The post David Hilbert’s 23 Fundamental Problems appeared first on SciHi Blog.

]]>On August 8, 1900 **David Hilbert**, probably the greatest mathematician of his age,** ** gave a speech at the Paris conference of the International Congress of Mathematicians, at the Sorbonne, where he presented 10 mathematical Problems (out of a list of 23), all unsolved at the time, and several of them were very influential for 20th century mathematics.

“Who of us would not be glad to lift the veil behind which the future lies hidden: to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?” (David Hilbert, 1900)

Those were the words spoken by Hilbert at probably the most important as well as famous mathematical lecture of all times. Several of the Hilbert problems have been resolved in ways that would have been profoundly surprising, and even disturbing, to Hilbert himself. Following Frege [1] and Bertrand Russell,[2] Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms. One of the main goals of Hilbert’s program was a finitistic proof of the consistency of the axioms of arithmetic (the 2nd problem). However, Kurt Gödel‘s second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is probably impossible.[9] Famously, Hilbert stated that the point is to know one way or the other what the solution is and he believed that we always can know this, i.e., that in mathematics there is no “*ignorabimus*” (Latin for “we will not know”or a statement that the truth can never be known).

David Hilbert was born on 23 January 1862 in Wehlau, near Königsberg, Prussia (now Kaliningrad, Russia), to Otto Hilbert, a county judge who had married Maria Therese Erdtmann, the daughter of a Königsberg merchant. After his high school graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, the “Albertina”. In early 1882, Hermann Minkowski returned to Königsberg and entered the university to whom. Hilbert developed a lifelong friendship.[5] Hilbert obtained his doctorate in 1885, with a dissertation titled *Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen* (“*On the invariant properties of special binary forms, in particular the spherical harmonic functions*“). In 1886 Hilbert habilitated himself in Königsberg with a thesis on invariant theoretical investigations in the field of binary forms and became a private lecturer. Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen [6]. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world. Hilbert remained there for the rest of his life.

Hilbert had been invited to give a lecture at the second international mathematicians’ congress in Paris in August 1900. He decided not to give a “ceremonial lecture” in which he would talk about what he had achieved so far in mathematics, nor to respond to Henri Poincaré‘s lecture at the first international mathematics congress in 1897, which had talked about the relationship between mathematics and physics.[10] Instead, his lecture should offer a programmatic view of the future of mathematics in the coming century. Of the 23 Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19, 20, and 21 have a solution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 15, 18+ (the “+” stands for a computer-generated proof), and 22 have solutions that have partial acceptance, but there exists some controversy as to whether it resolves the problem or not. That leaves 16, 8 (the Riemann hypothesis) and 12 unresolved. On this classification 4, 16, and 23 are too vague to ever be described as solved. The withdrawn 24 would also be in this class. 6 is considered as a problem in physics rather than in mathematics. Overall, some of the 23 famous problems continue to this day to remain a challenge for mathematicians.

“We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.” (David Hilbert, 1919)

Mathematics at the turn of the century was not yet well established. The tendency to replace words with symbols and vague concepts with strict axiomatics was not yet very pronounced and would only allow the next generation of mathematicians to formalise their subject more strongly. Hilbert could not yet fall back on the Zermelo-Fraenkel set theory, concepts such as topological space and the Lebesgue integral or the Church-Turing thesis. Functional analysis, which was founded among others by Hilbert himself with the introduction of the Hilbert space named after him, had not yet separated itself from the calculus of variations as a mathematical field. Many of the problems in Hilbert’s list are – partly also for this reason – not formulated so precisely and restrictedly that they could be clearly solved by the publication of a proof. Some problems are less concrete questions than calls to do research in certain areas; others are too vague to say exactly what Hilbert would have considered the solution.

Around 1909, Hilbert dedicated himself to the study of differential and integral equations. His work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on, Stefan Banach amplified the concept, defining Banach spaces. Hilbert spaces are an important class of objects in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century.

Among Hilbert‘s students were Hermann Weyl, the famous world chess champion Emanuel Lasker, and Ernst Zermelo. But the list includes many other famour names including Wilhelm Ackermann, Felix Bernstein, Otto Blumenthal, Richard Courant, Haskell Curry, Max Dehn, Rudolf Fueter, Alfred Haar, Georg Hamel, Erich Hecke, Earle Hedrick, Ernst Hellinger, Edward Kasner, Oliver Kellogg, Hellmuth Kneser, Otto Neugebauer, Erhard Schmidt, Hugo Steinhaus, and Teiji Takagi.

At yovisto academic search engine you can listen to Prof. Angus MacIntyre from Gresham College, London, asking ‘*What has become of Hilbert’s problems a century later?*‘ and ‘*How will the story continue?*`

**References and further Reading:**

- [1] Gottlob Frege and the Begriffsschrift, SciHi Blog, November 8, 2013.
- [2] The time you enjoy wasting is not wasted time – Bertrand Russell, Logician and Pacifist, SciHi Blog, July 11, 2012.
- [3] Gray, Jeremy J. (2000). The Hilbert Challenge. Oxford University Press.
- [4] O’Connor, John J.; Robertson, Edmund F., “David Hilbert“, MacTutor History of Mathematics archive, University of St Andrews.
- [5] Hermann Minkowski and the four-dimensional Space-Time, SciHi Blog, June 22, 2015.
- [6] Felix Klein and the Klein-Bottle, SciHi Blog, April 25, 2016.
- [7] David Hilbert at zbMATH
- [8] David Hilbert at Mathematics Genealogy Project
- [9] Kurt Gödel Shaking the Very Foundations of Mathematics, SciHi Blog
- [10] Henri Poincaré – the Last Universalist of Mathematics, SciHi Blog
- [11] David Hilbert at Wikidata
- [12] Timeline for David Hilbert, via Wikidata

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]]>The post William Hamilton and the Invention of Quaterions appeared first on SciHi Blog.

]]>On August 4, 1805, Irish physicist, astronomer, and mathematician **William Rowan Hamilton** was born. He made important contributions to classical mechanics, optics, and algebra, but is perhaps best known as the inventor of quaternions, a number system that extends the complex numbers.

‘This young man, I do not say will be, but is, the first mathematician of his age.’ (Astronomer Bishop Dr. John Brinkley about 18-year-old Hamilton)

William Hamilton was born as fourth of nine children of Archibald Hamilton, a solicitor originally from Duneboyne, and Sarah Hutton in Dublin. His father did not have time to teach William as he was often away in England pursuing legal business. By the age of five, William Hamilton had already learned Latin, Greek, and Hebrew. He was taught these subjects by his uncle, the Rev James Hamilton, a graduate of Trinity College. At a young age, Hamilton displayed an uncanny ability to acquire languages. At age 13, he had already acquired 13 languages, among them the classical and modern European languages, but also Persian, Arabic, Hindustani, Sanskrit, and even Marathi and Malay.

“Analysis and synthesis, though commonly treated as two different methods, are, if properly understood, only the two necessary parts of the same method. Each is the relative and correlative of the other.” (William Rowan Hamilton)

William Hamilton’s introduction to mathematics came at the age of 13 with algebra. Already two years later, he was studying the works of Newton and Laplace.[5] In 1822 Hamilton found an error in Laplace’s ‘*Mécanique célest*‘, which brought him to the attention of the scientists of his time. Hamilton was part of a small but well-regarded school of mathematicians associated with Trinity College, Dublin, which he entered at age 18. He studied both classics and mathematics, and was appointed Professor of Astronomy in 1827, prior to his graduation taking up residence at Dunsink Observatory where he spent the rest of his life. In 1827, Hamilton presented a theory of a single function, now known as Hamilton’s principal function. The theory brought together mechanics, optics and mathematics, thus helping establish the wave theory of light. The Royal Irish Academy paper was entitled* Theory of Systems of Rays*, with the first part being printed in 1828 in the Transactions of the Royal Irish Academy. According to this theory, a single ray of light entering a biaxial crystal at a certain angle emerged as a hollow cone of rays. This breakthrough is still known by its original name, “*conical refraction*” [1].

“Time is said to have only one dimension, and space to have three dimensions. […] The mathematical quaternion partakes of both these elements; in technical language it may be said to be ‘time plus space’, or ‘space plus time’: and in this sense it has, or at least involves a reference to, four dimensions.” (William Rowan Hamilton)

Hamilton had a deep interest in the fundamental principles of algebra. One thing common in all Hamilton’s research was that they were based on the principle of “*Varying Action*”. While the principle is based on the calculus of variation, it, however, revealed a detailed mathematical structure than that had been previously understood. Though Hamilton’s take on classical mechanics is based on the same physical principles of Newton and Lagrange, it provides a powerful new technique for working with the equations of motion.[6] Both Lagrangian and Hamiltonian approaches were initially developed to describe the motion of discrete systems and have proven to be critical in the study of continuous classical systems in physics, and even quantum mechanical systems. As such, the techniques are still in use in electromagnetism, quantum mechanics, quantum relativity theory, and quantum field theory [2].

In 1835 William Hamilton was knighted and from 1837 to 1846 he served as president of the Royal Irish Academy. The great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843. Originally, Hamilton was looking for ways of extending complex numbers (which can be considered as points on a 2-dimensional plane, where the x-axis are real numbers and the y-axis the imaginary numbers) to higher spatial dimensions. He failed to find a useful 3-dimensional system, but in working with four dimensions he created quaternions. According to Hamilton, he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation *i*^{2} = *j*^{2} = *k*^{2} = *i* *j* *k* = -1. suddenly occurred to him.

“Identity is a relation between our cognitions of a thing, not between things themselves.” (William Rowan Hamilton)

Hamilton then promptly carved this equation using his penknife into the side of the nearby Broome Bridge. Today, a plaque under the bridge, unveiled in 1958, marks the discovery of the quaternions. The quaternion involved abandoning the law of commutativity, a radical step for the time. Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the ‘scalar’ part, and the remaining three as the ‘vector’ part. Hamilton devoted the last 22 years of his life to the development of the theory of quaternions and related systems. For him, quaternions were a natural tool for the investigation of problems in three-dimensional geometry. Many basic concepts and results in vector analysis have their origin in Hamilton’s papers on quaternions. William Hamilton passed away on September 2, 1865 after a severe attack of gout.

Today, William Hamilton is recognized as one of Ireland’s leading scientists. The year 2005 was the 200th anniversary of Hamilton’s birth and the Irish government designated that the Hamilton Year, celebrating Irish science. Numerous concepts and objects in mathematics and mechanics are named after Hamilton, such as Hamilton’s equations, Hamilton’s principle, Hamilton’s principal function, and the Hamilton–Jacobi equation. The algebra of quaternions is usually denoted by a blackboard bold H in honor of Hamilton. Hamilton saw in the quaternions a revolution in theoretical physics and mathematics and tried for the rest of his life to propagate their use, being supported in the second half of the 19th century by other British mathematicians such as Peter Guthrie Tait. After his death he left behind an unfinished two-volume work on quaternions written with Euclid’s elements in mind. Finally, however, vector calculus and vector analysis became the language of description, represented by Hermann Graßmann, Josiah Willard Gibbs and Oliver Heaviside. Lord Kelvin wrote about it: *“Quaternions was invented by Hamilton after his truly important work had been completed. Although beautiful and of ingenious origin, they have been a curse on anyone who has come into contact with them in any way.”* In his own books Kelvin avoided both quaternions and vectors. Later it turned out that Olinde Rodrigues found the quaternions as early as 1840.

At yovisto academic video search you can learn about number theory and complex numbers in a funny little video by Dr. Mattew Weathers, where he explains complex numbers in a rather unusual way

**References and Further Reading Related:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Sir William Rowan Hamilton”,
*MacTutor History of Mathematics archive*, University of St Andrews.[2] Sir William Rowan Hamilton. 2013. The Famous People website. Available from: http://www.thefamouspeople.com/profiles/sir-william-rowan-hamilton-552.php. - [3] William Rowan Hamilton in Britannica Online
- [4] The Hamilton year 2005 web site
- [5] Pierre Simon de Laplace and his true love for Astronomy and Mathematics, SciHi Blog
- [6] Joseph-Louis Lagrange and the Celestial Mechanics, SciHi Blog
- [7] William Rowan Hamilton at Wikidata
- [8] Willian Rowan Hamilton at zbMATH
- [9] William Rowan Hamilton at Mathematics Genealogy Project
- [10] Timeline for William Rowan Hamilton, via Wikidata

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]]>The post Every Set can be Well Ordered – Ernst Zermelo appeared first on SciHi Blog.

]]>On July 27, 1871, German logician and mathematician **Ernst Zermelo** was born. Zermelo’s work had major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.

“…self-evidence … must not be confused with … provability.”

– Ernst Zermelo

Zermelo was the son of a grammar school professor and attended the Luisenstädtische Gymnasium in Berlin until he graduated from high school in 1889. He studied mathematics, physics and philosophy at the universities of Berlin, Halle (Saale) and Freiburg and received his doctorate in 1894 from the University of Berlin under Hermann Amandus Schwarz with honours and examinations of the calculus of variations(*Untersuchungen zur Variationsrechnung*), in which he extended Weierstrass‘ theory.[1] He studied in Berlin under Max Planck, whose assistant he was.[2] In 1896 and 1897 he was involved in a debate with Ludwig Boltzmann, as he saw a contradiction between Poincaré’s theorem and the second law of thermodynamics, which Boltzmann believed to have been derived from mechanics.[3]

In 1897 Zermelo went to Göttingen, then the world centre of mathematics, where he submitted his habilitation on a hydrodynamic topic (*Vortices on the Spherical Surface*). In 1900, in the Paris conference of the International Congress of Mathematicians, David Hilbert challenged the mathematical community with his famous Hilbert’s problems, a list of 23 unsolved fundamental questions which mathematicians should attack during the coming century.[4] The first of these, a problem of set theory, was the continuum hypothesis introduced by Georg Cantor in 1878, and in the course of its statement Hilbert mentioned also the need to prove the well-ordering theorem. The well-ordering theorem states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. This is also known as Zermelo’s theorem and is equivalent to the axiom of choice.

In 1904, he formulated the axiom of choice, thus proving the principle of well ordering that any quantity can be well ordered. The axiom of choice means that a selection function exists for each set of non-empty sets, i.e. a function that assigns an element to each of these non-empty sets and thus “selects” it. For finite sets this can also be concluded without this axiom, therefore the selection axiom is only interesting for infinite sets. This attracted so much attention that he was appointed professor in Göttingen in 1905. His proof, however, also produced fierce criticism, so that he gave a new proof in 1908. Subsequently, he founded axiomatic set theory with the axioms of Zermelo set theory 1907/08, the basis for Zermelo-Fraenkel set theory, which is now established as the standard approach. Zermelo set theory is the first published axiomatic set theory. To give set theory a solid formal basis, Bertrand Russell published his type theory in 1903,[5] which was difficult to access due to its syntactically complicated form. Zermelo therefore chose the more elegant way of the axiomatic structure of set theory. Its seven set axioms, which mainly secure the existence of sets, proved to be viable and allow the complete derivation of Cantor’s set theory in the extended form of Zermelo-Fraenkel set theory. Zermelo still formulated his axioms verbally; today, on the other hand, they are usually specified in predicate logical form.

In 1910 Zermelo was awarded the chair of mathematics at the University of Zurich. In 1913 he proved that finite games such as chess (there are certain abort conditions so that no game can last indefinitely) have a unique solution. This means that either white has a winning strategy, as in a chess composition (or chess problem), or black has one, or each of the two players can force at least one draw. This result was one of the first in mathematical game theory. In 1916 he was awarded the Ackermann-Teubner Memorial Prize.

Due to some health problems he gave up his professorship in Zurich in 1916 and took up residence in the Black Forest. He worked from 1926 with an honorary professorship at the Albert-Ludwigs-University in Freiburg im Breisgau, but had to give up this work again in 1935, because he refused to start the lectures with Hitler salute, which was denounced by colleagues (Gustav Doetsch and his assistant Eugen Schlotter). After the Second World War he resumed his position as honorary professor, but due to his health he was unable to give any more lectures.

Zermelo was known to be sharp-tongued. Wolfgang Pauli told the following anecdote about Ernst Zermelo: In a lecture on logic in Göttingen, Zermelo presented the following paradox: There would be two classes of mathematicians in Göttingen. The first class includes those who would do what Felix Klein wanted but did not like. Those who did what they liked belonged to the second class, but Felix Klein did not like that.[6] What class is Felix Klein in? When the students remained dumb (Felix Klein held an outstanding position in mathematics in Göttingen and elsewhere in Germany and was also the supervisor of Zermelo, who taught as a private lecturer in Göttingen), he thought the answer was terribly simple: Felix Klein was not a mathematician at all (in fact Klein was known in his later career for dealing a lot with physical applications, and he was accused of inadequate mathematical rigour, especially by the Berlin mathematics school from which Zermelo came). To the anecdote it can be added that Zermelo independently discovered Russell’s antinomy before the publication of Bertrand Russell (1903) and used it in lectures.

- [1] Karl Weierstrass – the Father of Modern Analysis, SciHi Blog
- [2] Max Planck and the Quantum Theory, SciHi Blog
- [3] Ludwig Boltzmann and Statistical Mechanics, SciHi Blog
- [4] David Hilbert’s 23 Problems, SciHi Blog
- [5] The time you enjoy wasting is not wasted time – Bertrand Russell, Logician and Pacifist, SciHi Blog
- [6] Felix Klein and the Klein-Bottle, SciHi Blog
- [7] O’Connor, John J.; Robertson, Edmund F., “Ernst Zermelo”,
*MacTutor History of Mathematics archive*, University of St Andrews - [8] Works by or about Ernst Zermelo at Internet Archive
- [9] Ernst Zermelo at Wikidata
- [10] Ernst Zermelo at zbMATH
- [11] Ernst Zermelo at Mathematics Genealogy Project
- [12] Timeline for Ernst Zermelo, via Wikidata

The post Every Set can be Well Ordered – Ernst Zermelo appeared first on SciHi Blog.

]]>The post John Dee and his World of Science and Magic appeared first on SciHi Blog.

]]>On July 13, 1527, Welsh mathematician, astronomer, astrologer, occultist, navigator, imperialist and consultant to Queen Elizabeth I, **John Dee** was born. He is considered one of the most learned men of his age. Besides being an ardent promoter of mathematics and a respected astronomer, in his later years he immersed himself in the worlds of magic, astrology and Hermetic philosophy. One of his aims was attempting to commune with angels in order to learn the universal language of creation.

“It is by the straight line and the circle that the first and most simple example and representation of all things may be demonstrated, whether such things be either non-existent or merely hidden under Nature’s veils.”

– John Dee, Monas Hieroglyphica (1564)

John Dee was born in 1527 in Tower Ward (City of London) as the son of the wealthy Rowland Dee of old nobility. Dee attended the Chelmsford Chantry School and from 1542 the St John’s College in Cambridge. In 1545 he received the Bachelor of Arts. In May 1547 he travelled to the Netherlands to study with the mathematician and astronomer Gemma Frisius [5] and his pupil, the cartographer Gerhard Mercator.[4] Equipped with Mercator’s astronomical instruments, which he acquired for Trinity College, Dee returned to Cambridge a few months later. The acquisition of such equipment and maps was of great importance for England’s role as an emerging colonial power in competition with Portugal and Spain. In 1548 Dee was appointed Master of Arts. He left Cambridge again and stayed in France and Leuven. During this time he also studied alchemy and the then branch of science magia naturalis and acquired an excellent scientific reputation, which gave him access to the highest circles.1550 Dee travelled to Paris, 1552 he met Gerolamo Cardano in London.[6] During their acquaintance they worked on a perpetual motion machine and examined a gemstone that was said to have magical properties. Dee gave lectures in Paris on Euclid.

In 1554 Dee was offered a chair at Oxford in mathematics, which he rejected because he felt that English universities were too rhetorically and grammatically oriented – these two subjects, together with logic, formed the academic trivium – while philosophy and science, the more advanced quadrivium, which included arithmetic, geometry, music and astronomy, were neglected. Dee presented Queen Mary I with a visionary plan for the preservation of old books, manuscripts and records and proposed the establishment of a national library in 1556, but his plan was not supported. Instead he built up a private library in his house in Mortlake, constantly buying books from England and the European continent. Thus his library with about 4000 volumes became the largest collection in England of his time and attracted many scholars. In around 1555, he was charged with having made the horoscopes of Queen Mary and Queen Elisabeth I. As an expansion of the case, Queen Mary was charged with treason. Dee was sent to a religious examination at Bishop Bonner’s, becoming one of his closest associates.

Elisabeth I took the throne shortly after and announced John Dee as one of her advisors in astronomy, astrology and other scientific matters. However, he never received an employment that secured him financial independence. From then on, Dee was occupied with numerous official tasks like navigation assistances and still managed to complete own works like *Monas Hieroglyphica*, his very successful Cabalistic interpretation of a glyph, which was highly admired by contemporary academics. His most read and best known work was published around 1570. It described the importance of mathematics in several other sciences as well as art. Its success is also caused by the writing style, designed for readers outside universities.

In 1570 he published a *mathematical preface* to Henry Billingsley’s English translation of Euclid’s *Elements*,[8] in which he emphasised the central importance of mathematics and its influence on the other arts and sciences. Although intended for the uneducated reader, it proved to be Dee’s most influential work and was reprinted frequently. Through the years, John Dee changed his views on science and his personal progress. In the early 1580s, Dee became increasingly dissatisfied as he made little progress in learning the secrets of nature, his plans for expeditions into North America failed, and his influence at court waned. He started to contact the supernatural using a crystal-gazer as a communication device. Soon, he began organizing spiritual conferences, influencing and impressing those he met. As his book collection (whose catalogue is known) proves, Dee had more than a casual interest in angels. He was very interested in Angelology and especially in the communication with angels; so he collected all written conversations between humans and angels. He studied the similarities of the angelic conversations with various texts, among others by Ficino, Agrippa von Nettesheim and Johannes Trithemius, as well as the widespread biblical apocrypha and pseudepigraphy.

Dee was able to write several books considering his new occupation and received many invitations by high ranked politicians. After six years of absence, which he lived as a nomad in Europe, Dee came back to see that his instruments and his impressive library were destroyed. When Elizabeth passed away, James I became her successor and found no reason to support Dee anymore, since he did not believe in anything supernatural. However, when Dee turned to him in 1604 for help with the charges brought against him, he promptly rejected him. So he spent his last years in poverty and died in late 1608 or early 1609 in Mortlake. Both the death register and Dee’s tombstone have been lost.

John Dee was an extraordinary thinker, believing in numbers delivering the real truth. After his death, many manuscripts and unpublished books were found and made public later on. He highly promoted mathematics to those not attending universities, which was often appreciated by the craftsmen and technical artists. In general, John Dee’s fans came from all parts of society, most politicians enjoyed his theories and advices, academics like his friend Tycho Brahe favored his scientific efforts, and the working class liked the way he transferred high knowledge to their uses and understandings.[7]

At yovisto academic video search, you may enjoy a webinar session on John Dee’s *Monas Hieroglyphica* by Peter Forshaw.

**References and Further Reading:**

- [1] The John Dee Society
- [2] Seeing the Word: John Dee and Renaissance occultism
- [3] Dr John Dee: Mathematician, Scientist, Magus and Conjuror
- [4] Gerardus Mercator – The Man who Mapped the Planet, SciHi Blog
- [5] The Most Accurate Instruments of Gemma Frisius, SciHi Blog
- [6] Gerolamo Cardano and the Mathematics of Chances, SciHi Blog
- [7] Tycho Brahe – The Man with the Golden Nose, SciHi Blog
- [8] Euclid – the Father of Geometry, SciHi Blog
- [9] John Dee at Wikidata
- [10] Timeline of English Alchemists, via DBpedia and Wikidata

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]]>The post Nicole Oresme – Polymath of the Late Middle Ages appeared first on SciHi Blog.

]]>On July 11, 1382, significant philosopher of the later Middle Ages Nicole Oresme passed away. As for many historic people of the middle ages, his actual birthdate is unknown and can only be fixed to a period between 1325 and 1330. Nicole Oresme besides William of Ockham [7] or Jean Buridan — a French priest who sowed the seeds of the Copernican revolution in Europe [6] — is considered as one of the most influential thinkers of the 14th century and he wrote influential works on economics, mathematics, physics, astrology and astronomy, philosophy, and theology. Besides his duties as clergyman and counselor of the king, he was very much interested in natural scientific problems. He was a decisive and eager opponent of astrology, which he attacked on religious and scientific grounds. Moreover, he also was a gifted teacher, who was able to communicate science to his contemporaries in a popular way.

“God in His infinite grandeur without any quantity and absolutely indivisible, which we call immensity, is necessarily all in every extension or space or place which exists or can be imagined.”

– Nicole Oresme, Le livre du ciel et du monde (1377), Book II, Ch. 2, p. 279.

We do not know much about Nicole Oresme’s life. He was born c. 1320-1325 in the village of Allemagne in the vicinity of Caen, Normandy, in the diocese of Bayeux. Practically nothing is known concerning his family. The fact that Oresme attended the royally sponsored and subsidized College of Navarre, an institution for students too poor to pay their expenses while studying at the University of Paris, makes it probable that he came from a peasant family. Oresme studied the “artes” in Paris, together with Jean Buridan, and there received the Magister Artium. Buridan was a philosopher and logician who made contributions to probability, optics and mechanics. He added to Aristotle’s theory of motion by understanding that motion was retarded by resistance from the air. Buridan had a major influence in interesting Oresme in natural philosophy and in encouraging him to question the ideas of Aristotle. [3]

Oresme was already a regent master in arts by 1342 and was probably teaching philosophy, during the crisis over William of Ockham‘s natural philosophy. In 1348, he was a student of theology in Paris, in 1356, he received his doctorate and in the same year he became grand master of the College of Navarre. Oresme held this position till 1362 and was teaching master in the faculty of theology during that time.[1] In 1364 he was appointed dean of the Cathedral of Rouen. The following year he took on the additional duties of canon at the Sainte-Chapelle in Paris. Around 1369 he began a series of translations of Aristotelian works at the request of Charles V, for whom he already served, when Charles was the dauphin of France, who granted him a pension in 1371. From 1370 he lived mainly in Paris, advising Charles on financial matters.[3] The friendship between Charles and Oresme was one which continued throughout their lives.[3] At Charles’s instance, too, Oresme pronounced a discourse before the papal court at Avignon, denouncing the ecclesiastical disorders of the time. Furthermore, with royal support, Oresme was appointed bishop of Lisieux in 1377, where he passed away in 1382.

“The heavenly bodies move with such regularity, orderliness, and symmetry that it is truly a marvel; and they continue always to act in this manner ceaselessly, following the established system, without increasing or reducing speed and continuing without respite, as the Scripture says: Summer and winter, night and day they never rest.”

– Nicole Oresme,Le livre du ciel et du monde (1377), Book II, Ch. 2, p. 283.

Nicole Oresme is best known as an economist, mathematician, and a physicist. His economic views are contained in a Commentary on the Ethics of Aristotle and a “*Treatise on Coins*“. These writings are considered among the earliest manuscripts devoted to an economic matter and account Oresme as the precursor of the science of political economy. In the middle ages, at a time when Aristotle’s ideas were accepted almost without question, Oresme did indeed question them. He rejected Aristotle’s definition of time, which was based on uniform motion, and proposed an original definition independent of motion. Similarly he rejected Aristotle’s definition of the position of a body, which was the boundary of the surrounding space, and replaced it with a definition in terms of the space which the body occupies.

“Since money belongs to the community … it would seem that the community may control it as it wills, and therefore may make as much profit from alteration as it likes, and treat money as its own property.”

– Nicole Oresme, Traictie de la Première Invention des Monnoies (1355), Ch. 22: Whether the community may alter money.

In mathematics, Oresme invented a type of coordinate geometry before René Descartes, finding the logical equivalence between tabulating values and graphing them in his manuscript *De configurationibus qualitatum et motuum*. He proposed the use of a graph for plotting a variable magnitude whose value depends on another variable. His attempt to apply mathematical concepts to scientific phenomena was groundbreaking. In particular, he applied his new approach to the following problem: the question of whether a litre of hot water is “warmer” than five litres of lukewarm water. Oresme presents both facts as two rectangles with different abscissa (here: water quantity) or ordinate (here: temperature) and solves the problem by comparing the areas. He is not interested in concrete measurements or comparisons, but rather in the basic solution.Oresme was also the first to prove Merton’s theorem, namely that the distance travelled in a fixed time by a body moving under uniform acceleration is the same as if the body moved at a uniform speed equal to its speed at the midpoint of the time period.

Also he anticipated the Copernican heliocentric model of the solar system. In his “*Livre du ciel et du monde*” Oresme discussed a range of evidence for and against the daily rotation of the Earth on its axis. From astronomical considerations, he maintained that if the Earth were moving and not the celestial spheres, all the movements that we see in the heavens that are computed by the astronomers would appear exactly the same as if the spheres were rotating around the Earth. He rejected the physical argument that if the Earth were moving the air would be left behind causing a great wind from east to west. However, in the end he refused his own argument. He concluded that none of his arguments were conclusive and “*everyone maintains, and I think myself, that the heavens do move and not the Earth.*“

Oresme provided the first modern vernacular translations of Aristotle’s moral works that are still extant today. Between 1371 and 1377 he translated Aristotle’s Ethics, Politics and Economics (nowadays considered to be pseudo-Aristotelian) into Middle French. He also extensively commented on these texts, thereby expressing some of his political views. Like his predecessors Albert the Great,[9] Thomas Aquinas [10] and Peter of Auvergne (and quite unlike Aristotle), Oresme favours monarchy as the best form of government. His criterium for good government is the common good. A king (by definition good) takes care of the common good, whereas a tyrant works for his own profit. A monarch can ensure the stability and durability of his reign by letting the people participate in government.

At yovisto academic video search, you can learn more about the work of Nicole Oresme as a mathematician in the lecture of Prof. John D. Borrow from Gresham College on ‘*Maths with Pictures*‘. There, Prof. Borrow speaks about the use of illustrations in ancient mathematics books, the invention of the first graphs and also the representation of probabilities.

**References and Further Reading:**

- [1] Nicole Oresme in the Stanford Encyclopedia of Philosophy
- [2] Nicole Oresme at the Catholic Encyclopedia
- [3] O’Connor, John J.; Robertson, Edmund F. “Nicole Oresme”. MacTutor History of Mathematics archive.
- [4] Nicole Oresme at Britannica.com
- [5] Nicole Oresme and the Latitide of Forms
- [6] Nicolaus Copernisus and the Heliocentric Model, SciHi Blog
- [7] Ockham’s Razor, SciHi Blog
- [8] Works by or about Nicole Oresme at Internet Archive
- [9] Albertus Magnus and the Merit of Personal Observation, SciHi Blog
- [10] Thomas Aquinas and the Tradition of Scholasticism, SciHi Blog
- [11] Nicole Oresme at Wikidata
- [12] Timeline of Medieval Mathematicians, via DBpedia and Wikidata

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]]>The post Augustus de Morgan and Formal Logic appeared first on SciHi Blog.

]]>On June 27, 1806, British mathematician and logician **Augustus De Morgan** was born. He formulated De Morgan‘s laws and introduced the term mathematical induction, a method of mathematical proof typically used to establish a given statement for all natural numbers. As a computer scientist, I am of course familiar with De Morgan‘s laws, which are fundamental for Boolean logic. De Morgan‘s laws are merely transformation rules for two of the basic operators of logic: conjunction and disjunction. The trick, how De Morgan succeeds to transform a conjunction into a disjunction is by applying negations. But, don’t worry. We won’t go deeper into that. If you are further interested in the subject, simply refer to the wikipedia article of De Morgan‘s Laws. Instead, let’s take a look at the man Augustus De Morgan.

“All existing things upon this earth, which have knowledge of their own existence, possess, some in one degree and some in another, the power of thought, accompanied by perception, which is the awakening of thought by the effects of external objects upon the senses.”

– Augustus de Morgan, Formal Logic (1847)

Augustus De Morgan was born in Madurai, India as son of Colonel Augustus De Morgan of the East India Company. Augustus De Morgan became blind in one eye a month or two after he was born and the family moved to England when Augustus was seven months old. At age ten, Augustus‘ father died and his mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses. His mother being an active and ardent member of the Church of England desired that her son should become a clergyman. In 1823,De Morgan entered Trinity College, Cambridge, where he came under the influence of George Peacock, from whom he derived an interest in the renovation of algebra, and William Whewell, who raised his interest in the renovation of logic. He received his Bachelor degree but, because a theological test was required for the Master, something to which De Morgan strongly objected despite being a member of the Church of England, he could go no further at Cambridge being not eligible for a Fellowship without his Master.

The two ancient universities of Oxford and Cambridge were so guarded by theological tests that no Jew or Dissenter outside the Church of England could ever be appointed to any position. Thus, De Morgan had to change plans and traveled to London to enter Lincoln’s Inn to study for the Bar, but found law unpalatable. At the age of 21, despite having no mathematical publications, on the strength of the strong recommendations of Peacock and Whewell he was unanimously elected to the chair of mathematics at the newly founded University of London. As being a new institution, the relations of the Council of management, the Senate of professors and the body of students were not well defined at the University of London. A dispute arose between the professor of anatomy and his students, and in consequence of the action taken by the Council, several professors resigned, headed by De Morgan. De Morgan regained his position five years later when his replacement accidentally drowned, and held it for thirty years until 1866.

In 1837 De Morgan married Sophia Elizabeth Frend, with whom he would have seven children. In 1838 he was the first to use the term “mathematical induction” in his publication Induction (Mathematics) in Penny Cyclopedia, for which he wrote a total of 712 articles. It also printed his famous work The Differential and Integral Calculus. He was best known for two rules named after him, De Morgan’s laws. These say that any conjunction can be expressed by a disjunction and vice versa. Since then, they have often been used in mathematical proofs and also in programming. De Morgan, together with George Boole,[6] is now regarded as the founder of formal logic. George Boole published in 1847 a small volume called *Mathematical Analysis of Logic*. The occasion to elaborate and publish it was the fierce priority dispute between William Rowan Hamilton and de Morgan on the quantification of predicates. In 1854, Boole’s second major work on algebra, *Laws of Thought*, was published. De Morgan commented:

“That the symbolic processes of algebra, originally invented for the purpose of numerical calculations, should be capable of expressing every act of thought and providing grammar and dictionary of an all-encompassing system of logic, nobody would have believed this before it was proved in Laws of Thought”.

De Morgan was a rather prolific writer. Were the writings of De Morgan published in the form of collected works, they would form a small library. Besides his general mathematical writings he wrote biographies of Newton and Halley and published *Arithmetical Books*, in which he described the work of more than 1,500 mathematicians and discussed the history of various mathematical ideas. De Morgan also plays a role in Quantitative Linguistics and Quantitative Stylistics in that he developed the idea that the problem of identifying anonymous authors could be solved by statistical means. He suggested, for example, that the problem of authorship in Paul’s letters should be tackled with the help of word length analyses, and suggested that the average word length could be revealing.

De Morgan never sought to become a Fellow of the Royal Society, and he never attended a meeting of the Society, holding the opinion that it was too much affected by social influences to be an effective scientific institution. His son George, who was a very able and promising mathematician, conceived the idea of founding a Mathematical Society in London. In difference to the Royal Society, the Mathematical Society should not only receive mathematics papers, but actually read and discuss. Founded in 1866 at University College, Augustus De Morgan became its first president. A few years later after the tragic loss of his son and his daughter, De Morgan’s health rapidly declined and he died of nervous exhaustion at his home on March 18, 1871.

By the way, one of Augustus De Morgan‘s most famous students was Ada Augusta King, Countess of Lovelace, the daughter of Lord Byron, who should become the assistant of mathematician and computer pioneer Charles Babbage, inventor of the first mechanical general purpose computer…but this is already another story.

At yovisto academic video search, you can learn more about formal logics and esp. also about De Morgan’s laws in my lecture ‘*Canonical Forms*‘ from the Spring 2013 OpenHPI course ‘Semantic Web Technologies’.

**References and Further Reading: **

- [1] Augustus De Morgan in Robert A Nowlan: A Chronical of Mathematical People
- [2]O’Connor, John J.; Robertson, Edmund F., “Augustus De Morgan”,
*MacTutor History of Mathematics archive*, University of St Andrews. - [3] Scott H. Brown: The Life and Work of Augustus De Morgan
- [4] Ada Lovelace – The World’s First Programmer, SciHi Blog
- [5] Charles Babbage – The Father of the Computer who hated Street Music, SciHi Blog
- [6] George Boole – The Founder of Modern Logics, SciHi Blog
- [7] Augustus de Morgan at Wikidata
- [8] Timeline of English Logicians, via DBpedia and Wikidata

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]]>The post The Phantastic Worlds of M. C. Escher appeared first on SciHi Blog.

]]>On June 17, 1898, Dutch graphic artist **Maurits Cornelis Escher**, better known as M. C. Escher, was born. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints, which feature impossible constructions, explorations of infinity, architecture, and tessellations.

“The ideas that are basic to [my work] often bear witness to my amazement and wonder at the laws of nature which operate in the world around us. He who wonders discovers that this is in itself a wonder. By keenly confronting the enigmas that surround us, and by considering and analyzing the observations that I had made, I ended up in the domain of mathematics.”

– Quote of Escher 1959 – in the introduction of M.C. Escher: The Graphic Work (1978)

M. C. Escher was born in 1898 as the youngest of five sons of the hydraulic engineer George Arnold Escher in the Princessehof in Leeuwarden. In 1903 the family moved to Arnhem, where the young Escher completed his basic school education. However, he was a rather bad pupil, had to repeat two classes and, despite his talent for drawing, even had poor marks in art.. After school, he enrolled at the Haarlem Schoof of Architecture and Decorative Arts, finishing in 1922. During these years he got in touch with and gained lots of experience in drawing as well as woodcuts. Very influencing to his artist career was the journey Escher made in 1922 through Europe. Escher visited Italy, and Spain, being mostly fascinated by the countrysides and the building’s designs. Due to the positive impressions and influences the artist gained, he visited Italy in the future more often, eventually living there, wherefore the country became an important fragment of his life and work. During his time in italy, Escher created ‘*Still Life and Street*‘, one of his most famous prints of an impossible reality. The image’s inspiration depicted a street in Savona, Italy. The picture was an early example of his playful way handling perspectives, as it looks like the books on his table leaned against the far away buildings. Escher travelled several times through Italy, mostly on foot or on a donkey, as well as Spain, where he studied Arabic ornamentation (Alhambra). In 1923 he met the Swiss Jetta Umiker, whom he married in Viareggio in 1924. The couple settled near Rome.

Due to the rise of fascism in Italy, M. C. Escher moved along with his family to Switzerland, where he was never really able to settle, while in search of the same beautiful landscaped he saw in Italy. In Belgium he could not stay because of World War II and he moved to the Netherlands. As depressing as this period seems, it was also the most productive in Escher’s life. The dark and cloudy weather made him focus on his work and he completed numerous pictures during his time there with almost no break. His teacher Mesquita was kidnapped by the German occupiers in 1944 and murdered in the Auschwitz concentration camp. Escher was able to save at least a large part of Mesquita’s work.

After the end of the war Escher learned the mezzotint technique and from 1946 he increasingly turned to perspective pictures *(top and bottom* 1947). He increasingly received well-paid commissions, sold many of his prints and was a sought-after artist in the USA in 1950. His great breakthrough in Europe came in September 1954, when the Stedelijk Museum granted him a solo exhibition in Amsterdam on the occasion of a mathematician congress held there at the same time.

Escher’s prints of impossible realities made him famous. Next to *Still Life and Street*, the *Drawing Hands*, published in 1948 in which the two hands draw each other. M. C. Escher knew how to portray mathematical relationships of shapes and bodies as well as playing with their perspectives brilliantly without having a higher mathematical education. The mathematical importance in his pictures evolved also during his stay in Italy, where he began gaining interest in orders of shapes and symmetry. Escher’s brother once sent him a paper by the mathematician George Pólya on plane symmetry groups that increased Escher’s attention and he decided to focus more on the mathematical issues in his works, wherefore the artist is liked by so many scientists. He spent more time figuring out how to combine infinity of objects with two-dimensional planes, which he began discussing with several mathematicians. By around 1956 Escher’s interests changed again taking regular division of the plane to the next level by representing infinity on a fixed 2-dimensional plane. Earlier in his career he had used the concept of a closed loop to try to express infinity as demonstrated in *Horseman*.By 1958 Escher had achieved remarkable fame. He continued to give lectures and correspond with people who were eager to learn from him. He had given his first important exhibition of his works and had also been featured in *Time* magazine. Escher received numerous awards over his career including the Knighthood of the Oranje Nassau (1955) and was regularly commissioned to design art for dignitaries around the world.[8]

His best-known works, which almost earned Escher the status of a pop star, deal with the representation of perspective impossibilities, optical illusions and multistable perceptual phenomena. One sees objects or buildings that seem natural at first glance, but are completely contradictory at second glance. In his works, Escher also devoted himself to themes such as Möbius bands, crystal shapes, reflections, optical distortions, fractals and approaches to infinity.

Through his works, Escher was able to open up whole new art creations and bring the mathematical component in art to a whole new level. At yovisto, you may enjoy the video lecture *Gödel, Escher, Bach* as part of a whole series by Justin Curry at MIT, bringing the character’s interests in art, mathematics, logics, physics and numerous other scientific fields together.

**References and Further Reading: **

- [1] Official Escher Website
- [2] Artful Mathematics: The Heritage of M. C. Escher
- [3] Escher at NGA
- [4] “Math and the Art of M.C. Escher”. USA: SLU.
- [5] Schattschneider, Doris (June – July 2010). “The Mathematical Side of M. C. Escher” (PDF).
*Notices of the American Mathematical Society*.**57**(6): 706–18. - [6] Escher-Museum, Den Haag
- [7] Escher for Real, Israel Institute of Technology
- [8] John J. O’Connor, Edmund F. Robertson:
*Maurits Cornelius Escher.*In:*MacTutor History of Mathematics archive.* - [9] M.C Escher at Wikidata
- [10] Timeline for M.C. Escher, via Wikidata

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]]>The post The Beautiful Mind of John Forbes Nash appeared first on SciHi Blog.

]]>On June 13, 1928, American mathematician **John Forbes Nash Jr.** was born. Nash made fundamental contributions to game theory, differential geometry, and the study of partial differential equations. His work has provided insight into the factors that govern chance and decision-making inside complex systems found in everyday life. John Nash is the only person to be awarded both the Nobel Memorial Prize in Economic Sciences and the Abel Prize.

“You don’t have to be a mathematician to have a feel for numbers.”

– John Forbes Nash Jr.

Nash was born on June 13, 1928, in Bluefield, West Virginia. His father, John Forbes Nash, was an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret Virginia (née Martin) Nash, had been a schoolteacher before she was married. From 1945 to 1948 Nash studied at the Carnegie Institute of Technology in Pittsburgh, where he received his Bachelor’s degree in 1945 and his Master’s degree in 1948. Originally he wanted to become an engineer like his father, but developed a great passion for mathematics. Still in Pittsburgh his interest in game theory began, whose solution John von Neumann and Oskar Morgenstern had left open in their book *Theory of Games and Economic Behavior,* 1944.[1, 2]

Nash received his doctorate from Princeton University in 1950 from the mathematician Albert W. Tucker. The work entitled *Non-cooperative Games* extended the game theory of Morgenstern and von Neumann to include the so-called Nash equilibrium. Nash proved that this equilibrium – deviating from the solutions – also exists for non-zero-sum games and for more than two players. The Nash equilibrium describes a combination of strategies in non-cooperative games, where each player chooses exactly one strategy from which it makes no sense for any player to deviate from his chosen strategy as the only one. In a Nash equilibrium every player agrees with his strategy choice afterwards, he would hit it again in the same way. The strategies of the players are therefore the best answers for each other. The Nash equilibrium is an elementary solution concept of game theory.

The importance of this work from 1950 was only later recognised in connection with the further development of game theory and earned him the Nobel Prize for Economics in 1994. Von Neumann himself was not very impressed at the time at a meeting with Nash; he considered the result to be trivial and mentioned it only indirectly in the introduction to the new edition of his book Morgenstern on game theory from 1953. Nash himself also regarded the work more as a by-product in comparison to his later works.

In 1952 his work on real algebraic manifolds appeared, which he himself regarded as his perfect work. The idea was to approximate each multiplicity by an algebraic variety (much easier to handle and describable by polynomials), possibly by moving to spaces of much higher dimension. In this context, Nash manifolds and Nash functions are named after him. After receiving his doctorate, Nash increasingly turned to analysis, in particular differential geometry and partial differential equations. He proved that every Riemann multiplicity isometric in the Euclidean vector space can be embedded (Nash Embedding theorem). The question of whether this is possible was already asked by Bernhard Riemann, and the common opinion in the 1950s was that this was not the case. The result of Nash came unexpectedly and had far-reaching consequences. [3]

From 1950 onwards, Nash worked for four years in the summer months at Rand Corporation on secret research related to applications of game theory to strategic situations in the Cold War. From 1951 to 1953, Nash was a Moore instructor at the Massachusetts Institute of Technology, where he was assistant professor from 1953 and associate professor from 1957 to 1959. In 1955, he submitted a proposal for an encryption procedure to the National Security Agency, but received a rejection. In 1958 he published (in parallel with Ennio De Giorgi, but independently of him) a solution to the regularity problem of partial differential equations, which David Hilbert had included in 1900 in his well-known list of the largest open problems in mathematics (19th problem).[4] The results became known as De Giorgi’s and Nash’s theorem and have far-reaching consequences for the theory of partial differential equations.

At the end of the 1950s Nash was widely recognised as a leading mathematician, which was also reflected in an article in Forbes Magazine, and he was nominated for the Fields Medal in 1958, particularly for his work on Hilbert’s 19th problem simultaneously with de Giorgi. He was third in the final evaluation behind Klaus Roth and René Thom, who finally received the Fields Medal in 1958. At MIT he was on the verge of a full professorship when the first signs of Nash’s disease became apparent in 1959. In May 1959 he was diagnosed with paranoid schizophrenia. According to Nash biographer Sylvia Nasar, Nash was now increasingly showing anti-Semitic tendencies and prone to violence. Nash gave up his position at MIT and after a short stay in hospital went first to Paris and Geneva in 1959/60, where he saw himself as a citizen of the world and exile. In 1961, his wife Alicia Lardé and his mother were forced to send Nash to a mental hospital (Trenton State Hospital). Here he was treated by insulin shock therapy, which put him in an artificial coma. He recovered and was able to attend a conference on game theory in 1961.

In 1964 his schizophrenia became so severe that he had to be admitted to a psychiatric clinic (the private Carrier Clinic) for a longer period of time. During the next 20 years he was repeatedly in clinics for relapses with interruptions. Between 1966 and 1996, as a result of his illness, he did not produce any publications. Before that, however, some outstanding works were published. The 1960s saw the emergence of an idea in the theory of the resolution of singularities in algebraic geometry known as *Nash Blowing Up* (so called by Heisuke Hironaka, to whom Nash gave the idea orally) and some influential papers on partial differential equations. In the 1970s to 1990s he lived in Princeton, where he was regularly seen on campus. While at first he attracted the students’ attention with strange messages he left behind, from the early 1990s mathematicians in Princeton (like Peter Sarnak) increasingly noticed that he had regained some of his old problem-solving skills. In his last years, he increasingly turned to monetary theory, arguing for index money.

Nash died with his wife in May 2015 in a traffic accident on the New Jersey Turnpike; they were in a taxi on their way home from receiving the Abel Award. Both were not strapped on and were thrown out of the vehicle. The 2001 feature *A Beautiful Mind*, starring Russell Crowe, tells the story of Nash’s ingenious designs, illness and recovery; the film won four Oscars in 2002. The script is based on Sylvia Nasar’s 1998 biography of the same name.

**References and Further Reading**:

- [1] John von Neumann – Game Theory and the Digital Computer, SciHi Blog
- [2] Oskar Morgenstern and the Game Theory, SciHi Blog
- [3] Bernhard Riemann’s novell approaches to Geometry, SciHi Blog
- [4] David Hilbert’s 23 Problems, SciHi Blog
- [5] O’Connor, John J.; Robertson, Edmund F., “John Forbes Nash Jr.“, MacTutor History of Mathematics archive, University of St Andrews.
- [6] Home Page of John F. Nash Jr. at Princeton
- [7] Fisher, Len (May 25, 2015). “John Nash obituary”.
*The Guardian*. - [8] Biography of John Forbes Nash Jr. from the Institute for Operations Research and the Management Sciences
- [9] John Forbes Nash Jr at Wikidata
- [10] Timeline of Game Theorists, via DBpedia and Wikidata

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]]>The post Please Don’t Ignite the Earth’s Atmosphere… appeared first on SciHi Blog.

]]>When in 1952 the world‘s first thermonuclear fusion bomb was ignited, mathematicians and physicists thought it would be rather unlikely that testing the newly developed device might result in burning all the nitrogen in the earth‘s atmosphere. However, the possibility could not be excluded completely. Nevertheless, they have have tested the bomb and fortunately for all of us not the like did happen. One of the key persons behind the development of the hydrogen bomb was **Stanislaw Ulam**, who together with physicist Edward Teller came up with the first successful design.[1]

On May 13, 1984, Stanislaw Ulam, Polish-American scientist in the fields of mathematics and nuclear physics passed away. Before developing the hydrogen bomb, he also participated in the Manhattan Project producing the very first atomic weapon based on thermonuclear fission. After the second World War scientists believed that maintaining nuclear fusion to support another kind of weapon should be rather unlikely. But in January 1951, Ulam came up with the decisive idea: put a fission bomb and fusion fuel together inside a massive casing. When the bomb detonated, the casing would contain the explosion long enough for mechanical shock to heat and compress the fusion fuel, and for fission neutrons to ignite nuclear fusion.

Ulam was born in Lemberg , Galicia, (today’s , Lviv in Poland) on 13 April 1909. Ulam came from a Polish-Jewish middle class family with members in the banking and wood processing industries; his father was a lawyer. His mathematics teacher was the Polish mathematician Stefan Banach, one of the leading minds of the Lviv School of Mathematicians. Ulam obtained his Ph.D. from the Polytechnic Institute in Lviv in 1933. He investigated a problem which originated with Lebesgue in 1902 to find a measure on [0,1] with certain properties.[6] Banach in 1929 had solved a related measure problem, but assuming the Generalised Continuum Hypothesis. Ulam, in 1930, strengthened Banach’s result by proving it without using the Generalised Continuum Hypothesis.[5]

In 1938, Ulam went to the United States at the invitation of George David Birkhoff as a Harvard Junior Fellow. In 1940 he became assistant professor at the University of Wisconsin. In 1943 he became a US citizen and in the same year his friend John von Neumann invited him to a secret project in New Mexico, the Manhattan Project.[3] Ulam had a number of specialties, including set theory, mathematical logic, functions of real variables, thermonuclear reactions, topology, and the Monte Carlo theory.

I don’t support any weapon technology and it might be controversial to acknowledge also a scientist like Stanislaw Ulam or his colleague Edward Teller, who spent a great deal their lives in the development of apocalyptic technology. But what really shocked me was the fact that both could not completely exclude the possibility of igniting the earth’s atmosphere when testing the first thermonuclear fusion bomb and thus possibly causing the end of the world:

“There remains the distinct possibility that some other less simple mode of burning may maintain itself in the atmosphere… the complexity of the argument and the absence of satisfactory experimental foundations makes further work on the subject highly desirable.”

(Report LA-602, Ignition of the Atmosphere with Nuclear Bombs [8])

Nevertheless, Stanislaw Ulam also has left us something more substantial: the Monte Carlo Method of Computation, a class of computational algorithms that rely on repeated random sampling and often used for computer simulations of mathematical or physical models. Through the use of electronic computers, this method became widespread, finding applications in weapons design, mathematical economy, and operations research.

Ulam, with J C Everett, also proposed the ‘Orion’ plan for nuclear propulsion of space vehicles. The aim of the Orion project was to develop a nuclear pulse engine to power spaceships. The project ran in the USA from 1957 to 1965. The design envisaged driving a spaceship with a nuclear pulse engine through a series of atomic bomb explosions, each taking place only a few metres behind the stern of the spaceship. Protected by a massive protective shield and a shock absorber system, the spaceship “rides” on the shock waves of the explosions. A nuclear pulse engine based on the Orion principle combines a high specific impulse with a high thrust. The project was abandoned in 1965 for political reasons and because of the 1963 Treaty banning nuclear testing in the atmosphere, in space and under water.

Ulam remained at Los Alamos until 1965 when he was appointed to the chair of mathematics at the University of Colorado. Stanislaw Ulam died of an apparent heart attack in Santa Fe on 13 May 1984 at age 75.

At yovisto academic video search, you might watch a documentary on ‘*Operation Ivy*‘, the detonation of the world’s first hydrogenic bomb back in 1952.

**Related Articles in the Blog: **

- [1] Edward Teller and Stanley Kubrick’s Dr. Strangelove, SciHi Blog
- [2] Stefan Banach and Modern Function Analysis, SciHi Blog
- [3] John von Neumann – Game Theory and the Digital Computer, SciHi Blog
- [4] Stanislaw Marcin Ulam, American Scientist,at Britannica Online
- [5] John J. O’Connor, Edmund F. Robertson: Stanisław Marcin Ulam. In: MacTutor History of Mathematics archive
- [6] Henri Léon Lebesgue and the Theory of Integration, SciHi Blog
- [7] Stanislaw Ulam at Wikidata
- [8] Report LA-602, Ignition of the Atmosphere with Nuclear Bomb
*s* - [9] Timeline with people related to nuclear weapon design, via DBpedia and Wikidata

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]]>The post Carl Friedrich Gauss – The Prince of Mathematicians appeared first on SciHi Blog.

]]>On April 30, 1777, German mathematician and physical scientist Carl Friedrich Gauss was born. He contributed significantly to many fields, including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics. He is often referred to as *Princeps mathematicorum* (Latin, “the Prince of Mathematicians”) as well as “greatest mathematician since antiquity”.

“Mathematics is the Queen of Science, and Arithmetic is the Queen of Mathematics” – handed down in Wolfgang Sartorius von Waltershausen, Gauss zum Gedächtniss, Verlag von S. Hirzel, Leipzig 1856, p.79

Carl Friedrich Gauss grew up as an only child, his mother could barely read but was known to be incredibly intelligent. Rumors about Gauss say that he could calculate before being able to speak and that he corrected his father on his wage accounting at the age of only three. No matter if these rumors are actually true, it indicates that Gauss’ talents and his love for complex calculations were detected very early. At the age of seven, he started to attend school and already designed formulas to easen his calculations during math class.

There is one famous telling about Carl Friedrich Gauss’s boyhood discovery of the “trick” for summing an arithmetic progression. The event occurred when Gauss was seven and attended the Katharina-school in Brunswick. The teacher, one Büttner, had set the class the task of calculating the sum 1 + 2 + 3 + …. + 100 – probably to get a bit of peace for himself, with instructions that each should place his slate on a table as soon as he had completed the task. Almost immediately Gauss placed his slate on the table, saying, “*There it is.*” The teacher looked at him scornfully while the others worked diligently. When the instructor finally looked at the results, the slate of Gauss was the only one to have the correct answer, 5050, with no further calculation. The ten-year-old boy evidently had computed mentally the sum of the arithmetic progression 1 + 2 + 3 + … + 99 + 100, presumably through the formula *m(m+1)/2*. His teachers soon called Gauss’ talent to the attention of the Duke of Brunswick [1].

At the age of 14, Gauss was introduced to Duke Karl Wilhelm Ferdinand von Braunschweig, who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems. His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.

“The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.”

— Gauss in a letter to Sophie Germain (30 April 1807)

Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss’s stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff, who was chosen to be his advisor. Gauss’s dissertation was a discussion of the fundamental theorem of algebra. With his stipend to support him, Gauss did not need to find a job so devoted himself to research. He published the book *Disquisitiones Arithmeticae* in the summer of 1801 with a special section dedicated to number theory.[7] In the following years, Carl Friedrich Gauss was offered several positions at foreign universities, but in loyalty to the Duke and the hope of getting his own observatory he stayed in Göttingen, where he had to give lectures. Despite the fact that he did not enjoy his teacher occupation, several famous future mathematicians were taught by him, like Richard Dedekind or Bernhard Riemann. [8,9]

Gauss’ contributions to the field of mathematics are numerous. At the age of only 16, he made first attempts leading to non-Eucleidean geometry. Two years later, Gauss began researching on properties of the distribution of prime numbers, which later on led him to calculate areas underneath graphs and to the Gaussian bell curve. Independent of Caspar Wessel and Jean-Robert Argand, Gauss found the geometrical expression of complex numbers in one plane.

In June 1801, Zach, an astronomer whom Gauss had come to know two or three years previously, published the orbital positions of Ceres, a new “small planet” which was discovered by Giuseppe Piazzi, an Italian astronomer on 1 January, 1801.[10] Unfortunately, Piazzi had only been able to observe 9 degrees of its orbit before it disappeared behind the Sun. Zach published several predictions of its position, including one by Gauss which differed greatly from the others. When Ceres was rediscovered by Zach on 7 December 1801 it was almost exactly where Gauss had predicted.[7]

“It may be true, that men, who are mere mathematicians, have certain specific shortcomings, but that is not the fault of mathematics, for it is equally true of every other exclusive occupation. So there are mere philologists, mere jurists, mere soldiers, mere merchants, etc. To such idle talk it might further be added: that whenever a certain exclusive occupation is coupled with specific shortcomings, it is likewise almost certainly divorced from certain other shortcomings.”

– Gauss-Schumacher Briefwechsel (1862)

At the age of 18, he discovered some properties of the prime number distribution and found the least squares method, which aims to minimize the sum of squares of deviations without first publishing anything about them. After Adrien-Marie Legendre published his “*Méthode des moindres carrés*” in a treatise in 1805 and Gauss only published his results in 1809, a dispute of priorities arose.

Gauss rejected an appointment to the Petersburg Academy of Sciences out of gratitude to his patron, the Duke of Braunschweig, and probably in the hope that he would build him an observatory in Braunschweig. After the sudden death of the Duke after the Battle of Jena and Auerstedt, Gauss became professor at the Georg August University of Göttingen and director of the observatory there in November 1807. There he had to hold courses, but he developed an aversion to them.

“I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of a mathematician, where ½ proof = 0, and it is demanded for proof that every doubt becomes impossible.”

— Gauss in a letter to Heinrich Wilhelm Matthias Olbers (14 May 1826)

Gauss had been asked in 1818 to carry out a geodesic survey of the state of Hanover to link up with the existing Danish grid. Gauss was pleased to accept and took personal charge of the survey, making measurements during the day and reducing them at night, using his extraordinary mental capacity for calculations. Because of the survey, Gauss invented the heliotrope which worked by reflecting the Sun’s rays using a design of mirrors and a small telescope. However, inaccurate base lines were used for the survey and an unsatisfactory network of triangles.[7]

Gauss began working in the field of astronomy after finishing his famous ‘*Disquisitiones Arithmeticae*‘ and managed to calculate planetary orbits through his method of least squares. He shared his experiences in the work ‘*Theoria motus corporum coelestium in sectionibus conicis solem ambientium*. His achievements in this field made Gauss internationally famous and several of his astronomical methods are still in use today.

Together with Wilhelm Eduard Weber Gauss worked in the field of magnetism starting in 1831. Together with Weber, Gauss invented the magnetometer, thus connecting his observatory with the Institute of Physics in 1833. He exchanged messages with Weber via electromagnetically influenced compass needles: the world’s first telegraph connection. Together with him he developed the CGS unit system, which was determined in 1881 at an international congress in Paris as the basis for electrotechnical units.

Gauss worked in many fields, but only published his results when he believed a theory was complete. As a result, he occasionally pointed out to colleagues that he had long proved this or that result, but had not yet presented it because of the incompleteness of the underlying theory or the lack of carelessness necessary for rapid work. Numerous mathematical methods and formulas carry Gauss’ name today and throughout his lifetime and beyond he earned himself the reputation as one of the most genius and productive mathematicians of all times.

He was still scientifically active towards the end of his life and gave lectures on the least squares method in 1850/51. Two of his most important students, Bernhard Riemann (who received his doctorate from Gauss in 1851 and impressed Gauss in 1854 with his habilitation lecture on the basics of Riemann geometry) and Richard Dedekind, were only at the end of his career. Gauss suffered in his last years from heart failure (diagnosed as dropsy) and insomnia. In June 1854 he travelled with his daughter Therese to the construction site of the railway line from Hanover to Göttingen, where the passing railway made the horses shy and overturned the carriage, the coachman was seriously injured, Gauss and his daughter remained unharmed. Gauss took part in the inauguration of the railway line on 31 July 1854, after which he was increasingly restricted to his house due to illness. He died on 23 February 1855 in Göttingen in his armchair.

At yovisto academic video search, you may enjoy a video lecture by Prof. Ramamurti Shankar on Gauss’s Law at Yale University.

**References:**

- [1] Boyer, Carl B. 1968, 1991. A History of Mathematics. Second edition. Revised by Uta C. Merzbach; foreword by Isaac Asimov. New York: Wiley. (p. 497)
- [2] Carl Friedrich Gauss Biography
- [3] Carl Friedrich Gauss Info Website
- [4] Carl Friedrich Gauss at Wolfram Research
- [5] Carl Friedrich Gauss at Britannica
- [6] Carl Friedrich Gauss at zbMATH
- [7] O’Connor, John J.; Robertson, Edmund F., “Carl Friedrich Gauss“, MacTutor History of Mathematics archive, University of St Andrews.
- [8] Richard Dedekind and the Real Numbers, SciHi Blog
- [9] Bernhard Riemann’s novell approaches to Geometry, SciHi Blog
- [10] Giuseppe Piazzi and the Dwarf Planet Ceres, SciHi Blog
- [11] Carl Friedrich Gauss at Wikidata
- [12] Carl Friedrich Gauss at the Mathematics Genealogy Project
- [13] Timeline for Carl Friedrich Gauss

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