibly have been the smartest Englishman since John Stuart Mill, who, not coincidentally, was Russell's godfather and a close friend of his father's. And yet, brilliant as he was, Russell was a powerfully flawed figure, eloquently self-absorbed and self-dramatizing, capable of remarkable callousness, especially in matters of love.

Russell's philosophical writings, moreover, unlike those of his celebrated godfather, have not stood the test of time especially well. He was, to be sure, the founder of analytical philosophy, which still dominates philosophical thought in England and America, and that is an important legacy. But even many analytical philosophers, as the University of Virginia's Richard Rorty wrote in a recent essay in The New Republic, have been asking ''how anybody as smart as Russell could ever have been foolish enough to hold his absurd views.''

Brilliant and foolish but exceedingly interesting in both of those qualities, that would be a pretty good summing up of the Bertrand Russell that emerges from this new, ambitious and enfolding biography by Ray Monk, a philosopher himself, who teaches at the University of Southampton in England. Following by five years Mr. Monk's luminous biography of Ludwig Wittgenstein, this new book, ''Bertrand Russell: The Spirit of Solitude,'' consolidates Mr. Monk's standing as a chronicler of the great intellects of the century.

Mr. Monk's book, the first of two projected volumes, takes Russell from his birth in 1872 into one of England's most distinguished and well-connected families to his second marriage in 1921 and the birth of his son. Following the technique he used in his work on Wittgenstein, Mr. Monk weaves together Russell's mathematical and philosophical investigations with his private life, devoting rather more time to the intense passions and sorrows of the latter than to the former. Mr. Monk states at the outset that to understand Russell is to understand ''the role played in his life and in his imagination by his hopes for philosophy.''

Despite Mr. Monk's commendable effort to explain Russell's technical work in language that lay readers can understand, you will probably not become proficient in analytical philosophy as a result of reading this book. Mr. Monk glosses rather too quickly over the various conundrums that engaged Russell's mind -- like how a statement can be false even when its opposite is also false -- to be readily graspable.

''In Meinong's theory,'' Mr. Monk writes at one point, providing some background to one of the logical problems Russell faced, ''objects had different kinds of being; particular objects which occupied space and time existed, abstract objects (like numbers) subsisted, but some objects (unreal particulars like the characters of fiction and mythology) neither existed nor subsisted.'' It would not hurt to be an analytical philosopher yourself to understand that sentence, or many others like it in this book.

And yet, Mr. Monk's narrative never flags, in part because the basic idea, that Russell wanted to establish mathematics (and, after mathematics, all of science) on ''hard, dry intellect,'' rather than on intuition or some other kind of ''mystic illumination,'' is not that difficult to comprehend. Sometimes, moreover, Mr. Monk's handling of matters is deft enough at least to produce some intellectual enjoyment for the mathematically challenged. The famous conundrum known as Russell's paradox, for example, Mr. Monk likens to ''defining the village barber as 'the man who shaves all those who do not shave themselves' and then asking if he shaves himself or not.''

But Mr. Monk's main effort in this volume is not on Russell's thought but on the first 48 years of his private life and, especially in the later chapters, his political activities. Here Mr. Monk reaches high levels of brilliance in biographical writing, his narrative informed by a complete mastery of his subject and by penetrating psychological insight. Russell, who lost both of his parents when he was still a small boy, led private, political and philosophical lives linked together, in Mr. Monk's vision, by a powerful, driving effort to overcome two lifelong fears: solitude and madness.

This, Mr. Monk writes, ''was the origin of that chilling detachment of which he was capable in his relations with people close to him, and also of his tendency to form intense and stormy relationships, in which his hopes of overcoming his detachment and of finally making human contact fought a losing battle with his fears that he would once again fail.''

What makes this especially fascinating is the intensity of Russell's own self-examination, even when he was too blinded by priggishness and self-importance for self-examination to be of much use. The thousands of letters that Russell exchanged with lovers and friends makes for a remarkably intimate portrait of his inner life.

Among his main characteristics, he was prone to falling in love with women married to other men, while denigrating the women who were available to him. Among these married women was Vivien Eliot, the wife of the poet T. S. Eliot; Evelyn Whitehead, the wife of his collaborator on ''Principia Mathematica,'' and, most important, Ottoline Morrell, whose husband, Philip Morrell, Russell had supported in a Parliamentary election in 1908.

Russell's relations with men, while never sexual, were also often stormy and frequently unsuccessful. Mr. Monk's account of his disastrous friendship with D. H. Lawrence is searing to read. Lawrence, breaking with Russell over political matters, wrote him a furious, caustic, stinging letter that left Russell in a suicidal frame of mind. ''Your will is false and cruel,'' Lawrence wrote. ''You are too full of devilish repressions to be anything but lustful and cruel.'' Russell, reading the letter, ''sat stunned for a whole day -- he was deeply horrified,'' Ottoline Morrell wrote in her memoirs, ''for his belief in Lawrence's insight was still unshaken, and he thought it must be true.''

It is hard to imagine Russell retaining heroic stature after Mr. Monk is finished with him, but it is important to bear in mind not only his brilliance, but also his courageous iconoclasm and his unwillingness to bend his principles even if it meant going to prison, as Russell did for six months during World War I. ''Our hearts build precious shrines for the ashes of dead hopes,'' Russell once wrote to Mrs. Morrell, giving expression to a complex, dense vision brilliantly captured in this penetrating book.