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

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]]>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.

]]>The post Do You Speak Polish… Or Maybe Reverse Polish? appeared first on SciHi Blog.

]]>I guess almost nobody except a few mathematicians and computer scientists have ever heard of the Australian computer scientist **Charles Leonard Hamblin**, who passed away on May 14, 1985. And also most of my fellow computer scientists might not have heard of him. But, one of his major contributions to computer science was the introduction of the so-called **Reverse Polish Notation**. Does that ring a bell?

Interrupted by the Second World War and radar service in the Australian Air Force, Charles Leonard Hamblin’s studies included mathematics, physics, and philosophy at the University of Melbourne, and he obtained a doctorate in 1957 at the London School of Economics. From 1955, he was lecturer at N.S.W. University of Technology, and later professor of philosophy at the same place, until his death in 1985, during which time the organization had been renamed The University of New South Wales.

In the second half of the 1950s, Hamblin worked with the third computer available in Australia, a DEUCE computer manufactured by the English Electric Company. For the DEUCE, he designed one of the first programming languages, later called GEORGE, which was based on Reverse Polish Notation. His associated compiler (language translator) translated the programs formulated in GEORGE into the machine language of the computer, in 1957. Hamblin’s work is considered to be the first to use Reverse Polish Notation, and this is why he is called an inventor of this representation method. Regardless of whether Hamblin independently invented the notation and its usage, he showed the merit, service, and advantage of the Reverse Polish way of writing programs for the processing on programmable computers and algorithms to make it happen. The second direct result of his work with the development of compilers was the concept of the push-pop stack (previously invented by Alan M. Turing for the ACE in 1945), which Hamblin developed independently of Friedrich Ludwig Bauer and Klaus Samelson, and for which in 1957 he was granted a patent for the use of a push-pop stack for the translation by programming languages

Hamblin became aware of the problem of computing mathematical formulae containing brackets results in memory overhead, which was rather critical at these times, because memory was rather small and expensive. One solution to the problem has already been prepared by the famous Polish mathematician Jan Lukasiewicz’s, inventor of the original Polish notation, which enables a writer of mathematical notation to instruct a reader the order in which to execute the operations (e.g. addition, multiplication, etc) without using brackets. Polish notation achieves this by having an operator (+, *, etc) precede the operands to which it applies, e.g., +ab, instead of the usual, a+b. Hamblin, with his training in formal logic, knew of Lukasiewicz’s work. Hamblin improved this principle to save additional storage by putting the operator behind the operands and thus, enabling the computer to make use of a storage, which did not require an address.

In the 1960s Charles Leonard Hamblin began to turn more and more to philosophical questions. In addition to an influential introductory book on formal logic, Fallacy’s work, which is still regarded as a standard work and in print today, is dedicated to the treatment of misconceptions by traditional logic and with which he brought formal dialectics to life. Hamblin later contributed to the development of modern temporal logic. In 1972 he independently rediscovered a form of duration calculus (interval logic).

This might sound rather weird to you, but 30 years ago using one of those sophisticated HP calculators that forced you to use and — more important -think RPN, made you the undisputed number one among all the other geeks.

You might learn more about Reverse Polish Notation at yovisto academic video search by watching ‘The Joys of RPN‘

**References and Further Reading:**

- [1] Churchill’s Best Horse in the Barn – Alan Turing, Codebreaker and AI Pioneer, SciHi Blog
- [2] “Everything you’ve always wanted to know about RPN but were afraid to pursue – Comprehensive manual for scientific calculators – Corvus 500 – APF Mark 55 – OMRON 12-SR and others” (PDF). T. K. Enterprises. 1976.
- [3] Parsing/RPN calculator algorithm at rosettacode.org
- [4] Online implementation to translate standard notation to RPN
- [5] Charles Leonard Hamblin at Wikidata

The post Do You Speak Polish… Or Maybe Reverse Polish? appeared first on SciHi Blog.

]]>The post Ramon Llull and the Tree of Knowledge appeared first on SciHi Blog.

]]>Probably in 1232, philosopher, logician, Franciscan tertiary and Catalan writer Ramon Llull (Anglicised Raymond Lully, Raymond Lull; in Latin Raimundus or Raymundus Lullus or Lullius) was born. He is credited with writing the first major work of Catalan literature Recently surfaced manuscripts show his work to have predated by several centuries prominent work on elections theory. He is also considered a pioneer of computation theory, especially given his influence on Leibniz.[2]

“For a long time I have struggled to seek the truth in this and other ways, and by God’s grace I have come to a good end and to the knowledge of the truth which I longed to know and which I laid down in my books, but I am without consolation because I could not finish what I so much desired and for which I have worked for thirty years, and also because my books are little valued, yes – I tell you also – because many people even consider me a fool”. — Ramon Llull

Ramon Llull was the son of a Catalan knight who fought under James I of Aragon for the Reconquista of the Saracen-controlled Balearic Islands. He grew up at court and was appointed prince educator at an early age. He led a courtly, secular life and devoted himself to poetry as a troubadour. In 1263 a vision in which he saw the crucified Christ beside him led Llull to a radical change in his life. He undertook pilgrimage and educational trips, also to the Arab world, continued his education, learned Arabic and placed his poetry at the service of the Catholic faith.

Llull soon became a famous scholar and confidante of James II of Mallorca, who he raised, taught at the Sorbonne in Paris and took part in the Council of Vienne. There he supported the establishment of chairs for Hebrew, Arabic and Chaldean (= Old Church Syrian) at the universities of Paris, Oxford, Bologna and Salamanca, which made him a founder of Western European Orientalism. In 1276 Ramon Llull founded a mission school in Miramar Monastery in Valldemossa. 1314 he went on a journey to Tunis on behalf of James II. In North Africa, in addition to his diplomatic and literary activities, he continued his evangelization. The circumstances of his death are unclear. According to a tradition, for which there is no evidence, he was stoned by angry Muslims in Bougie (Algeria) and died on his way back to Mallorca, where he was buried in La Palma, as a result of the stoning.

The special interest of King Philip II of Spain in the writings of the universal scholar led to the first canonization proceedings. In the middle of the 18th century it was stopped because of the writings of the Catalan Dominican and Grand Inquisitor Nicolás Aymerich (1322-1399), who wanted more than a hundred heresies to be found in Lull’s writings. At the beginning of the 20th century, Catalan theologians demanded Lull’s rehabilitation. The Catalan theologian and historian Josep Perarnau was able to prove that Aymerich had misquoted Lull’s writings in order to expose him to suspicion of heresy. The canonization process was resumed at the end of the 20th century.

Llull was active as a missionary throughout the Mediterranean due to his visions of Christ. He also taught at the universities of Paris and Montpellier. He was influenced by three cultures: Christian, Islamic and Jewish. He wrote a large part of his over 280 works in Latin and Catalan. This made Llull the founder of Catalan literature.His Arabic works have been lost.

Llull called logic the art and science of distinguishing truth and lie with the help of the mind, of accepting truth and rejecting lies.

This art, which simultaneously became the title for his work *Ars magna* (“*Great Art*“), came down to the idea of mechanically combining concepts with the help of a logical machine, thereby simultaneously creating the traditional algorithmic line of heuristics. Llull himself constructed such a “logical machine”, which consisted of seven discs rotating around a center. On each of these discs were noted words denoting different terms, e.g. man, knowledge, truth, fame, well-being and quantity, logical operations, e.g. difference, agreement, contradiction and equality. The rotation of these concentric disks resulted in various combinations of terms that corresponded to final forms of the syllogistic principle.

During Llull’s lifetime and also in the following period his ideas were received with suspicion. His arsenic was based on a Neo-Platonic system, which contradicted the mainstream of contemporary scholasticism. Through his language skills Llull had a direct access to the Arabic world of thought and adopted an attitude towards Islam that was unusually tolerant for his time. Towards the end of his life, however, Llull certainly represented a “mission with the sword” (*missio per gladium*).

Nevertheless, his works have a great history and in the centuries in which Llull was officially banned, his works were secretly studied and copied. Llull’s followers are called lullists. The philosopher Nikolaus Cusanus can also be counted among them. There are also some pseudo-lullistic writings that mainly deal with alchemy.[3] His theology was also taken up by Giordano Bruno.[4] In the 15th century Giovanni Pico della Mirandola quoted Llull’s writings in detail.[5]

For Ernst Bloch, Lullus is “*the strangely rationalistic scholastics*,” who attempted to refute the Koran with his *ars inveniendi.*

In the 17th century, his works *Ars magna* and *Ars brevis* gained greater influence through the system of a perfect philosophical, universal language described therein. This system is based on a combination of philosophical basic concepts. Llull’s thoughts were taken up by Gottfried Wilhelm Leibniz, the founder of mathematical logic. In the 19th century William Stanley Jevons tried to realize the idea of a logical machine. Llull studied both syllogism and induction. He was the first to devote himself to the systematic study of material implication, which is one of the fundamental operations of mathematical logic, analyzing logical operations with the copula “and” (conjunction) and the copula “or” (disjunction).

Llull systematically structured the sciences in *L’arbre de ciència* (c. 1295/96, published in Latin 1482), for which he used the allegory of the tree; this metaphor was first introduced to the history of science around 1240 by Petrus Hispanus under the term *Arbor porphyriana* (Tree of Knowledge). In Llull, fourteen trees represent the areas of being such as elements, botany, animals, sensory perception, imagination, morality, social science, etc.; in two other trees, these areas are illustrated by examples (copies) and proverbs (bonmots). Each tree has a seven-part internal structure, consisting of root, trunk, branches, twigs, leaves, flowers and fruits.

“The human intellect in itself has the nature of cognition, for cognition is its activity.”

— Ramon Llull, De modo naturali intelligendi

Llull’s teachings were already controversial during his lifetime. Later the movement of lullism developed around his students. The Roman church put him on the index of forbidden books for a long time and only rehabilitated him later.

In 1314, at the age of 82, Llull traveled again to North Africa where he was stoned by an angry crowd of Muslims in the city of Bougie. Genoese merchants took him back to Mallorca, where he died at home in Palma the following year. Though the traditional date of his death has been 29 June 1315, his last documents, which date from December 1315, and recent research point to the first quarter of 1316 as the most probable death date.

**References and Further Reading**

- [1] O’Connor, John J.; Robertson, Edmund F., “Ramon Llull“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog
- [3] Nikolaus of Cusa and the Learned Ignorance, SciHi Blog
- [4] Giordano Bruno and the Wonders of the Universe, SciHi Blog
- [5] Pico della Mirandola and the 900 Theses, SciHi Blog
- [6] Ramon Llull at Wikidata

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]]>The post Kurt Gödel Shaking the Very Foundations of Mathematics appeared first on SciHi Blog.

]]>On April 28, 1906, **Kurt Gödel** was born. He was one of the most significant logicians of all time. Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

“Either mathematics is too big for the human mind, or the human mind is more than a machine.”

— Kurt Gödel as quoted in Topoi : The Categorial Analysis of Logic (1979) by Robert Goldblatt, p. 13

Kurt Gödel came from a wealthy upper middle-class family in Brno, Moravia. His parents were Marianne (née Handschuh) and Rudolf August Gödel from Brno. The father was a wealthy textile entrepreneur. Caused by rheumatic fever, Gödel often suffered from poor health in his childhood. However, he did his best at school. After the First World War Brno became part of the newly founded Czechoslovak Republic in 1918/1919 and in 1923 Gödel, who felt like an “Austrian exile in Czechoslovakia” took Austrian citizenship. In autumn 1924, after graduating from high school, Gödel moved to Vienna and enrolled at the University of Vienna, initially in the course of studies in theoretical physics. He also attended Heinrich Gomperz’s philosophical lecture and Philipp Furtwängler’s lecture on number theory. These two professors gave Gödel the decisive impetus to work intensively on the fundamentals of mathematics, which are based on formal logic and set theory. Furtwängler was an outstanding mathematician and teacher, but in addition he was paralysed from the neck down so lectured from a wheel chair with an assistant who wrote on the board. This would make a big impact on any student, but on Gödel who was very conscious of his own health, it had a major influence. [5]

He began to visit the Wiener Kreis (Vienna Circle), an academic circle founded by Moritz Schlick, which dealt with the methodological foundations of thinking and thus with the foundations of any philosophy. Conversations with the other members of the group, of which Hans Hahn, Karl Menger and Olga Taussky were of particular importance to Gödel, also led to the expansion of his mathematical knowledge. Fascinated by the conversations in the Vienna Circle, Gödel attended Karl Menger‘s *Mathematische Kolloquium* and became familiar with the fundamental problems of mathematics and logic of his time. In particular, he got to know Hilbert’s program, which was to prove the consistency of mathematics.[2] He was awarded his doctorate on 6 February 1930 for his dissertation entitled *Über die Vollständigkeit des Logikkalküls* (1929) under the supervision of Hans Hahn.

He became a member of the faculty of the University of Vienna in 1930, where he belonged to the school of logical positivism until 1938. Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old in *Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme*. He proved fundamental results about axiomatic systems, showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system. In particular the consistency of the axioms cannot be proved. This ended a hundred years of attempts to establish axioms which would put the whole of mathematics on an axiomatic basis. One major attempt had been by Bertrand Russell with Principia Mathematica (1910-13).[3] Another was Hilbert’s formalism which was dealt a severe blow by Gödel’s results. The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than that envisaged by Hilbert. Gödel’s results were a landmark in 20th-century mathematics, showing that mathematics is not a finished object, as had been believed. It also implies that a computer can never be programmed to answer all mathematical questions.[2]

“But every error is due to extraneous factors (such as emotion and education); reason itself does not err.”

Attributed as a remark of Gödel of 29th November 1972, in Incompleteness (2005) by Rebecca Goldstein

Submitting his paper on incompleteness to the University of Vienna for his habilitation, this was accepted by Hahn on 1 December 1932. Gödel became a Privatdozent at the University of Vienna in March 1933. Gödel’s groundbreaking work on completeness and the logic of provability earned him recognition as one of the leading logicians of his time. His American colleague Oswald Veblen invited him to Princeton to join the newly founded Institute for Advanced Study.[6] In 1933/1934 he travelled to America for the first time. Together with James Alexander, John von Neumann and Oswald Veblen he became a founding member of the faculty and gave a number of lectures.[7] When Gödel returned to the now dictatorial Vienna in the spring of 1934, he had already received the invitation to continue as a lecturer. Gödel was not interested in the political situation in Europe.

Travel and work exhausted Gödel. Now the mental illness, which he probably had latent in him since childhood, became noticeable as depression. In autumn 1934 he had to go to a sanatorium for a week. In 1935 he spent several months in a psychiatric clinic. When the philosopher Moritz Schlick, one of the leading heads of the Vienna circle, was murdered by a former student at Vienna University in June 1936, Gödel suffered a nervous breakdown. He developed hypochondriac obsessions, especially a pathological fear of being poisoned. Since Gödel did not eat enough, his physical health suffered increasingly. His condition deteriorated over the years. Since he fell ill with rheumatic fever as a child, he was convinced that he had a weak heart and developed distrust of the medical profession, which could find nothing like this in him. He avoided doctors and almost died of an untreated duodenal ulcer in the 1940s.

When the war started Gödel feared that he might be conscripted into the German army. Of course he was also convinced that he was in far too poor health to serve in the army, but if he could be mistaken for a Jew he might be mistaken for a healthy man. He was not prepared to risk this, and after lengthy negotiation to obtain a U.S. visa he was fortunate to be able to return to the United States, although he had to travel via Russia and Japan to do so.[2] Gödel was now more concerned with philosophy, especially with Gottfried Wilhelm Leibniz, later also with Edmund Husserl. In Princeton he began to deal more and more with philosophical problems and to turn away from formal logic.[8,9]

In 1942 Gödel got to know Albert Einstein better and began to discuss physical problems such as relativity and philosophical topics with him.[10] A close friendship developed between Einstein and Gödel, which lasted until Einstein’s death in 1955. Together they used to walk to the institute and home. Einstein once said that he would only come to the Institute “*to have the privilege of walking home with Gödel*“. In 1947 Gödel became a citizen of the USA. The naturalisation process required a judicial hearing in which he had to show knowledge of the country and the constitution. In his preparations for this, Gödel discovered that the country’s constitution was incomplete to the extent that it would have been possible to establish a dictatorship within the framework of this constitution despite its individual provisions protecting democracy. Two friends, Albert Einstein and the economist Oskar Morgenstern, accompanied him during the proceedings. Thanks to their help and an enlightened judge it was possible to prevent Gödel from getting himself into trouble at the hearing.[11]

It was not until 1953 that he was appointed professor at Princeton, as Hermann Weyl and Carl Ludwig Siegel in particular considered him unsuitable because of his strange behaviour.[12] In 1955 he was elected to the National Academy of Sciences, in 1957 to the American Academy of Arts and Sciences. He stopped giving lectures in the 1960s. His illness made him less and less able to work and participate in social life. Nevertheless, he was still considered one of the leading logicians, and he was granted academic recognition in the form of awards. Gödel’s condition did not improve. In 1970 he tried to publish for the last time. However, the scripture had to be withdrawn because he had overlooked many errors due to the effect of psychotropic drugs.

Gödel spent the last years of his life at home in Princeton or in various sanatoria, from which he fled several times. Only the care of his wife Adele, who made sure that he ate at least halfway normally, kept him alive. When Adele Gödel was admitted to hospital in 1977 due to a stroke, she was helplessly forced to watch her husband lose weight. When she was released after six months – now dependent on a wheelchair – she immediately delivered him to a hospital with a body weight of about 30 kg. Kurt Gödel died a few weeks later of malnutrition and exhaustion.

Unfortunately, at yovisto academic video search, we don’t have original video footage of Gödel. But we have some lectures from Princeton honoring Kurt Gödel and his works to mark the centenary year of his birth back in 2006:

- John W. Dawson: ‘At Odds with the Zeitgeist: Kurt Gödel‘

- [1] It’s Computable – thanks to Alonzo Church, SciHi Blog
- [2] David Hilbert’s 23 Problems, SciHi Blog
- [3] The time you enjoy wasting is not wasted time – Bertrand Russell, Logician and Pacifist, SciHi Blog
- [4] Kurt Gödel at zbMATH
- [5] O’Connor, John J.; Robertson, Edmund F., “Kurt Gödel“, MacTutor History of Mathematics archive, University of St Andrews.
- [6] Oswald Veblen and modern Topology, SciHi Blog
- [7] John von Neumann – Game Theory and the Digital Computer, SciHi Blog
- [8] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog
- [9] Edmund Husserl’s Phenomenology, SciHi Blog
- [10] Albert Einstein revolutionized Physics, SciHi Blog
- [11] Oskar Morgenstern and the Game Theory, SciHi Blog
- [12] Hermann Weyl – between Pure Mathematics and Theoretical Physics, SciHi Blog
- [13] Kurt Gödel at Mathematics Genealogy Project
- [14] Kurt Gödel at Wikidata
- [15] Timeline for Kurt Gödel, via Wikidata

The post Kurt Gödel Shaking the Very Foundations of Mathematics appeared first on SciHi Blog.

]]>The post Kolmogorov and the Foundations of Probability Theory appeared first on SciHi Blog.

]]>On April 25, 1903, Soviet mathematician **Andrey Nikolaevich Kolmogorov** was born. He was one of the most important mathematicians of the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity.

“The epistemological value of probability theory is based on the fact that chance phenomena, considered collectively and on a grand scale, create non-random regularity.”

Andrey Kolmogorov, Limit Distributions for Sums of Independent Random Variables (1954)

Andrey Kolmogorov was born in Tambov in 1903 and raised by his aunts. Fortunately, one of them was occupied as a teacher, wherefore Kolmogorov was able to receive a good basic school education. Already in his teenage years, he began designing perpetual motion machines and after moving to Moscow, Kolmogorov graduated from high school. In 1920, he enrolled at Moscow State University, more specifically at the Chemistry Technological Institute.

The young scientist became widely known for his wide ranging knowledge. During his undergraduate years he published papers on the science of history and began gaining interest in set theory and the theory of Fourier series. Kolmogorov’s decision to become a mathematician occurred around that time. He was able to construct a Fourier series diverging almost everywhere, which was noticed internationally.

Right after his graduation at Moscow State University, Kolmogorov started his research under Nikolai Luzin, a famous mathematician active in the field of set theory, mathematical analysis and point-set topology. There he got to know Pavel Alexandrov and established a great friendship with him. By the way, Alexandrov was also a close friend to the German mathematician Emmy Noether, who worked with Alexandrov at the University of Moscow for a few years.[8]

However, Andrey Kolmogorov became highly interested in probability theory along with Aleksandr Khinchin. In 1925, he published his famous paper ‘*On the principle of the excluded middle*‘ , proving that all statements of classical formal logic could be expressed of intuitionistic logic.

The great breakthrough came in 1933. Kolmogorov published the book *Foundations of the Theory of Probability*, axiomatizing the probability theory in a rigorous way from fundamental axioms in a way comparable with Euclid’s treatment of geometry. It changed Andrey Kolmogorov’s status to the world’s leading expert in the field, wherefore he became the first chairman of the department of probability theory at the Moscow State University. Kolmogorov later extended his work to study the motion of the planets and the turbulent flow of air from a jet engine. He thus demonstrated the vital role of probability theory in physics. Next to his efforts in mathematics, Kolmogorov was always willing to establish a better relationship between mathematics and philosophy in aspects of probability. Kolmogorov had many interests outside mathematics, in particular he was interested in the form and structure of the poetry of the Russian author Alexander Pushkin.

In his further years of study, Kolmogorov worked on stochastic processes and classical mechanics. In his study of stochastic processes, especially Markov processes, Kolmogorov and the British mathematician Sydney Chapman independently developed the pivotal set of equations in the field, which have been given the name of the Chapman–Kolmogorov equations. Later, Kolmogorov focused his research on turbulence, where his publications (beginning in 1941) significantly influenced the field. In classical mechanics, he is best known for the Kolmogorov–Arnold–Moser theorem, first presented in 1954 at the International Congress of Mathematicians. In 1957, working jointly with his student Vladimir Arnold, he solved a particular interpretation of Hilbert’s thirteenth problem. Around this time he also began to develop, and was considered a founder of, algorithmic complexity theory – often referred to as Kolmogorov complexity theory.[9]

“Every mathematician believes that he is ahead of the others. The reason none state this belief in public is because they are intelligent people.”

attributed to Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov passed away on October 20, 1987.

At yovisto academic video search, you may enjoy a video lecture on probability theory by professor Faber at ETH Zurich.

**References and Further Reading:**

- [1] Andrey Kolmogorov Website
- [2] The origins and legacy of Kolmogorov’s Grundbegriffe [PDF]
- [3] Kolmogorov: Foundations of the Theory of Probability, [PDF]
- [4] History of Mathematics Website
- [5] Interpretations of Probability
- [6] Andrey Kolmogorov at zbMATH
- [7] Andrey Kolmogorov at Mathematics Genealogy Project
- [8] Emmy Noether and the Love for Mathematics, SciHi Blog
- [9] David Hilbert’s 23 Problems, SciHi Blog
- [10] Andrey Kolmogorov at Wikidata
- [11] Timeline for Andrey Kolmogorov, via Wikidata

The post Kolmogorov and the Foundations of Probability Theory appeared first on SciHi Blog.

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