The post What’s your Erdös Number? – The bustling Life of Mathematician Paul Erdös appeared first on SciHi Blog.

]]>On September 20, 1996, Hungarian mathematician **Paul Erdös** passed away. He published more scientific papers than any other mathematician in history, with hundreds of collaborators. Thus, he even created a ‘small world’ of its own, the famous club of people that posess an ‘*Erdös Number*‘. BTW, my Erdös number is 3, i.e. I have published a paper together with a co-author whose Erdös number is 2. In this little game of numbers, Paul Erdös has the Erdös number 0, his direct co-authors have the Erdös number 1. Thus, the Erdös number gives the distance – according to co-authorship of published scientific papers – to the famous Paul Erdös. But who was Paul Erdös and what else is he famous for….

Paul Erdös was born in Budapest in 1913 into a jewish family and showed an early interest in mathematics. Both of his parents happened to be mathematicians themselves, providing Erdös with the necessary support. While his mother was teaching, Paul was brought up by a German governess. He could already count on three years and with four friends of the family in his head he could calculate how many seconds they already lived. Their genius son was introduced to set theory as well as infinite series at the age of 16 and was awarded the doctorate in mathematics at 21. In 1920 his father returned from Siberian captivity. He had taught himself English in captivity, but without mastering pronunciation, and transferred this accent to his son. Since anti-Semitism was on the rise, he went to Harold Davenport in Manchester with a scholarship in the same year, but travelled far and wide within England and met Hardy [7] in Cambridge, among others.

In 1938 he took his first position in the USA, as a scholarship holder, in Princeton (New Jersey). However, he did not keep them for long, as the Princeton directors considered him to be “peculiar and unconventional,” and he accepted Stanislaw Ulam‘s invitation to Madison. Around this time he began to develop the habit of traveling from campus to campus. He never lasted long in one place and travelled back and forth between mathematical institutes until his death.

In 1941 Paul Erdős made a trip with his colleagues Arthur Stone and Shizuo Kakutani. They wanted to look out at the sea from an elevation with a tower. Just thinking about math, they missed a sign saying “no admission.” They took some souvenir photos and were later arrested and interrogated by the FBI for espionage. The misunderstanding soon cleared up, but the entry in an FBI file hurt him later in the McCarthy era. Only after the war did he learn of the fate of his relatives in Hungary, many of whom died in the Holocaust. He was very worried about his mother, who had survived the Holocaust. His father had died of a heart attack in 1942. When he visited his mother and friends in Hungary in December 1948 after a break of ten years, he only managed to leave Hungary again in February 1949, as Stalin had sealed off the borders in the beginning Cold War. Then he commuted back and forth between England and the USA for three years before accepting a position at the US University of Notre Dame in 1952.

When he wanted to travel to a conference in Amsterdam in 1954, he was told after an investigation before a McCarthy commission that if he left the USA, he would not be allowed to re-enter, which did not stop Erdős from going to that conference. Since the Netherlands and England also imposed travel and residence restrictions on him, he accepted a position at the Hebrew University of Jerusalem in the 1960s. Despite many attempts, he only received a permit to enter the USA again in 1963. Officially no reason was given, from the files it follows that his arrest in 1941 and his contacts to the Chinese number theorist Loo-Keng Hua were the cause.

Paul Erdös loved and lived the mathematics like no other, he made major contributions to the Ramsey theory as well as the probabilistic method. He has discovered a proof for the prime number theorem and Bertrand’s postulate. Erdös published more than 1,500 articles, but even though he has built up a tremendous reputation in the field of mathematics, he never won the greatest mathematical prize, the Fields Medal. The great scientist used mathematics for problem solving as well as socializing, sometimes combining the two of these. He would offer prizes for people, who finished some of his unsolved problems. The prizes ranged from 25$ up to several thousands, but there is no official number on how many prizes he actually gave away. Up to this day you can work on Erdös’ problems, they are administrated by Ronald Graham, a famous mathematician himself.

He slept only four to five hours a day and putsched himself up with coffee, caffeine tablets and amphetamine, which he was prescribed due to depression after his mother’s death. In 1979, his friend Ronald Graham offered him a $500 bet because he worried that Erdős was dependent: he would not be able to last 30 days without stimulants. He lasted the 30 days, but said that the bet had set mathematics back a month because he could not put a thought on paper. After the bet, he resumed amphetamine use.

You may also be wondering, how Erdös managed to work together with 511 collaborators and publish this many papers. Actually it’s simple. He lived a life of a vagabond, traveling from place to place, teaching, writing, researching. Often enough he would just appear at other mathematicians homes saying “my brain is open” and not leaving until the work was done.

Erdös was a man seeking the entire freedom, which he found in mathematics. He called men ‘slaves’, women the ‘bosses’, and children were ‘epsilons’, that is how he rolled, never stopping to focus on mathematics, considering all kinds of commitment as distractions. He truly believed in what he was doing all his life, he understood mathematical lectures as prayers and mathematicians as devices for “*turning coffee into theorems*“. Paul Erdös is to be considered as one of the most active and the most productive mathematicians in history.

In September 1996 Erdős participated in a conference on graph theory in Warsaw. He died there on September 20 as a result of two heart attacks.

At yovisto academic video search you can watch Prof. John Borrowman from Gresham College explaining the ‘small world phenomenon’ on Erdös numbers.

**References and Further Reading:**

- [1 ]Man Who Loved Only Numbers: Story of Paul Erdos and the Search for Mathematical Truth, Paul Hoffmann, 1999
- [2] Combinatorics, geometry and probability. A tribute to Paul Erdös, Béla Bollobás, 1997
- [3] Paul Erdös in the Wikipedia
- [4] Paul Erdös Article by L´aszló Babai on the American Mathematical Society Website
- [5] Paul Erdös at zbMATH
- [6] Paul Erdös at Wikidata
- [7] Timeline for Paul Erdös via Wikidata
- [8] G. H. Hardy and the aesthetics of Mathematics, SciHi Blog
- [9] Please Don’t Ignite the Earth’s Atmosphere…, SciHi Blog

The post What’s your Erdös Number? – The bustling Life of Mathematician Paul Erdös appeared first on SciHi Blog.

]]>The post Charles Sanders Peirce – One of the Founders of Semiotics appeared first on SciHi Blog.

]]>On September 10, 1839, mathematician, philosopher and logician **Charles Sanders Peirce**, the founder of philosophical ‘pragmatism’ was born.

“Few persons care to study logic, because everybody conceives himself to be proficient enough in the art of reasoning already.”

— Charles Sanders Peirce, [10]

Peirce was born in Cambridge, Massachusetts, the second of five children of Sarah and Benjamin Peirce (1809-1880). His father was professor of astronomy and mathematics at Harvard University and proved to be the first seriously researching mathematician in the USA. His living environment was that of a well-off educated middle class. Even as a boy, Peirce was given a chemistry laboratory by an uncle. His father recognized his talent and tried to give him a comprehensive education. At the age of 16 he began to read the Kant‘s *Critique of Pure Reason*.[4] He needed three years for the study of the work, with which he dealt daily several hours, after which, according to his own statement, he almost knew the book by heart. Peirce studied at Harvard University and Lawrence Scientific School. He passed the Master of Arts in 1862 and was one of the first (1863) to graduate with a Bachelor of Science in Chemistry.

From 1859 to 1891 he worked with interruptions at the United States Coast and Geodetic Survey. From 1861 he had a regular post, so he did not have to take part in the American Civil War. He received this post through his father’s mediation, who was one of the founders of this authority and acted as a board member there. Peirce’s tasks in the field of geodesy and gravimetry were the further development of the application of pendulums for the determination of local deviations in earth gravity. At Harvard between 1864 and 1870 Peirce gave part-time lectures on the history and theory of science. Even at this point in his life, the manuscripts of the lectures contained almost all of the fundamental themes of philosophy that occupied him throughout his life. At the beginning he was very strongly influenced by Kant, but he intensively dealt with questions of logic and first developed his own theory of categories.

In the first years, the logical work was in the foreground. Thus in 1865 he dealt with the new logic of George Boole and Augustus De Morgan, which gave his mental development a substantial impulse. His first articles were published in 1868 and soon he was giving lectures on logic at Harvard University. In the late 1860’s and early 1870’s, Peirce started researching at Harvard’s astronomical observatory and joined a club of young scientists, where he got to know Alexander Bain, who influenced Peirce enormously. He was also able to present his thoughts about pragmatism, this led to him being able to publish his ideas and counts as the birth of the pragmatism.

In the 1860s Peirce accompanied the astronomical research of his peer George Mary Searle, who also worked for the Coast Survey and the Harvard Observatory during this time, who was interested in astronomical research. From 1869 to 1872 Peirce then worked himself at the astronomical observatory of Harvard as an assistant on questions of photometry to determine the brightness of stars and the structure of the Milky Way. 1870 saw the publication of a small, for Peirce and logicians but important paper on the *Logic of Relatives*, which was also published as a lecture to the American Academy of Arts and Sciences under the title *Description of a Notation for the Logic of Relatives, Resulting from the Amplification of Boole’s Calculus of Logic*. Important for Peirce and also for William James was a circle of young scientists of different disciplines at the beginning of the 1870s, which was called “metaphysical club”. Here Peirce got to know the philosophy of Alexander Bain, from whom he adopted the principle of doubt and the convictions that determine people’s actions. Peirce presented his basic ideas on pragmatism and put them up for debate, which later gave rise to his important series of essays from 1877/78. This publication in Popular Science is usually referred to as the birth of pragmatism.

Between 1871 and 1888 Peirce was able to undertake a total of five research trips, each lasting several months, to Europe as part of his geodetic task, where he met a number of prominent scientists. In 1879 Peirce was appointed “half-time lecturer of logic” at Johns Hopkins University in Baltimore, his only permanent academic position. In 1887 Peirce used the inheritance of his parents to buy a farm near Milford, Pennsylvania, where he spent the rest of his life – with the exception of a few trips, especially to lectures – writing incessantly. In the late 1880s Peirce made a major contribution to *The Century Dictionary and Cyclopedia*, an encyclopaedia of 450,000 terms and names in mechanics, mathematics, astronomy, astrology and philosophy edited by James Mark Baldwin. After he had delivered an extensive scientific report about his pendulum tests to the US Coast Survey, but this report had been rejected by Thomas C. Mendenhall, who had been Superintendent for only a short time, Peirce gave up his position with this authority after more than 30 years at the end of 1891. He had thus lost his secure economic livelihood and now had to earn his money exclusively through teaching, translations, lectures and publications.

In the course of time he got into ever greater financial difficulties, which accompanied him until the end of his life. Often enough the money was lacking to procure only food or fuel for the heating. In 1898, through William James, with whom he had been friends since the time he studied chemistry, Peirce was able to give a series of lectures at Cambridge on the general topic of reasoning and the logic of things. In 1903 James was again able to help James, so that Peirce was given the opportunity of a lecture series at Harvard on Pragmatism as a Principle and Method of Right Thinking. Thus Peirce, at a relatively mature stage of his thinking, presented essential cornerstones of his philosophy in a closed context, but did not publish them.

In the course of time he got into ever greater financial difficulties, which accompanied him until the end of his life. Often enough the money was lacking to procure only food or fuel for the heating. In 1898, through William James, with whom he had been friends since the time he studied chemistry, Peirce was able to give a series of lectures at Cambridge on the general topic of reasoning and the logic of things. In 1903 James was again able to help James, so that Peirce was given the opportunity of a lecture series at Harvard on Pragmatism as a Principle and Method of Right Thinking. Thus Peirce, at a relatively mature stage of his thinking, presented essential cornerstones of his philosophy in a closed context, but did not publish them. Peirce had no children and died of cancer in 1914.

Only after the publication of the *Collected Papers* did systematic cataloguing and microfilming begin. The microfilming was only completed in 1966 (provisionally). Again and again additions were found in the archives, most recently in 1969, so that the microfilm files and the catalogues had to be updated. The current cataloguing is based on the year 1971, when it became clear that Peirce had left about 1650 unpublished manuscripts with about 80,000 handwritten pages, most of which have not yet been published. Some of the documents that had not gone to Harvard were lost because they were burned after the death of Peirce’s wife Juliette.

“It is the man of science, eager to have his every opinion regenerated, his every idea rationalized, by drinking at the fountain of fact, and devoting all the energies of his life to the cult of truth, not as he understands it, but as he does not yet understand it, that ought properly to be called a philosopher.”

— Charles Sanders Peirce, [11]

Alongside Ferdinand de Saussure,[9] Peirce is one of the founders of semiotics, his preferred term being “semeiotic”, while Saussure described his own approach as “sémiologie” (semiology). In contrast to Saussure’s concept of the sign, which refers exclusively and formally to language, so that essential impulses for linguistics arose from it, Peirce’s concept of the sign is holistic: in addition to its representational function, it also contains a cognitive function of the signs. Likewise, the semiotics of Peirce must not be mixed with the subdivisions of Charles W. Morris (syntax, semantics, and pragmatics) (although Morris refers to Peirce).

Peirce defined semiosis (see also Semiose) as *“… a process or an influence which is or contains the interaction of three objects, namely the sign, its object and its interpreter; a threefold influence which in no case can be resolved in pairs.”* He divided semiotics into speculative grammar, logical criticism and speculative rhetoric. For him, the word “speculative” was synonymous with “theoretical”.

- Speculative grammar examines the possible types of characters and their possible combinations.
- Logical criticism is directed at the question of correct justification.
- Speculative rhetoric is the study of the effective application of signs (the question of the economics of research).

As with other subjects, Peirce never wrote an exact determination of his semiotics. Rather, he has been involved with the subject throughout his life, often changing his perception of the definition of key terms.

At yovisto academic video search you may enjoy the lecture ‘Semiotics and Structuralism’ by Professor Paul Fry at Yale University.

**References and Further Reading:**

- [1] Charles Sanders Peirce Society Website
- [2] Charles Sanders Peirce in The Information Philosopher
- [3] Charles Sanders Peirce at Wikidata
- [4] Immanuel Kant – Philosopher of the Enlightenment, SciHi Blog
- [5] The Time You Enjoy Wasting Is Not Wasted Time – Bertrand Russell – Logician and Pacifist, SciHi blog
- [6] Charles Sanders Peirce at zbMATH
- [7] Charles Sanders Peirce at Mathematics Genealogy Project
- [8] Timeline for Charles Sanders Peirce, via Wikidata
- [9] Ferdinand de Saussure and the Study of Language, SciHi Blog
- [10] “Illustrations of the Logic of Scence” First Paper — The Fixation of Belief”, in
*Popular Science Monthly*, Vol. 12 (November 1877) - [11] “The Century’s Great Men in Science” in The 19th Century : A Review of Progress During the Past One Hundred Years in the Chief Departments of Human Activity (1901), published by G. P. Putnam’s Sons.

The post Charles Sanders Peirce – One of the Founders of Semiotics appeared first on SciHi Blog.

]]>The post How to Calculate Fortune – Jakob Bernoulli appeared first on SciHi Blog.

]]>The Swiss Bernoulli family is well known for their many offsprings who gained prominent merits in mathematics and physics in the 18th century.** Jakob Bernoulli**, born in 1654 (or 1655 according to the new Gregorian calendar), is best known for his work *Ars Conjectandi* (*The Art of Conjecture*). In this work, published 8 years after his death in 1713 by his nephew Nicholas, Jakob Bernoulli described the known results in probability theory and in enumeration, including the application of probability theory to games of chance.

“[P]robability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers…”

— Jacob I Bernoulli, Ars Conjectandi (1713)

Jacob I Bernoulli was born in Basel, Switzerland**, ** the son of the merchant Niklaus Bernoulli and his wife Margaretha Schönauer, as well as brother of the mathematician Johann Bernoulli. After attending school and receiving his first lessons from his father, Jakob studied philosophy and theology at the University of Basel at his request. In 1671 he received his Master of Arts and in 1676 his lic. theol. against his father’s will and almost autodidactically he deepened his knowledge in mathematics and astronomy. It is worth remarking that this was a typical pattern for many of the Bernoulli family who made a study of mathematics despite pressure to make a career in other areas.

From 1676 to 1680 Jakob held various positions as a tutor in Geneva. During this time he also travelled several times to France. Between 1681 and 1682 Jacob I undertook a kind of cavalier tour through Holland, Great Britain and Germany. During these journeys he not only got to know Cartesian mathematics, but also Hudde, Boyle and Hooke, among others. Many of his later contacts with leading mathematicians of the time emerged from this time. Thus, he also became familiar with calculus through a correspondence with Gottfried Wilhelm Leibniz, then collaborated with his brother Johann on various applications, notably publishing papers on transcendental curves and isoperimetry. In 1690, Jacob Bernoulli became the first person to develop the technique for solving separable differential equations. Upon returning from his European travels to Basel in 1682, Jakob held private lectures on experimental physics at the University of Basel from 1683. During this time he studied among others the geometry of René Descartes.[4] From 1686 Jakob used the complete induction, examined important power series with the help of the Bernoulli numbers, and co-founded the probability theory (see Bernoulli distribution). In 1687, he was appointed professor of mathematics at the University of Basel, remaining in this position for the rest of his life.

Until 1689 Jakob had published important works on power series and probability calculation, among others on the law of large numbers. He formulated Bernoulli’s law of large numbers, which is considered the first weak law of large numbers. In the early 1690s he worked mainly in the field of calculus of variations, where he studied important curves and differential equations. In 1697, after many years of rivalry, Jacob quarreled with his brother Johann.

Jakob Bernoulli‘s *Ars conjectandi* was also the first substantial treatise on probability and was only published in Basel in 1713, eight years after his death. It contained the general theory of permutation and combination, the weak law of large numbers (that states that the sample average converges in probability towards the expected value), as well as the binomial theorem (for which the first adequate proof for positive integers was given) and multinomial theorem. In the Dictionary of Scientific Biografies, J.E. Hofmann summarizes Jakob Bernoulli`s contributions to mathematics in the following way:

“Bernoulli greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability. He was self-willed, obstinate, aggressive, vindictive, beset by feelings of inferiority, and yet firmly convinced of his own abilities. With these characteristics, he necessarily had to collide with his similarly disposed brother. He nevertheless exerted the most lasting influence on the latter.”

Jakob Bernoulli died on August 16, 1713 at age 59. His professorship in Basel was then taken over by his brother Johann. One of Bernoulli’s favourite toys was the logarithmic spiral, which he dealt with extensively. According to the story, Bernoulli wanted such a spiral on his tombstone. Instead, after Bernoulli’s death (probably out of ignorance or to save himself work), the stonemason in charge carved an Archimedean spiral into the epitaph, which can now be seen in the cloister of Basel Cathedral.

At yovisto academic video search you might find many video lectures related to the works of Jakob Bernoulli. We want to recommend you Prof. Jerzy M. Wrobel’s lecture on ‘The Archimedes Principle and Bernoulli’s Equation’ from the University of Missouri, Kansas City, where he presents types of interactions occurring in fluids.

**References and further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Jacob Bernoulli”, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Let us calculate – The last universal academic Gottfried Wilhelm Leibniz, from SciHi blog, Juli 1, 2012
- [3] J. E. Hofmann, Biography in Dictionary of Scientific Biography (New York 1970-1990).
- [4] Cogito Ergo Sum – The Philosophy of René Descartes, SciHi Blog
- [5] Gottfried Leibniz and Jakob Bernoulli Correspondence Regarding the Art of Conjecturing”
- [6] Jakob I Bernoulli at Wikidata
- [7] Jakob I Bernoulli at zbMATH
- [8] Jakob I Bernoulli at Mathematics Genealogy Project

The post How to Calculate Fortune – Jakob Bernoulli appeared first on SciHi Blog.

]]>The post It’s Computable – thanks to Alonzo Church appeared first on SciHi Blog.

]]>You know, the fact that you can read your email on a cell phone as well as on your desktop computer or almost any other computer connected to the internet, in principle is possible thanks to mathematician **Alonzo Church**, who gave the proof (together with Alan Turing) that everything that is computable on the simple model of a Turing Machine, also is computable with any other ‘computer model’.

Church studied at Princeton University and graduated with a doctorate. After stays at the University of Chicago, the Georg August University of Göttingen and the University of Amsterdam he became Princeton Professor of Mathematics in 1929. He became known to his mathematical-logical colleagues for his development of the Lambda Calculus, to which he wrote in a report published in 1936 (Church-Rosser’s theorem) in which he demonstrated that there are undecidable problems (i.e. the answer to a question cannot be calculated mathematically).

In mathematics and computer science, the ‘Entscheidungsproblem‘ is one of the challenges posed by mathematician David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm that takes as input a statement of a first-order logic and answers “Yes” or “No” according to whether the statement is universally valid, i.e., valid in every structure satisfying the underlying axioms.

Actually, the origin of the Entscheidungsproblem goes back to Gottfried Wilhelm Leibniz, who in the 17th century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements.[10] Leibniz realized that the first step would have to be a clean formal language, and much of his subsequent work was directed towards that goal.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic. In 1936 and 1937, Alonzo Church and his student Alan Turing, respectively, published independent papers showing that a general solution to the Entscheidungsproblem is impossible. To achieve this, Alonzo Church applied the concept of “effective calculability” based on his Lambda calculus, while Alan Turing based his proof on his concept of Turing machines. Church and then found that the Lambda calculus and the Turing machine were equal in expressiveness, and were able to give some more equivalent mechanisms for calculating functions. A thesis for the intuitive calculability concept derived from this is known as the Church-Turing thesis. The lambda calculus also influenced the design of the LISP programming language and functional programming languages in general.

It was recognized immediately by Turing that these two concepts are equivalent models of computation. Both authors were heavily influenced by Kurt Gödel‘s earlier work on his incompleteness theorem, especially by the method of assigning numbers (also-called Gödel numbering) to logical formulas in order to reduce logic to arithmetic. Church’s Theorem, showing the undecidability of first order logic, appeared in A note on the Entscheidungsproblem published in the first issue of the Journal of Symbolic Logic. This, of course, is in contrast with the propositional calculus which has a decision procedure based on truth tables. Church’s Theorem extends the incompleteness proof given of Gödel in 1931.[11]

Church was a founder of the *Journal of Symbolic Logic* in 1936 and was an editor of the reviews section from its beginning until 1979. In 1960 Church was elected to the American Academy of Arts and Sciences, in 1978 to the National Academy of Sciences. In 1962 he gave a plenary lecture at the International Congress of Mathematicians in Stockholm (*Logic, Arithmetic and Automata*).

Alonzo Church died on August 11, 1995, aged 92.

At yovisto academic video search you can learn more about Alonzo Church in the lecture ‘At odds with the Zeitgeist: Kurt Gödel’ by Prof. John W. Dawson from the Institute of Advanced Studies in Princeton.

**References and further Reading:**

- [1] Church’s Theorem in Wikipedia
- [2] Alonzo Church, “A note on the Entscheidungsproblem“, Journal of Symbolic Logic, 1 (1936), pp 40–41.
- [3] Alan Turing, “On computable numbers, with an application to the Entscheidungsproblem“, Proceedings of the London Mathematical Society, Series 2, 42 (1937), pp 230–265.
- [4] O’Connor, John J.; Robertson, Edmund F., “Alonzo Church”, MacTutor History of Mathematics archive, University of St Andrews.
- [5] Alonzo Church at the Mathematics Genealogy Project
- [6] Alonzo Church at zbMATH
- [7] Alonzo Church at Wikidata
- [8] Churchill’s Best Horse in the Barn – Alan Turing, Codebreaker and AI Pioneer, SciHi Blog
- [9] David Hilbert’s 23 Problems, SciHi Blog
- [10] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog
- [11] Kurt Gödel Shaking the Very Foundations of Mathematics, SciHi Blog

The post It’s Computable – thanks to Alonzo Church appeared first on SciHi Blog.

]]>The post The time you enjoy wasting is not wasted time – Bertrand Russell, Logician and Pacifist appeared first on SciHi Blog.

]]>On July 11, 1906, mathematician and philosopher **Bertrand Russell** was suspended from Trinity College, Cambridge due to his engagement in pacifist activities. The remarkable Bertrand Russell, a philosopher, logician, mathematician, historian, and social critic was best known for the famous ‘*Principia Mathematica*‘, which he published along with Alfred North Whitehead between 1910 and 1913.

“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. I.”

— Bertrand Russel [9]

Bertrand Russel was born into a prominent family in Britain and differentiated thinking was passed on to him by his parents, who were very much ahead of their time and often positioned themselves politically against any expectancy. Unfortunately, his parents passed away soon. He grew up with his grandfather, who was the Prime Minister of Great Britain and his more dominant grandmother, who in contrast to his father John was truly religious, but still had liberal views on many topics, such as Darwinism. Her believes and principles had a great impact on Russell, such as her admiration for rules and definitions, which reflect Russel’s later interest in logic.

Also he could benefit (at least scientifically) from the many home teachers his grandmother hired, especially from his math teacher who was the first to introduce him to Euclid’s theories.[2] Because of him being rather lonely in his childhood, Bertrand Russell had much time to educate himself and made mathematics to his first priority in life. Russell received a scholarship from Cambridge University, his father’s alma mater, and studied mathematics there from 1890 to 1894. However, he always missed finding the real truth of mathematics, i.e. the one thing in mathematics, you can rely on and you can build upon. Later he was awarded a Fellowship, which enabled him to conduct research from 1895 to 1901 without any teaching duties.

Through his later first wife Alys Pearsall Smith he also occupied himself with philosophical studies. He is now seen as one of the founders of analytic philosophy and was greatly influenced by Gottfried Wilhelm Leibniz.[3] Several meetings with George Edward Moore cleared up his mind and introduced him to the field of logic. At a mathematical congress in 1900, Russell met the Italian logician Giuseppe Peano and his work. Russell adopted Peano’s methods, expanded them and thus laid the foundation for Principia Mathematica, an attempt to trace all mathematics back to a limited set of axioms and concluding rules. Work on this monumental work lasted from 1902 to 1913, when the third and last volume appeared. Russell wrote Principia Mathematica together with Whitehead, who temporarily lived with his family at Russell’s house.

In the early 1900’s, Russell began his study on the foundations of mathematics, became a member of the Royal Society and published the first of the three books of ‘*Principia Mathematica*‘ along with Whitehead. The first book dealt with set theory, cardinal numbers, ordinal numbers, and real numbers. Towards the end of their work the authors made clear that all known mathematical principles can be developed from the formalism depicted above.

In 1911 Russell first met the Viennese philosopher Ludwig Wittgenstein, who had studied at Cambridge, and made friends with him. Through the years, Russell could make up a great reputation for himself in many scientific areas, always questioning and seeking for the ‘real’ truth. He developed theories on society and distributed his clear religious convictions. A decisive event in Russell’s life was the First World War. From 1914 Russell suspended his mathematical research and began to work as an activist and author for peace and conscientious objection. The fact that he had been fined for a leaflet prompted Cambridge University to withdraw his professorship. He was later sentenced to six months in prison for considering in an anti-war service magazine the possibility of US soldiers being used as strikebreakers in England. However, Russell was allowed to read and write in prison, and so he wrote several books during his imprisonment. After the book *Introduction to Mathematical Philosophy* (1919), written in prison, in which he mainly explains earlier works and their philosophical significance, Russell turned away from problems of mathematics and logic.

Unlike World War I, Russell did not take a pacifist position in World War II. Shortly after the end of the war he even spoke out in favour of a preventive war against the Soviet Union, which did not yet have nuclear weapons. He wanted to prevent a nuclear war that would destroy humanity. In 1949 Russell received the Order of Merit, and in 1950 he was awarded the Nobel Prize for Literature, especially for *marriage and morality*, for which he had been strongly criticized a few years earlier.

“Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs”

— Bertrand Russel, The Study of Mathematics” (November 1907)

Russell, who was 78 years old and has received many awards worldwide, did not withdraw from the public after 1950. It was above all a possible Third World War that moved him as a great danger for mankind. He was the driving force behind the Russell-Einstein Manifesto and acted as mediator between the heads of state in various political crises during the Cold War. In 1963 he founded the Bertrand Russell Peace Foundation. At the Russell Tribunal, he investigated US war crimes in Vietnam.

Bertrand Russell died of influenza on 2 February 1970 at the age of 97 in Penrhyndeudraeth (Wales).

At yovisto academic video search, you have the possibility the see an extraordinary lecture about a graphic novell about the life of the great philosopher Bertrand Russel entitled ‘Logicomix – The Epic Search for Truth’ by Prof. Christos Papadimitriou.

**References and further Reading:**

- [1] A. Papadatos, A. Doxiadis, C. Papadimitriou: Logicomix – The Epic Search for Truth, Bloomsbury (2009)
- [2] Euclid – the Father of Geometry, SciHi Blog
- [3] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog
- [4] The Philosophy of Ludwig Wittgenstein, SciHi Blog
- [5] Bertrand Russel at Wikidata
- [6] Bertrand Russel at zbMATH
- [7] O’Connor, John J.; Robertson, Edmund F., “Bertrand Russell“, MacTutor History of Mathematics archive, University of St Andrews.
- [8] Bertrand Russel at Mathematics Genealogy Project
- [9] Bertrand Russel, Recent Work on the Principles of Mathematics, published in International Monthly, Vol. 4 (1901), later published as “Mathematics and the Metaphysicians” in Mysticism and Logic and Other Essays (1917)
- [10] Timeline for Bertrand Russel, via Wikidata

The post The time you enjoy wasting is not wasted time – Bertrand Russell, Logician and Pacifist appeared first on SciHi Blog.

]]>The post Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz appeared first on SciHi Blog.

]]>On July 1, 1646, one of the last universally interdisciplinary academics, active in the fields of mathematics, physics, history, politics, philosophy, and librarianship was born. **Gottfried Wilhelm Leibniz** counts as one of the most influential scientists of the late 17th and early 18th century and impersonates a meaningful representative of the Age of Enlightenment. Moreover, he is also the namesake of the association to which the institute I am working for is a member of, the Leibniz Association (Leibniz Gemeinschaft).

Leibniz made up his interests concerning philosophy and law studies in his early years, following his father’s footsteps. He even decided to acquire Latin auto-didactically at the age of eight, which is impossible to imagine for today’s Latin students, who experience this language more as a constant torture. But Leibniz sticked to it and was therefore able to attend the famous Thomasschule in Leipzig. His later years at the University of Leipzig and the University of Jena were filled with studies in philosophy, law, mathematics, physics, and astronomy. Because of his widely spread field of education he is now titled as the ‘last universal academic’. He was able to establish a great reputation, working for archbishop Johann Phillip von Schönborn in the 1670‘s. During his time in Mainz he published his first work of great reception ‘Nova methodus discendae docendaeque jurisprudentiae’, a new method to teach and study jurisprudence. He also became a member of the British Royal Society due to his achievement of creating a calculating machine with a stepped reckoner. Another contribution to the field of mathematics was his (and Newton’s) development of infinitesimal calculus, revolutionary then and a basis of many calculations in mathematical, physical, stochastic and economical problems today. In philosophy, Leibniz got famous with the phrase of the ‘best of all possible worlds’. It pictures the correlation between the good and the evil, meaning that the world has a huge potential of development and that even God cannot realize the good things on earth without a certain amount of the evil.

Leibniz’s achievements are far too many to be mentioned all in one small blog post [1]. Thus we will focus here only on a small episode. Also for computer scientists, Leibniz anticipated the use of formal logic for automated reasoning and decision making. Besides inventing the binary system, which is the basis of nowadays computers, Leibniz argued that if we would be able to find a formal (logic) language to express problems instead of our ambiguous natural language, we should be able to solve arguments simply performing a calculation. *Let us calculate!* (in Latin: *Calculemus*!) he requested, to solve every argument or dispute. He believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion:

“The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right”

— Gottfried Wilhelm Leibniz in a letter to Philip Spener, The Art of Discovery 1685, Wiener 51

Leibniz’s *calculus ratiocinator*, which resembles symbolic logic, can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda that can now be read as groping attempts to get symbolic logic – and thus his calculus – off the ground. These writings remained unpublished until the appearance of a selection edited by C.I. Gerhardt (1859). L. Couturat published a selection in 1901; by this time the main developments of modern logic had been created by Charles Sanders Peirce and by Gottlob Frege.[7]

Another highlight in Leibniz‘ career probably was becoming the first president of the Prussian Academy of Sciences in Berlin. His achievements and contributions to the world’s development are numerous and therefore he was honored several times during his lifetime and has not been forgotten today. Since a big part of his scientific work is documented in letters, the collection of these papers have been inscribed on UNESCO‘s Memory of the World Register in 2007.

At yovisto academic video search, you may learn about the Highlights of Calculus, a lecture by Professor Strang, who shows how calculus applies to ordinary life situations, such as: driving a car or climbing a mountain.

**References and Further Reading:**

- [1] Leibniz and the Integral Calculus, SciHi Blog
- [2] O’Connor, John J.; Robertson, Edmund F., “Gottfried Wilhelm Leibniz“, MacTutor History of Mathematics archive, University of St Andrews.
- [3] Gottfried Wilhelm Leibniz at Wikidata
- [4] Timeline for Gottfried Wilhelm Leibniz, via Wikidata
- [5] Gottfried Wilhelm Leibniz at zbMATH
- [6] Gottfried Wilhelm Leibniz at Mathematics Genealogy Project
- [7] Gottlob Frege and the Begriffsschrift, SciHi Blog
- [8] Charles Sanders Peirce and Semiotics, SciHi Blog

The post Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz appeared first on SciHi Blog.

]]>The post It is not Certain that Everything is Uncertain – Blaise Pascal’s Thoughts appeared first on SciHi Blog.

]]>“*It is not certain that everything is uncertain.*” is one of the many profound insights that philosopher and mathematician Blaise Pascal (1623-1662) published in his seminal work entiteled “*Pensées*” (*Thoughts*, published in 1669, after his death). He literally had versatile scientific interests, as he provided influential contributions in the field of mathematics, physics, engineering, as well as in religious philosophy. Blaise Pascal was a child prodigy educated by his father Étienne Pascal, a tax collector in Rouen. His mother died when he was only three years old. As the father did not like the way school was taught at that time he decided to instruct Blaise and his three siblings at home by himself. In particular he put emphasis on learning the classic languages Latin and Greek. But he considered not to teach geometry to Blaise because he felt the topic was too enticing and attractive.

Geometry is the branch of mathematics that deals with points, lines, angles, surfaces, and solids. Blaise’s father thought that if exposed to geometry and mathematics too soon, Blaise would abandon the study of classics. But, this ban on mathematics made Blaise even more curious. On his own he experimented with geometrical figures. He invented his own names for geometrical terms because he had not been taught the standard terms. He discovered that the sum of the angles of a triangle are two right angles and, when his father found out, he relented and allowed Blaise a copy of Euclid‘s standard textbook on mathematics. Some people believe that Blaise was twelve years old when he started attending meetings of a mathematical academy together with his father. Others suppose that he did not attend the meetings until he was about sixteen. In any case he was far younger than the adults who were there.[4]

Blaise Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on it for three years between 1642 and 1645. The device, called the Pascaline, resembled a mechanical calculator of the 1940s. This, almost certainly, makes Pascal the second person to invent a mechanical calculator for Wilhelm Schickard had manufactured one in 1624.[2] First, the Pascaline was only able to do additions, but was subsequently improved also to provide the means for subtractions. There were additional problems faced by Pascal in the design of his calculator due to the French currency at that time. There were 20 sols in a livre and 12 deniers in a sol. This monetary system remained in France until 1799 (in the UK a system with similar multiples lasted even until 1971). Thus, Pascal had to solve much harder technical problems to work with this division of the livre into 240 than he would have had if the division had been 100. Although Blaise Pascal filed a patent for his machine, he was not able to make a fortune out of it. The elaborate and complex design of the machine made it’s construction much too expensive to be sold in significant numbers.

In correspondence with the famous hobby mathematician Pierre de Fermat (you might remember Fermat’s last theorem) Pascal laid the foundation for the modern theory of probability.[3] This correspondence started in 1654 and consisted of five letters about the dice problem that asks how many times one must throw a pair of dice before one expects a double six.

Following a mystical experience in late 1654, Pascal had his “second conversion”, abandoned his scientific work, and devoted himself to philosophy and theology. Two of his most famous works date from this period: the *Lettres provinciales* and the *Pensées. *The later consists of a collection of personal thoughts on human suffering and faith in God. This work contains also ‘*Pascal’s wager*‘ which claims to prove that belief in God is rational with the following argument.

“If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.”

With his already poor health it went downhill faster and faster during these years, certainly also because of his extremely ascetic way of life, which weakened him additionally. So he could not work for many weeks in 1659. Nevertheless, in the same year he was a member of a committee that tried to initiate a new Bible translation. In 1660 he spent several months as a convalescent at a small castle of his older sister and his brother-in-law near Clermont. In early 1662, together with his friend Roannez, he founded a hackney carriage company (“*Les carrosses à cinq sous*“), which marked the beginning of public transport in Paris. In August he fell seriously ill, had his household (still quite respectable) sold for charitable purposes and died at the age of only 39.

At yovisto academic video search you might watch a brief introduction into Pascal’s Pensées or also a short description of his famous Pascaline.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Blaise Pascal“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Wilhelm Schickard and his Calculator Machine, SciHi Blog
- [3] Pierre de Fermat and his Last Problem, SciHi Blog
- [4] Euclid – the Father of Geometry, SciHi Blog
- [5] Blaise Pascal at zbMATH
- [6] Blaise Pascal at Mathematics Genealogy Project
- [7] “Blaise Pascal“. Catholic Encyclopedia. 1913.
- [8] Blaise Pascal at Wikidata

The post It is not Certain that Everything is Uncertain – Blaise Pascal’s Thoughts appeared first on SciHi Blog.

]]>The post Although I Cannot Prove it… – The Famous Goldbach Conjecture appeared first on SciHi Blog.

]]>On the 7th of June in the year of our Lord 1742, Prussian mathematician **Christian Goldbach** wrote a letter to his famous colleague Leonard Euler, which should make history. Well, at least in the mathematical world. In this letter Christian Goldbach refined an already previously stated conjecture from number theory concerning primes to his friend Euler, which by today is known as the famous **Goldbach conjecture**. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

And still today this conjecture holds, but could never be formally proven. Even Leonard Euler himself was not able to give a proof. By the end of june he replied in a letter to Goldbach:

“Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe necht demonstriren kann.” (“every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it.”)

Progress has been made on this problem, but slowly – it may be quite awhile before the work is complete. All attempts to solve the problem remained unsuccessful for a long time. G.H. Hardy and J.E. Littlewood in 1923 succeeded in showing that if certain theorems concerning Dirichlet L-functions (which have not been proved till now) are valid, then any sufficiently large odd number is the sum of three prime numbers.[3,5] Vinogradov in 1937 showed that every sufficiently large odd integer can be written as the sum of at most three primes, and so every sufficiently large integer is the sum of at most four primes.[4] One result of Vinogradov’s work is that we know Goldbach’s theorem holds for almost all even integers. In 1973 Chen Jing-run proved that every sufficiently large even number is a sum of a prime and a number composed of at most two primes.[3] Some years ago, Jean-Marc Deshouillers, Yannick Saouter, and Herman te Riele have verified Goldbach’s conjecture with brute force computing with the help of a Cray C90 super computer up to 10^{14}. And on April 4, 2012 Thomás Oliveira e Silva has finished a distributed computer search that has extended over years and has verified the conjecture for up to 4*10^{18}.

Born the son of a Protestant pastor, Christian Goldbach studied medicine and law at the Albertus University in his home town Königsberg, Prussia (today’s Kaliningrad, Russia) . From 1710-1724 he went on longer study trips through Germany, England, the Netherlands, Italy and France. He came into contact with many well-known mathematicians such as Gottfried Wilhelm Leibniz, Leonhard Euler, Nicolas I. Bernoulli and acquired thorough mathematical knowledge. Back in Königsberg he met Georg Bernhard Bilfinger and Jakob Hermann. Both had been appointed by Tsar Peter the Great to the newly founded St. Petersburg Academy. Goldbach then applied to the President of the Academy Lorenz Blumentrost (1692-1755) in July 1725 and was appointed professor of mathematics and history. At the constitutive first meeting on 27 December 1725 he acted as secretary of the Academy. In 1727 Goldbach was appointed teacher to the young Tsar Peter II and moved with him to Moscow. From 1727 onwards, regular academic correspondence began with Leonhard Euler, who had been appointed to the Petersburg Academy, which lasted for several decades.

After the death of Peter II from smallpox in 1730, the entire court moved again from Moscow to St. Petersburg with the new Tsarina Anna. Goldbach followed and resumed his activities at the Academy. In 1737, together with Johann Daniel Schumacher (1690-1761), he was appointed managing director of the academy. Despite the unstable and changeable political conditions in tsarist Russia, Goldbach managed to remain in the favor of the powerful throughout. In the 1740s he discontinued his activities at the Academy and took up a well-paid position in the Russian Foreign Ministry. Later he was commissioned to draw up principles for the education of princes of royal blossom.

At yovisto you might watch a rather interesting introduction into ‘The History of Primes’ presented by Manindra Agrawal, where he gives insight into number theoretic results concerning primes and is also referring to Goldbach’s famous conjecture…

**Further Reading:**

- [1] Apostolos K. Doxiadis: Uncle Petros and Goldbach’s Conjecture: A Novel of Mathematical Obsession, Bloomsberry (2001)
- [2] Goldbach’s original letter to Euler
- [3] Hazewinkel, Michiel, ed. (2001) [1994], “Goldbach problem“, Encyclopedia of Mathematics
- [4] Ivan Matveevich Vinogradov and the Goldbach Conjecture, SciHi Blog
- [5] G. H. Hardy and the aesthetics of Mathematics, SciHi blog
- [6] Euler’s Correspondence with Christian Goldbach
- [7] Christian Goldbach at zbMATH
- [8] O’Connor, John J.; Robertson, Edmund F., “Christian Goldbach“, MacTutor History of Mathematics archive, University of St Andrews.
- [9] Christian Goldbach at Wikidata

The post Although I Cannot Prove it… – The Famous Goldbach Conjecture appeared first on SciHi Blog.

]]>The post Can you solve Rubik’s Cube? appeared first on SciHi Blog.

]]>On June 2, 1980, the world’s most famous puzzle – **Rubik’s Cube** – started to spread all over the world, infecting the population with addiction and curiosity about its solving.

The Rubik’s Cube started to take over Germany in 1980, but its birth lays back in Hungary in the mid 1970’s. The magic cube is named after its creator Ernő Rubik, who is a Hungarian architect and inventor. Between 1971 and 1975 he worked as a professor at the Budapest College of Applied Arts in the department of interior design and during these years, the magic cube was born. Ernő Rubik invented the cube to show his classes how 3D objects move and to teach structural design problems like “*How could the blocks move independently without falling apart?*” Back then he didn’t realize he had just created the best selling puzzle game in history. The first time he twisted the cube for a few times he began noticing, that it is even harder to twist the blocks back in order and it took him a whole month to figure out a right solution.

After Rubik was granted Hungarian Patent No. 170062 for the Cube on 28 October 1976, the Cube entered the “capitalist world” in December 1977 when a copy of the Cube was sent to the British company Pentangle. This company then acquired the licence to distribute the cube in Great Britain. In 1979, however, the Hungarian government granted the worldwide sales rights for the cube to the US company Ideal Toy Corporation (also known in Europe as Arxon). This also included the rights for the United Kingdom in breach of the Treaty. Ideal Toy Corporation allowed Pentangle to sell the cube to gift shops, but not to toy stores.

In 1981, the demand for mechanical patience play reached its peak. Ideal Toy Corporation was unable to meet demand, allowing cheap Far Eastern products to flood the market. Altogether, about 160 million cubes were sold until the peak of the boom. In early 1982 the demand for the dice collapsed and with it the demand for many other puzzles.

Ernő Rubik was not the first to deal with the theme of a game of this kind. As early as 1957, the chemist Larry Nichols developed a similar cube, which, however, consisted of only 2×2×2 parts and was held together by magnets. He had his design patented in 1972. In 1984 Nichols won a patent lawsuit against the company that sold the Rubik’s Cube in the USA. However, this judgment was partially annulled in 1986, so that it only concerned the 2×2×2 Pocket Cube.

Nowadays many algorithms are widely spread all over the Web. After the big hype in the 1980’s the magical cube was gone for a while, but in the past years it came back to households and even classrooms. Now the focus lies not on solving the cube, but rather on speedcubing. Speedcuber can solve any twisted Rubik’s Cube with 45 to 60 movements. Speedcubing depends on the quick recognition of positions, the internalization of a high number of algorithms, planning ahead and dexterity. In Speedcubing national, continental and world championships are held by the World Cube Association (WCA).

The first World Championship, organized by the Guinness Book of Records, took place in Munich on March 13, 1981. The cubes were twisted 40 times and rubbed with vaseline. Winner of the championship was Jury Fröschl from Munich with a record time of 38 seconds. The current world record for a 3×3×3 cube is 4.22 seconds and was set by Feliks Zemdegs at Cube for Cambodia 2018.

Other challenges these days include the blind-folded solving, underwater solving, and solving the cube with the feet. Also, many variations of the original cube were created, for example the V-Cube 7 or the Pocket Cube.

**References and Further Reading:**

- [1] Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys

Published by Johns Hopkins University Press, 2008 - [2] Rubik’s Cube History
- [3] Rubik’s Cube at Britannica
- [4] Rubik’s Cube at Wikidata

The post Can you solve Rubik’s Cube? appeared first on SciHi Blog.

]]>The post Only the Good Die Young – the Very Short Life of Évariste Galois appeared first on SciHi Blog.

]]>

On June 1st, 1832, famous French mathematician Évariste Galois was killed in a duel. He was only 20 years of age. And why did he have to die so young? Just because of a girl…

Her name was Stéphanie-Félicie Poterine du Motel, the daughter of a physician. But, there have been rumors that the duel has been set up by his political opponents while this could never be proven. His opponent was a well known shootist and also some people say that Galois has committed suicide because of the unhappy romance. So in the night before his duel Galois demanded his friend Auguste Chevalier to forward his mathematical writings to Carl Friedrich Gauss and Carl Gustav Jacobi, the two leading mathematicians of the time, and commented his writings with the note *“Je n’ai pas le temps” (I do not have the time…)*.[2,3] With a shot in the abdomen, he died the next morning in the Cochin hospital. His last words to his younger brother Alfred were:

“Ne pleure pas, Alfred! J’ai besoin de tout mon courage pour mourir à vingt ans.” (Don’t cry, Alfred! I need all my courage to die at twenty.)

Galois was born on 25 October 1811 to Nicolas-Gabriel Galois and Adélaïde-Marie (born Demante). His father was a Republican and was head of Bourg-la-Reine’s liberal party. His father became mayor of the village after Louis XVIII returned to the throne in 1814. His mother, the daughter of a jurist, was a fluent reader of Latin and classical literature and was responsible for her son’s education for his first twelve years. At the age of 10, Galois was offered a place at the college of Reims, but his mother preferred to keep him at home.

Galois attended the College Louis-le-Grand in Paris, failed the entrance examination to the École polytechnique twice and began his studies at the École normale supérieure. At the age of 17 he published his first work on continued fractions; shortly afterwards he submitted a paper to the Académie des Sciences on the resolution of equations, which contained the core of the Galois theory named after him today. The Academy rejected the manuscript but encouraged Galois to submit an improved and expanded version. This process was repeated twice with the participation of Augustin-Louis Cauchy, Joseph Fourier and Siméon Denis Poisson. Galois reacted bitterly, accusing the Academy of embezzling manuscripts and deciding to have his work printed at his own expense.[5,6,7]

As a Republican, Galois was disappointed by the outcome of the July Revolution and became politically increasingly exposed; he was expelled from his college and arrested twice. The first arrest for a toast to the new King Louis-Philippe, made at a banquet with a bare knife in his hand, as a hidden death threat, was followed by an acquittal on 15 June 1831. Only a month later, Galois took part in a demonstration on 14 July in the uniform of the artillery guard, now disbanded for political unreliability and heavily armed, was arrested again and sentenced to six months in prison in Sainte-Pélagie after three months in custody. In March 1832 he was transferred with other prisoners to Sieur Faultrier Sanatorium due to a cholera epidemic. He was released from custody on 29 April.

Galois founded the Galois Theory named after him for the solution of algebraic equations. While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word group as understood today, making him among the founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. He also introduced the concept of a finite field also known as a Galois field in his honor.

Joseph Liouville, who saw the connection with Cauchy’s theory of permutations and published it in his journal, only recognized the significance of the writings in 1843. Because of the concepts and sentences he found, Galois is one of the founders of group theory. In recognition of his fundamental work, the mathematical structures Galois body and Galois compound were named after him. Like other particularly famous mathematicians, a symbol is dedicated to him: *GF(q)* stands for Galois Field (galoy body) with q elements and is as established in literature as the Gaussian bracket or the Kronecker symbol.

He also provided the basis for proofs of the general unsolvability of two of the three classical problems of ancient mathematics, the tripartition of the angle and the doubling of the cube (each with compass and ruler, i.e. with square roots and linear equations). The third problem, squaring the circle, was solved by proving the transcendence of pi by Ferdinand Lindemann.

At yovisto academic video search the short life of Évariste Galois is briefly recounted in a Ted talk ‘Symmetry – Reality’s Riddle‘ by Marcus du Sautoy.

**Related Articles in the yovisto Blog:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Évariste Galois“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Carl Friedrich Gauss – The Prince of Mathematicians, SciHi Blog
- [3] Carl Jacobi and the Elliptic Functions, SciHi Blog
- [4] Works by or about Évariste Galois at Internet Archive
- [5] Augustin-Louis Cauchy and the Rigor of Analysis, SciHi Blog
- [6] Joseph Fourier and the Greenhouse Effect, SciHi Blog
- [7] Siméon Denis Poisson’s Contributions to Mathematics, SciHi Blog
- [8] Evariste Galois at Wikidata

The post Only the Good Die Young – the Very Short Life of Évariste Galois appeared first on SciHi Blog.

]]>