An eternity of infinities: the power and beauty of mathematics

The biggest intellectual shock I ever received was in high school. Someone gifted me a copy of the physicist George Gamow’s classic book “One two three...infinity”. Gamow was not only a brilliant scientist but also one of the best science popularizers of the late twentieth century. In his book I encountered the deepest and most utterly fascinating pure intellectual fact I have ever known; the fact that mathematics allows us to compare ‘different infinities’. This idea will forever strike awe and wonder in me and I think is the ultimate tribute to the singularly bizarre and completely counter-intuitive worlds that science and especially mathematics can uncover.

Gamow starts by alerting us to the Hottentot tribe in Africa. Members of this tribe cannot formally count beyond three. How then do they compare commodities such as animals whose numbers are greater than three? By employing one of the most logical and primitive methods of counting- the method of counting by one-to-one correspondences or put more simply, by pairing objects with each other. So if a Hottentot has ten animals and she wishes to compare these with animals from a rival tribe, she will pair off each animal with its counterpart. If animals are left over in her own collection, she wins. If they are left over in her rival’s collection, she has to admit the rival tribe’s superiority in sheep.

What is remarkable is that this simplest of counting methods allowed the great German mathematician Georg Cantor to discover one of the most stunning and counter-intuitive facts ever divined by pure thinking. Consider the set of natural numbers 1, 2, 3… Now consider the set of even numbers 2, 4, 6…If asked which set is greater, commonsense would quickly point to the former. After all the set of natural numbers contains both even and odd numbers and this would of course be greater than just the set of even numbers, wouldn’t it? But if modern science and mathematics have revealed one thing about the universe, it’s that the universe often makes commonsense stand on its head. And so it is the case here. Let’s use the Hottentot method. Line up the natural numbers and the even numbers next to each other and pair them up.

1 2 3 4 5…
2 4 6 8 10…

So 1 pairs up with 2, 2 pairs up with 4, 3 pairs up with 6 and so on. It’s now obvious that every natural number n will always pair up with an even number 2n. Thus the set of natural numbers is equal to the set of even numbers, a conclusion that seems to fly in the face of commonsense and shatters its visage. We can extend this conclusion even further. For instance consider the set of squares of natural numbers, a set that would seem even ‘smaller’ than the set of even numbers. By similar pairings we can show that every natural number n can be paired with its square n², again demonstrating the equality of the two sets. Now you can play around with this method and establish all kinds of equalities, for instance that of whole numbers (all positive and negative numbers) with squares.

But what Cantor did with this technique was much deeper than amusing pairings. The set of natural numbers is infinite. The set of even numbers is also infinite. Yet they can be compared. Cantor showed that two infinities can actually be compared and can be shown to be equal to each other. Before Cantor infinity was just a place card for ‘unlimited’, a vague notion that exceeded man’s imagination to visualize. But Cantor showed that infinity can be mathematically precisely quantified, captured in simple notation and expressed more or less like a finite number. In fact he found a precise mapping technique with which a certain kind of infinity can be defined. By Cantor’s definition, any infinite set of objects which has a one-to-one mapping or correspondence with the natural numbers is called a ‘countably’ infinite set of objects. The correspondence needs to be strictly one-to-one and it needs to be exhaustive, that is, for every object in the first set there must be a corresponding object in the second one. The set of natural numbers is thus a ruler with which to measure the ‘size’ of other infinite sets. This countable infinity was quantified by a measure called the ‘cardinality’ of the set. The cardinality of the set of natural numbers and all others which are equivalent to it through one-to-one mappings is called ‘aleph-naught’, denoted by the symbol . The set of natural numbers and the set of odd and even numbers constitute the ‘smallest’ infinity and they all have a cardinality of . Sets which seemed disparately different in size could all now be declared equivalent to each other and pared down to a single classification. This was a towering achievement.

The perplexities of Cantor’s infinities led the great mathematician David Hilbert to propose an amusing situation called ‘Hilbert’s Hotel’. Let’s say you are on a long journey and, weary and hungry, you come to a fine-looking hotel. The hotel looks like any other but there’s a catch: much to your delight, it contains a countably infinite number of rooms. So now when the manager at the front desk says “Sorry, but we are full”, you have a response ready for him. You simply tell him to move the first guest into the second room, the second guest into the third room and so on, with the n^th guest moving into the (n+1)^th room. Easy! But now what if you are accompanied by your friends? In fact, what if you are so popular that you are accompanied by a countably infinite number of friends? No problem! You simply ask the manager to move the first guest into the second room, the second guest into the fourth room, the third guest into the sixth room…and the n^th guest into the 2n^th room. Now all the odd-numbered rooms are empty, and since we already know that the set of odd numbers is countably infinite, these rooms will easily accommodate all your countably infinite guests, making you even more popular. Mathematics can bend the laws of the material world like nothing else.

But the previous discussion leaves a nagging question. Since all our infinities are countably infinite, is there something like an ‘uncountably’ infinite set? In fact, what would such an infinity even look like? The ensuing discussion probably constitutes the gem in the crown of infinities and it struck infinite wonder in my heart when I read it.

Let’s consider the set of real numbers, numbers defined with a decimal point as a.bcdefg... The real numbers consist of the rational and the irrational numbers. Is this set countably infinite? By Cantor’s definition, to demonstrate this we would have to prove that there is a one-to-one mapping between the set of real numbers and the set of natural numbers. Is this possible? Well, let’s say we have an endless list of rational numbers, for instance 2.823, 7.298, 4.001 etc. Now pair up each one of these with the natural numbers 1, 2, 3…, in this case simply by counting them. For instance:

S1 = 2.823
S2 = 7.298
S3 = 4.001
S4 = …

Have we proved that the rational numbers are countably infinite? Not really. This is because I can construct a new real number not on the list using the following prescription: construct a new real number such that it differs from the first real number in the first decimal place, the second real number in the second decimal place, the third real number in the third decimal place…and the nth real number in the nth decimal place. So for the example of three numbers above the new number can be:

S0 = 3.942

(9 is different from 8 in S1, 4 is different from 9 in S2 and 2 is different from 1 in S3)

Thus, given an endless list of real numbers counted from 1, 2, 3…onwards, one can always construct a number which is not on the list since it will differ from the 1^st number in the first decimal place, 2^nd number in the second decimal place…and from the nth number in the nth decimal place.

Cantor called this argument the ‘diagonal argument’ since it really constructs a new real number from a line that’s diagonally drawn across all the relevant numbers after the decimal points in each of the listed numbers. The image from the Wikipedia page makes the picture clearer:


In this picture, the new number is constructed from the red numbers on the diagonal. It’s obvious that the new number Eu will be different from every single number E1…En on the list. The diagonal argument is an astonishingly simple and elegant technique that can be used to prove a deep truth.

With this comparison Cantor achieved something awe-inspiring. He showed that one infinity can be greater than another, and in fact it can be infinitely greater than another. This really drives the nail in the coffin of commonsense, since a ‘comparison of two infinities’ appears absurd to the uninformed mind. But our intuitive ideas about sets break down in the face of infinity. A similar argument can demonstrate that while the rational numbers are countably infinite, the irrational numbers are uncountably so. This leads to another shattering comparison; it tells us that the tiny line segment between 0 and 1 on the number line containing real numbers (denoted by [0, 1]) is ‘larger’ than the entire set of natural numbers. A more spectacular case of David obliterating Goliath I have never seen.

The uncountably infinite set of reals comprises a separate cardinality from the cardinality of countably infinite objects like the naturals which was denoted by . Thus one might logically expect the cardinality of the reals to be denoted by ‘’. But as usual reality thwarts logic. This cardinality is actually denoted by ‘c’ and not by the expected . Why this is so is beyond my capability to understand, but it is fascinating. While it can be proven that 2^ = c,the hypothesis that c = is actually just a hypothesis, not a proven and obvious fact of mathematics. This hypothesis is called the ‘continuum hypothesis’ and happens to be one of the biggest unsolved problems in pure mathematics. The problem was in fact the first of the 23 famous problems for the new century proposed by David Hilbert in 1900 during the International Mathematical Congress in France (among others on the list were the notorious Riemann hypothesis and the fond belief that the axioms of arithmetic are consistent, later demolished by Kurt Gödel). The brilliant English mathematician G H Hardy put the continuum at the top of his list of things to do before he died (he did not succeed). A corollary of the hypothesis is that there are no sets with cardinality between and c. Unfortunately the continuum hypothesis may be forever beyond our reach. The same Gödel and the Princeton mathematician Paul Cohen damned the hypothesis by proving that, assuming the consistency of the basic foundation of set theory, the continuum hypothesis is undecidable and therefore it cannot be proved nor disproved. This is assuming that there are no contradictions in the basic foundation of set theory, something that itself is 'widely believed' but not proven. Of course all this is meat and drink for mathematicians wandering around in the most abstract reaches of thought and it will undoubtedly keep them busy for years.

But it all starts with the Hottentots, Cantor and the most primitive methods of counting and comparison. I happened to chance upon Gamow’s little gem yesterday, and all this came back to me in a rush. The comparison of infinities is simple to understand and is a fantastic device for introducing children to the wonders of mathematics. It drives home the essential weirdness of the mathematical universe and raises penetrating questions not only about the nature of this universe but about the nature of the human mind that can comprehend it. One of the biggest questions concerns the nature of reality itself. Physics has also revealed counter-intuitive truths about the universe like the curvature of space-time, the duality of waves and particles and the spooky phenomenon of entanglement, but these truths undoubtedly have a real existence as observed through exhaustive experimentation. But what do the bizarre truths revealed by mathematics actually mean? Unlike the truths of physics they can’t exactly be touched and seen. Can some of these such as the perceived differences between two kinds of infinities simply be a function of human perception, or do these truths point to an objective reality ‘out there’? If they are only a function of human perception, what is it exactly in the structure of the brain that makes such wondrous creations possible? In the twenty-first century when neuroscience promises to reveal more of the brain than was ever possible, the investigation of mathematical understanding could prove to be profoundly significant.

Blake was probably not thinking about the continuum hypothesis when he wrote the following lines:

To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.

But mathematics would have validated his thoughts. It is through mathematics that we can hold not one but an infinity of infinities in the palm of our hand, for all of eternity.

A hermitian operator in self-imposed exile

Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century
Masha Gessen (Houghton Mifflin Harcourt, 2009)

Pure mathematicians have the reputation of being otherworldly and divorced from practical matters. Grisha or Grigory Perelman, the Russian mathematician who at the turn of this century solved one of the great unsolved problems in mathematics, the Poincare Conjecture, is sadly or perhaps appropriately an almost perfect specimen of this belief. For Perelman, even the rudiments of any kind of monetary, professional or material rewards resulting from his theorem were not just unnecessary but downright abhorrent. He has turned down professorships at the best universities in the world, declined the Fields Medal, and will probably not accept the 1 million dollar prize awarded by the Clay Mathematics Institute for the solution of the some of the most daunting mathematical problems of all time. He has cut himself off from the world after seeing the publicity that his work received and has become a recluse, living with his mother in St. Petersburg. For Perelman, mathematics should purely and strictly be done for its own sake, and could never be tainted with any kind of worldly stigma. Perelman is truly a mathematical hermit, or what a professor of mine would call using mathematical jargon, a "hermitian operator".

In "Perfect Rigor", Masha Gessen tells us the story of this remarkable individual, but even more importantly tells us the story of the Russian mathematical system that produced this genius. The inside details of Russian mathematics were cut off from the world until the fall of the Soviet Union. Russian mathematics was nurtured by a small group of extraordinary mathematicians including Andrey Kolmogorov, the greatest Russian mathematician of the twentieth century. Kolmogorov and others who followed him believed in taking latent, outstanding talent in the form of young children and single-mindedly transforming them into great problem solvers and thinkers. Interestingly in the early Soviet Union under Stalin's brutal rule, mathematics flourished where other sciences languished partly because Stalin and others simply could not understand abstract mathematical concepts and thus did not think they posed any danger to communist ideology. Soviet mathematics also got a boost when its great value was recognized during the Great Patriotic War in building aircraft and later in work on the atomic bomb. Mathematicians and physicists thus became unusually valued assets to the Soviet system.

Kolmogorov and a select band of others took advantage of the state's appreciation of math and created small, elite schools for students to train them for the mathematical olympiads. Foremost among the teachers was a man named Sergei Rukshin who Gessen talks about at length. Rukshin believed in completely enveloping his students in his world. In his schools the students entered a different universe, forged by intense thought and mathematical camaraderie. They were largely shielded from outside influences and coddled. The exceptions were women and Jews. Gessen tells us about the rampant anti-Semitism in the Soviet Union which lasted until its end and prevented many bright Jewish students from showcasing their talents. Perelman was one of the very few Jews who made it, and only because he achieved a perfect score in the International Mathematical Olympiad.

Perelman's extreme qualities were partly a result of this system, which had kept him from knowing about politics and the vagaries of human existence and insulated him from a capricious world where compromise is necessary. For him, everything had to be logical and utterly honest. There was no room for things such as diplomacy, white lies, nationalism and manipulation to achieve one's personal ends. If a mathematical theorem was proven to be true, then any further acknowledgment of its existence in the form of monetary or practical benefits was almost vulgar. This was manifested in his peculiar behavior in the United States. For instance, when he visited the US in the 90s as a postdoctoral researcher he had already made a name for himself. Princeton offered the twenty nine year old an assistant professorship, a rare and privileged opportunity. However Perelman would settle for nothing less than a full professorship and was repulsed even by the request that he officially interview for the position (which would have been simply a formality) and submit his CV. Rudimentary formalities which would be normal for almost everyone were abhorrent for Perelman.

After being disillusioned with what he saw as an excessively materialistic academic food chain in the US, Perelman returned to Russia. For five years after that he virtually cut himself off from his colleagues. But it was then that he worked on the Poincare Conjecture and created his lasting achievement. Sadly, his time spent intensely working alone in Russia seemed to have made him even more sensitive to real and perceived slights. However, he did publicly put up his proofs on the internet in 2002 and then visited the US. For a brief period he even seemed to enjoy the reception he received in the country, with mathematicians everywhere vying to secure his services for their universities. He was unusually patient in giving several talks and patiently explaining his proof to mathematicians. Yet it was clear he was indulging in this exercise only for the sake of clarifying the mathematical concepts, and not to be socially acceptable.

However, after this brief period of normalcy, a series of events made Perelman reject the world of human beings and even that of his beloved mathematics. He was appalled by the publicity he received in newspapers like the New York Times which could not understand his work. He found the rat race to recruit him, with universities climbing over each other and making him fantastic offers of salary and opportunity, utterly repulsive. After rejecting all these offers and even accusing some of his colleagues of being traitors who gave him undue publicity, he withdrew to Russia and definitively severed himself from the world. The final straw may have been two events; the awarding of the Fields Medal which, since his work was still being verified, could not explicitly state that he had proven the Poincare conjecture, and the publication of a paper by Chinese mathematicians which in hindsight clearly seems to have been written for stealing the limelight and the honors from Perelman. For Perelman, all this (including the sharing of the Fields with three other mathematicians) was a grave insult and unbecoming of the pursuit of pure mathematics.

Since then Perelman has been almost completely inaccessible. He does not answer emails, letters and phone calls. In an unprecedented move, the president of the International Mathematical Congress which awards the Fields Medals personally went to St. Petersburg to talk him out of declining the prize. Perelman was polite, but the conversation was to no avail. Neither is there any indication that he would accept the 1 million dollar Clay prize. Gessen himself could never interview him, and because of this the essence of Perelman remains vague and we don't really get to know him in the book. Since Gessen is trying to somewhat psychoanalyze her subject and depends on second-hand information to draw her own conclusions, her narrative sometimes lacks coherence and meanders off. As some other reviewers have noted, the discussion of the actual math is sparse and disappointing, but this book is not really about the math but about the man and his social milieu. The content remains intriguing and novel.

Of course, Perelman's behavior is bizarre and impenetrable only to us mere mortals. For Perelman it forms a subset of what has in his mind always been a perfectly internally consistent and logical set of postulates and conclusions. Mathematics has to be done for its own sake. Academic appointments, prizes, publicity and professional rivalries should have no place in the acknowledgement of a beautiful mathematical proof. While things like applying for interviews and negotiating job offers may seem to us to be perfectly reasonable components of the real world and may even seem to be necessary evils, for Perelman they are simply evils interfering with a system of pure thought and should be completely rejected. He is the epitome of the Platonic ideal; where pure ideas are concerned, any human association could only be a deeply unsettling imposition.

A journey with the good Lord

A lively Ludwig Wittgenstein tries to gently persuade Bertrand Russell of the veracity of his ideas in "Logicomix: An Epic Search for Truth"


In the autumn of 1939, as the world in its tormented restlessness perched on the edge of conflict, a tall, lanky man was slowly making his way towards the steps of a prominent American university, his ubiquitous pipe in hand. His face was one of the most recognizable in the world, his mind among the most respected minds of his time. He was slated to give a talk at this august institution, and as he approached he was accosted by protestors who pleaded to him to not give his talk and instead join them in their pacifist protests. Only two decades back the gentleman had been a famously eloquent pacifist, spending time in prison during World War 1 after invoking the ire of his countrymen. Now he hesitated for a moment, and then told the protestors that he would be happy to take up their cause, but only if they first attended his talk, a talk on the role of logic in human affairs.

Such begins "Logicomix: An Epic Search for Truth", a remarkable, unique and highly original graphic novel about Bertrand Russell and the foundations of mathematics. Who would have thought that something as profound and deep as the search for certainty in that most crystalline of human endeavors could be turned into a comic book. And yet Berkeley computer scientist Christos H. Papadimitriou and artist and author Apostolos Doxiadis have achieved this feat. Logicomix traces the history of the foundations of mathematics through the extraordinary life of perhaps its most famous and colorful proponent. The history of the foundations of mathematics is one of the most fascinating expositions of human thought in history, partly because it is the epitome of that ultimate human desire, a search for certainty in a world full of changes of fortune. The field itself is as old as the Greeks, but one would be hard pressed to find anyone else other than Lord Russell who witnessed so much development in the field, and who through his long and eventful life contributed so much to it. I thought that Logicomix, deftly making use of the modern medium of the comic book, provides the best introduction to the foundation of mathematics and its most famous philosopher that I have come across.

The comic book traces the foundations of mathematics through the life of Russell and begins with Russell retelling the story of his life to an audience in America on the eve of the Second World War. Russell's journey toward truth began as a child with a remarkable stay at Pembroke Lodge, where his grandfather, a past Prime Minister of Great Britain, held court with his puritanical and strict grandmother. The amiable and docile grandfather was stark contrast to his wife; sadly he did not live long enough to shield young Bertie from his overbearing grandmother. Privately schooling him at home through hand-picked teachers and drumming Bible lessons into his head, the grandmother actively tried to stifle every inquiry by Russell about his parents, which Russell later found was because of their rather scandalous lifestyle involving a three-way relationship which elicited howls in Victorian England. To escape from his grandmother Russell sought refuge in logic and mathematics. An inspiring teacher introduced Bertie to Euclid's theorems, and the airtight logic inherent in the structure of geometry took hold of Russell's mind like a spirit.

This refuge quickly turned into an all-consuming quest to find absolute certainty in an uncertain world. Russell became utterly fascinated by the fact that different "infinities" could be compared (as did I when I first read George Gamow's "One, two, three, infinity"). Logicomix has him visiting the great mathematicians Gottlob Frege and Georg Cantor who were responsible for crucial developments in the field. Mulling over their work in set theory and logic and inspired by Leibniz, Russell came up with Russell's paradox, a wondrous construct which haunted him to no end and further fueled his intense desire to end contradiction in mathematics. As a student at Cambridge Russell came in contact with Alfred North Whitehead. The result of his friendship and collaboration with Whitehead was one of the most famous works ever produced in the history of human thought, the Principia Mathematica, a book which Russell toiled over like a madman and which used 362 pages to prove 1 + 1 = 2. The book was partly a result of Russell listening to a talk by David Hilbert at the turn of the century in which Hilbert posed 23 famous unsolved fundamental problems in mathematics. Perhaps the loftiest goal advanced by Hilbert was to make mathematics completely self-consistent, so that there would be absolutely no contradiction and paradox and the entire grand edifice would follow precisely and logically from a few axioms. During the process Russell married and also fell for Whitehead's wife. Throughout his life Russell was an impetuous man, and like many other deep thinkers was often cruel and indifferent to his spouses and family.

The tome that Russell and Whitehead laboriously produced was so dense that nobody could critique it, and the two had to embarrassingly publish it themselves. To this day it is doubtful whether more than a handful have digested its contents; one person who Russell was convinced was the only person he had met who had read the work, was to prove his undoing. After this monumental work Russell felt a little burnt out and, as the drums of war sounded in Europe, took refuge again in pure thought and contemplation. He also became interested in social activism. He spoke against the war and went to jail for his protests. He decided on a bicycle ride that he no longer wanted to live with his wife. His greatest discovery however was yet to come.

Just before World War 1 a young man walked into his rooms at Cambridge. Ludwig Wittgenstein had an intensity and interest in philosophy that bordered on madness. A singular character of the twentieth century who gave away all his great wealth, deliberately asked to go to the front, and composed his greatest work lying in the trenches of the Great War, Wittgenstein saw the essence of philosophy and our perception of our world lying in language. With his arrival Russell started a new chapter in his life, but his admiration and buoyant enthusiasm for his disciple soon turned to chagrin when Wittgenstein who would never mince words started telling him that the entire basis of his Principia Mathematica was flawed. Deeply shaken, Russell emerged from the war more of a writer and a public intellectual, with his best years behind him

He still commanded respect of the highest degree, and was the patron saint of the famous Vienna Circle of the 1930s, whose philosophy of logical positivism strove to put the entire world into the constraints of science, and whose luminaries included Moritz Schlick, John von Neumann, and for a time as a sort of external admirer, Karl Popper. However, it was a diffident young member of the circle who was to signal the coup de grace for Bertrand Russell. In 1932, Austrian logician Kurt Gödel produced what is perhaps the single most profound and remarkable idea of the twentieth century, and perhaps of the entire history of intellectual thought. Gödel showed that even arithmetic, the basis of all mathematics and everything that follows, that purest of pure constructs, is essentially full of unprovable statements and paradoxes; it is essentially an incomplete system. Gödel's Incompleteness Theorem dealt a death knell to hundreds of years of fond hope that mathematics could be put on a complete, secure and logically perfectly consistent foundation. Along with Heisenberg's uncertainty principle and Einstein's principle of relativity, it signaled the end of certainty for human beings. In one fell swoop Gödel dashed the hopes of Hilbert, Russell and countless others away to oblivion. Right after Godel's talk, the famous polymath von Neumann simply said "It's over"; a more literal interpretation of the phrase has rarely been expressed in history. Godel's talk signaled the most significant break in Russell's life. After this, he never produced a single significant work in mathematics and spent the rest of his life campaigning against war and nuclear weapons, pontificating on everything from happiness to Christianity to marriage, writing best-selling books on all these topics, and becoming a signature symbol of the rational life for countless in the world. He remains one of the most important intellectuals in history.

So how is all of this captured in a comic book? The answer is, most impressively. There are moving passages where Russell literally goes insane trying to search for certainty in mathematics. The book also explores the troubling connection between genius and insanity, and especially between mathematical genius and insanity. Consider the evidence; Russell's son descended into schizophrenia and his brother committed suicide, Russell himself constantly worried that he was descending into madness in the great toil of his labors, Wittgenstein was nothing short of bona fide crazy and so was Georg Cantor, Gödel became so paranoid toward the end of his life in Princeton that he starved himself to death, convinced that the doctors were trying to poison him through his food. All these men were also acknowledged geniuses. Perhaps it is not a coincidence that John Nash, after he won the Nobel Prize, indicated that there was a deep connection somewhere between his schizophrenia and his remarkable mathematical achievements, accurately observing that neuroses and genius have been commonly connected with each other throughout history; according to Nash there might even be a necessary condition between genius and what we may perceive as irrational obsession.

"...rationality of thought imposes a limit on a person's concept of his relation to the cosmos. For example, a non-Zoroastrian could think of Zarathustra as simply a madman who led millions of naive followers to adopt a cult of ritual fire worship. But without his "madness" Zarathustra would necessarily have been only another of the millions or billions of human individuals who have lived and then been forgotten

Perhaps there is indeed a price that one pays for daring to soar into the highest heights of abstract human thought.

Logicomix explores this troubling association. The illustrations in it are endearing, although they could perhaps have been better (I was reminded of the marvelous artwork in "Watchmen"). Also importantly, although almost all the characters and events are real, their placement and chronology is often fictional. For instance Russell never actually met Hilbert, nor did he attend the devastating talk by Gödel. Plus, some of the dialogue is rather unlikely; for instance did Wittgenstein really call Hilbert a "bloody ass"? The book also perhaps lacked the kind of depth that I expected at first, but then it is geared toward a popular audience after all, and concepts like Russell's paradox, countable and uncountable infinity and Gödel's theorems are difficult for laymen to understand even when simplified. Nonetheless, these are minor quibbles which don't detract from the uniqueness and substance in the book.

Keeping true to its emphasis on logic, the book follows a recursive self-referential kind of theme and cycles between two stories. The major story is of Bertrand Russell and the foundations of mathematics, and the side story is the story of the writers planning the book. Both authors are Greek and their story takes a stroll, both literally and figuratively, through the scenic gardens of Athens and through the great plays of Sophocles and Aeschylus. The writers' and artists' story is interspersed with the main story line, even as they grapple with the concepts and their presentation.

The book ends with the artists and writers attending a performance of Aeschylus's The Eumenides in the great Acropolis of Athens. Aeschylus's play extolled logic and reason, qualities that Russell and many of the book's protagonists held dearer than life. In the end, logic and reason are not enough. But they are candles in the dark, threads of Ariadne, ephemeral wisps of the human mind that keep us from wandering too far from the straight line. For this we can be grateful to the men who spent their lives struggling for certainty, and we can continue to join them in their quest.

ANALOGIES BETWEEN ANALOGIES



A post about metaphors and analogies on a fellow blogger's page reminded me of a quote by a remarkable mathematician whose name is known only to afficionados now, but who stands in the front rank of brilliant mathematicians and physicists of the twentieth century- Stanislaw Ulam. Here's a quote from him about analogies:

"Great scientists see analogies between theorems or theories. The very best ones see analogies between analogies"

Indeed. And Stan Ulam could very well put himself into the second category, although his modest nature would have not made him do so.

Ulam was born in Poland and grew up in a romantic time in the 20s and 30s, when great discoveries in mathematics and physics were being made in small, enchanting roadside cafes by small groups of people working intensely together. One of those, the Scottish Cafe in Lwow, Poland, was a focal point for meeting of great minds, the best pure mathematicians in Europe. Equations used to be scribbled on tables there, and the waiters were told never to erase them. Marathon sessions used to be common, fueled by black coffee, and interrupted only by occasional meals and trips to the bathroom; one non-stop session lasted 17 hours. The mathematician Rota said this about Ulam's fascinating mind:

"Ulam's mind is a repository of thousands of stories, tales, jokes, epigrams, remarks, puzzles, tounge-twisters, footnotes, conclusions, slogans, formulas, diagrams, quotations, limericks, summaries, quips, epitaphs, and headlines. In the course of a normal conversation he simply pulls out of his mind the fifty-odd relevant items, and presents them in linear succession. A second-order memory prevents him from repeating himself too often before the same public."

Ulam was invited to visit the US as a lecturer several times during the 1930s by his fellow famous emigre from Europe, and admittedly the smartest man of his generation; John Von Neumann. Within a short time, the romantic days were at a tragic end. Ulam held out in Poland much longer than many other brilliant European scientists and mathematicians, and in 1939, on the eve of World War 2, escaped to America with his brother Adam. The rest of the Ulam family perished in the Holocaust.

After coming to the US, Ulam was secretly invited to join the Manhattan Project in Los Alamos, where he was known to be a problem solver and jovial team worker. In Los Alamos, he tried to recreate the idyllic atmosphere of his young years in Europe by installing a coffee machine outside his office where scientists could talk shop. You can get to see Ulam in The Day after Trinity. Here is a photo of three prodigies from those days, (From L to R) Ulam, Richard Feynman, and John Von Neumann



While at Los Alamos, Ulam made what was probably the most important contribution of his career- the Monte Carlo method, a way of calculating the result of complex processes through random numbers. This method is now so important and deep-rooted in physics, chemistry, and engineering, that many students have forgotten that somebody invented it. The method is now implemented as a black box in many computer programs, such as those which I use for calculating the structure of organic molecules, and so people tend to sometimes use it without knowing that they are using it.

In 1946, Ulam suffered an attack of encephalitis; he could not remember events after the attack, and after the operation, federal agents asked him questions to make sure that he may not have given away atomic secrets during his loss of recollection. After the operation, Ulam seemed to some to become even more brilliant than he had been before.

However, Ulam probably became best-known to a greater audience through his participation in the development of the hydrogen bomb. After the war, he and fellow scientist Cornelius Everett embarked on a series of tedious calculations to prove that the then accepted and widely touted design of the hydrogen bomb would not work. This was a significant result, as President Harry Truman had been earlier prodded to announce a crash effort to develop the bomb based on this design. WIthin a short time however, Ulam came to the essential breakthrough that encouraged the infamous Edward Teller to develop the most widely used design of the h-bomb. The breakthrough involved separating the fission and fusion parts of the weapons, and using compression from the fission bomb to activate the fusion bomb. After this design was invented, everybody assumed that the Soviets were doing it too, and the program was purused with vigour. Every country afterwards that developed thermonuclear weapons has used this so-called "Teller-Ulam" design or a variant of it.

The imperious Teller essentially took much of the credit for the invention, and later tried to expunge Ulam's name from that part of history. Hans Bethe liked to joke that Ulam was really the "father of the h-bomb" while Teller was the mother since he carried the baby for so long. Ulam for his own part, an unassuming and docile man, stayed away from these disputes, when he rightly could have done more for asserting his claim to fame. Ulam and Teller parted ways after the discovery, Ulam returning to his world of pure science, and Teller becoming increasingly belligerent and disliked by his fellow scientists, and pushing for new and "better" nuclear weapons, thus becoming what Richard Rhodes calls the "Richard Nixon of American science". Till the end of his life in 2004 at the age of 95, he gave hawkish and wrong advice to Presidents (famously about "Star Wars" to Ronald Reagan) and believed that he was doing the right thing in advancing peace by building more hydrogen bombs.

During his professional career, Ulam spent time at the Universities of Wisconsin, UCLA, and Boulder. His wife, Francoise, was always a loving support as well as an admirer of him. She remembers one defining moment from their lives, when she found her husband staring out the window after he had had the idea for a successful hydrogen bomb. "I have just discovered the idea that will change history", he presciently said.

Ulam died in 1984. An astonishingly versatile scientist, he had been equally at home with the most abstruse reaches of set theory and with the details of thermonuclear fusion. His memoirs, Adventures of a Mathematician, paints a fascinating and delightful portrait of the golden age of physics and mathematics, as well as the dawn of the nuclear age. In this book, we get to hear anecdotes about famous mathematicians and physicists, many of whom were good friends of Ulam.

Ulam once said:

"It is still an unending source of surprise for me how a few scribbles on a blackboard or on a piece of paper can change the course of human affairs."

Ulam was certainly one of the select few who scribbled.