Mapping the Universe 2 page

My chief deficiency was arithmetic. Here my understanding was far beyond my manipulation, which was definitely poor. My father saw quite correctly that one of my chief difficulties was that manipulative drill bored me. He decided to take me out of school and put me on algebra instead of arithmetic, with the purpose of offering a greater challenge and stimulus to my imagination.

From this time on Wiener's father took over his education and he made rapid progress for so young a child. However, Wiener had problems relating to his movements and was obviously very clumsy. This stemmed partly from poor coordination but also partly for poor eyesight. Advised by a doctor to stop reading for six months to allow his eyes to recover, he still had regular lessons from his father who now taught him to do mathematics in his head. After the six months were up Wiener went back to reading but he had developed some fine mental skills during this period which he retained all his life.

In the autumn of 1903, at age nine, he was sent to school again, this time to Ayer High School. The school agreed to experiment and to find the right level for Wiener who was soon put into senior third year class with pupils who were seven years older than he was. The school only formed part of his education, however, for his father continued to coach him. He graduated in 1906 from Ayer at the age of eleven and celebrated with his eighteen year old fellow students:-

I owe a great deal to my Ayer friends. I was given a chance to go through some of the gawkiest stages of growing up in an atmosphere of sympathy and understanding.

In September 1906, still only eleven years old, Wiener entered Tufts College. Socially a child, he was an adult in educational terms so his student days were not easy ones. Although taking various science courses, he took a degree in mathematics. Wiener's father continued to coach him in mathematics showing complete mastery of undergraduate level topics. In 1909 Wiener graduated from Tufts at age fourteen and entered Harvard to begin graduate studies.

Rather against his father's advice, Wiener began graduate studies in zoology at Harvard. However things did not go too well and by the end of a year a decision was taken, partly by Wiener partly by his father, that he would change topic to philosophy. Having won a scholarship to Cornell he entered in 1910 to begin graduate studies in philosophy. Taking mathematics and philosophy courses, Wiener did not have a successful year and before it was finished his father had made the necessary arrangements to return to Harvard to continue philosophy.

Back at Harvard Wiener was strongly influenced by the fine teaching of Edward Huntington on mathematical philosophy. He received his Ph.D. from Harvard at the age of 18 with a dissertation on mathematical logic supervised by Karl Schmidt. From Harvard Wiener went to Cambridge, England, to study under Russell who told him that in order to study the philosophy of mathematics he needed to know more mathematics so he attended courses by G H Hardy. In 1914 he went to Göttingen to study differential equations under Hilbert, and also attended a group theory course by Edmund Landau. He was influence by Hilbert, Landau and Russell but also, perhaps to an even greater degree, by Hardy. At Göttingen he learned that:-

... mathematics was not only a subject to be done in the study but one to be discussed and lived with.

Wiener returned to the United States a couple of days before the outbreak of World War I, but returned to Cambridge to study further with Russell. Back in the United States he taught philosophy courses at Harvard in 1915, worked for a while for the General Electric Company, then joined Encyclopedia Americana as a staff writer in Albany. While working there he received an invitation from Veblen to undertake war work on ballistics at the Aberdeen Proving Ground in Maryland. Taking about mathematics with his fellow workers while undertaking this war work revived his interest in mathematics. At the end of the war Osgood told him of a vacancy at MIT and he was appointed as an instructor in mathematics.

His first mathematical work at MIT led him to examine Brownian motion. In fact, as Wiener explained, this first work would provide a connecting thread through much of his later studies:-

... this study introduced me to the theory of probability. Moreover, it led me very directly to the periodogram, and to the study of forms of harmonic analysis more general than the classical Fourier series and Fourier integral. All these concepts have combined with the engineering preoccupations of a professor of the Mathematical Institute of Technology to lead me to make both theoretical and practical advances in the theory of communication, and ultimately to found the discipline of cybernetics, which is in essence a statistical approach to the theory of communication. Thus, varied as my scientific interests seem to be, there has been a single thread connecting all of them from my first mature work ...

He attended the International Congress of Mathematicians at Strasbourg in 1920 and while there worked with Fréchet. He returned to Europe frequently in the next few years, visiting mathematicians in England, France and Germany. Especially important was his contacts with Paul Lévy and with Göttingen where his work was seen to have important connections with quantum mechanics. This led to a collaboration with Born.

In 1926 Wiener married Margaret Engemann, and after their marriage Wiener set off for Europe as a Guggenheim scholar. After visiting Hardy in Cambridge he returned to Göttingen where his wife joined him after completing her teaching duties in modern languages at Juniata College in Pennsylvania. Another important year in Wiener's mathematical development was 1931-32 which he spent mainly in England visiting Hardy at Cambridge. There he gave a lecture course on his own contributions to the Fourier integral but Cambridge also provided a base from where he was able to visit many mathematical colleagues on the Continent. Among these were Blaschke, Menger and Frank who invited him to make a visit, while he also met Hahn, Artin and Gödel.

Wiener's papers were hard to read. Sometimes difficult results appeared with hardly a proof as if they were obvious to Wiener, while at other times he would give a lengthy proof of a triviality. Freudenthal writes:-

All too often Wiener could not resist the temptation to tell everything that cropped up in his comprehensive mind, and he often had difficulty in separating the relevant mathematics neatly from its scientific and social implications and even from his personal experiences. The reader to whom he appears to be addressing himself seems to alternate in a random order between the layman, the undergraduate student of mathematics, the average mathematician, and Wiener himself.

Despite the style of his papers, Wiener contributed some ideas of great importance. We have already mentioned above his work in 1921 in Brownian motion. He introduced a measure in the space of one dimensional paths which brings in probability concepts in a natural way. From 1923 he investigated Dirichlet's problem, producing work which had a major influence on potential theory.

Wiener's mathematical ideas were very much driven by questions that were put to him by his engineering colleagues at MIT. These questions pushed him to generalise his work on Browian motion to more general stochastic processes. This in turn led him to study harmonic analysis in 1930. His work on generalised harmonic analysis led him to study Tauberian theorems in 1932 and his contributions on this topic won him the Bôcher Prize in 1933. He received the prize from the American Mathematical Society for his memoir Tauberian theorems published in Annals of Mathematics in the previous year. The work on Tauberian theorems naturally led him to study the Fourier transform and he published The Fourier Integral, and Certain of Its Applications (1933) and Fourier Transforms in 1934.

Wiener had an extraordinarily wide range of interests and contributed to many areas in addition to those we have mentioned above including communication theory, cybernetics (a term he coined), quantum theory and during World War II he worked on gunfire control. It is probably this latter work which motivated his invention of the new area of cybernetics which he described in Cybernetics: or, Control and Communication in the Animal and the Machine (1948). Freudenthal writes:-

While studying anti-aircraft fire control, Wiener may have conceived the idea of considering the operator as part of the steering mechanism and of applying to him such notions as feedback and stability, which had been devised for mechanical systems and electrical circuits. ... As time passed, such flashes of insight were more consciously put to use in a sort of biological research ... [Cybernetics] has contributed to popularising a way of thinking in communication theory terms, such as feedback, information, control, input, output, stability, homeostasis, prediction, and filtering. On the other hand, it also has contributed to spreading mistaken ideas of what mathematics really means.

Wiener himself was aware of these dangers and his wide dealings with other scientists led him to say:-

One of the chief duties of the mathematician in acting as an adviser to scientists is to discourage them from expecting too much from mathematics.

Some of Wiener's publications which we have not mentioned include Nonlinear Problems in Random Theory (1958), and God and Golem, Inc.: A Comment on Certain Points Where Cybernetics Impinges on Religion (1964).

We have mentioned above Freudenthal's comments on Wiener's poor writing style. His most famous work Cybernetics comes in for special criticism by Freudenthal:-

Even measured by Wiener's standards "Cybernetics" is a badly organised work -- a collection of misprints, wrong mathematical statements, mistaken formulas, splendid but unrelated ideas, and logical absurdities. It is sad that this work earned Wiener the greater part of his public renown, but this is an afterthought. At that time mathematical readers were more fascinated by the richness of its ideas than by its shortcomings.

Freudenthal describes both Wiener's appearance and his character:-

In appearance and behaviour, Norbert Wiener was a baroque figure, short, rotund, and myopic, combining these and many qualities in extreme degree. His conversation was a curious mixture of pomposity and wantonness. He was a poor listener. His self-praise was playful, convincing and never offensive. He spoke many languages but was not easy to understand in any of them. He was a famously bad lecturer.

D G Kendall writes:-

As a human being Wiener was above all stimulating. I have known some who found the stimulus unwelcome. He could offend publicly by snoring through a lecture and then asking an awkward question in the discussion, and also privately by proffering information and advice on some field remote from his own to an august dinner companion. I like to remember Wiener as I once saw him late at night in Magdalen College, Oxford, surrounded by a spellbound group of undergraduates, talking, endlessly talking.

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

3.3.5. Johannes Kepler. An act of Divine Providence

"Kepler, in his inquiries, asked questions that none before him, including Copernicus, has asked.... [They were] questions in physics—not in some preconceived geometrical framework." — S. Chandrasekhar, Truth and Beauty

By Yuli Danilov

IT HAPPENED ON THE NINTH OF JULY IN 1595. Johannes Kepler, a young teacher at the Lutheran high school in the Austrian town of Graz, was solving a geometrical problem and drew an equilateral triangle with inscribed and circumscribed circles on the blackboard. At that very moment an idea hit him — an idea that seemed to be the key to solving the secret of the universe's structure: the ratios of the radii of the planetary orbits are determined by the ratios of the radii of inscribed and circumscribed circles of certain regular polygons. But Kepler encountered some difficulties on the path to discovering the Creator's intentions. The main problem was that he couldn't explain the number of planets. At that time there were six known planets (including the Earth), but there were infinitely many regular polygons—why should some be preferable to others? And then Kepler turned his attention to solid bodies. As you know, there are only five regular (convex) polyhedrons—just as many as there are intervals between the six planets. These are the so-called Platonic bodies: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. And here, in Kepler's mind, was the solution to the "cosmographic mystery":

The Earth is the measure of all orbits. Let us circumscribe a dodecahedron around its orbit. The sphere circumscribed about the dodecahedron is the sphere of Mars. Let us circumscribe a tetrahedron around the sphere of Mars. The sphere circumscribed about the tetrahedron is the sphere of Jupiter. Let us circumscribe a cube around the sphere of Jupiter. The sphere circumscribed about the cube is the sphere of Saturn. Let us insert an icosahedron in the sphere of Earth. The sphere inscribed in it is the sphere of Venus. Let us insert an octahedron in the sphere of Venus. The sphere inscribed in it is the sphere of Mercury.

All that remained was to adjust the thickness of the spheres, correct the remaining discrepancies, and the like. To this end Kepler needed observational data.¹ At that time only one person in Europe possessed such data: Tycho Brahe (1546-1601). But in vain did Kepler send the famous astronomer a copy of his Mysterium Cosmo-graphicum. He even paid him a call at the Benatek castle near Prague. Tycho stubbornly refused to share his precious observations, compiled over many years. He harbored the dream of creating his own theory of the design of the universe (according to which the Sun revolves around the Earth and the planets revolve around the Sun). Nevertheless, he undoubtedly noticed his guest's talent and grasp of the facts. How else can one explain the sudden about-face? Tycho soon appointed Kepler his new assistant in the most difficult problem facing him: a theory of Mars, which had withstood the efforts of his other assistant Longomontanus.

So various circumstances forced Kepler to leave Graz and settle at Benatek, where he became an assistant to Tycho. After his patron died he inherited the title (and duties) of Mathematician of His Imperial Majesty and — more to the point, perhaps — twenty precious volumes of the most exact astronomical observations.

"I think," wrote Kepler, "it was an act of Divine Providence that I arrived just when Longomontanus was busy with Mars. Only Mars gives us the opportunity to penetrate the secrets of astronomy, which oth erwise would forever remain hidden from us."

Tycho gave Kepler the task of developing a theory of Mars and assigned Longomontanus the simpler problem of a theory of the Moon. The culmination of Kepler's work was his Astronomia Nova, a treatise written in Latin (as was customary at the time) and published in 1609. ²

When people speak about the Astronomia Nova, people usually emphasize Kepler's "beelike industry" (Einstein), overlooking or forgetting (Bertrand Russell) the audacity of his ideas and the resoluteness with which he broke with the age-old tradition of circular motion in astronomy. There is no doubt that such a step called for the highest sort of courage — courage of mind and spirit. Not without reason did Kepler, in his allegorical dedication to the emperor Rudolf II, liken the development of his theory of Mars to a battle with the terrible god of war himself. The victor was awarded a pair of trophies: two laws of planetary motion, now known as Kepler's first and second laws.

¹Kepler never was able to make this scheme work.

²The title page reads in English: A New Astronomy, Causally Justified, or Celestial Physics, together with Commentaries on the Movements of the Planet Mars in Accordance with Observations Made by the Eminent Tycho Brahe, by Decree and at the Expense of Rudolf II, Emperor of the Holy Roman Empire etc. Written in Prague during Many Years of Persistent Investigations by the Mathematician of His Most Holy Imperial Majesty, Johannes Kepler.

3.3.6. Karl Theodor Wilhelm Weierstrass

Born: 31 Oct 1815 in Ostenfelde, Westphalia (now Germany)

Died: 19 Feb 1897 in Berlin, Germany

Karl Weierstrass's father, Wilhelm Weierstrass, was secretary to the mayor of Ostenfelde at the time of Karl's birth. Wilhelm Weierstrass was a well educated man who had a broad knowledge of the arts and of the sciences. He certainly was well capable of attaining higher positions than he did, and this attitude may have been one of the reasons that Karl Weierstrass's early career was in posts well below his outstanding ability. Weierstrass's mother was Theodora Vonderforst and Karl was the eldest of Theodora and Wilhelm's four children, none of whom married.

Wilhelm Weierstrass became a tax inspector when Karl was eight years old. This job involved him in only spending short periods in any one place so Karl frequently moved from school to school as the family moved around Prussia. In 1827 Karl's mother Theodora died and one year later his father Wilhelm remarried. By 1829 Wilhelm Weierstrass had become an assistant at the main tax office in Paderborn, and Karl entered the Catholic Gymnasium there. Weierstrass excelled at the Gymnasium despite having to take on a part-time job as a bookkeeper to help out the family finances.

While at the Gymnasium Weierstrass certainly reached a level of mathematical competence far beyond what would have been expected. He regularly read Crelle's Journal and gave mathematical tuition to one of his brothers. However Weierstrass's father wished him to study finance and so, after graduating from the Gymnasium in 1834, he entered the University of Bonn with a course planned out for him which included the study of law, finance and economics. With the career in the Prussian administration that was planned for him by his father, this was indeed a well designed course. However, Weierstrass suffered from the conflict of either obeying his father's wishes or studying the subject he loved, namely mathematics.

The result of the conflict which went on inside Weierstrass was that he did not attend either the mathematics lectures or the lectures of his planned course. He reacted to the conflict inside him by pretending that he did not care about his studies, and he spent four years of intensive fencing and drinking. As Biermann writes:-

... the conflict between duty and inclination led to physical and mental strain. He tried, in vain, to overcome his problems by participating in carefree student life ...

He did study mathematics on his own, however, reading Laplace's Mécanique céleste and then a work by Jacobi on elliptic functions. He came to understand the necessary methods in elliptic function theory by studying transcripts of lectures by Gudermann. In a letter to Lie, written nearly 50 years later, he explained how he came to make the definite decision to study mathematics despite his father's wishes around this time:-

... when I became aware of [a letter from Abel to Legendre] in Crelle's Journal during my student years, [it] was of the utmost importance. The immediate derivation of the form of the representation of the function given by Abel ..., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester ...

Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration. After his decision, he spent one further semester at the University of Bonn, his eighth semester ending in 1838, and having failed to study the subjects he was enrolled for he simply left the University without taking the examinations. Weierstrass's father was desperately upset by his son giving up his studies. He was persuaded by a family friend, the president of the law courts at Paderborn, to allow Karl to study at the Theological and Philosophical Academy of Münster so that he could take the necessary examinations to become a secondary school teacher.

On 22 May 1839 Weierstrass enrolled at the Academy in Münster. Gudermann lectured in Münster and this was the reason that Weierstrass was so keen to study there. Weierstrass attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies. Leaving Münster in the autumn of 1839, Weierstrass studied for the teacher's examination which he registered for in March 1840. By this time, however, Weierstrass's father had moved jobs yet again, becoming director of a salt works in January 1840, and the family was now living in Westernkotten near Lippstadt on the Lippe River, west of Paderborn.

At Weierstrass's request he was given a question on the paper he received in May 1840 on the representation of elliptic functions and he presented his own important research as an answer. Gudermann assessed the paper and rated Weierstrass's contribution:-

... of equal rank with the discoverers who were crowned with glory.

When, in later life, Weierstrass learnt of Gudermann's comments he said that he would have published his results had he known. Weierstrass also commented on how generous Gudermann had been in his praise, particularly since he had been highly critical of Gudermann's methods.

By April 1841 Weierstrass had taken the necessary oral examinations and he began one year probation as a teacher at the Gymnasium in Münster. Although he did not publish any mathematics at this time, he wrote three short papers in 1841 and 1842:-

The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann. The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity.

Weierstrass began his career as a qualified teacher of mathematics at the Pro-Gymnasium in Deutsch Krone in West Prussia (now Poland) in 1842 where he remained until he moved to the Collegium Hoseanum in Braunsberg in 1848. As a teacher of mathematics he was required to teach other topics too, and Weierstrass taught physics, botany, geography, history, German, calligraphy and even gymnastics. In later life Weierstrass described the "unending dreariness and boredom" of these miserable years in which:-

... he had neither a colleague for mathematical discussions nor access to a mathematical library, and that the exchange of scientific letters was a luxury that he could not afford.

From around 1850 Weierstrass began to suffer from attacks of dizziness which were very severe and which ended after about an hour in violent sickness. Frequent attacks over a period of about 12 years made it difficult for him to work and it is thought that these problems may well have been caused by the mental conflicts he had suffered as a student, together with the stress of applying himself to mathematics in every free minute of his time while undertaking the demanding teaching job.

It is not surprising that when Weierstrass published papers on abelian functions in the Braunsberg school prospectus they went unnoticed by mathematicians. However, in 1854 he published Zur Theorie der Abelschen Functionen in Crelle's Journal and this was certainly noticed. This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series.

With this paper Weierstrass burst from obscurity. The University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1855 Weierstrass applied for the chair at the University of Breslau left vacant when Kummer moved to Berlin. Kummer, however, tried to influence things so that Weierstrass would go to Berlin, not Breslau, so Weierstrass was not appointed. A letter from Dirichlet to the Prussian Minister of Culture written in 1855 strongly supported Weierstrass being given a university appointment. After being promoted to senior lecturer at Braunsberg, Weierstrass obtained a year's leave of absence to devote himself to advanced mathematical study. He had already decided, however, that he would never return to school teaching.

Weierstrass published a full version of his theory of inversion of hyperelliptic integrals in his next paper Theorie der Abelschen Functionen in Crelle's Journal in 1856. There was a move from a number of universities to offer him a chair. While universities in Austria were discussing the prospect, an offer of a chair came from the Industry Institute in Berlin (later the Technische Hochschule). Although he would have prefered to go to the University of Berlin, Weierstrass certainly did not want to return to the Collegium Hoseanum in Braunsberg so he accepted the offer from the Institute on 14 June 1856.

Offers continued to be made to Weierstrass so that when he attended a conference in Vienna in September 1856 he was offered a chair at any Austrian university of his choice. Before he had decided what to do about this offer, the University of Berlin offered him a professorship in October. This was the job he had long wanted and he accepted quickly, although having accepted the offer from the Industry Institute earlier in the year he was not able to formally occupy the University of Berlin chair for some years.

Weierstrass's successful lectures in mathematics attracted students from all over the world. The topics of his lectures included:- the application of Fourier series and integrals to mathematical physics (1856/57), an introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics.

In his lectures of 1859/60 Weierstrass gave Introduction to analysis where he tackled the foundations of the subject for the first time. In 1860/61 he lectured on the Integral calculus.

We described above the health problems that Weierstrass suffered from 1850 onwards. Although he had achieved the positions that he had dreamed of, his health gave out in December 1861 when he collapsed completely. It took him about a year to recover sufficiently to lecture again and he was never to regain his health completely. From this time on he lectured sitting down while a student wrote on the blackboard for him. The attacks that he had suffered from 1850 stopped and were replaced by chest problems.

In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers. In his 1863 lectures he proved that the complex numbers are the only commutative algebraic extension of the real numbers. Gauss had promised a proof of this in 1831 but had failed to give one.

In 1872 his emphasis on rigour led him to discover a function that, although continuous, had no derivative at any point. Analysts who depended heavily upon intuition for their discoveries were rather dismayed at this counter-intuitive function. Riemann had suggested in 1861 that such a function could be found, but his example failed to be non-differentiable at all points.

Weierstrass's lectures developed into a four-semester course which he continued to give until 1890. The four courses were

1. Introduction to the theory of analytic functions,

2. Elliptic functions,

3. Abelian functions,

4. Calculus of variations or applications of elliptic functions.

Through the years the courses developed and a number of versions have been published such as the notes by Killing made in 1868 and those by Hurwitz from 1878. Weierstrass's approach still dominates teaching analysis today and this is clearly seen from the contents and style of these lectures, particularly the Introduction course. Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.

Date: 2015-12-18; view: 794