Frank Rill Chen: Mathematics

In honor of John von Neumann

Thu, 08 Feb 2007 00:58:54 GMT

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

Anyone who considers arithmetical methods of producing random numbers is, of course, in a state of sin. (Quoted in Knuth, 1968, Vol. 2, also in Goldstine, 1972, p. 297.)

John von Neumann was born János von Neumann. He was called Jancsi as a child, a diminutive form of János, then later he was called Johnny in the United States. His father, Max Neumann, was a top banker and he was brought up in a extended family, living in Budapest where as a child he learnt languages from the German and French governesses that were employed. Although the family were Jewish, Max Neumann did not observe the strict practices of that religion and the household seemed to mix Jewish and Christian traditions.

It is also worth explaining how Max Neumann's son acquired the "von" to become János von Neumann. In 1913 Max Neumann purchased a title but did not change his name. His son, however, used the German form von Neumann where the "von" indicated the title.

As a child von Neumann showed he had an incredible memory. Poundstone, in [8], writes:-

At the age of six, he was able to exchange jokes with his father in classical Greek. The Neumann family sometimes entertained guests with demonstrations of Johnny's ability to memorise phone books. A guest would select a page and column of the phone book at random. Young Johnny read the column over a few times, then handed the book back to the guest. He could answer any question put to him (who has number such and such?) or recite names, addresses, and numbers in order.

In 1911 von Neumann entered the Lutheran Gymnasium. The school had a strong academic tradition which seemed to count for more than the religious affiliation both in the Neumann's eyes and in those of the school. His mathematics teacher quickly recognised von Neumann's genius and special tuition was put on for him. The school had another outstanding mathematician one year ahead of von Neumann, namely Eugene Wigner.

World War I had relatively little effect on von Neumann's education but, after the war ended, Béla Kun controlled Hungary for five months in 1919 with a Communist government. The Neumann family fled to Austria as the affluent came under attack. However, after a month, they returned to face the problems of Budapest. When Kun's government failed, the fact that it had been largely composed of Jews meant that Jewish people were blamed. Such situations are devoid of logic and the fact that the Neumann's were opposed to Kun's government did not save them from persecution.

In 1921 von Neumann completed his education at the Lutheran Gymnasium. His first mathematics paper, written jointly with Fekete the assistant at the University of Budapest who had been tutoring him, was published in 1922. However Max Neumann did not want his son to take up a subject that would not bring him wealth. Max Neumann asked Theodore von Kármán to speak to his son and persuade him to follow a career in business. Perhaps von Kármán was the wrong person to ask to undertake such a task but in the end all agreed on the compromise subject of chemistry for von Neumann's university studies.

Hungary was not an easy country for those of Jewish descent for many reasons and there was a strict limit on the number of Jewish students who could enter the University of Budapest. Of course, even with a strict quota, von Neumann's record easily won him a place to study mathematics in 1921 but he did not attend lectures. Instead he also entered the University of Berlin in 1921 to study chemistry.

Von Neumann studied chemistry at the University of Berlin until 1923 when he went to Zurich. He achieved outstanding results in the mathematics examinations at the University of Budapest despite not attending any courses. Von Neumann received his diploma in chemical engineering from the Technische Hochschule in Zürich in 1926. While in Zurich he continued his interest in mathematics, despite studying chemistry, and interacted with Weyl and Pólya who were both at Zurich. He even took over one of Weyl's courses when he was absent from Zurich for a time. Pólya said [18]:-

Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper.

Von Neumann received his doctorate in mathematics from the University of Budapest, also in 1926, with a thesis on set theory. He published a definition of ordinal numbers when he was 20, the definition is the one used today.

Von Neumann lectured at Berlin from 1926 to 1929 and at Hamburg from 1929 to 1930. However he also held a Rockefeller Fellowship to enable him to undertake postdoctoral studies at the University of Göttingen. He studied under Hilbert at Göttingen during 1926-27. By this time von Neumann had achieved celebrity status [8]:-

By his mid-twenties, von Neumann's fame had spread worldwide in the mathematical community. At academic conferences, he would find himself pointed out as a young genius.

Veblen invited von Neumann to Princeton to lecture on quantum theory in 1929. Replying to Veblen that he would come after attending to some personal matters, von Neumann went to Budapest where he married his fiancée Marietta Kovesi before setting out for the United States. In 1930 von Neumann became a visiting lecturer at Princeton University, being appointed professor there in 1931.

Between 1930 and 1933 von Neumann taught at Princeton but this was not one of his strong points [8]:-

His fluid line of thought was difficult for those less gifted to follow. He was notorious for dashing out equations on a small portion of the available blackboard and erasing expressions before students could copy them.

In contrast, however, he had an ability to explain complicated ideas in physics [3]:-

For a man to whom complicated mathematics presented no difficulty, he could explain his conclusions to the uninitiated with amazing lucidity. After a talk with him one always came away with a feeling that the problem was really simple and transparent.

He became one of the original six mathematics professors (J W Alexander, A Einstein, M Morse, O Veblen, J von Neumann and H Weyl) in 1933 at the newly founded Institute for Advanced Study in Princeton, a position he kept for the remainder of his life.

During the first years that he was in the United States, von Neumann continued to return to Europe during the summers. Until 1933 he still held academic posts in Germany but resigned these when the Nazis came to power. Unlike many others, von Neumann was not a political refugee but rather he went to the United States mainly because he thought that the prospect of academic positions there was better than in Germany.

In 1933 von Neumann became co-editor of the Annals of Mathematics and, two years later, he became co-editor of Compositio Mathematica. He held both these editorships until his death.

Von Neumann and Marietta had a daughter Marina in 1936 but their marriage ended in divorce in 1937. The following year he married Klára Dán, also from Budapest, whom he met on one of his European visits. After marrying, they sailed to the United States and made their home in Princeton. There von Neumann lived a rather unusual lifestyle for a top mathematician. He had always enjoyed parties [8]:-

Parties and nightlife held a special appeal for von Neumann. While teaching in Germany, von Neumann had been a denizen of the Cabaret-era Berlin nightlife circuit.

Now married to Klára the parties continued [18]:-

The parties at the von Neumann's house were frequent, and famous, and long.

Ulam summarises von Neumann's work in [35]. He writes:-

In his youthful work, he was concerned not only with mathematical logic and the axiomatics of set theory, but, simultaneously, with the substance of set theory itself, obtaining interesting results in measure theory and the theory of real variables. It was in this period also that he began his classical work on quantum theory, the mathematical foundation of the theory of measurement in quantum theory and the new statistical mechanics.

His text Mathematische Grundlagen der Quantenmechanik (1932) built a solid framework for the new quantum mechanics. Van Hove writes in [36]:-

Quantum mechanics was very fortunate indeed to attract, in the very first years after its discovery in 1925, the interest of a mathematical genius of von Neumann's stature. As a result, the mathematical framework of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analysed by one single man in two years (1927-1929).

Self-adjoint algebras of bounded linear operators on a Hilbert space, closed in the weak operator topology, were introduced in 1929 by von Neumann in a paper in Mathematische Annalen . Kadison explains in [22]:-

His interest in ergodic theory, group representations and quantum mechanics contributed significantly to von Neumann's realisation that a theory of operator algebras was the next important stage in the development of this area of mathematics.

Such operator algebras were called "rings of operators" by von Neumann and later they were called W*-algebras by some other mathematicians. J Dixmier, in 1957, called them "von Neumann algebras" in his monograph Algebras of operators in Hilbert space (von Neumann algebras). In the second half of the 1930's and the early 1940s von Neumann, working with his collaborator F J Murray, laid the foundations for the study of von Neumann algebras in a fundamental series of papers.

However von Neumann is know for the wide variety of different scientific studies. Ulam explains [35] how he was led towards game theory:-

Von Neumann's awareness of results obtained by other mathematicians and the inherent possibilities which they offer is astonishing. Early in his work, a paper by Borel on the minimax property led him to develop ... ideas which culminated later in one of his most original creations, the theory of games.

In game theory von Neumann proved the minimax theorem. He gradually expanded his work in game theory, and with co-author Oskar Morgenstern, he wrote the classic text Theory of Games and Economic Behaviour (1944).

Ulam continues in [35]:-

An idea of Koopman on the possibilities of treating problems of classical mechanics by means of operators on a function space stimulated him to give the first mathematically rigorous proof of an ergodic theorem. Haar's construction of measure in groups provided the inspiration for his wonderful partial solution of Hilbert's fifth problem, in which he proved the possibility of introducing analytical parameters in compact groups.

In 1938 the American Mathematical Society awarded the Bôcher Prize to John von Neumann for his memoir Almost periodic functions and groups. This was published in two parts in the Transactions of the American Mathematical Society, the first part in 1934 and the second part in the following year. Around this time von Neumann turned to applied mathematics [35]:-

In the middle 30's, Johnny was fascinated by the problem of hydrodynamical turbulence. It was then that he became aware of the mysteries underlying the subject of non-linear partial differential equations. His work, from the beginnings of the Second World War, concerns a study of the equations of hydrodynamics and the theory of shocks. The phenomena described by these non-linear equations are baffling analytically and defy even qualitative insight by present methods. Numerical work seemed to him the most promising way to obtain a feeling for the behaviour of such systems. This impelled him to study new possibilities of computation on electronic machines ...

Von Neumann was one of the pioneers of computer science making significant contributions to the development of logical design. Shannon writes in [29]:-

Von Neumann spent a considerable part of the last few years of his life working in [automata theory]. It represented for him a synthesis of his early interest in logic and proof theory and his later work, during World War II and after, on large scale electronic computers. Involving a mixture of pure and applied mathematics as well as other sciences, automata theory was an ideal field for von Neumann's wide-ranging intellect. He brought to it many new insights and opened up at least two new directions of research.

He advanced the theory of cellular automata, advocated the adoption of the bit as a measurement of computer memory, and solved problems in obtaining reliable answers from unreliable computer components.

During and after World War II, von Neumann served as a consultant to the armed forces. His valuable contributions included a proposal of the implosion method for bringing nuclear fuel to explosion and his participation in the development of the hydrogen bomb. From 1940 he was a member of the Scientific Advisory Committee at the Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland. He was a member of the Navy Bureau of Ordnance from 1941 to 1955, and a consultant to the Los Alamos Scientific Laboratory from 1943 to 1955. From 1950 to 1955 he was a member of the Armed Forces Special Weapons Project in Washington, D.C. In 1955 President Eisenhower appointed him to the Atomic Energy Commission, and in 1956 he received its Enrico Fermi Award, knowing that he was incurably ill with cancer.

Eugene Wigner wrote of von Neumann's death [18]:-

When von Neumann realised he was incurably ill, his logic forced him to realise that he would cease to exist, and hence cease to have thoughts ... It was heartbreaking to watch the frustration of his mind, when all hope was gone, in its struggle with the fate which appeared to him unavoidable but unacceptable.

In [5] von Neumann's death is described in these terms:-

... his mind, the amulet on which he had always been able to rely, was becoming less dependable. Then came complete psychological breakdown; panic, screams of uncontrollable terror every night. His friend Edward Teller said, "I think that von Neumann suffered more when his mind would no longer function, than I have ever seen any human being suffer."
Von Neumann's sense of invulnerability, or simply the desire to live, was struggling with unalterable facts. He seemed to have a great fear of death until the last... No achievements and no amount of influence could save him now, as they always had in the past. Johnny von Neumann, who knew how to live so fully, did not know how to die.

It would be almost impossible to give even an idea of the range of honours which were given to von Neumann. He was Colloquium Lecturer of the American Mathematical Society in 1937 and received the its Bôcher Prize as mentioned above. He held the Gibbs Lectureship of the American Mathematical Society in 1947 and was President of the Society in 1951-53.

He was elected to many academies including the Academia Nacional de Ciencias Exactas (Lima, Peru), Academia Nazionale dei Lincei (Rome, Italy), American Academy of Arts and Sciences (USA), American Philosophical Society (USA), Instituto Lombardo di Scienze e Lettere (Milan, Italy), National Academy of Sciences (USA) and Royal Netherlands Academy of Sciences and Letters (Amsterdam, The Netherlands).

Von Neumann received two Presidential Awards, the Medal for Merit in 1947 and the Medal for Freedom in 1956. Also in 1956 he received the Albert Einstein Commemorative Award and the Enrico Fermi Award mentioned above.

Peierls writes [3]:-

He was the antithesis of the "long-haired" mathematics don. Always well groomed, he had as lively views on international politics and practical affairs as on mathematical problems.

Article by: J J O'Connor and E F Robertson

Grigori Perelman

Wed, 30 Aug 2006 15:11:04 GMT

只要提出的证明本身是对的，就不需要其他认可。

——格里高利.佩雷尔曼（Grigori Perelman）

“只要我不惹人注目，我还有权选择做某些事情，如今我成为一位非常有名的人，我不能像宠物般什么也不说，因此我决定退出。”

——格里高利.佩雷尔曼（Grigori Perelman）

"It looks pretty clear that whether or not the complete proof has been demonstrated, what Perelman did is a major breakthrough," said Arthur Jaffe, a mathematician at Harvard University.

Even by the standards of troubled maths virtuosos such as John Nash, portrayed in the film A Beautiful Mind, Dr Perelman is widely considered "unconventional".

He has already publicly refused a $1m (£520,000) prize offered by a private maths research institute in the US if his proof turns out to be correct.

"He's totally focused on mathematics," said Prof Jaffe. "He does not worry at this stage of his life about personal things like wealth and position. But he carries it to an extreme which people might describe as a little crazy"

牛津大学马库斯-杜-桑托伊(Marcus Du Sautoy)教授说：“格里高利-佩雷尔曼有几分故意疏远数学界的意思，他因为数学成就一生的梦想，如果这样的话，就太遗憾了。钱财乃身外之物，他对此没有一点兴趣。在他看来，最大的褒奖莫过于验证自己的破解之道。”

Poincaré Conjecture

If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is "simply connected," but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

所谓的庞加莱猜想是指：在任何一个封闭的三维空间中，只要所有的封闭曲线都可以收缩成一点，那么这个空间就一定是一个三维圆球。它涉及多维空间几何学，是通向拓扑学领域的一把钥匙。在研究拓扑学的学者看来，一只兔子也是一个球体。

Is Mathematics Beautiful?

Mon, 28 Aug 2006 07:12:03 GMT

Bertrand Russell (1872-1970), Autobiography, George Allen and Unwin Ltd, 1967, v1, p158

It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect but true.

Bertrand Russell (1872-1970), The Study of Mathematics

Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.

Aristotle (384 B.C.-322 B.C.), Poetics

Beauty depends on size as well as symmetry.

J.H.Poincare (1854-1912), (cited in H.E.Huntley, The Divine Proportion, Dover, 1970)

The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.

J.Bronowski, Science and Human Values, Pelican, 1964.

Mathematics in this sense is a form of poetry, which has the same relation to the prose of practical mathematics as poetry has to prose in any other language. The element of poetry, the delight of exploring the medium for its own sake, is an essential ingredient in the creative process.

J.W.N.Sullivan (1886-1937), Aspects of Science, 1925.

Mathematics, as much as music or any other art, is one of the means by which we rise to a complete self-consciousness. The significance of Mathematics resides precisely in the fact that it is an art; by informing us of the nature of our own minds it informs us of much that depends on our minds.

G. H. Hardy (1877 - 1947), A Mathematician's Apology, Cambridge University Press, 1994.

The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics.

Lawrence University catalog, Cited in Essays in Humanistic Mathematics, Alvin White, ed, MAA, 1993

Born of man's primitive urge to seek order in his world, mathematics is an ever-evolving language for the study of structure and pattern. Grounded in and renewed by physical reality, mathematics rises through sheer intellectual curiosity to levels of abstraction and generality where unexpected, beautiful, and often extremely useful connections and patterns emerge. Mathematics is the natural home of both abstract thought and the laws of nature. It is at once pure logic and creative art.

I.Newton, Letter to H.Oldenburg, the Secretary of the Royal Society, October 24, 1676, in A Source Book in Mathematics, D. J. Struik, ed, Princeton University Press, 1990

I can hardly tell with what pleasure I have read the letters of those very distinguished men Leibniz and Tschirnhaus. Leibniz's method for obtaining convergent series is certainly very elegant...

Jane Muir, Of Men & Numbers, Dover, 1996.

Gauss: You have no idea how much poetry there is in the calculation of a table of logarithms!

F.Dyson, in Nature, March 10, 1956

Characteristic of Weyl was an aesthetic sense which dominated his thinking on all subjects. He once said to me, half-joking, "My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful." (Herman Weyl (1885-1955))

O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956

To Goethe again we owe the profound saying: "the mathematician is only complete in so far as he feels within himself the beauty of the true."

O. Spengler, in J. Newman, The World of Mathematics, Simon & Schuster, 1956

"A mathematician," said old Weierstrass, "who is not at the same time a bit of a poet will never be a full mathematician."

Jakob Bernoulli, Tractatus de Seriebus Infinitis, 1689 (quoted in From Five Fingers to Infinity, F.J.Swetz (ed), Open Court, 1996)

So the soul of immensity dwells in minutia.
And in narrowest limits no limits inhere.
What joy to discern the minute in infinity!
The vast to perceive in the small, what divinity!

S.Lang, The Beauty of Doing Mathematics, Springer-Verlag, 1985

Last time, I asked: "What does mathematics mean to you?" And some people answered: "The manipulation of numbers, the manipulation of structures." And if I had asked what music means to you, would you have answered: "The manipulation of notes?"

Kurt Gödel

Wed, 23 Aug 2006 16:06:30 GMT

This year marks the 100th anniversary of the birth of Kurt Gödel (1906-1978), the foremost mathematical logician of the twentieth century. Looking back over that century in the year 2000, TIME magazine included Gödel among its top 100 most influential thinkers. Gödel was associated with the Institute for Advanced Study from his first visit in the academic year 1933–34, until his death in 1978. He was Professor in the School of Mathematics from 1953 until 1976, when he became Professor Emeritus.

The Early Years

Kurt Friedrich Gödel was born on April 28, 1906, in what is now Brno in the Czech Republic. His father, Rudolf Gödel, was originally from Vienna; his mother, Marianne Handschuh, came from the German Rhineland. Rudolf Gödel managed and was part owner of one of Brno’s major textile companies and the family lived in middle-class comfort with servants and a governess for Kurt and his older brother, Rudolf, born in 1902.

The young Gödel was known affectionately by the nickname, der Herr Warum (Mr. Why). In Logical Dilemmas: The Life and Work of Kurt Gödel, biographer John W. Dawson, Jr. describes him as “an earnestly serious, bright, and inquisitive child who was sensitive, often withdrawn and pre-occupied, and who, already at an early age, exhibited certain signs of emotional instability.” At age eight, after reading a medical book, Gödel became convinced that he had a weak heart, a possible complication of the rheumatic fever that he had recovered from at age six. Hypochondriac concerns for his health would become a lifelong preoccupation.

Academic Life

In 1923, Gödel enrolled in the University of Vienna with the intention of studying physics. After attending lectures on number theory by the charismatic professor Philip Furtwängler, brother of famed German conductor Wilhelm Furtwängler, he switched to mathematics. Furtwängler was paralyzed from the neck down and lectured from his wheel chair with an assistant writing his formulae on the board. He made an impression much like that of Stephen Hawking today.

As a student, Gödel attended meetings of what would later become Der Wiener Kreis (Vienna Circle), a group of mainly philosophers who met to discuss foundational problems and who were inspired by Ludwig Wittgenstein’s Tractatus Logico-Philosophicus. The group focused on questions of language and meaning and logical relations such as entailment, and originated Logical Positivism. Led by Moritz Schlick, a professor at the University who was later murdered by a deranged former student as he climbed the steps to the main lecture hall (1936), its members included Rudolf Carnap, Otto Neurath, Carl Hempel, Hans Reichenbach, and others.

In 1927, at age 21, Gödel met dancer Adele Nimbursky (née Porkert), in the Viennese night club, Der Nachtfalter (The Moth). Because Adele had been married and was six years older than Kurt, his parents disapproved of the match (his mother Marianne was 14 years younger than his father, Rudolf). This was the second time they had disapproved of Kurt’s liaison with an older woman and it was not until the autumn of 1938 that Kurt and Adele were married.

Gödel pursued studies in mathematics and logic with Hans Hahn and Karl Menger. His doctoral thesis was completed in 1929, the year in which his father Rudolf died, leaving the family in comfortable circumstances. Gödel’s mother bought an apartment in Vienna where she lived with both of her sons and enjoyed the cultural life of the city, especially musical theater. Gödel developed a lifelong love of operetta.

After receiving his doctorate, Gödel became a privatdozent (unpaid lecturer) at the University of Vienna. Like many of the young scholars who later found their way to the Institute for Advanced Study from Europe in the 1930s, Kurt Gödel was brilliant. Unlike many, he was not Jewish, although he moved in circles of Jewish intellectuals and was sometimes thought to be Jewish. He had once been attacked as such by a gang of youths while walking with Adele on a street in Vienna. During the 1930s it was not unusual for university students who were Jewish or had socialist leanings to be forcibly removed from classes. Many of Gödel’s contemporaries were fleeing Europe.

Incompleteness Theorems

In 1931, Gödel published results in formal logic that are considered landmarks of 20th-century mathematics. Gödel demonstrated, in effect, that hopes of reducing mathematics to an axiomatic system, as envisioned by mathematicians and philosophers at the turn of the 20th century, were in vain. His findings put an end to logicist efforts such as those of Bertrand Russell and Alfred North Whitehead and demonstrated the severe limitations of David Hilbert’s formalist program for arithmetic.

In his introduction to his 1931 paper, Gödel stated: “It is well known that the development of mathematics in the direction of greater precision has led to the formalization of extensive mathematical domains, in the sense that proofs can be carried out according to a few mechanical rules.... It is reasonable therefore to make the conjecture that these axioms and rules of inference are also sufficient to decide all mathematical questions, which can be formally expressed in the given systems. In what follows it will be shown that this is not the case.”

By the age of 25 Kurt Gödel had produced his famous “Incompleteness Theorems.” His fundamental results showed that in any consistent axiomatic mathematical system there are propositions that cannot be proved or disproved within the system and that the consistency of the axioms themselves cannot be proved. In addition to his proof of the incompleteness of formal number theory, Gödel published proofs of the relative consistency of the axiom of choice and the generalized continuum hypothesis (1938, 1940). His findings strongly influenced the (later) discovery that a computer can never be programmed to answer all mathematical questions.

In 1938, Gödel’s application for a paid position at the University of Vienna was turned down. Fearing conscription into the German army, he applied for a visa to the United States. In late 1939, Kurt and Adele fled Nazi Germany, traveling via the trans-Siberian railway and ship to San Francisco, where they arrived on March 4, 1940. They settled in Princeton where Gödel’s position at the Institute was renewed annually until 1946, when he became a permanent Member until appointed to the Faculty.

The Later Years

After suffering from severe bleeding from a duodenal ulcer, Gödel maintained an extremely strict diet that led to severe weight loss. By several accounts, Adele Gödel was a loving support to her husband, whom she addressed as strammer bursche (strapping lad). Mathematical logician Georg Kreisel, a Member in the School of Mathematics (1955–57), records their relationship in Biographical Memoirs of Fellows of the Royal Society [1980, Volume 26]: “I visited them quite often in the fifties and sixties. It was a revelation to see him relax in her company. She had little formal education, but a real flair for the mot juste, which her somewhat critical mother-in-law eventually noticed too, and a knack for amusing and apparently quite spontaneous twists on a familiar ploy: to invent—at least, at the time—far-fetched grounds for jealousy. On one occasion, she painted the I.A.S., which she usually called Altersversorgungsheim (home for old-age pensioners), as teeming with pretty girl students who queued up at the office doors of permanent professors. Gödel was very much at ease with her style.”

When Gödel became convinced that he was being poisoned, Adele became his food taster. His digestive ailments and, particularly, his refusal to eat led ultimately to his death on January 14, 1978. He died in Princeton at age 72 and is buried in the Princeton Cemetery.

Gödel's Awards

One of the most significant acknowledgements of Gödel's accomplishmentscame in 1974, when he received the National Medal of Science in the discipline of mathematics and computer science from President Ford in a ceremony at the White House.

The award citation read: “For laying the foundation for today’s flourishing study of mathematical logic.”

Prior to the National Medal of Science, Gödel received the Institute’s Einstein Award in 1951, which consisted of a gold medal (pictured) and the sum of $15,000. The gift of Institute Trustee Lewis L. Strauss, it was presented to Gödel by Einstein at a ceremony in Princeton. Following this award, several other accolades came to Gödel, including honorary doctorates from Yale, Harvard, and Rockefeller universities and from Amherst College. Gödel was a Member of the National Academy of Sciences of the United States, a Foreign Member of the Royal Society of London, a Corresponding Member of the Institut de France, a Corresponding Fellow of the British Academy, and an Honorary Member of the London Mathematical Society.

Quotations by Proclus

Fri, 04 Aug 2006 06:23:58 GMT

According to most accounts, geometry was first discovered among the Egyptians, taking its origin from the measurement of areas. For they found it necessary by reason of the flooding of the Nile, which wiped out everybody's proper boundaries. Nor is there anything surprising in that the discovery both of this and of the other sciences should have had its origin in a practical need, since everything which is in process of becoming progresses from the imperfect to the perfect.
On Euclid

The Pythagoreans considered all mathematical science to be divided into four parts: one half they marked off as concerned with quantity, the other half with magnitude; and each of these they posited as twofold. A quantity can be considered in regard to its character by itself or in relation to another quantity, magnitudes as either stationary or in motion. Arithmetic, then, studies quantity as such, music the relations between quantities, geometry magnitude at rest, spherics magnitude inherently moving.
A Commentary on the First Book of Euclid's Elements

Wherever there is number, there is beauty.
Quoted in M Kline, Mathematical Thought from Ancient to Modern Times

This, therefore, is mathematics:
she reminds you of the invisible form of the soul;
she gives light to her own discoveries;
she awakens the mind and purifies the intellect;
she brings light to our intrinsic ideas;
she abolishes oblivion and ignorance which are ours by birth.
Quoted in M Kline, Mathematical Thought from Ancient to Modern Times

It is well known that the man who first made public the theory of irrationals perished in a shipwreck in order that the inexpressible and unimaginable should ever remain veiled. And so the guilty man, who fortuitously touched on and revealed this aspect of living things, was taken to the place where he began and there is for ever beaten by the waves.
Scholium to Book X of Euclid V.

http://www.goddess-athena.org/Encyclopedia/Friends/Proclus/

http://www-history.mcs.st-and.ac.uk/Mathematicians/Proclus.html