| - Topology Atlas Document # topc-02.txt WWW url: http://at.yorku.ca/t/o/p/c/02.txt Topology Atlas, atlas@yorku.ca http://www.unipissing.ca/topology/ IN THE NEIGHBORHOOD OF MATHEMATICAL SPACE (an interview with Alexander V. Arhangelskii) by Karen Shenfeld - Exclusive reprint in Topology Atlas from the Summer 1993 issue of The Idler with kind permission of the publisher. - Alexander Vladimirovich Arhangel'skii walked briskly up a tree-lined avenue toward the tiered ``Stalinist Gothic'' towers of Moscow State University. ``Topology,'' he said, slightly out of breath, ``is the study of absolute nearness . . . the absolute, infinite nearness of `a point' to `a set.' Consider, if you will, the present moment as a topologist would: it is, simply put, `a point' in time, which is infinitely near to `the set of all points' in the past, and infinitely near, without the slightest gap, to `the set of all points' in the future. A topologist can represent, with mathematical rigour, the essence of the present moment. He has only to recall the topological structure of `the real line.' '' It seemed ironic that, even as he was speaking of the infinite nearness of points in time, the eminent Russian mathematician should be running 15 minutes late for the seminar he teaches at six o'clock on Monday evenings. An hour earlier, he had been standing at the door of his apartment, changing out of his slippers and into his oxfords, when the telephone had rung. An old friend was on the line, and Alexander Vladimirovich had sat down on a rickety wooden chair in the entrance hall for a chat. He might have cut the conversation short had he known that he would have to wait longer than usual for the Lomonosovskii-Prospekt bus. ``Don't worry,'' he said now, as we hurried across the verdant top of Lenin Hill. ``My students expect me to be at least fifteen minutes late.'' Approaching the university's soaring central tower, we passed a paved, hedged-in courtyard and joined a crush of students entering the building, one by one, through a swinging glass door. We filed quickly by three uniformed guards, then jostled our way through a grand, marble-clad lobby to take the elevator. M.V. Lomonosov Moscow State University opened in 1755, upon the issue of an edict by Empress Elizabeth Petrovna, daughter of Peter the Great. Its official title bears the name of its founder, Mikhail Vasilevich Lomonosov: a scientist, historian, linguist, and poet who, in Russia, possesses the stature of Leonardo da Vinci. Originally housed in a building in Red Square, M.S.U. is now one of the largest higher-educational institutes in the former U.S.S.R., employing roughly 3,500 instructors and 4,000 researchers, and with an enrollment of more than 32,000 undergraduate and graduate students. Dominating the city's skyline south-west of the Kremlin, its principal modern edifice rises 32 storeys, the tip of its red-starred spire reaching a height of 994 feet. The central tower is flanked on both sides by 18-storey wings, each of which in turn branches off into two 12-storey wings. The labyrinthine structure comprises a total of 45,000 rooms: students of mathematics like to tell you that to spend just a minute in each would take 37 days and require walking 90 miles. The Chair of General Topology and Geometry is located in Room 1222 on the 12th floor, the first of four floors occupied by the Department of Mechanics and Mathematics. Its heavy door opens into a narrow chamber whose furnishings are sparse, worn with age, and ``low-tech'': a long wooden table and, at right angles to it, a big square desk with the secretary's manual typewriter resting on its top. A chess set and two timers sit permanently at one end of the room: members of the Chair drop by, on occasion, for a couple of matches of ``Blitz.'' A large, finely wrought portrait hangs in the centre of the north wall. It depicts an old man with a completely bald head and round, black spectacles. Seated calmly behind an ancient desk, he leans on his elbows, palms facing each other, fingertips lightly touching. His eyebrows are slightly raised, as if he were listening to a student explain a difficult mathematical proof. ``That's Pavel Sergeevich,'' said Arhangel'skii, nodding to the portrait. He was referring to his mentor, P.S. Alexandroff: one of the greatest figures in Russia's mathematical history and one of the world's foremost pioneers of that branch of modern mathematics known as topology. Surrounding the painting are a number of quick, gestural pencil sketches; most of these also portray Alexandroff, caught in the act of lecturing. One drawing, however, is of someone else: a man in his late 20s, perhaps, viewed in profile and slightly from behind, seated at a small desk, hunched in concentration over some papers. The sketch accentuates the subject's long, wavy mane of hair and the massive bulk of his shoulders. There is something vaguely Grecian in his profile, which reveals thin lips, a straight nose, and a slightly hooded eye. I did not need my host to tell me who was depicted in this sketch; for although it was made in his youth, it was still a good likeness. Alexander Vladimirovich has evidently changed little over the past 25 years. A few wrinkles have etched themselves about his hooded eyes, which, in real life, flash a wintry blue. His hair, once strawberry blond, has thinned slightly and turned snow white, and he now keeps it closely cropped. He has put little excess weight on his athletic, six-foot-tall frame. When the drawing was made, in 1966 (more than 10 years after its subject had enrolled as an undergraduate student of mathematics at Moscow State University and had decided to concentrate his thinking in the field of topology, under Alexandroff's tutelage), Arhangel'skii had already won substantial recognition for introducing the concept of ``a network,'' which has since become fundamental to the study of the field. (He devised it in 1959, as part of the thesis he wrote in order to receive his diplom: an undergraduate degree awarded after five years of study, which is, however, more akin to an American master's than a bachelor's degree). Already, too, he had been accepted as an aspirant (graduate student) at Moscow State University and had been awarded his kandidat nauk (candidate of sciences degree, equivalent to an American Ph.D.). He had also been granted the position of docent (associate professor) at the Chair of Higher Geometry and Topology. It was in 1966, at the unusually young age of 28, that he was awarded his doktor nauk (doctor of sciences degree, which is of higher rank than an American Ph.D., and for which there is no American equivalent); he was correspondingly granted the title of professor. Three years later, Arhangel'skii proved that you cannot construct ``a compact, first-countable, topological space'' whose size, or ``cardinality,'' is greater than that of ``the real line.'' In doing so, he solved a fundamental question, regarding the possible size of such a topological space, that had been posed by Alexandroff in 1922. He thus dramatically strengthened his reputation and launched a new area of interest for general topologists to do with ``cardinal functions.'' I first met A.V. Arhangel'skii in August 1987, at York University in Toronto. He had been invited there to speak at a conference on set theory and its applications, an event attended by 80 mathematicians from 12 countries. The organizers were particularly pleased by Arangel'skii's presence: it was the first time in 22 years that he had been permitted to leave the Soviet Union to visit North America. Our meeting was fortuitous: I am not a mathematician, but rather the wife of one (which is, at times, an art unto itself), and thus had no reason to attend the mathematical lectures. But because my husband, topologist Stephen Watson, was one of the organizers of the conference, Arhangel'skii was one of many renowned set theorists and topologists whom I had the opportunity to meet socially. It was at a small dinner party held in an elegant Thai restaurant in downtown Toronto that I first spoke to Arhangel'skii. The party had been organized by Frank Tall, a sociable professor of mathematics at the University of Toronto, who relishes Oriental food. (Besides being a noted topologist and an editor of the Proceedings of the American Mathematical Society, he is a practitioner of a kind of psychotherapy from California called ``neurolinguistic programming'' which he says involves ``the study of the structure of subjective experience.'') Tall had invited an illustrious group of mathematicians to dine that evening. Seated about the round table with him, Arhangel'skii, and my husband were American set theorist Lee Stanley; Dutch topologist Jan van Mill, who, among other things, has a special interest in ``ultrafilters''; and Mary Ellen Rudin, a professor emeritus at the University of Wisconsin at Madison, who is universally respected by her peers as North America's greatest living set-theoretic topologist. Seated beside Arhangel'skii, I noticed that he was neatly attired in a jacket, dress shirt, and tie. (While there are exceptions to the rule, mathematicians are not generally known for their sartorial flair.) He proved to be an interesting and charming conversationalist whose manner, when speaking to women, could at times be construed as mildly flirtatious. His English was surprisingly fluent and colloquial: he explained that in the 1970s he had spent several years working at the University of Islamabad, in Pakistan, where he had habitually spoken English. An accomplished linguist, he also speaks some French, Italian, and Spanish, and is an avid reader of foreign literature. In the course of the evening, he conversed thoughtfully and eloquently not only about mathematics but also about politics, classical music, literature, and art. I detected in the manner of his speech an unhurried politeness that I couldn't help but associate with a past age. I felt, too, that the list of purchases he hoped to make while visiting Toronto teas, coffees, tobacco, and parfum could have been stolen from the moneybag of a 19th-century gentleman traveller. I was most intrigued by his questions about Canada and Canadians. He wanted to know which native poets Canadians were currently reading; what issues concerned them; in what styles did they write. Who were the local artists currently in fashion? Where could he go to view some recent Canadian art? His mother, he told me in passing, was a painter. I was also intrigued by a brief exchange on mathematics that passed between him and my husband. He wanted to know not only which problems Stephen was in the process of trying to solve but why Stephen had chosen to work on those particular ones. It seemed to me that Arhangel'skii was, at least on the surface, more overtly philosophical than the North American and European mathematicians that I had previously encountered. He seemed to be more the scion of that long line of venerated mathematicians, from the time of the ancient Greeks up until the 20th century, who had also been philosophers. I began to wonder whether he would be able, willing, or even perhaps keen to convey to a general audience some of the central notions, intellectual rigour, and beauty inherent in his chosen field. I found, too, that Arhangel'skii, like many mathematicians, was not without a sense of humour and competitive spirit. At one point during the evening, he mentioned to Dutch topologist Jan van Mill that, while living in Pakistan, he had acquired a taste for very hot curries. Van Mill countered that he too liked very spicy food and often dined in Indonesian restaurants. Overhearing their conversation, Frank Tall called over our waiter and whispered something in his ear. With a quick nod, the waiter scurried off to the kitchen, returning several moments later with two large bowls brimming with small, shiny, deep red peppers, each of them curled like half moons. ``These are the chef's hottest Thai peppers,'' he said. Tall bade the waiter to set down one bowl in front of Arhangel'skii and the other in front of van Mill. He then asked if the two mathematicians would care to partake in a hot-pepper-eating competition: he who finished his bowl first would be declared winner. Rising to the challenge, Arhangel'skii and van Mill turned to look one another squarely in the face, as Tall counted down: ``On your mark. . . . Get set. . . . Go!'' As the men devoured the peppers, their faces turned pale, then several shades of red. At last, with both bowls almost empty and sweat pouring from both men's brows, van Mill conceded defeat. Arhangel'skii, unable to resist a dramatic flourish, popped one last pepper into his mouth. Frank Tall once conveyed to me a piece of conventional wisdom regarding the evaluation of a given work of mathematics by mathematicians. A mathematical work, he held, could be judged fairly on the basis of its depth and its breadth both of which criteria could be applied to either a discussion of a single theorem or the accumulated achievements of a lifetime. The terms require some explanation. When speaking about the ``depth'' of a piece of mathematics, mathematicians are, to some extent, making reference to how far that piece penetrates to the essence of a known problem an essence that may be hidden to all but the finest of minds. They may also be making reference to how far it ventures from known territory into uncharted realms. The term deep, therefore, can be used by mathematicians to describe a mathematical result that is deemed original, and thus most likely difficult. It is, however, quite possible for mathematicians to debate about the level of originality and difficulty inherent in a given result even when the veracity of that result is beyond any doubt. Breadth, on the other hand, is used to say something about the range of mathematical ideas present in a given discovery, or the range of ideas to which it naturally connects. (Note that neither the depth nor the breadth of a mathematical work and here I am referring only to a work of pure mathematics, such as topology, which is pursued for its own sake, as opposed to applied mathematics, which is pursued for the purposes of science is related in any way to its potential or realized practical applications.) Not surprisingly, the work of most great mathematicians exhibits both breadth and depth and this is certainly true of Arhangel'skii's. Over the past three decades, his mathematical discoveries have reflected a broad range of topological concerns, as well as those that stand outside the domain's traditional bounds. Influenced, no doubt, by the struggles of his teacher, Alexandroff, to unify the field, he has shown interest in such varying matters as ``cardinal functions,'' ``mappings,'' ``topological groups,'' and, most recently, ``function spaces.'' The breadth of his mathematical reflection has given him an overall perspective on topology that is rare. It has enabled him to pose problems of relevance and difficulty for other topologists to work on. In doing so, he has wielded considerable influence upon the way in which the field has developed, especially in the former Soviet Union. It is also, perhaps, this breadth of knowledge that has helped him make his deepest mathematical discoveries. Certainly, the depth of Arhangel'skii's thinking has resulted from an idiosyncratic bent that is best described as philosophical. He has won for himself an eternal place in the history of topology, not so much through the resolution of difficult problems (though he has solved his fair share) but rather through the creation of concepts. These concepts such as that of a ``network,'' or the ``tightness'' of a topological space, or the presence of a ``free sequence'' within it have become fundamental to the ways in which topologists think about space. ``Mathematics is, for me, an investigation of the reality of thought,'' Arhangel'skii once said. ``It investigates the mind's thinking, perfects the thinking, creates new modes of thinking. For me, the truth that is revealed by mathematics is, above all, the truth about the working of the mind.'' The word topology is derived from the Greek words topos and logos and means ``the science of place.'' Adapted from the French topologie, the word came into use in English in the 1600s. Its original meaning the branch of botany that deals with the localities of particular plants has fallen into disuse, along with a host of later ones. (In the 1860s, the word primarily referred to the art of assisting the memory by associating the thing to be remembered with some well-known place.) Topology was introduced as a mathematical term by the German mathematician Johann Benedict Listing, whose Vorstudien zur Topologie, published in 1847, was the first systematic treatise on the subject. The branch of mathematics known today as topology divides into two connected but separate divisions: ``algebraic'' topology and Arhangel'skii's area of expertise ``set- theoretic'' topology. Set-theoretic topology comprises a modern, abstract geometry that deals with the construction, classification, and description of the properties of ``topological spaces.'' Its development into a distinct mathematical realm was dependent upon the 19th-century creation of a geometry that was non-Euclidean and the subsequent re- evaluation of the relationship between geometry and the reality of the physical space. While the name geometry is derived from two Greek words ge, meaning ``earth'' or ``ground,'' and metrein, meaning ``to measure'' the science was invented, at some unknown date, by the ancient Egyptians. The architecture of the great pyramids, which were constructed between the 30th and 26th centuries B.C., indicates that their builders possessed some geometric knowledge. As its name suggests, geometry probably arose as a tool for the accurate surveying of land. The writings of Herodotus in the fifth century B.C. and Proclus in the fifth century A.D. recount that flooding of the Nile river made it difficult for royal surveyors, gathering information for the levying of taxes, to measure agricultural lands along its banks. The priests were thus called upon to invent a system of measurement to assist them. Ancient Egyptian surveyors were known as ``rope stretchers,'' because they used ropes in the taking of their measurements. There is considerable evidence that they were aware of some fundamental geometric theorems: it seems they knew, for example, that when three ropes of lengths three, four, and five units respectively are brought together to form a triangle, the particular triangle that is formed is a right-angled triangle. Knowledge of the rise of mathematics in Egypt, gleaned from Greek sources, was corroborated and supplemented by the information brought to light by the Rhind Papyrus: a scroll named after the young Scottish antiquary, A. Henry Rhind, who purchased it in Luxor in 1858. It was written around 1700 B.C. by a scribe named Aahmes and rather mysteriously entitled Directions for Knowing All Dark Things. The science of geometry was developed further by the classical Greeks. From a variety of sources, scholars date the beginnings of Greek mathematics (and philosophy) at roughly 600 B.C. Both disciplines developed, simultaneously, in several successive centres where the civilization of the Greeks had established itself. At each centre, a group of scholars gathered about one or two influential leaders to form an informal or formal school. The first of these was founded by Thales (c.640 - c.546 B.C.) in the wealthy Mediterranean port city of Miletus, Ionia. The second school of Greek mathematical thought was founded by Pythagoras (who may have been a student of Thales) in Croton, southern Italy, in the sixth century B.C. The Pythagoreans are credited with having discovered many geometric theorems, concerning triangles, regular polyhedra, polygons, circles, spheres, etc. The geometric discoveries of the Greeks were compiled and recorded by Euclid in the Elements. Unfortunately, all that is known of Euclid is that he lived and taught in Alexandria, Egypt, around 300 B.C. Unfortunately, too, no original manuscript of his famous treatise has survived; standard English versions, such as T.L. Heath's The Thirteen Books of Euclid's ``Elements'' (1908), are derived from Arab and Latin translations. Nevertheless, no other work has had greater influence than the Elements on mathematical and scientific thought. Sir Isaac Newton, for example, patterned his Philosophiae Naturalis Principia Mathematica after it. In setting down the Elements, Euclid employed, in its nascent form, the axiomatic method. Probably invented by the Pythagoreans, the axiomatic method manifests the Greeks' unparalleled recognition of the power of deductive reasoning to discover truth. It remains the means by which logicians of all stripes, be they mathematicians or philosophers, formulate their ideas. It consists of setting forth certain statements, which are believed but not demonstrated to be true, then inferring other statements from the first, in a logical chain. (Bertrand Russell argued that the axiomatic method was the essence of mathematics. In his 1929 essay ``Mathematics and the Metaphysicians'' he wrote, ``Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing.'') Euclid had indubitably looked to Aristotle's writings for guidance on the mode of deductive reasoning. Aristotle had argued persuasively that ``every demonstrative science must start from indemonstrable principles; otherwise, the steps of demonstration would be endless.'' He had divided these indemonstrable principles into (a) axioms, which are common to all sciences, and (b) postulates, which are particular to a specific subject such as geometry. Euclid thus began the Elements with a set of postulates and axioms (which he called common notions), as well as a set of definitions. The first five of each of these three sets reads as follows: Definitions 1. A point is that which has no part. 2. A line is breadthless length. 3. The extremities of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. Postulates 1. It is possible to draw a straight line from any point to any point. 2. It is possible to extend a finite straight line continuously in a straight line. 3. It is possible to describe a circle with any centre and distance. 4. That all right angles are equal to one another. 5. That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. Common Notions 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. Based upon his set of axioms and postulates, Euclid deduced the truth of further propositions, called theorems. Many of these concern the relationships between various distances and angles embodied in abstract, ideal figures. The most famous theorem among them, known as the Pythagorean theorem, states that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. For more than 2,000 years, mathematicians held the truth of Euclidean geometry to be absolute and incontrovertible. The theorems put forward in the Elements were believed to be ``certain'' and ``necessary,'' because they were demonstrated to be true without appeal to empirical fact. (It is the deductive, as opposed to inductive, nature of mathematical thinking that inspires trust in the validity of mathematical discoveries and accounts for the historic acclaim of mathematics as ``the queen of sciences.'' The findings of all other sciences derived from testing the accuracy of reasoned hypotheses through careful experimentation and observation do not, and cannot, elicit the same faith. A scientific hypothesis can be regarded as the truth only within a high degree of probability, on the basis of available evidence. A scientist must accept that, no matter how many tests he conducts, any given hypothesis may have to be thrown out in the light of new evidence. A mathematician, however, can feel confident that once he has proven the truth of a given hypothesis, it will stand as proven, once and for all.) The geometric theorems of Euclid's Elements the truths of which were established without address to experiential evidence were imbued with that particular ``mathematical certainty,'' and, for this reason, esteemed. Paradoxically, non-empirical Euclidean geometry was also believed to describe and reflect the properties of physical space. The roots of this self-contradictory posture are profound, bedded in the interlocking history of Greek philosophy and mathematics (in which the discovery of geometry plays a crucial part). Greek philosophic inquiry began in the seventh century B.C. with questions regarding the physical world. Filled with wonder, the first Greek philosophers (who were also the civilization's first mathematicians), Thales of Miletus and his pupils Anaximander and Anaximenes, asked themselves: What are things really like? How can we explain the process of change in things? How can we distinguish what things seem to be, from what they are, between their ``appearance'' and their ``reality''? They put forth their replies in the belief that nature was patterned by a coherent design that could be apprehended by the reasoning mind. By the fifth century B.C., the Pythagoreans recognized that nature's design was mathematical, that mathematics the truths of which were discovered by means of deductive reasoning was therefore the key to unlocking her secrets. The importance of mathematics was thus linked, from one of its earliest inceptions, to its applicability in the search to comprehend the physical world. Furthermore, Euclidean geometry had its specific origins in the generalization and systematization of ``real-life'' discoveries made in connection with the practice of surveying, the physical measurement of areas and volumes, and the development of astronomy. In the 18th century, Immanuel Kant, the philosopher of the enlightenment, contemplated the truths of Euclidean geometry and their relationship to physical space. Kant, whose major work is the Critique of Pure Reason (1781), held that the true task of philosophy was the critical appraisal of the capacities of human reason. Rejecting the arguments of the radical empiricist David Hume that all our knowledge consists of a series of impressions which we derive through our senses, he maintained, in a somewhat paradoxical fashion, that ``though our knowledge begins with experience, it does not follow that it all arises out of experience.'' He called this knowledge, which arises out of our faculty of rational judgment, a priori; and, to provide an example of a priori knowledge, he offered up ``any proposition in mathematics.'' Kant then asked how is it possible that such a priori judgments, which do not arise out of experience, can apply to the external world of sense-perceptions, the world of science. His answer was revolutionary: a priori judgments of the mind do not conform to external reality; the mind, he argued, is so structured that it acts upon the world of sense-perceptions, imposing itself, controlling and organizing, so that the world of sense-perceptions conforms to the mind. Kant also maintained that the mind possesses, innately, what he called ``forms of intuition,'' through which the mind acts upon the objects of the world of sense-perceptions. Among these forms of intuition were space and time. Thus we inevitably perceive the objects of the world as existing in space and time. He argued, too, that the mind's ``form of space'' was Euclidean. Despite Kant's influential arguments, and in the face of age-old beliefs, mathematicians began to question the relationship between the truths of Euclidean geometry and the properties of physical space. Their doubts were raised, in the 19th century, by the founding of non-Euclidean geometry which is considered by historians to be the most consequential and revolutionary step in the development of mathematics since Greek times. Due west of the Kremlin, beyond the ring road that separates Moscow's inner and outer cities, the east-west Kalinina Prospekt bends slightly to the south, heading across the River Moskva, just where it snakes around the gargantuan Hotel Ukraine. Stretching away from the Kalinina Bridge, on the western bank of the river, the avenue continues, now called Kutuzovski Prospekt. It was given this name in 1957 in honour of Mikhail Illarionovich Kutuzov, who led the Russian forces against Napoleon in the War of 1812. Following the course of the ancient road to Smolensk, Kutuzovski Prospect veers gently toward a rise of land called the Poklonnaya Gora; the Russian name translates as ``the hill you bow to,'' recalling a custom once observed by travellers arriving in Moscow from this direction. It was on this height, which years ago commanded a view of the city, that Napoleon stood, waiting to receive official word of Moscow's surrender. The building of Kutuzovski Prospekt began in the 1930s, entailing the destruction of smaller streets of old wooden houses, and was not completed until the early 1960s. Sixteen lanes wide, its scale is emblematic of the grand proportions (some might think monstrously grand) of modern Moscow, planned and constructed during Stalin's era. It is, to some extent, Moscow's Champs Elysees, imposing and attractive, lined on both sides by large apartment houses built in a neo-classical style. Mainly six to 12 storeys high, and stretching the length of a longer-than-average city block, they look like high-rises tipped on their sides. Their monolithic facades are broken up by large windows, balustrades, and decorative reliefs; stuccoed pastel shades of peach and gold, they grow more roseate when struck by the sun. Ground floors are occupied by shops, purveying bread, milk, fruits, vegetables, meats, fish, canned goods, and clothes. (They were, when I was in Moscow, well-stocked one day and near empty the next, depending upon the vicissitudes of an economic system in the process of change.) Babushkas, who have come into the city from nearby villages, sit out on the wide, tree-lined sidewalks, wearing flowered scarves and printed dresses. They totter on small stools before their baskets of tomatoes, cucumbers, and dill, on sale at black-market rates. Alexander Vladimirovich Arhangel'skii lives on Kutuzovski Prospekt with his wife, Olga Constantinovna, and their 27-year-old son, Vladimir Alexandrovich. (The Arhangel'skiis also have a 31-year-old daughter, Tatiana Alexanderovna; she is married and lives in Moscow with her husband and eight-year-old son.) Arhangel'skii enjoys its relative proximity to M.S.U. He also appreciates the fact that from the Kutuzovskaya Metro Station, near the western end of Kutuzovski Prospekt, it is only a two-stop subway ride to the Kievskaya Station. From there, he can board an Elektrishke, an electrical commuter train, and, in less than a half an hour, be walking (or, in winter, cross-country skiing) through a forest of birch. It is not much farther from the Kutuzovskaya station to the Kurski station, from where he can catch another train, and, in roughly an hour's time, arrive in the village of Saltykovskaya, where his mother resides. ``It is an important street,'' Alexander Vladimirovich told me as we drove down Kutuzovski Prospekt one time. Its historic name and associations obviously pleased him. His flat is situated on the top floor of a six-storeyed apartment house. Below average height and lacking balconies, it's slightly less grandiose than many of the other apartments on the street. Its distinguishing feature is a small grove of birch and poplar trees growing out front in a sandy patch of earth and shading its neo-classical facade. Alexander Vladimirovich likes these trees. He finds them not only beautiful but a source of pride. ``They are a sign that this is a special place,'' he said. ``In the old days, during festivals, people would hang lamps on their branches.'' He also pointed out that the only other apartment house on the street to be graced by such trees was that standing almost directly across from his own, which housed both Brezhnev and Andropov during their terms of power. During my two-week stay in Moscow, in July of 1991, Arhangel'skii and his family graciously allowed me to stay in their home, affording me an intimate view of their lives that was unexpected and delightful. Arhangel'skii was a charming, accommodating, thoughtful, and proud host. He went to some effort to meet my flight which, landed in the early evening at Sheremetyevo, Moscow's international airport. His car, a white 1980s Volga, had been in a state of disrepair for several months. (Spare parts, he later explained, were available only on the black market at prices he either couldn't afford or refused to pay.) So he convinced a colleague to make the 45-minute drive to the airport from the city centre. Standing in Sheremetyevo's dingy arrival hall, Arhangel'skii stood out, tall and erect, among a crowd of his compatriots. He looked dapper in a tweed cap, khaki-green Gore-Tex jacket, jeans, and walking shoes all of which he had purchased on a recent trip to Oxford, England. On his wrist flashed a Swatch watch. He greeted me with a broad smile and firm handshake, and introduced me to V.V. Tkachuk, a former pupil and professor at Moscow State University. He then politely insisted upon carrying my bags. ``I am sorry it is gray for your arrival,'' he fretted. ``It has been sunny and warm for two weeks straight. I hope this bad weather will not continue for too long.'' We left the airport, weaving in and out of traffic, on a rain-slicked eight-laned highway, along which stood shimmering birch trees, interrupted by the odd billboard advertising Goldstar Electronics. After several minutes, Arhangel'skii pointed out a gigantic monument built of World War II tank traps, saying it marked the spot where the German army was halted in its advance toward Moscow in October 1941. Having passed through the city's ever-sprawling suburbs of high-rise towers, we reached the ring road that encircles Moscow's centre. Arhangel'skii commented that officials had rechristened many streets, returning to them their pre-revolutionary names. ``Soon the tourist will require a map from the age of Czar Nicholas!'' he said. To reach his flat, the visitor must first exit off Kutuzovski Prospect, through a square portal large enough to accommodate a car, and enter a courtyard. In the centre is a treed, fenced-in, sandy park, with benches, swings, and slides most of them in need of repair and a fresh coat of paint. (On fine days, people hang their laundry to dry on lines strung between the trees.) Leading off the courtyard, an entranceway opens into a small, damp, dark passage, lined with cracked white and red tiles, leading to a gloomy, circular stairwell and grilled elevator shaft. Arhangel'skii's apartment is on the sixth and final floor. The heavy, padded door is double locked: Muscovites have started to worry about break-ins. Inside, Arhangel'skii's wife and son are waiting to greet us. Olga Constantinovna shyly shakes hands. She is a short, stout, no-nonsense, but kindly-looking woman with brown, grey-streaked hair, clipped boyishly short; a rounded, slightly Oriental face with gently slanted eyes; a flattened nose; and a gold-toothed smile. She calls her husband ``Shurie'' or ``Shurika,'' and speaks to him in a high-pitched, sing-song voice. She too pursued mathematics at Moscow State University and was awarded with a kandidat nauk. She teaches at an institute devoted to the training of high-school teachers. Arhangel'skii 's son, Vladimir Alexanderovich, or ``Volodya,''as his father calls him, is a strapping, big-boned young man with dark brown hair parted in the middle, reaching past his shoulders. He is dressed all in black: a black T-shirt, black corduroy pants, a black bandanna across his forehead. Olga Constantinovna turns to her husband and speaks quietly with him in Russian for several moments. (While she is able to understand simple conversational English, she is not familiar enough with the language to speak it.) He explains that his wife had come home not long before our arrival and wishes to excuse herself to finish cooking dinner. She promptly disappears through a side door. Arhangel'skii then tells me to leave my bags for now in the entrance hall. He beckons me to change out of my shoes into slippers, which he provides from a pile just inside the front door, and he and his son lead me deeper into the apartment. Inside, Arhangel'skii's apartment exudes a dusty, imperial charm, a faded glamour, romantic and sad. It is larger than the average Moscow flat (one room, plus kitchen and bathroom) that many families must remain content to live in. It has a lovely layout: four rooms a number of them serving several purposes plus a big kitchen and bathroom, lying off a central entrance hall. The rooms are graced by 15-foot moulded ceilings, the paint peeling in places; aged silken wallpapers; broad-stripped herringbone parquet floors, in need of a new wood finish and wax; and tall windows, their glass slightly grimy, facing east and west. Throughout, the apartment is furnished with a capricious, cluttered mix of things: old and new, exotic and native, precious and worthless, inherited and acquired. Arhangel'skii ushers me into the apartment's most spacious room, serving as living room, dining room, and guest bedroom, and leaves me there with Volodya while he goes off in search of a bottle of Georgian champagne. The room's furnishings include several glass-fronted bookcases, an assortment of wooden and upholstered chairs, a table, and two divans the newest of which Arhangel'skii says was manufactured in Romania and, though purchased less than two years ago, has already collapsed. Among the room's smaller treasures are an end-table with a carved wooden base and hand-worked brass top, bought in Pakistan; a turn-of-the-century inkwell of wood and copper encrusted with agate and lapis lazuli; and numerous framed oil paintings: landscapes and family portraits. I chat with Volodya, whose English, acquired at school, is heavily accented but comprehensible. Looking and acting the part of the rebel son, he seems to be in the process of ``trying to find himself.'' He recently attempted to set up a business distributing videotapes of American films in Moscow, but the venture fell through. He is an avid fan of Sylvester Stallone, whom he describes as ``a beautiful actor.'' He also admires Anatoly Sobchak, the radical mayor of St. Petersburg, and Boris N. Yeltsin, then president of the Russian Federated Republic. (In August of this same year, during the attempted coup against Mikhail Gorbachev, Vladimir Alexandrovich stands guard, day and night, with the small band of men defending the Russian parliament building. Arhangel'skii, who has been away from Moscow during the coup, calls to tell me of his son's heroism. ``Perhaps Volodya was born just for this moment,'' he says.) Arhangel'skii returns with the champagne, which he pours into three crystal glasses. ``Let us toast your visit,'' he says. Olga calls out from the kitchen and Arhangel'skii informs me that dinner will be served shortly. He and his son lift up a long wooden table, standing below the room's west-facing windows, and move it into the centre. He covers it with a white linen cloth, and then puts down bottles of champagne, wine, beer, cognac, and, of course, vodka. (I wonder if there will be any room left for the food.) He lays out old, gold-trimmed, cobalt-blue and white china, silverware, and sets of crystal glasses. Olga carries in the food: fresh breads, both white and brown; plates of sliced cucumbers, salamis, and smoked fish; a bowl of steaming, boiled potatoes; and kabachok, a peppery stew of carrots and vegetable marrow smothered in sunflower oil. ``In Russia,'' Arhangel'skii says as we sit down to eat, ``our shops may be empty, but our fridges are full.'' A.V. Arhangel'skii rises at nine o'clock each morning ten if he has taken more vodka than he is accustomed to the evening before. He does some calisthenics, then washes and shaves. (In Moscow, morning ablutions can demand a little more time than one might expect. Each summer, the authorities turn off the city's supply of hot water in order to overhaul the system; the repair period usually lasts one month, possibly two. At such times the Arhangel'skiis set three large, mineral-encrusted pots of water to boil each morning on the stove, carrying them into the bathroom as needed.) After dressing and, perhaps, splashing his freshly-shaven face with cologne, Arhangel'skii heads into the kitchen for breakfast. On summer mornings, the wondrously cluttered kitchen, facing east toward the courtyard, is flooded with light. It is furnished with a gas stove, two refrigerators, a round table with plastic cloth, wooden chairs and stools. Its walls are covered with a green-and-white striped paper, peeling in places and splattered with age-old grease. Every available surface window sills, counter tops, cupboard-tops, refrigerator-tops, much of the green linoleum floor is smothered and weighted down. Dusty stacks of mathematical books and articles are piled up to the ceiling. Empty tins and canisters are strewn about, as well as lidless glass pickle jars. The rest of the available space is occupied by dozens of empty, sticky liquor bottles: I counted 40 vodka bottles on the three window sills alone. ``If you return these bottles for a deposit, they will give you fifty kopecks, which is not a bad amount,'' Arhangel'skii explains one morning. ``However, sometimes you might go to the store for a bottle of vodka, and they won't give you one unless you present them with an empty. I can solve many difficult math problems, but what to do with these bottles is a problem I cannot solve.'' Arhangel'skii's breakfast is hearty, ritualistic, and languidly paced. ``I move slowly in the morning,'' he admits. He begins by making coffee grinding beans in an electric grinder, then carefully heating the fresh grounds in an ancient hand-engraved, copper dgezava. In traditionally Turkish fashion, he heats the coffee just to the boiling point, until it froths above the utensil's copper lip. He drinks one or two strong demi-tasse cups, then gets up from the kitchen table to prepare tea, which he does equally fastidiously, using a strikingly decorated ceramic pot purchased by his father at an industrial arts exhibition in 1939 and treasured ever since. Then it is time to think about food. Arhangel'skii loads the kitchen table with an earthy smorgasbord: plates of sliced tomatoes, cucumbers, salami; sheep's milk cheese from Bulgaria, sour cream, and curds; smoked-fish pate; cheese buns and, of course, dark, chewy rye. He will now spend an hour or two easing himself into the new day, slowly digesting the meal and the morning paper. It will then be time to head into his study to think about mathematics. ``I cannot resist the temptation to do mathematics,'' he says. ``It is an obsession, perhaps a disease. Sometimes I feel that I should spend more time with my family, but mathematics draws me away. There have been times when I haven't done any math for six months, but then I have been completely drawn into it again. I cannot calculate when those periods of activity or inactivity will be. ``It used to be, when I was younger and I had many duties, I had to look for the time for mathematics, so I had a plan to sit down every day at eleven o'clock, whether or not I had an idea, or even knew what I was going to think about. I discovered, when I sat down in the quiet, ideas would come. When I was younger, I also found that I could work anywhere: on the street, on the bus. I remember I was sitting on a bus, talking to a colleague about something unimportant, when I suddenly got my idea for a `p-space.' ``Now I don't have so many duties, so I don't have to struggle to find time for mathematics. But I am very driven by my desire, so I don't get distracted.'' During the school year, which runs from September to mid-June, Arhangel'skii's teaching duties take him to the campus of Moscow State University several times a week. He leads two seminars: the first, for his graduate students, takes place at six o'clock on Monday nights; while formally scheduled to last two hours, it often stretches into three, possibly four. The second, a higher-research-level seminar, is reserved for members of the Chair of General Topology and Geometry. It is scheduled to take place on Thursdays from four p.m. to six. On top of this, he may be asked to deliver a first-year unified course on linear algebra and analytic geometry, or a beginner's course in topology. These usually demand four hours per week. ``I sometimes like to give these undergraduate lectures, because I might directly or indirectly inspire students to study topology,'' he says. Generally, however, undergraduate-level courses at Moscow State University are taught by docents. Except for the hours that he is expected to lecture, and the time it takes to prepare for them, Arhangel'skii's time is free to conduct his own research. As is the case for all mathematics professors employed by M.S.U., he has not been allocated a private office there and thus must work in the library or at home. ``Maybe this is a problem,'' Arhangel'skii says. ``But when P.S. Alexandroff and A.N. Kolmogorov set up the department, they did not think it was important for us to have private offices. They believed that we could think while walking about the corridors, as monks do walking about quadrangles.'' Arhangel'skii's study, the most beautifully appointed and traditional room in the apartment, is the primary locus of his mathematical thinking. It is decorated with a gold wallpaper, patterned with stars, and with three utterly different Oriental carpets: one, in reds and blues, on the parquet floor; another, geometrically patterned in reds, black, and white, thrown, according to Russian custom, over the settee; and a third, in shades of gold and brown with touches of red and black, hanging on the nothern wall. These dizzily contrasting patterns give the study the atmosphere of a mosque. An old square wooden desk with a deeply scratched top stands by a wall of windows that look out eastwards on a view of birch trees and spires. Against the southern wall leans a lacquered bookcase, its lower shelves crammed with books and journals on mathematics, most of them in Russian, a few in English. The upper shelves are lined with literature, mainly American and French classics. Madame Bovary, Le Rouge et Le Noir, and several volumes of A la Recherche du Temps Perdu stand side by side with Life on the Mississippi and For Whom The Bell Tolls. There are also numerous language dictionaries: English, French, Italian, Spanish, and Portuguese. On the same wall hangs a collection of photographs and paintings, the latter the work of Arhangel'skii's maternal grandfather, Pavel Alexandrovich Radimov. Among the photographs is a round-framed, sepia-toned portrait of an aristocratic- looking woman with narrow eyes, aquiline nose, fine lips, and dark hair combed loosely back in a bun. This was Arhangel'skii's paternal grandmother, Anastasi Yevgrafovna Arhangel'skaya. It was she who commissioned the magnificent turn-of-the-century chandelier of bronze, crystal, and cobalt-blue glass that illuminates the room; and she, too, who began the education of its present occupant. ``She was a teacher,'' says Arhangel'skii, leaning back in the settee. ``When I was four or five, she taught me how to read and write. She also taught me a bit of German. Mostly she was reading to me, religious stories from the Bible. She was a religious woman; she took me to church when I was young she even had me baptized. My father also taught me when I was young. He gave me mathematical problems to solve, merchant's problems [word problems]. He also gave me geometry problems. He cut out paper triangles and asked me which ones were congruent. My father had a passion for mathematics. ``I think that it was always in my own character to be a mathematician. I believe I was always interested in this striving for the perfection of thought. I do remember clearly that, from a very young age, eight or nine, perhaps, I began to ask myself philosophical questions, which rightly belong to the study of the foundations of mathematics. I remember, for example, asking myself why the answer to 2 + 3 is the same as the answer to 3 + 2. Not many young boys would ask themselves why.'' A.V. Arhangel'skii was born on March 13, 1938, in an old hospital which still stands on Kalinskii Prospect near Arbat Street. His father, Vladimir Alexandrovich Arhangel'skii, was a concert pianist who had begun his career as an aeronautical engineer, rising to become director of the prestigious Central Aerohydro Dynamical Institute in Moscow. ``He told me that if he had studied to be a mathematician,'' recalls Arhangel'skii, ``he would not have switched careers.'' His mother, Marija Pavlova Radimova, was a celebrated painter. His godmother (and his father's first wife), Marija Alexandrovna, was the daughter of the famous Russian composer Scriabin. During his mother's pregnancy, Arhangel'skii's father was arrested and accused of being a German spy. ``My father had travelled a lot during the 1930s. He went to Germany, Austria, maybe even France, playing concerts on the piano. He also travelled all over the Soviet Union, even to the very far east. He wanted to play for simple people. He put his Steinway in the back of a truck, stabilizing it with special shock absorbers he designed himself, so he could drive on very bad roads. ``He was arrested in Khabarovsk [a city in the far east of the Soviet Union, on the Amur River, just north of the border with China] and held for some months. The security police were suspicious of this Russian concert pianist, travelling by himself so far from Moscow, with all these foreign stamps in his passport.'' Vladimir Alexandrovich was lucky to be released. ``It just so happened that around the time of my father's arrest, my uncle, Alexander Alexandrovich, who had become an important aviator, was being awarded a prestigious prize for the design of a Russian fighter-bomber. Stalin even spoke with him personally to congratulate him. During their phone conversation, Stalin evidently asked my uncle if there was anything he could do for him. For a moment, my uncle wondered whether or not he should tell Stalin about my father's arrest and ask for his release. My uncle decided that he could not predict how Stalin would react, and so never said a word. My uncle did tell his superior, the chief of the Ministry of Aviation, who promised to investigate. He found out that the decision to arrest my father had been made only by the local authorities in Khabarovsk. These authorities were not following any orders from Moscow. So my uncle's superior went to see a minister of law, and not long after my father was set free.'' When Germany declared war on the Soviet Union in June of 1941, A.V. Arhangel'skii's parents feared that Moscow might become dangerous; and so in October of that year they sent him off to live with Alexander Alexandrovich, who had already been evacuated, along with his airplane factory, to Omsk, in the heart of Siberia. On the western wall of his study, tucked between the door frame and a Romanian-made cabinet, hangs a black-and-white photo of the boy and his uncle taken during that period of exile. Both are posed with their legs apart, hands thrust into their pockets. The boy wears a cap, giving him an oddly adult look. ``My uncle always loved this photo,'' Arhangel'skii says. ``He had no children of his own and so we were very close to one another. We remained close even when I reached adulthood, partly because he lived twenty years longer than my father.'' Arhangel'skii returned to Moscow sometime after his fifth birthday. From then on, he was raised in a fashionable, 1930s, high-ceilinged, two-roomed apartment: number 54, 5/7 Nemirovich-Dunchenko, centrally located between Red Square and Pushkin Square, off Gorky Street. He lived in this same apartment for 42 years, even after he married and had children of his own right up until the time that he and his family moved to Kutuzovskii Prospekt by means of a private trade. Thirteen storeys high, the apartment house used to be considered a skyscraper, and was known throughout Moscow as the ``Artist's House.'' ``Many famous Soviet artists lived there,'' Arhangel'skii says. ``The famous theatre producer Nemirovich Dunchenko, after whom the street was named, lived two floors below us. So did the famous actor Shtrauch, who played the role of Lenin in many movies. The widow of Chekhov lived just through the walls.'' Arhangel'skii lived on Nemirovich-Dunchenko alone with his father; for by the time he was four, his parents had parted ways. ``Mother wanted to keep me, but she thought my father's life was better organized than her own. She also thought that the schools were better in the centre of Moscow than where she was living and working as an artist in the village of Saltykovska, and no doubt she was right.'' Not long after Arhangel'skii's parents divorced, his mother met and married the painter Constantin Pavelovich Radimov. For the most part, young Alexander Vladimirovich was content living in Moscow with his father. ``I know that, at some moments, I would have liked to live with my mother,'' he says. ``My father had a temper and he sometimes got very angry with me. He wanted me to do my duties in school. But I sometimes wasn't as well organized and disciplined as he wanted me to be. I didn't want to do my homework; I wanted to play soccer and volleyball in the courtyard. I wanted to read books by Walter Scott. He would yell at me for my bad marks. At those moments, I wanted to go and live with Mother. But I would see her often. She would come to Moscow to visit me every week or two. In 1953, Vladimir Alexandrovitch Arhangel'skii married a young pianist, Varvara Gryaznaya, with whom he had two children. ``My half-sister lives in Moscow,'' A.V. Arhangel'skii says, ``My half-brother did, but he moved away. I don't see them very much, not because of any ill feelings, but because of lack of time. In fact, I miss my half-brother. My own mother had no other children but me.'' V.A. Arhangel'skii died in 1958, one year after his third wife. ``I think that living with my father made me self-sufficient at a very young age,'' Arhangel'skii says after a pause. While contemplating, he delicately bites his right fingernails and gently chews his upper lip. ``I am not saying that I was well-organized, just self-sufficient. At six, I would go to my grandmother's house all by myself. She lived near Arbat and I had to travel through all Moscow to reach her house. I remember, also, that the primary school that I went to was a little distance from our apartment. The school had no name, it was just called School Number 135. It was across Gorky Street, which had a lot of traffic. Crossing this street was a bit of an experience. I remember I was always late and would cross it in a sportive manner. I was proud of this.'' Listening to him speak, I try to imagine the life of V.A. and A.V. alone together in the Artist's House on Nemirovich-Dunchenko. I imagine the father, seated at his Steinway grand piano, practising concertos in one room, while the son sits at his desk in another, day-dreaming over his school books. I begin to wonder if the father imparted to the son his love of classical music; if it was being reared in a house filled with music that would later influence the son to relate the moment of his discovering a mathematical truth to being overwhelmed by ``a feeling of harmony.'' Arhangel'skii's memories, however, present a somewhat less romantic picture. ``When I was young, music students were coming to the house all the time to be taught by my father. They would practise scales, or parts of a piece, over and over again. Actually, I found this annoying to listen to. I did develop a love of classical music, much later, that was my own. I was influenced by my teacher P.S. Alexandroff. He loved classical music. Periodically, perhaps once a month, he gave talks on a chosen piece of music for any members of the Chair of Higher Geometry and Topology at Moscow State University who wanted to listen. These took place in the lounge of a campus dormitory. After a talk, he would play a recording of the music he spoke about for us to hear. He also would invite his close students to come to his house to listen to music.'' At this mention of P.S. Alexandroff, I turn the discussion back to mathematics and ask Arhangel'skii to talk about his formal education. ``I remember that I always found geometry more satisfying than algebra,'' he recalls. ``Geometry is logical, rather than algorithmic. I liked that, when we started to learn geometry in school, we began with concepts and assumptions that seemed clear. I also liked that it wasn't experimental, like physics. For example, I remember that when we started to study electricity, it seemed that we did not really understand what we were speaking about. The study of geometry seemed to present the correct way of thinking, and it was a pleasure for me.'' The Soviet public school system, into which young Alexander Vladimirovich entered was, and remains, structured in the following fashion. From the ages of four to six, a child could, if his or her parents wished and there was enough space, attend kindergarten. At seven, he began his primary education, which consisted of the first three or four grades, or ``forms.'' The fourth or fifth form, depending on the region, marked the start of his secondary education, which was subdivided into two consecutive parts, the second of which was optional and varied. The first four to five years constituted the ``secondary incomplete'' program that all Soviet children were required to finish. After its completion, however, most furthered their education with two more years of secondary study. Doing so, they could attend a technical or vocational school, or they could attend an academic school and, in this case, finish the ``secondary complete'' program. ``By the time I started my secondary-school education, I was interested in subjects other than mathematics,'' Arhangel'skii admits. ``I liked literature, very much. I also liked to write compositions. I remember that one of my teachers was urging me to study humanities. By then I had also developed some liking of physics, as well as biology. But only girls went into biology in those days. I had even wanted to become a movie director for a while, but this was hard to achieve! In the end, I decided to pursue mathematics.'' Arhangel'skii entered Moscow State University in 1954. Then, as now, only a chosen few were accepted into its undergraduate mathematics program. The application process has also remained unchanged. An applicant, called an abiturient, must apply to the Faculty of Mechanics and Mathematics, or Mekhmat, and must enter a nation-wide competition that entails the submission of grades and the taking of four entry examinations: two mathematics exams (one written and one oral), one physics exam (oral), and one Russian composition exam (written). Each year, roughly 275 students are admitted into the mathematics, and 175 into the mechanics (mathematical physics) program, selected from candidates from across the Soviet Union. The undergraduate mathematics program has always been rigorous. There is, at present, no three- or four-year bachelor's-degree program in place. (The administration is, however, planning to introduce a four-year undergraduate-studies program.) All students enroll for at least five years and work towards their diplom. During the first two years, they are required to attend six to eight hours of scheduled lectures and exercise sessions, six days a week. On top of this, they are expected to work on assigned problems at home. In the third year, mandatory class hours are reduced somewhat, and, in each of the following years, they are reduced again. By the fourth year, however, the better students will try to produce a piece of original mathematical research. ``It is not necessary for a student to do so,'' Arhangel'skii explains, ``but every student will try, and it is by judging this work that we can decide whether or not to recommend that a student go on for their Ph.D. If a boy is really good, he will, in the fourth year, get a simple, original result. By the fifth year, he will produce something worth publishing.'' All undergraduate students must pass an oral exam covering the fundamentals of higher mathematics to attain their diplom. According to official procedures, students wishing to continue studying mathematics must work for two or three years at a scientific institute or an industrial establishment, to which they are assigned by the Ministry of Higher Education, before they may apply to enter a graduate program. A university's Academic Council can, however, (and did in Arhangel'skii's case) recommend that a given student be allowed to apply for graduate studies immediately after attaining the diplom. Each year, roughly 80 students of mathematics are permitted to enter the program at Moscow State University. Graduate students, called aspirants, must study for at least three years in order to receive their kandidat nauk. During this time, they attend seminars, work on a dissertation, and prepare for exams. There are two final stages in the process of obtaining the degree of kandidat nauk: first, a student must defend his dissertation (which must, at this point, have been published in a Russian journal of mathematics) before an academic council; second, a central government body, called the Vysshaya Attestatsionnaya Kommissiya, or VAK, must approve the degree. A.V. Arhangel'skii was awarded the degree of kandidat nauk in June of 1966. It was due, in part, to P.S. Alexandroff that he had decided to specialize in topology. ``P.S. Alexandroff influenced me a lot. He had an impressive, charismatic personality. He was a humanist as well as a mathematician, with a gift for writing and speaking. During my first year as a student at Moscow University, he taught a course in analytic geometry. It was a required first-year course, and so I took it. Alexandroff, I remember, was an interesting lecturer, not always brilliant he would sometimes make a mess of the blackboard! but always emotional. He was also very attentive and polite when speaking with students after class. He would shake your hand and talk very kindly.'' Arhangel'skii's interest in topology was also inspired by an introductory seminar on the theory of functions that he attended in his first year. It was given by the blind mathematician Vytushkin. ``He began introducing the study of `functions of a real variable,' `subsets of the real line and of the plane,' and some `set theory' [all of which provide the necessary background for the study of topology]. He gave us many problems, which we had to solve on our own, without referring to any books. ``I actually found this seminar very difficult and did not attend much. But I had my inclination for the foundations of mathematics. I was interested in the notion of `the cardinality of infinities,' of the idea of `countable' and `uncountable sets,' of the `convergence of functions.' All these ideas were being discussed in this seminar. Though I did not attend much, I followed what was being done through discussions with my friends on the bus.'' (Vytushkin's method of teaching mathematics encouraging students to solve problems independently, without the aid of textbooks was pioneered in Russia by Nikolai Nikolaevich Luzin, a founder of what is known locally as ``The Moscow School of Mathematics.'' Luzin was born in 1883 in Siberia, the son of a merchant and grandson of a serf of Count Stroganoff (after whom the gourmet dish was named). He studied mathematics at Moscow University under Boleslav Kornelievich Mlodzeyevskii (who, in 1901, taught the university's first course on Georg Cantor's revolutionary ``theory of sets'') and Dmitrii Fedorovich Egorov. In 1917, he began teaching at Moscow University, giving a course on the theory of functions and the theory of sets. Luzin possessed an enthusiastic and charismatic character; his personal magnetism and the liveliness and theatricality of his lectures attracted an ardent group of brilliant young students including P.S. Urysohn and P.S. Alexandroff, founders of the Russian school of topology. Bonded together by their devotion to Luzin, his students formed a secret society, which they called ``The Order of Luzitania,'' in ironic reference to the American ship Lusitania, sunk by the Germans during World War I.) At the end of Arhangel'skii's first year at university, P.S. Alexandroff began to conduct an introductory seminar in set-theoretic topology. ``This was not usual,'' he explains, ``for most seminars began in September, not in April or May. But Alexandroff had just returned to working in the area after a break. I came to Alexandroff, at the end of the first session, and told him I wanted to specialize in topology. I told him what I had already learned about functions and set theory from following Vytushkin's seminar on my own. At that time, Aleksei Serpionovich Parhomenko was an assistant professor in topology. Alexandroff went to Parhomenko, told him of my interest in topology, and asked him to test my knowledge. Parhomenko met with me, and asked me if I could provide him with a set which is neither open or closed. And so I did. Parhomenko then went back to Alexandroff and told him, `Okay, this boy is serious.' ``In September of 1955, Alexandroff's seminar in topology began again. There were only ten students left out of fifty. I enjoyed it immensely, and was treated in this seminar as one who is well advanced. I liked that one's knowledge of topology proceeded from fundamental principles, and not a mixture of structures. Some people like to pull together a combination of things to make a system. But I am not a quick thinker, so I don't like to work with too many instruments at once.'' In his book Mathematical Thought from Ancient to Modern Times, American mathematician Morris Kline notes that no branch of mathematics, or even a major result, has arisen from the work of one man; at best, some decisive step may be credited to a single individual. The cumulative nature of the development of mathematics is especially evident in the history of non-Euclidean geometry. A complete account, such as Robert Bonola's Non-Euclidean Geometry, would have to consider the accomplishments of Gerolamo Saccheri (1667-1733), Georg S. Klugel (1739-1812), Heinrich Lambert (1728-1777), Ferdinand Karl Schweikart (1780-1859), and Carl Friedrich Gauss (1777-1855). By the age of 15, Gauss had apparently grasped the idea that there could be invented a logically consistent geometry that was different from Euclid's. He began work on the creation of such a geometry around 1813, and there is evidence that he was successful. But because he never published any fully developed mathematical exposition of his work, historians of mathematics do not credit him as the discoverer of non-Euclidean geometry. That honour is usually reserved for two mathematicians who independently achieved results about the same time: Janos Bolyai and N.I. Lobacevskii. There is some indication that Bolyai, a Hungarian, had realized his ideas on non-Euclidean geometry by 1825. In a letter to his father, the mathematician Wolfgang Farkas Bolyai, dated November 23, 1823, he wrote, ``I have made such wonderful discoveries that I myself am lost in astonishment.'' He did not, however, publish his results encapsulated in a 26-page paper entitled ``The Science of Absolute Space'' until 1832. The first mathematician to publish a definitive work on non-Euclidean geometry was Nikolai Ivanovich Lobacevskii. His paper, ``On the Foundations of Geometry,'' appeared in the Journal of the University of Kazan in 1829. He continued to develop and propagate his ideas in a series of papers, culminating in the ``Pangeometrie'' (1855), which he dictated as a blind old man who still retained his energy and strength of mind. He was, without doubt, Russia's first great mathematician. N.I. Lobacevskii was born on November 2, 1792, in the district of Makarief, roughly 200 miles due east of Moscow. He was the second of three sons of a minor government official. His father died when he was aged seven, leaving his mother, Praskovia Ivanovna, in extreme poverty. Soon after, she moved the family to the city of Kazan, several hundred miles further east, on the River Volga. In 1802, Lobacevskii was allowed to enter, as a free scholar, the local gymnasium. (Founded in the 18th century during the enlightened reign of Catherine I, Russia's gymnasia were classically-oriented, state-run schools whose curricula were designed to prepare young men for university. Their teaching programs emphasized the natural sciences and mathematics.) Five years later, at the precocious age of 14, he entered Kazan University, originally to study medicine. He was to spend the next 40 years of his life there, as a student, assistant professor, professor, and, ultimately, rector. Kazan University had opened in 1804, during the reign of Alexander I, only three years before the arrival of Lobacevskii. While geographically isolated, in a location farther east than that of any Russian university, Kazan was nevertheless a thriving intellectual and cultural centre, tuned to modern European ideas and scientific developments. The university's sophistication and cosmopolitanism was due, in no small part, to the presence on staff of a number of well-recognized professors from Germany. It was Johann Martin Bartels (1769-1836) who had the most profound influence on Lobacevskii. He introduced the young mathematician to the most recent advances in ``differential and integral calculus,'' ``analytical mechanics,'' and ``the applications of analysis to geometry,'' as well as to the ideas of Gauss. Many years earlier, Bartels had tutored Gauss, and some scholars believe that Bartels may have relayed his former pupil's special interest in geometry to Lobacevskii. They are also certain, however, that the Russian came up with the mathematical ideas that founded his non-Euclidean geometry on his own. The non-Euclidean geometry of N.I. Lobacevskii emerged out of a struggle to perfect the logical exposition of Euclid's Elements. As noted previously, Euclid began the Elements with a set of postulates and axioms (which he called ``common notions''), as well as a preliminary set of definitions, which introduced his fundamental concepts, such as ``point,'' ``line,'' ``surface, '' etc. Modern geometers have recognized that Euclid's definitions are devoid of real meaning and serve no logical purpose: fundamental terms, such as ``point'' and ``line,'' are not made more fundamental, or rigorously defined, by such phrases as ``that which has no part,'' and ``breadthless length,'' even though the phrases are non-technical in nature. As Aristotle advised, a logical system must have a starting point and must, therefore, begin with concepts that stand undefined. While Euclid's definitions sparked critical debate, it was concern over his fifth postulate that inspired the invention of non-Euclidean geometry. It states ``that if a straight line falling on two straight lines [see Figure 1, below] makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.'' Geometers, over the ages, never doubted the truth of the above postulate; they felt, however, that it lacked the simplicity and elegance of statement displayed by Euclid's other preliminary assumptions. From Euclid's own time, they struggled to restate it, or to render it superfluous by deducing it as a theorem from his other preliminary assumptions. In 1795, John Playfair restated Euclid's fifth postulate in a simpler but logically equivalent fashion. His postulate, the one still taught in high-school geometry courses, states: ``Through a given point P, not on line l [see Figure 2, below], there is one, and only one, line in the plane of P and l, which does not meet l. The line through point P, which does not meet line l, is that which is parallel to line l.'' N.I. Lobacevskii agreed with John Playfair and others that Euclid's fifth postulate also called the parallel postulate was not superfluous, but necessary for the founding of classical geometry. Lobacevskii realized, however, that replacing Euclid's parallel postulate with a contrary assertion was the key to the creation of a new geometry just as rich, complete, and logically consistent as that developed by the Greeks. The Russian mathematician thus set down an assertion opposing the parallel postulate, adjoined it to Euclid's remaining axioms and postulates, and began to develop his own geometrical theorems. His parallel postulate was somewhat complex in both its form and content. As recorded in his paper ``New Foundations of Geometry with a Complete Theory of Parallels'' (1835-37), it asserted, generally speaking, that through a given point P not on line l, there is more than one line in the plane of P and l that, no matter how far extended, will remain parallel to l. To reiterate Lobacevskii's parallel postulate more precisely: given a line l and a point P not on line l (see Figure 3, below), there are two lines, q and r, such that, given any other line s through point P, s will either meet l, if it does not lie between lines q and r, or s will not meet l, if it does lie between lines q and r. It's true that our figurative representation of Lobacevskii's postulate is not accurate: the dotted lines that we have drawn lying between lines q and r would, if extended further, in our figure, meet line l as would lines q and r themselves. Our difficulty in representing Lobacevskii's postulate does not, however, show that it is false or that its assumption will lead to illogical consequences. It reflects the fact that Lobacevskian geometry is not in accordance with our intuitive picture of space. Indeed, Lobacevskian geometry contains a number of fundamental theorems that, upon first encounter, seem strange and impossible. For example, in Euclidean geometry, the sum of the angles of every triangle equals 180 degrees. In Lobacevskii's geometry, the sum of the angles of every triangle is less than 180 degrees. Furthermore, as the area of a triangle increases in size, the sum of its angles decreases. The great English geometer William Kingdom Clifford commented eloquently in 1872: ``What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobacevskii to Euclid. There is, indeed, a somewhat instructive parallel between the last two cases. Copernicus and Lobacevskii were both of Slavic origin. Each of them has brought about a revolution in scientific ideas so great that it can only be compared with that wrought by the other. And the reason of the transcendent importance of these two changes is that they are changes in the conception of the cosmos.'' The invention of non-Euclidean geometry meant that the system perfected by Euclid could no longer stand with absolute authority. Mathematicians became convinced that the truths of Euclidean geometry, as they relate to space, are not ``self-evident,'' or ``necessary,'' or given ``a priori,'' as the philosopher Kant had held. They are merely founded upon experience and, therefore, their veracity could be tested by means of scientific experiment; although Euclidean geometry seemed to reflect accurately the properties of space, further investigations in astronomy might reveal otherwise. The question of the relevance of non-Euclidean geometry to the study of space gained prominence after the publication of ``On the Hypotheses which Lie at the Foundations of Geometry'' (Ueber die Hypothesen welche der Geometrie zu Grunde liegen) by Bernhard Riemann. The ground-breaking treatise was originally delivered as a probationary lecture before the Philosophical Faculty of the University of Gottingen in 1854. At the time, the brilliant German mathematician was twenty-eight years of age and attempting to qualify for the position of privatdozent. Startling in its originality and scope, the content of his lecture would serve to revolutionize the treatment of space in the realms of philosophy, physics, and mathematics. Riemann attempted, in his exploration of the foundations of geometry, to uncover what facts, derived through the experience of physical space, are presupposed by the geometer (Euclidean and non-Euclidean alike), before he can begin to lay down the axioms and postulates of his science. He argued that intrinsic to the pursuit of geometry is a primary presupposition of an abstract concept of ``a space,'' mathematical in nature, which had not yet been elaborated upon at all. The topic for Riemann's Probe-Vorlesung had been chosen by none other than the eminent Gauss. Though Riemann had produced no previous work on geometry, Gauss intuited that the subject would spark the imagination of an eclectic mathematician whose interests extended into regions of physical science. Riemann showed a passion for physics that is uncommon to the vast majority of pure mathematicians. ``From this distance,'' E.T. Bell wrote in Men of Mathematics (1937), ``it seems as though Riemann's real interest was in mathematical physics and it is quite possible that had he been granted twenty or thirty more years of life he would have become the Newton or Einstein of the nineteenth century.'' Aware that the audience for his lecture ``On the Hypotheses which Lie at the Foundations of Geometry'' included those with extensive and little mathematical training, Riemann tried to present his ideas with few technical calculations or proofs. (He provided these in a second paper, published in Volume 13 of the Abbandlundgen of the Royal Society of Sciences of Gottingen.) In a philosophical manner, Riemann first and foremost introduced what he called ``the concept of manifolds, multiply, or n-fold, extended.'' He defined, that is to say, his notion of ``a mathematical space'' of n, of one, two, four, eight, or any finite number of dimensions, elaborating upon the means by which any such given space could be constructed and subjected to measurement. (Note that Riemann employed the term manifold to denote his concept of a mathematical space. It is the translation into English of the German mannigfaltigkeit, used by Immanuel Kant in the Critique of Pure Reason to refer to the ``sensible world of experience.'' Prior to Riemann, Gauss to whom Riemann is intellectually indebted borrowed the term from the German philosopher, employing it in a mathematical sense in 1831.) Riemann's concept of a mathematical space or manifold was born of the fundamental workings of ``analytic geometry'' (which represents curves by means of algebraic equations). He defined a manifold as an abstract geometric object composed of points, whose positions could be uniquely determined by a given number of ``coordinates.'' To understand something of the above, let us look at the Euclidean plane as Riemann would have done. We must first set aside Euclid's definition of the plane as ``that which has length and breadth only,'' and define it as a ``manifold'' composed of points, each of whose positions can be determined by two numbers, x and y. (See Figure 4, below.) It is evident that the position of points A, B, and C can each be determined by two numbers, or ``coordinates,'' one horizontal and one vertical. The above representation of the plane is, in fact, no different from that introduced by Rene Descartes and Pierre Fermat in the 17th century. Riemann, however, envisioned the Cartesian plane in a new light as a two-dimensional manifold or space unto itself. He considered it two-dimensional by virtue of the fact that every point in it can be located by means of two measures. With this loose understanding of the nature of dimension, it was then possible for him to envision the possibility of three-, four-, five- or n-dimensional manifolds or spaces all of whose points could be located by three, four, five, or n coordinates. He further explored the reality of n-dimensional manifolds by introducing the means by which the distance between points of any such given manifold could be measured. At its simplest, his method employed a formula for measuring the distance between two points of the Cartesian plane. That formula states that the distance between any two points, A and B, of the plane (each of which can be located by two coordinates, x and y, and x' and y', respectively) equals EQUATION HERE. Such a formula involves a translation of the Pythagorean theorem into the language of analytic geometry. Riemann, however, extended the above formula. For example, to measure the distance between two points, A and B, of a five-dimensional manifold (each of which could be located by five coordinates, x, y, z, w, v and x' y' z' w' v', respectively) he used the following formula: Distance AB = EQUATION HERE. Riemann also closely considered a class of n-dimensional manifolds for which the above formula for measuring the distance between points was insufficient. Such abstract spaces, Riemann envisioned as curved in the way, for example, that the surface of a sphere is curved. In fact, just as Riemann could, through a leap of the imagination, see the plane as a ``flat'' two-dimensional space, he also could see the surface of a sphere as a ``curved'' two-dimensional space. To be sure, this characterization of the surface of a sphere was already implicit in Gauss's General Investigations of Curved Surfaces (1827). Riemann, however, made his master's implicit assumptions explicit. He also extended Gauss's methods (which involved the calculus, as well as analytic geometry) to construct and measure the distance between points of curved spaces of n-dimensions. (Riemann regarded the physical space of our perceptions as but one possible three-dimensional manifold, whose properties could be determined only through experiment. He recognized that, while any given n-dimensional manifold, is, in its essence, an abstract, imaginary, mathematical object, it could, perhaps, serve as a ``model'' of physical space; indeed, a crucial area of modern mechanics entails the struggle to determine through experimentation which ``mathematical model'' most closely approximates the reality of physical space. Riemann's thoughts on the nature of space were prophetic: 62 years after the delivery of his famous treatise, his definition of an n-dimensional manifold proved indispensible to the work of 20th-century physicists and cosmologists, such as Albert Einstein. In fact, the mathematical model of physical space employed by Einstein, in his General Theory of Relativity (1916) was, in precise terms, a curved Riemannian manifold of four dimensions. Einstein interpreted the term dimension in the same way that Riemann did: as the number of coordinates needed to locate each point in a given space. He also employed Riemann's formula for measuring the distance between points in space.) The concept of ``a multiply-extended manifold'' forever transformed and broadened the mathematical meaning of the term space. Mathematicians, pursuing Riemann's ideas at the turn of the 19th century, began to consider a variety of unified collections of geometric and non-geometric objects (such as points, lines, spheres, or functions), together with a means by which the objects could be shown to relate to one another (such as a means for measuring the distance between them) as ``spaces'' unto themselves. Through a process of abstraction, they thus began to construct and explore a variety of mathematical spaces Riemannian, projective, metric, function, and, of course, topological. They classified such spaces into their respective types according to which relations were specified as binding together their component parts. The constructing of abstract spaces which entails using the concepts and methods of geometry proved highly advantageous to mathematicians (and scientists) for the solving of diverse problems. Geometric ideas, however abstract, retain their intuitive appeal, sometimes allowing complexities to be grasped at once, without the need for long calculations. Mathematicians accomplish the construction of abstract spaces through definitions: philosophically speaking, the definition and construction of an abstract space is one and the same thing. The definition of a topological space is both similar to and different from that of a Riemannian space. It, too, is defined as an abstract geometric object, composed of points, in conjunction with a specified means of showing how the points relate to one another. Essential to Riemann's definition of an n-dimensional manifold is the means by which the distance between its points (located by n coordinates) can be measured. The definition of a topological space, however, does not specify that its points are to be located by means of Cartesian coordinates. It also does not specify how to measure the distance between them. It merely specifies how the points can be shown to relate to one another in terms of neighborhoods. The notion of a neighborhood describes the relationship between one of a space's points to a sub-collection of its points; it describes, that is, the state of one point in a topological space being near infinitely near, in fact to a sub-collection of its points, without the measurement of any distances whatsoever. (Hence a topologist might tell you, without a trace of irony or whimsy, that he is consumed, above all, with the study of ``nearness without distance.'') The definition of one particular topological space differs from the definition of another in terms of which points, and which sub-collections of points, or ``neighborhoods,'' are specified. The importance of the notion of neighborhood and its utility in the definition of a topological space was propagated by the German mathematician Felix Hausdorff. Born at Breslau in 1868, Hausdorff attained his doctoral degree at the University of Leipzig in 1891. His early scientific work concerned the physics of light; however, he turned his attentions to pure mathematics soon after 1900. He was eventually appointed a professor of mathematics at the University of Bonn. While not primarily a topologist, Hausdorff published in 1914 a book, Essentials of Set Theory (Grundzuge der Mengenlehre), that established him as a major figure in the newly developing field. In it he gave the first popular definition of a topological space (known today as ``a Hausdorff space'') as a collection or ``set'' of elements (points), to each of which correspond certain subsets of the set called neighborhoods, such that: (A) To each point x, there corresponds at least one neighborhood Ux, and Ux contains x. (B) If Ux and Vx are neighborhoods of x, there exists another neighborhood of x, Wx, which is a subset of Ux and of Vx. (C) If y is in Ux, there is a neighborhood Uy of y such that Uy is a subset of Ux. (D) For two distinct points, x and y, there exist two neighborhoods, Ux and Uy, with no point in common. Hausdorff's definition of a topological space, along with the theory he began to build in chapters seven and eight of his Essentials of Set Theory, caught the interest of a number of brilliant mathematicians who lived primarily in Russia and Poland; these included Pavel Sergeevich Alexandroff (A.V. Arhangel'skii's mentor), Pavel Samuelovich Urysohn, and Andre Nicholaevich Tychonoff (founders of the Moscow school of topology); as well as Waclaw Sierpinski and Casimir Kuratowski (founders of the Warsaw school of topology). They produced many of the formative works on set-theoretic topology, published in the years following World War I. The most significant and influential of these was, perhaps, Alexandroff's and Urysohn's Memoir on Compact Topological Spaces (Memoir sur les espaces topologiques compactes), written during the summer of 1922. The two young Russians had rented a room in a dacha on a bank of the Klyaz'ma River, opposite the village of Burkovo, not far east of Moscow. They passed the unusually hot, sunny months, bathing, boating, walking, and fervently contemplating and conversing about mathematics. Their joint intellectual effort produced their famous monograph, in which they defined a number of famous spaces, all of them topological, including ``the compact space,'' ``the double-arrow space,'' ``the Alexandroff duplicate,'' and ``the lexicographically-ordered square.'' On the morning of a warm, sunny Sunday, A.V. Arhangel'skii and I set off by train for the village of Abramtsevo, some 50 miles north-east of Moscow. Abramtsevo is well known to scholars of Russian architecture and design. In the late 1870s, Savva Mamontov, a Russian railroad magnate and philanthropist, transformed his country estate there into an artists' retreat. Mamontov also commissioned artist Viktor Vasnetsov to design and supervise the construction of an estate church. Completed in 1882, the graceful, white, golden-domed structure, playfully modelled on medieval architectural forms and decorated with abstract mosaics of coloured stones, represented the beginning of the Russian style moderne, a national variation of art nouveau. In 1934, Arhangel'skii's maternal grandfather, Pavel Alexanderovich Radimov, sought and won state permission for the building of a ``painters' settlement'' of dachas on lands adjacent to the Mamontov estate (now a public museum). There is no station to welcome our arrival at Abramtsevo just a cracked concrete platform and rusted sign affixed to an iron railing much in need of paint. Immediately upon stepping off the narrow platform, we are walking on a dirt path under a canopy of leaves. Drawn, as if by force, into a forest of green light and shadow, where the air is cool and marvellously fragrant, we soon come upon a small wooden bridge, and cross over a stream. ``This piece of land symbolizes Russia for me,'' Arhangel'skii says. ``It is where I spent the summer months when I was a child swimming, fishing, hunting for mushrooms.'' He pauses for a moment to look down at the stream and up at the tall white birches, poplars, oaks, and spruce sheltering both banks. ``The river is not as clear as it was back then, '' he says with disapproval. ``I can see there is more algae growing in it. I think they have built some small factory a little ways upstream.'' We continue in silence along the path, until Arhangel'skii stops and pulls open a battered fence gate. ``Welcome to my dacha,'' he says, leading me toward a two-storeyed log cabin, screened from view by a stand of birches. Hearing our voices, Arhangel'skii's wife, Olga Constantinovna, comes out front to welcome us, exchanges a few words with her husband, and heads off again to peel potatoes. Since the weather has been warm and dry for days, she has set up a wooden work table outside, laden with enamelled iron pails, pitchers, and pots. The dacha has no indoor plumbing, so Olga Constantinovna uses water from a hose its one end fastened round a nearby tree-trunk, its other attached to the settlement's main water line and from large iron drums placed below eavestroughs to catch the rain. Arhangel'skii suggests that while his wife cooks dinner, we take a look at the dacha and then, perhaps, have a snack. Stepping through the front door into a small entrance hall, we turn right to enter a narrow kitchen, furnished with a rustic wooden table and stools, an old refrigerator and hot plate, and open shelves stacked with well-worn dishes, pots, and pans. It is cool and dark: ceiling, walls, and floor are all finished in a rough, deeply stained wood. Off the kitchen is a small bedroom, with two settees, and what would be a spacious living area, were it not cluttered with aged sofas, cabinets, tables, chairs, and piles of raw lumber. Apologizing for the mess, Arhangel'skii explains that he has been trying to stock up on materials available only on the black market for the construction of a second floor to accommodate guests. Last summer, he had a wooden staircase built as a preliminary. ``It was made by two drunkards from the next village who, together, consumed three bottles of vodka a day. I had to provide the vodka as part of their payment.'' The staircase looks surprisingly solid, nonetheless. We decide to have our snack on a side porch. Unpacking a bag of food he has brought from home, Arhangel'skii cuts each of us a slice of dark Russian rye, Bulgarian feta, and tomato. He opens a bottle of vodka and pours out two small glasses. ``This is all one needs to be happy,'' he says. ``Just a bit of vodka and some small snack.'' We eat and drink, looking on to the treed yard and listening to the wind, the hum of crickets, and the crowing of a rooster. ``When I was a boy, my grandfather kept pigs and cows at his dacha,'' Arhangel'skii recalls. ``I remember drinking fresh milk. Nowadays, people only keep a few chickens.'' He gets up and paces about the yard for a few moments, muttering that he must cut its long, tangled grass. He inspects the health of a number of trees, looking with satisfaction at a pine. ``There were no pine trees here when we had this dacha built. I wanted pines, so I transplanted a few from the nearby forest.'' Arhangel'skii brings his right hand to his forehead to shield his eyes from the sun, then draws my attention to a dacha standing behind his own. ``That was the dacha my father built and that I visited as a boy. Today it is occupied by the president of the Soviet Union of Painters, the grandson of Viktor Vasnetsov, who designed the Mamontov church.'' The construction of Arhangel'skii's own dacha was overseen by his wife in 1975, while Arhangel'skii was in Pakistan, teaching at the University of Islamabad under a program sponsored by UNESCO. ``I was the official UNESCO expert on topology, which I thought was quite an original title.'' Although Arhangel'skii has eagerly embraced such rare opportunities for travel as were afforded to Soviet citizens (in the 1960s he participated in an experimental cultural-exchange program, staying for three weeks in the U.S. and Canada), it was not wanderlust that prompted his trip to Pakistan. ``My material conditions at that time were not very good. I had to support a young family and affording anything presented a problem. While pure mathematicians are well respected in Russian society, that respect has not been reflected in our salaries. At the time that I decided to go to Pakistan, I couldn't afford such things as a dacha, or even a car. ``I left Russia two months before my family did and found and rented a house in the capital. When I arrived, I discovered that the University of Islamabad was on strike, so I didn't have to teach for my first half a year. Afterwards, I taught undergraduate courses in topology and set theory. Roughly five students out of thirty were talented and worth working with. Unfortunately, I did not find it easy in Pakistan to conduct my own research. I just couldn't establish a favourable routine. I also found it difficult to live amongst people who had to suffer such impoverished conditions, conditions that degraded the human spirit.'' I ask Arhangel'skii why the Soviet authorities would allow his wife and children to leave the U.S.S.R. and accompany him to Pakistan. ``Weren't they afraid that you would defect?'' ``No,'' Arhangel'skii explains with a grin. ``The authorities believed that the family would keep you tied to the state. If a man goes away from his family for too long, he may have some small romantic adventure and forget them, and his country too. But I could never leave Russia as long as my mother is alive.'' In 1959, the year he married Olga Constantinovna, Arhangel'skii produced his first important mathematical result. His definition of the concept of ``a network'' distills within it the ``topology'' or the inherent structure of ``compact topological spaces.'' Since that time he has published roughly 120 papers, both research and survey articles, in such journals as Doklady Akademii Nauk SSSR, Mat. Zametki, Uspekhi Mat. Nauk, and Izvestiya Akademii Nauk SSSR. He has also written four books: Cantorian Set Theory, Linear Algebra (both textbooks for undergraduates), General Topology (co-written with V.I. Ponmarev and consisting entirely of questions and answers), and CP Theory (one of Arhangel'skii's more recent interests, which discloses the topological structure of ``a function space''). ``It has never been more important for me to solve a problem than to create a new concept,'' he says. ``A major part of my work, therefore, has been to introduce new notions through which links can be found among seemingly disparate things. It would not have pleased me to have added one, two, or three more isolated pieces of information to the field. I would rather concentrate upon, understand, and develop a systematic body of knowledge encompassing some part of topology.'' He has been described by a colleague as possessing ``an almost mystical intuition.'' In an effort to describe the nature of mathematical creativity, Arhangel'skii refers to his sensitivity to beauty, his tangible awareness of the harmonious. ``Beauty is, for me, a sign of the truth. . . . When I think about a mathematical problem or theorem, my intuition suggests to me what should be true: I will see relationships between properties or results, and these relationships will suggest something new. I guess at the truth, even before I know the truth. When I believe something will be true, there is also usually suggested to me a way to prove it. And when I try to prove it, I will sometimes feel things fitting together harmoniously. It is from this feeling of harmony that I know my way will work.'' Fitting words for the son of a concert pianist and painter. He is, of course, not the first mathematician to link mathematics and beauty. The 20th-century English mathematician G.H. Hardy wrote: ``A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas. . . . The mathematician's patterns, like the painter's or poet's, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.'' Hardy argues that the beauty of a mathematical theorem lies in its seriousness measured not in terms of its practical consequences but in terms of the significance of the mathematical ideas with which it connects. Arhangel'skii would add that the beauty of a mathematical theorem lies in its unexpectedness. ``You may come upon a theorem that is not, in itself, difficult to prove,'' he says. ``In fact, it may be very simple, yet unexpected, and because of this it brings delight.'' He compares this delight to that felt by the first explorers ``to go around the world in one direction and suddenly find themselves returning to their starting place.'' A mathematician's power, according to Arhangel'skii, lies mainly in his ability to envision a highly abstract concept by means of ``an effective image.'' (I can understand what he means: my husband has told me that he sometimes thinks of topological spaces as living creatures, imagining them to be divided up into different species.) ``When a mathematician begins to ponder such things as `a compact topological space,' he thinks about it in a way that seems to him quite simple. I might, for example, imagine it as a line segment, or a circle, or a sphere. I have in my head a kind of encyclopedia of images, mainly simple geometric constructions, such as those I just mentioned, that I use to represent the various known topological spaces. You must remember that topology is, in its essence, a kind of abstract geometry. To construct a new topological space, you have got to first imagine that space, to envision its points and neighbourhoods. You have got to be inspired. You cannot create a space using formulas or arguments of logic. To create a new space, I might try to combine, or `glue together,' a number of existing spaces, like building blocks. Geometric images can be very helpful here.'' Images are crucial for mathematicians, such as topologists, who work with infinity. ``The infinite exists simply because we imagine it to, because we invoke its existence. The mathematics of the infinite can never be reduced to algorithms or calculations; it can never be formalized. It is inherently fraught with paradoxes and contradictions. Mathematics teaches a person the laws of the infinite. It domesticates the infinite, makes it familiar. But our desire to work with the infinite comes from our understanding that it can never be reduced to the finite. It is the desire to be in touch with something mystical. The infinite exists like God exists. We can try to understand Him, knowing we can never see Him completely.'' One of Arhangel'skii's earliest survey papers on topology ``Mappings and Spaces,'' published in Russian Mathematical Surveys in 1966 is prefaced by two lines that I thought he had borrowed from a Russian poet: On the Edges of Darkness I sing of Your Galaxies. ``Many people have asked me about this quote,'' he says. ``But I did not take it from any famous poem; I wrote it myself. When I was younger I wrote many poems, perhaps thirty or more. They were often very short, just a few lines. I wrote this one three years before the paper. I don't really understand where it, or the others, came from.'' Arhangel'skii, who refers to his mathematical results as his ``children,'' finds it difficult to single out one result as standing above the rest. If gently pressed, however, he will admit to being most pleased with his resolution of P.S. Alexandroff's problem regarding the possible ``cardinality'' of ``a compact, first-countable topological space.'' ``Yes, that was probably my best work,'' he says. ``Solving that question really meant something to me because I kept leaving and returning to it for a period of several years. It was, for me, a favourite problem. When I saw the answer, I saw it very clearly. It seemed very simple, so I was afraid it must be wrong. ``I obtained the result sometime between the first and tenth of May, 1969, and it was published three months later. I solved it sitting like Archimedes in the bathtub. I had a habit of thinking in the bath, where I could lie and relax for three or four hours at a stretch. The sound of water dripping from the faucet would block out all other sounds and put me into a meditative state. That night, I lay in the bath thinking from about ten-thirty to three a.m. Then I went to sleep. In the morning, when I woke up, I felt the problem was solved.'' ``Did you shout `Eureka'?'' I ask. ``I wanted to,'' Arhangel'skii replies, smiling. ``But being too modest, I couldn't.'' In February 1969, three months before he obtained his most significant result, A.V. Arhangel'skii was permitted to join the Communist Party. He did so, he said, because it was expedient. ``Not all professors at Moscow State University had wished to become members of the Party. Some professors did so in order to increase their chances for a promotion or for the opportunity to travel; others because they had wanted to gain a position of influence to enable them to help their students. I simply recognized that, to play a public role in the life of the university, I had to become a member.'' He slowed down the pace at which we had been walking. Having spoken for several hours on the side porch of his dacha, Arhangel'skii had stood up to stretch his long limbs, and suggested we take a short hike through the surrounding forest. Perhaps, he had said, we might also take a swim in a pond in which he had bathed as a child. We could, of course, continue our discussions as we walked. So we packed suits and towels into a string bag and set off alone from the dacha, following a dirt path in the dappled shade of birch and poplar trees. Arhangel'skii wielded an old, twisted walking stick in his left hand. ``Many more people actually wanted to become members of the Party than were allowed to,'' he continued as we walked. ``One's opportunity to join was supposed to have depended upon whether or not a position had become vacant. The availability of positions, however, did not arise in a `regular' way: if certain people wanted a friend, say, to be admitted, then, suddenly, `a vacancy' would open up whether or not anyone had actually left the Party. Furthermore, in order to become a member, you had to fill out a certain form; if, however, you were not issued such a form in the first place, then you could not become one.'' Arhangel'skii submitted his application form in April 1967, the year after he had been awarded his doctoral degree. ``It took me more than a year to enter the Party,'' he said, ``because I had signed a letter of protest, regarding the treatment of [A.S.] Esenin-Volpin. ``Esenin-Volpin was a topologist, older than myself, who had done mathematical research in the Forties. He was also the son of the famous Russian poet [Sergei Aleksandrovich] Esenin. I had known that my colleague had suffered a number of nervous breakdowns and had spent some time in mental hospitals. But one day, a friend of mine came to me and told me that Esenin-Volpin had been sent to a mental hospital against his will. He asked me to sign a letter of protest. ``Ninety-nine professors at Moscow State University signed this letter. Of those ninety-nine, only two were members of the Party. P.S. Alexandroff, who was a member, did not. I don't think I realized that it would be read aloud over the BBC and Voice of America. ``I was punished after this. I was not allowed to enter the Party when I applied, and I was reprimanded by officials in person, and in a written letter.'' ``Couldn't your punishment have been far worse?'' ``Yes, at another time, it might have been. But I didn't hesitate when my friend asked me to sign that letter. I signed in one minute. There was no deliberation. I wanted to state that there were some things that I could not accept.'' ``Were you involved in any other political protests?'' ``No that was the only serious step that I ever took. I was never again asked to take such a step. Actually, I do not think it was a bad move for my career. It may have even helped my career.'' I asked Arhangel'skii about his activities as a member of the Communist Party and about the role that the Party had played on the campus of Moscow State University. ``Being a Party member meant you had to fulfil certain obligations,'' he said. ``You had to join the `primary cell' of your workplace and pay a small due, roughly two to three per cent of your salary. You also had to allow your name to be listed on announcements of meetings, which were posted on a bulletin board near the faculty office. The primary cell of the university would meet twice per semester. At these meetings which were like sessions of a parliament all matters and questions concerning the life of the university or of a certain department were considered. It was discussed, for example, how to buy computers, or equipment for scientific research, or, perhaps, what direction in research a department should take. Such things as which students should be allowed to study for their Ph.D., and who should be appointed chairman of a certain department were also debated. It was understood that no final decisions about these questions could be made. However, the Party could, and did, make recommendations to different administrative bodies. ``As a member of the Party you also had to fulfil some duty, additional to those demanded of you by the university administration. For example, I became an editor of our faculty newspaper for a while. I was also elected to a kind of council that was responsible for finding people who would tutor students, free of charge students from socially backward areas of the U.S.S.R., who were not so well prepared for university. Other members were responsible for organizing sports activities, or picnics, or some other social events. ``None of these activities, however, have taken place for the past two years. They were stopped before the ban.'' (On the 20th of July, 1991, a few weeks before the attempted coup to overthrow Gorbachev, Yeltsin had issued a decree banning Communist Party organizations from operating in government offices and workplaces in the Russian Federated Republic.) I asked Arhangel'skii whether or not he had been happy to see the Party banned from Moscow State University. ``Yes,'' he replied. ``The university should not be an arena for Communist Party activities. It should be free from all politics. But there should be some organization that is responsible for arranging social and cultural activities for the students.'' On several other occasions, I had heard Arhangel'skii refer to the Party as ``a Mafia.'' I had heard him complain that because of it, the wrong people were promoted into positions, for the wrong reasons. I had also heard him complain that corruption and sloth had become endemic in Soviet society. In the past, he had said, people worked because they were afraid. Now that they were no longer afraid, they did not want to work. It was only under a capitalistic system, he had said, that people had the motivation to work. Despite these comments, I had found Arhangel'skii's private stance toward the regime of the former U.S.S.R. complicated, if not paradoxical. He had been, for example, openly supportive of Mikhail Gorbachev, right up until the August coup, and expressed a lack of trust and faith in Boris Yeltsin. Furthermore, before the Communist Party's final demise, he had seemed hopeful that it would evolve and become a vehicle for social, political, and economic reform. He had remained a daily reader of Pravda, the Communist Party organ, until the day it ceased publication. Walking down Kutuzovskii Prospekt one day, he had pointed out, ruefully, that in the recent past Moscow's shops could at least provide sustenance for its population. Yes, he had admitted, foods were not always of the finest quality or the most varied; nevertheless, no one had to do without. No one went homeless in Moscow either. He had seemed to wish to remind me that the forgotten and perverted egalitarian ideals of the Bolshevik revolution had, after all, been worthy. He had seemed to feel sorrow for the death of the 19th-century utopian vision that had fuelled the Russian Revolution a vision, he, at the same time, believed was impossible to realize. Speaking about the Russian Revolution, Arhangel'skii had said, ``In the nineteenth century, there existed a contrast between our intelligentsia and the common man. Our intelligentsia felt the greatness of this contrast, and it was this that led them to want to persuade others of the truth of their ideas or even to impose them. A society, however, should never be founded upon one single ideology. For there is always the possibility that the reasoning upon which one bases one's belief in the truth of that ideology will ultimately be proven false. ``Being a mathematician trains you to recognize the meaning of truth; it also trains you to recognize what is not the truth. Mathematicians, like philosophers, social scientists, or historians, work with abstract ideas. They are necessary in most disciplines for the developing of conclusions. But the abstractions that we work with in mathematics are rigorously defined. We can, therefore, be certain of the truth of the conclusions that we draw from them. When a mathematician reads philosophy, or social or political theory, he is often struck by the vagueness of the ideas and the muddiness of the logic. ``When you found a society upon one ideological system, you can end up with something that is intrinsically `double-faced.' Here in Russia we strove to create a society that was supposed to be good for the people, but we ended up with something that was not. We created a society in which a kind of schizophrenia became systemic: in which we learned to say one thing to each other, when we really meant to say the opposite. And this pathological condition began to seem normal.'' Arhangel'skii's political views reminded me of those expressed by Alexander Herzen, a Russian writer of the 19th century. Best known for his arresting autobiography, My Past and Thoughts, Herzen produced some of the deepest of modern writings on the subject of human liberty. Although he began his intellectual career with a utopian faith in an archaic form of socialism, he refused to prescribe it as a remedy for social problems. He refused to condone the sacrifice of liberty, ``the liberty of living individuals with their own individual ends, the ends for which they move and fight and perhaps die,'' in the name of any abstraction. We had been walking along the path for some time in complete silence. We had followed it, first, through the forest surrounding the settlement of dachas, back in the direction in which we had come from the train. We turned, however, away from the tracks, and crossed a wooden bridge over a narrow, muddy river, continuing for a mile or two along its reedy bank. Now and again, we looked across the sluggish water to the opposing bank to catch sight of a family at rest under the trees, the contents of a picnic laid out upon a moth-eaten blanket. In a few places, kids were swimming where the river had deepened and was less choked by reeds. After some steady walking, we came out of the forest into brilliant sunshine, turning away from the river and tramping through a field of tall, yellowy grass until we reached a country lane. We headed down this lane, passing a decrepit wooden cottage, surrounded by a small field of corn. At the bottom of the field, the land fell away, and there, before us, lay the dark, still waters of a pond, a few hundred yards across. ``Here we are,'' Arhangel'skii said, leading us off the road toward a grassy bank, where a few people were lying on towels, tanning themselves. We sat down for a minute, then took turns changing into our swimsuits behind a nearby clump of bushes. After changing, Arhangel'skii stood on the bank, doing vigorous deep-knee bends, toe-touches, and army-style stride jumps. Without pausing to catch his breath, he dove into the pond and began swimming the crawl toward the far side. I jumped in after him and gasped, stung by the coldness of spring water. We swam for 10 minutes or so, dried off in the sun, then changed back into our clothes. Arhangel'skii said we had better head home; he did not wish to be late for dinner. Walking along the deserted path back to the dacha, we spoke of the changes that had just begun to transform Russia. While Arhangel'skii neither wholeheartedly lauded nor condemned communism, he also expressed no unbridled optimism for Russia's future. He hoped that all economic and social changes though necessary and desirable would be introduced slowly and carefully; he feared that sudden upheaval would give rise to chaos, poverty, suffering, and a possible return to Stalinism and repression. Despite his undeniably privileged position, he appeared worried about the well-being of his own family. ``In Russia,'' he remarked, ``we stand with our heads in the heavens, but our feet in the swamp. We are always well aware of our goals, but not the practical means by which we can achieve them.'' I asked Arhangel'skii what he hoped to accomplish in the future. ``My main goal,'' he said, ``is to understand topology as a whole, to understand better the relationship between different branches, the relationship between new concepts and old. I have, as you know, done work in a number of different directions. I would like to return to them and improve upon what I have done. I would also like to continue to develop some new directions. However, above all, I would like my knowledge to be universal in set-theoretic topology. Nowadays, this is a very difficult task. There are more and more bright young people coming up with deeper results. It is difficult to keep up.'' ``Do you believe, as G.H. Hardy believed, that `mathematics, more than any other art or science, is a young man's game?' '' Arhangel'skii considered this question carefully, pausing a while before he replied. ``No, I disagree with Hardy. A mathematician does not necessarily do his best work when he is young. It is true that he doesn't need to learn to use sophisticated equipment. He also doesn't need to spend years in a lab working on an experiment before he can achieve an important result. To some extent, it may be true that it is easier for a young person to show his force, to solve a specific problem. However, one's intuition improves with age and experience, the experience of thinking about mathematics. It becomes easier to see how things are connected. . . .'' He paused again, then returned to the question, taking a different perspective. ``As a person ages, psychologically, he is inclined to become more realistic, to stand with his feet more firmly planted on the earth. A good mathematician, however, never grows old. He remains a child. He remains a dreamer: curious, imaginative, free of concrete purpose.'' We had retraced the path back from the road, through the field, and along the bank of the river into the forest once again. The sun was beginning to set, and a diffused, ochrous light spilled through overhead leaves, polishing the slender, black and white trunks of birch trees. I stood still for a while to watch Arhangel'skii lightly threading his way between them. Catching up with him again, I asked if he ever found the effort to solve a difficult mathematical problem, frustrating or disagreeable, if the satisfaction came only fleetingly, when a problem had been finally solved. ``No, for me the satisfaction in doing mathematics is not with the result, but with the process. The joy that comes to me after I have attained a result is only the joy in remembering how I struggled to achieve it.'' Stephen Watson Department of Mathematics and Statistics York University e-mail: stephen.watson@mathstat.yorku.ca http://www.unipissing.ca/topology/ | |
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