The post Girard Desargues and Projective Geometry appeared first on SciHi Blog.

]]>On February 21, 1591, French mathematician and engineer **Girard Desargues **was born. Desargues is considered one of the founders of projective geometry. Desargues‘ theorem, the Desargues graph, and the crater Desargues on the Moon are named in his honour. In his later years, he designed an elaborate spiral staircase, and an ingenious new form of pump, but the most important of Desargues‘ interests was Geometry. He invented a new, non-Greek way of doing geometry, now called ‘projective’ or ‘modern’ geometry. As a mathematician he was highly original and completely rigorous. However, he was far from lucid in his mathematical style.

“I freely confess that I never had taste for study or research either in physics or geometry except in so far as they could serve as a means of arriving at some sort of knowledge of the proximate causes… for the good and convenience of life, in maintaining health, in the practice of some art,… having observed that a good part of the arts is based on geometry, among others that cutting of stone in architecture, that of sundials, that of perspective in particular.”

– Girard Desargues, (ca. 1640) as quoted in [13]

Little is known about Girard Desargues’ personal life. Born in Lyon, Desargues came from a wealthy family devoted to service to the French crown. His father was a royal notary, an investigating commissioner of the Seneschal’s court in Lyon (1574), the collector of the tithes on ecclesiastical revenues for the city of Lyon (1583) and for the diocese of Lyon, then the second most important city in France. Desargues seems to have made several extended visits to Paris in connection with a lawsuit for the recovery of a huge debt. Despite this loss, the family still owned several large houses in Lyon, a manor house at the nearby village of Vourles, and a small chateau surrounded by the best vineyards in the vicinity. Thus, it is clear that Desargues had every opportunity of acquiring a good education, could afford to buy what books he chose, and had leisure to indulge in whatever pursuits he might enjoy.

Girard Desargues worked as an architect from 1645. Prior to that, he had worked as a tutor and may have served as an engineer and technical consultant in the entourage of Richelieu. As an architect, Desargues planned several private and public buildings in Paris and Lyon. As an engineer, he designed a system for raising water that he installed near Paris. It was based on the use of the at the time unrecognized principle of the epicycloidal wheel. When in Paris, Desargues became part of the mathematical circle surrounding Marin Mersenne, which also included Rene Descartes,[4] Étienne Pascal and his son Blaise Pascal.[5] It was probably essentially for this limited readership of friends that Desargues prepared his mathematical works, and had them printed.

Desargues wrote on ‘practical’ subjects such as perspective (1636), the cutting of stones for use in building (1640) and sundials (1640). His writings are, however, dense in content and theoretical in their approach to the subjects concerned. His research on perspective and geometrical projections can be seen as a culmination of centuries of scientific inquiry across the classical epoch in optics that stretched from al-Hasan Ibn al-Haytham (Alhazen) to Johannes Kepler,[6] and going beyond a mere synthesis of these traditions with Renaissance perspective theories and practices. Desargues conceived projective geometry as a natural extension of Euclidean geometry [7] in which parallel lines at infinity, sizes can vary as long as proportions are kept, and shapes are considered to be one with the totality of shadows they can cast. This is exactly what is needed in perspective design, where each object appears deformed according to the point of observation. Thus the plane sections of a cone are nothing but the different images projected by a light source on a wall when its inclination varies. In this framework, a circle is equivalent to an ellipse, which becomes a parabola as soon as the intersection point of the axis of the light cone with the wall ends up in infinity.

Desargues’ most important work, the one in which he invented his new form of geometry, *Rough draft for an essay on the results of taking plane sections of a cone *was published in 1639 only in a small number. Just one is now known to survive. The theorem of Desargues of projective geometry states that the intersection points of corresponding sides of two triangles lie on a straight line when the connecting lines of corresponding vertices intersect in one point (and vice versa). The painter Laurent de La Hire and the engraver Abraham Bosse found Desargues’s method attractive. Bosse, who taught perspective constructions based on Desargues’s method at the Royal Academy of Painting and Sculpture in Paris, published a more accessible presentation of this method in 1648.

In the 17th century Desargues’s new approach to geometry, i.e. studying figures through their projections, was appreciated by a few gifted mathematicians, such as Blaise Pascal and Gottfried Wilhelm Leibniz,[8] but it did not become rather influential. Rene Descartes’s algebraic way of treating geometrical problems, published in *Discours de la méthode* (1637) came to dominate geometrical thinking and Desargues’s ideas were forgotten. Desargues’ work, however, was rediscovered and republished in 1864. Late in his life, Desargues published a paper with the cryptic title of *DALG*. The most common theory about what this stands for is *Des Argues, Lyonnais, Géometre* (proposed by Henri Brocard). He died in Lyon in 1661.

At yovisto academic video search you can learn more about Desargues and his projective geometry in the lecture series of Yale Prof N.J. Wildberger on the History of Mathematics

**References and Further Reading**

- [1] O’Connor, John J.; Robertson, Edmund F., “Girard Desargues”,
*MacTutor History of Mathematics archive*, University of St Andrews. - [2] Girard Desargues at Britannica.com
- [3] Girard Desargues at Scienceworld.wolfram.com
- [4] Cogito Ergo Sum – René Descartes, SciHi Blog, March 31, 2013.
- [5] It is not Certain that Everything is Uncertain – Blaise Pascal’s Thoughts, SciHi Blog, June 19, 2012.
- [6] And Kepler Has His Own Opera – Kepler’s 3rd Planetary Law, SciHi Blog, May 15, 2012.
- [7] Euclid – the Father of Geometry, SciHi Blog, January 30, 2015.
- [8] Let Us Calculate – the Last Universal Academic Gottfried Wilhelm Leibniz, SciHi Blog, July 1, 2012.
- [9] Girard Desargues in Wikidata
- [10] Girard Desargues at Reasonator
- [11] Girard Desargues at Mathematics Genealogy Project
- [12] Timeline for geometers, via Wikidata
- [13] William Thompson Sedgwick, Harry Walter Tyler,
*A Short History of Science*(1917)

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]]>The post Wilhelm Weinberg and the Genetic Equilibrium appeared first on SciHi Blog.

]]>On January 13, 1908, German physician and obstetrician-gynecologis **Wilhelm Weinberg** delivered an exposition of his ideas on the principle of genetic equilibrium in a lecture before the Verein für vaterländische Naturkunde in Württemberg. He developed the idea of genetic equilibrium independently of British mathematician G. H. Hardy.

Wilhelm Weinberg studied medicine at the Universities of Berlin, Tübingen, and Munich, Germany. He returned to his birth town, Stuttgart in 1889, where he remained running a large practice as a gynecologist and obstetrician. He is known to have been a physician to the poor and delivered around 3500 babies in his life. Still, he managed to write over 160 scientific papers as well as numerous reviews and comments in addition. The fact that his recognition outside of the German speaking area was so little, was according to contemporary scientists highly noticeable in his writings. His criticism was often very personal and his reviews very argumentative.

However, the scientist is mainly known for his contributions to the Hardy–Weinberg principle. It is also known as the Hardy–Weinberg equilibrium, model, theorem, or law. It states that allele and genotype frequencies in a population will remain constant from generation to generation in the absence of other evolutionary influences. These influences include non-random mating, mutation, selection, genetic drift, gene flow and meiotic drive. Because one or more of these influences are typically present in real populations, the Hardy–Weinberg principle describes an ideal condition against which the effects of these influences can be analyzed. The principle was developed by the British mathematician Godfrey Harold Hardy and Weinberg independently. Weinberg’s work was published in a lecture at the Society for the Natural History of the Fatherland in Württemberg, which was six months before the publishing of Hardy’s paper in English.

Since Weinberg’s scientific work was published in German, it remained completely unrecognized for over 35 years. Curt Stern was a contemporary German geneticist and he immigrated to the United States just before World War II. When he found out, that the achievement of both scientists was named “Hardy’s law” or “Hardy’s formula”, he pointed out Weinberg’s work in a scientific paper. Another reason, why Weinberg’s achievement was ignored for so many years may be the fact that it was written very difficultly, even for native speakers. Still, he used only elementary mathematics and avoided calculus as much as he could.

Wilhelm Weinberg also pioneered in the studies of twins. He developed techniques to analyze the phenotypic variation that partitioned this variance into genetic and environmental components. Weinberg recognized that ascertainment bias was affecting many of his calculations, and he produced methods to correct for it. Weinberg observed that proportions of homozygotes in familial studies of classic autosomal recessive genetic diseases generally exceed the expected Mendelian ratio of 1:4, and he explained how this is the result of ascertainment bias. He discovered the answer to several seeming paradoxes caused by ascertainment bias and he recognized that ascertainment was responsible for a phenomenon known as anticipation, the tendency for a genetic disease to manifest earlier in life and with increased severity in later generations.

Wilhelm Weinberg passed away on November 27, 1937.

At yovisto academic video search, you may be interested in a short introduction lecture to the Hardy-Weinberg Principle.

**References and Further Reading:**

- [1] Hardy, Weinberg and Language Impediments. James F. Crow, 1999 [PDF]
- [2] Weinberg, W., 1908: Über den Nachweis der Vererbung beim Menschen, Jahreshefte des Vereins für Vaterländische Naturkunde in Württemberg 64: 369-82. Digitalisat
- [3] Hardy–Weinberg Equilibrium Calculator
- [4] G. H. Hardy and the aesthetics of Mathematics, SciHi Blog, December 1, 2016.
- [5] Crick and Watson decipher the DNA, SciHi Blog
- [6] The Avery-McLeod-McCarthy Experiment, SciHi Blog
- [7] Max Delbrück and the Genes, SciHi Blog
- [8] Wilhelm Weinberg at Wikidata

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]]>The post Ada Lovelace – The World’s First Programmer appeared first on SciHi Blog.

]]>On November 27, 1852, Augusta Ada King**, Countess of Lovelace** passed away, who is considered to be the world’s very first computer programmer. Every student of computer science has most probably heart of Ada Countess of Lovelace, assistant to mathematician Charles Babbage, [1] inventor of the very first programmable (mechanical) computer, the analytical engine. Although probably not widely known to the general public, there are Ada Lovelace tuition programs for girls, a programming language called ‘*ADA*‘, as well as numerous references in popular culture, literature, and even a graphic novel.

“The Analytical Engine] might act upon other things besides number, were objects found whose mutual fundamental relations could be expressed by those of the abstract science of operations, and which should be also susceptible of adaptations to the action of the operating notation and mechanism of the engine…Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.”

— Ada Lovelace Notes in Menabrea, Luigi (1842). Sketch of the Analytical Engine invented by Charles Babbage Esq..

Augusta Ada King, Countess of Lovelace, was born on December 10, 1815, as Ada Augusta Byron, daughter of the famous as well as notorious English poet Lord Byron, to a wealthy family of nobility. Still, her childhood was pretty unfortunate. Byron had numerous affairs and three children of three women, only Ada was born legitimate. Her mother moved to Kirkby Mallory on January 16, 1816, together with the one-month-old Ada, to her parents in Kirkby Mallory due to ongoing disputes with Lord Byron. On April 21, 1816 Lord Byron signed a deed of separation and left England a few days later. Lord Byron had no relationship with his daughter, she never met him. When Ada was eight years old, he died

Ada’s mathematically interested mother, who in her youth had also been taught natural sciences and mathematics by house teachers, gave Ada an education in natural sciences, rather uncommon for girls in her age. Luckily, Ada’s talents and her brilliance were quickly detected and would basically determine the rest of her life.

Ada’s love to mathematics and her admiration for the scientist Mary Somerville led to a meeting of Charles Babbage and herself in 1833, which was to change her life critically. Babbage had published a paper on his famous difference engine about 10 years earlier, a calculating machine designed to tabulate polynomial functions. Augusta Ada Byron was highly interested in Charles Babbage’s work and especially in his machine, which many scientists were talking about. After some scientific debates with Babbage, he was deeply in love with her writing abilities as well as her mathematical skills, wherefore he called her the ‘*enchantress of numbers*‘. Augustus De Morgan, professor at University College London, who himself made fundamental contributions to the development of mathematical logic, had a major influence on her later education and on her main work – *the Notes*. Lovelace took lessons with him from 1841 on his own initiative. At the age of 19 Ada Byron married William King, 8th Baron King, who in 1838 became the 1st Earl of Lovelace. He, too, had a mathematical education and, since women were forbidden to enter libraries and universities at that time, had himself accepted into the Royal Society for her sake, where he copied articles for her.

Ada’s ascent to being a recognized scientist was hard due to her family’s public attention as well as to the fact that women in science and technology were still rare in the middle of the 19th century. Still, her chance came with Babbage’s publication of the ‘analytical engine’, a successor of the prior ‘difference engine’ and the very first general purpose programmable mechanical computer. In 1843 she translated the description of Babbages Analytical Engine, written in French by the Italian mathematician Luigi Menabrea, into English. Encouraged by Babbage, she added her own notes and reflections on the construction of this planned machine adding numerous notes explaining and commenting the machine’s function. Ada’s notes turned out being longer than the original work itself, because most scientists were not able to understand the difference between the two machines of Babbage.

Ada also explained an algorithm for calculating a sequence of Bernoulli numbers with the new machine, wherefore she is now mostly known for being the world’s first computer programmer. Many scholars belief that Babbage must have written programs for the Analytical Engine beforehand. However, Ada Lovelace’s program was the first published computer program. As many researchers read Ada’s work over the years, they recognized her being even more visionary than Babbage himself. Babbage’s machine was never built during his lifetime. On the one hand, precision mechanics had not yet been developed far enough to produce the machine parts with the necessary precision; on the other hand, the British Parliament refused to finance Babbage’s research programme, having already supported the development of its predecessor – the Difference Engine – with 17,000 British pounds (a value of around 3.4 million British pounds in 2014). Unfortunately, Ada was recognized for her work only over a century after the first publication, when the engine was proven to be an early model of the computer.

Ada Lovelace has critically influenced early achievements on programming but also faced lifelong difficulties with her family as well as with society, that rather emphasized her antics with alcohol, men or gambling than paying attention to her mathematical brilliance. Lovelace’s *Notes* contains a number of concepts that are far ahead of the state of research around 1840. While her contributions to computer architecture and the fundamentals of programming were largely forgotten until their rediscovery in the 1980s, her views on artificial intelligence played a certain role in epistemological debates as “*Lady Lovelace’s Objection*” even when this field of computer science research was founded. She wrote that

“

The Analytical Engine has no pretensions whatever tooriginate anything. It can do whatever we know how to order itto perform. It can follow analysis; but it has no power of anticipating any analytical relations or truths.”

This objection has been the subject of much debate and rebuttal, for example by Alan Turing in his paper “*Computing Machinery and Intelligence*“

Babbage’s motivation for the Analytical Engine was the calculation of numerical tables for use in science and engineering. Lovelace, on the other hand, had recognized the far greater potential of the machine: It would not only be able to perform numerical calculations, but would also combine letters and compose music, which was based on the relations of tones that could be expressed as combinations of numbers. Ada Lovelace also recognized that the machine has a physical part, namely the copper wheels and punched cards, and a symbolic part, i.e. the automatic calculations coded in the punched cards. She thus anticipated the subdivision into hardware and software.

Ada Lovelace died at the age of 36 – the same age that her father had died – on 27 November 1852, from uterine cancer probably exacerbated by bloodletting by her physicians.

At yovisto academic video search you can learn more about Ada Lovelace in the video report from the Institution of Engineering and Technology (IET) explaining the contribution and importance of the Countess of Lovelace in the field of engineering.

**References and further Reading:**

- [1] Charles Babbage – The Father of the Computer who hated Street Music, SciHi Blog
- [2] The Thrilling Adventures of Lovelace and Babbage – a web comic
- [3] Augustus de Morgan and Formal Logic, SciHi Blog
- [4] Ada Lovelace at San Diego Supercomputer Center Website
- [5] Ada Byron, Lady Lovelace at Agnes Scott College
- [6] The Babbage Engine at Computer History Museum
- [7] H. Sack,
*Programmieren*, in: Historisches Wörterbuch des Mediengebrauchs. Band 2, hrsg. von Heiko Christians, Matthias Bickenbach und Nikolaus Wegmann, Böhlau Verlag, Köln, Weimar, Wien, 2018, 363-375.*[in German]* - [8] Churchill’s Best Horse in the Barn – Alan Turing, Codebreaker and AI Pioneer, SciHi Blog
- [9] Ada Lovelace at Wikidata
- [10] O’Connor, John J.; Robertson, Edmund F., “Ada Lovelace“, MacTutor History of Mathematics archive, University of St Andrews.

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]]>The post Benoît Mandelbrot and the Beauty of Mathematics appeared first on SciHi Blog.

]]>On November 20, 1924, French American mathematician** Benoît B. Mandelbrot** was born. Mandelbrot worked on a wide range of mathematical problems, including mathematical physics and quantitative finance, but is best known as the popularizer of fractal geometry. He was the one who coined the term ‘fractal’ and described the Mandelbrot set named after him.

“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

— Mandelbrot, in his introduction to The Fractal Geometry of Nature

Raise your hands, who of you has a set that is named after you? Ok, there are (were) quite some famous mathematicians who own a set: the Julia set, the Cantor set, Borel sets, the Zermelo–Fraenkel set (theory), …ok and then, I run out of set names. If you are searching at Google for ‘*sets named after mathematicians*‘ you will soon end up with a lot of ‘Mandelbrot’ results. Moreover in the 1980s the graphical representation of this set raised popular public interest in mathematics in general and especially in computer graphics. BTW, this worked also for me in high school and I developed a first interest in computer science when analyzing graphical representations of the Mandelbrot set for a school project.

Benoît Mandelbrot was born in Warsaw into a Jewish family from Lithuania with strong academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. It was his two uncles introduced him to the wonderworld of mathematics. Mandelbrot’s family emigrated to France in 1936, where he studied at the Lycée du Parc in Lyon and from 1945 to 1947 he attended the École Polytechnique, where he studied under Gaston Julia (sic! here is another man with a set named after him…. the famous Julia sets). From 1947 to 1949 he worked on a master degree in aeronautics at the California Institute of Technology. Returning to France, he obtained his Ph.D. degree in Mathematical Sciences at the University of Paris in 1952. From 1949 to 1958 Mandelbrot was a staff member at the Centre National de la Recherche Scientifique. During this time he spent a year at the Institute for Advanced Study in Princeton, New Jersey, where he was sponsored by the famous computer pioneer John von Neumann.[10]

From the time of the ancient Greeks, geometry had studied a restricted range of objects, i.e. lines, circles, planes, cylinders and so on. Euclid‘s geometry [11] did hold until the 19th century, where under the impact of new developments in physics, especially electricity and magnetism, mathematician scientists like Carl Friedrich Gauss [12] and Bernhard Riemann [13] extended geometry into higher dimensions. They examined properties such as curvature that were often far beyond our daily experience. This was the type of geometry that Einstein could apply to his theories of relativity. Nevertheless, even these new geometries relate to things that are smooth and even “tame” in some sense.

“Being a language, mathematics may be used not only to inform but also, among other things, to seduce.”

— Benoit Mandelbrot, Fractals : Form, chance and dimension (1977)

Mandelbrot began looking at many different branches of science – turbulence in fluid mechanics and meteorology, price fluctuations in economics, the growth of cells in physiology, the clustering of galaxies, the shape of trees and plants – and realized that traditional smooth geometry was often at best a crude approximation. Mandelbrot wanted to show that what often appeared to be wild or random fluctuations that did not obey the laws of statistics could in fact be explained by simple mathematical laws or rules. His fundamental idea was that of “self similarity.” This approach enabled a geometry to be applied to things where it did not previously seem possible, as e.g., to clouds, coastlines or mountains.

Mandelbrot found out that the price fluctuations of the financial markets cannot be described by a normal distribution, but by a Lévy distribution, which theoretically has an infinite variance. Thus he also provided a possible explanation for the Equity Premium Puzzle, a paradox that occurs in financial markets. There is, according to economic theory, an excessively high difference between returns on risky securities (especially equities) and those considered relatively safe (e.g. government bonds). Although economic theory predicts the existence of a difference between these two categories of securities based on the assumption of risk aversion, its extent contradicts theoretical predictions. Mandelbrot also applied these ideas to cosmology. In 1974 he proposed a new explanation for the Olbers’ paradox of the dark night sky. He showed that the paradox can also be avoided without recourse to the Big Bang theory, if one assumes a fractal distribution of the stars in the universe, in analogy to the so-called Cantor dust.

At IBM – Mandelbrot spent most of his career at IBM’s Thomas J. Watson Research Center, and was appointed as an IBM Fellow – Mandelbrot was able to use computers in a new way that has since become central to many branches of science. “*I made them [the computer and computer graphics], not a tool to be called on only if needed, but a constant and integral part of my process of thinking*,” he wrote. During his appointment as visiting professor of mathematics at Harvard University in 1979, Mandelbrot began studying fractal Julia sets that are invariant to certain transformations in the complex plane, previously studied by Gaston Julia and Pierre Fatou. These sets are generated by the iterative formula *z _{n+1}=z_{n}^{2}+c . *Mandelbrot used computer plots of this set to study their topology depending on the complex parameter

In 1982 Mandelbrot expanded his ideas and published them in his probably best-known book *The Fractal Geometry of Nature*. This influential book made fractals known to a wider public and silenced many of the critics who had previously dismissed fractals as programming artifacts. Mandelbrot left IBM in 1987 after 35 years with the company after IBM decided to dissolve its basic research department. He then worked in the mathematics department at Yale University, where he took his first permanent professorship in 1999 at the age of 75. When he retired in 2005, he was Sterling Professor of Mathematics. His last position was as Battelle Fellow at the Pacific Northwest National Laboratory in 2005.

Over the decades, Mandelbrot’s ideas have become incorporated into a new mathematical area called chaos theory and are now widely respected. The small asteroid 27500 Mandelbrot was named in his honor. In November 1990, he was made a Knight in the French Legion of Honor. Mandelbrot died from pancreatic cancer at the age of 85 in Cambridge, Massachusetts on 14 October 2010.

“My life seemed to be a series of events and accidents. Yet when I look back I see a pattern.”

— Benoit Mandelbrot, from “A Fractal Life” by Valerie Jamieson in New Scientist (November 2004)

At yovisto academic video search you can listen to Benoite Mandelbrot himself explaining ‘*Fractals and the art of roughness*‘.

**References and further Reading:**

- [1] Mandelbrot’s Introduction to Fractals for the Classroom: Part One, by Heinz-Otto Pietgen, Harmut Jürgens, Dietmar Saupe, Springer, 1992.
- [2] Chaos, James Glieck, Penguin,1988.
- [3] Chris Talbot: Fractal visionary dies: Benoit Mandelbrot, 1924-2010, at wsws.org
- [4] Benoit Mandelbrot Biography, at jewage.org
- [5] Benoit Mandelbrot at zbMATH
- [6] Benoit Mandelbrot at Mathematics Genealogy Project
- [7] O’Connor, John J.; Robertson, Edmund F., “Benoit Mandelbrot“, MacTutor History of Mathematics archive, University of St Andrews.
- [8] Benoit Mandelbrot at Wikidata
- [9] Timeline for Benoit Mandelbrot, via Wikidata
- [10] John von Neumann – Game Theory and the Digital Computer, SciHi Blog
- [11] Euclid – the Father of Geometry, SciHi Blog
- [12] Carl Friedrich Gauss – The Prince of Mathematicians, SciHi Blog
- [13] Bernhard Riemann’s novell approaches to Geometry, SciHi Blog

**Related Articles in the Blog:**

- Humphry Davy and the Electrolysis
- Charles Lyell and the Principles of Geology
- Lise Meitner – The Misjudged Genius
- Antonie van Leeuwenhoeck – The Father of Microbiology
- A Life of Discoveries – the great Michael Faraday
- The bustling Life and Publications of Mathematician Paul Erdös
- How to Calculate Fortune – Jakob Bernoulli
- David Hilbert’s 23 Problems
- Ernst Haeckel and the Phyletic Museum
- Henry Moseley and the Atomic Numbers
- Mary Leakey and the Discovery of the false ‘Nutcracker Man’
- The Time you enjoy wasting is not wasted Time – Bertrand Russel, Logician and Pacifist
- Sir Isaac Newton and the famous Principia

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]]>The post Jean Baptiste le Rond d’Alembert and the Great Encyclopedy appeared first on SciHi Blog.

]]>On November 16, 1717, French mathematician, mechanician, physicist, philosopher, and music theorist **Jean Baptiste le Rond d’Alembert**** **was born. He was one of the most important mathematicians and physicists of the 18th century and a philosopher of the Enlightenment. Probably he is best known as co-editor with Denis Diderot of the famous Encyclopédie, edited between 1751 and 1772.[5]

“Nothing is more incontestable than the existence of our sensations; …”

— Jean Baptiste le Rond d’Alembert, in the Discours préliminaire de l’Encyclopédie (1759)

D’Alembert was the illegitimate son of Duc d’Arenberg (1690-1754) and Marquise de Tencin (1682-1749), who became known as Salonnière. His mother had him abandoned on the steps of the northern side chapel St-Jean-le-Rond of Notre Dame de Paris. He was adopted as foundling by Madame Rousseau, née Etiennette Gabrielle Ponthieux (ca. 1683-1775), wife of the glassmaker Alexandre Nicolas Rousseau, at the instigation of General Louis Camus Destouches (1668-1726); he remained there until the age of 48. His biological father, however, made it possible for him to have a comprehensive education and training.

At the age of twelve he entered the Collège des Quatre Nations and completed it in 1735 with the baccalauréat en arts. His educators noticed d’Alembert’s talent and supported him to begin an ecclesiastical career, which he rejected. However, D’Alembert understood theology as “*rather unsubstantial fodder*“. D’Alembert later enrolled at the École de Droit under the surname Daremberg, which he later changed to d’Alembert. He first studied law, then medicine, before finally turning to mathematics and physics in an autodidactic manner.

His first success was reached at the age of 22, when d’Alembert noticed mathematical errors in Charles René Reynaud’s work ‘*L’analyse démontrée*‘, which he made public. He was instantly known widely across the scientific community, because of improving a standard work, learned by every student of mathematics. His reputation continued to improve through publishing his work on fluid mechanics and refraction. D’Alembert’s most famous work (besides the encyclopedia) was published in 1743, it was called ‘*Traité de dynamique*‘, and explained the laws of motion, which he developed himself.

The D’Alembert principle of mechanics is named after him, which allows the equations of motion of a mechanical system to be established with constraints. In 1747 he solved the (one-dimensional) wave equation of the vibrating string, named after him today, and thus became the founder of mathematical continuum physics. He was also the founder of the D’Alembert operator, with which the wave equation can be written in a particularly compact way. D’Alembert also worked in the field of the convergence of series and found the quotient criterion, which is also called D’Alembert’s criterion after him. The reduction method of d’Alembert is important here. Further work was done on probability theory; a popular, certainly useless game system for the roulette game, the Progression d’Alembert, is attributed to him.

In 1752 he published the *Éléments de la musique théorique et pratique* (*Elements of theoretical and practical music*) and two years later *Réflexions sur la musique en général et sur la musique française en particulier* (*Reflections on music in general and French music in particular*).

The probably most important and most influential work of d’Alembert was his contribution to the famous ‘*Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers*‘. An encyclopedia in French language. The masterpiece was published by d’Alembert along with Denis Diderot and counts as one of the most important works of the Age of Enlightenment.

The idea of the encyclopedia was to collect all current human knowledge and to make it accessible to everyone. Special was though, that the publishers saw the necessity to make visible all relations between the single articles. Through cross-references, it was made easier to look for and understand the content and therefore get broadly based information on a specific topic. The work is based on the ‘arbor porphyriana‘ (tree of knowledge) by Francis Bacon.[6] Bacon himself was a widely known philosopher, scientist and a pioneer of empiricism. The encyclopedia by Diderot and d’Alembert is seen as a turning point in epistemology, they supported Locke’s theory of the development of knowledge though experience and rejected the methods by Carthesius and Aquinas. The work followed the principle of social equality, it rejected the church’s believes, and measured all human interactions on the theory of reason. The work was accepted well by society due to its easy understanding and its good usability due to the cross-references, a new way of the freedom of thought was developed.

D’Alembert edited more than a thousand articles in the encyclopedia in which he was mainly responsible for mathematical problems. Besides the encyclopedia, d’Alembert researched on further mathematical and philosophical problems, such as his laws of motion, wherefore he counts as the father of mathematical physics. D’Alembert’s achievements were essential to the scientific and cultural development of society during the enlightenment and his contributions count as indispensable for this period.

D’Alembert had been in correspondence with Frederick II of Prussia since 1746.[8] He had taken the initiative for the postal exchange of ideas in response to the prize competition held by the Royal Prussian Academy of Sciences, for which d’Alembert wrote *Réflexions sur la cause générale des vents*. With her he also endeavoured to become a member of the Berlin Academy. Pierre-Louis Moreau de Maupertuis advised him on his project, and so he wrote a dedication poem to the Prussian king about his writing in Latin. However, Frederick II’s direct answer failed to materialize, rather he answered Jean-Baptiste de Boyer, Marquis d’Argens, instead. When d’Alembert published his *Discours préliminaire de l’Encyclopédie* in 1751, Frederick II became aware of him.

The Prussian king offered d’Alembert a position as president of the Royal Prussian Academy of Sciences, and although d’Alembert was at times undecided as to whether he should move his centre of life to Prussia, he refrained from the offer. For his life’s work, highly esteemed by Frederick, d’Alembert nevertheless received from 1754 a Prussian pension of 1200 livres. In the summer of 1763 d’Alembert travelled to Sanssouci Castle for a three-month stay. During his stay in Potsdam he visited Leonhard Euler in Berlin.[7] Euler had been appointed in 1741 by Frederick II to the Royal Prussian Academy of Sciences, which he left in 1766 and went back to St. Petersburg, where Catherine the Great resided since 1762 as Empress of Russia. However, d’Alembert’s mistrust of the rulers was always awake. In his 1759 *Essai sur la société des gens de lettres et des grands* (*Essay on the Society of the Literary and the Great*), he called on intellectuals to liberate themselves from their humiliating role as courtiers of noble patrons. In his relationship to Frederick II d’Alembert differed from Diderot, who from the Seven Years’ War (also called the Third Silesian War from the Prussian point of view) at the latest had an antipathy towards the Friderician state and its first representatives.

At the end of 1757 and the beginning of 1758, the *Encyclopédie* experienced a severe crisis under the editorship of Denis Diderot and d’Alembert. This was due to d’Alembert’s articles on the city of Geneva, which were written at Voltaire’s suggestion. The entry led to extensive discussions and numerous letters of protest and finally broke the relationship between the two editors, which had in part already been strained. D’Alembert was also a brilliant Tacitus translator. He was a member or honorary member of the Russian Academy of Sciences (Petersburg, 1764), the Prussian Academy of Sciences, the American Academy of Arts and Sciences (1781), the Académie des sciences and the Académie française, of which he became General Secretary for life in 1772. He was a member of the Paris Masonic Lodge Les Neuf Sœurs.

D’Alembert died on 29 October 1783 at the age of 65 from the consequences of a urinary bladder disease.

At yovisto academic video search, you may enjoy the mathematical lecture on d’Alembert’s famous wave equation by Dr. Chris Tisdell.

**References and Further Reading:**

- [1] D’Alembert at Wolfram’s Scienceworld
- [2] D’Alembert at Britannica
- [3] O’Connor, John J.; Robertson, Edmund F., “Jean le Rond d’Alembert“, MacTutor History of Mathematics archive, University of St Andrews.
- [4] Jean le Rond d’Alembert at zbMATH
- [5] Denis Diderot’s Encyclopedia, or a Systematic Dictionary of the Sciences, Arts, and Crafts, SciHi Blog
- [6] Sir Francis Bacon and the Scientific Method, SciHi Blog
- [7] Read Euler, he is the Master of us all…, SciHi Blog
- [8] Frederick the Great’s Cunning Plan to Introduce the Potato, SciHi Blog
- [9] D’Alembert at Wikidata
- [10] Timeline for Jean le Rond d’Alembert, via Wikidata

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]]>The post Wilhelm Schickard and his Calculating Clock appeared first on SciHi Blog.

]]>On October 23, 1635, German astronomer and mathematician **Wilhelm Schickard**, who constructed the very first mechanical calculator, passed away. His famous calculator was able to perform additions and subtractions. For more complicated operations, it provided so-called Napier bones, named after the Scottish mathematician John Napier,[1] who came up with the idea of logarithms. Although it is widely believed that the first mechanical calculating device was created by the French mathematician Blaise Pascal in 1642.[2] However, that distinction actually belongs to Wilhelm Schickard.

Born in Herrenberg, Germany, on April 22, 1592, Wilhelm Schickard was a brilliant student. Little is known about his early life. In 1611 he obtained a master’s degree at the University of Tübingen and then studied theology. From 1613 he was vicar at several places in Württemberg until he was appointed deacon in Nürtingen in 1614. Johannes Kepler, who had come to Tübingen to defend his mother in a witch trial, met him there in 1617. For Kepler’s work *Harmonice mundi* he made several copper engravings and woodcuts.[3]

In 1619 he was appointed professor of Hebrew at the University of Tübingen. In his teaching activities, he looked for simple methods to make learning easier for his students. Thus he created the *Rota Hebræa*, a representation of the Hebrew conjugation in the form of two rotating discs which are placed on top of each other and allow the respective forms to appear in windows. To study the Hebrew language, he created the *Horologium Hebræum*, the Hebrew clock, a textbook of Hebrew in 24 chapters, each to be learned in one hour. This book was Schickard’s best-known and was reprinted repeatedly until 1731. In 1627 he wrote a textbook for learning Hebrew in German, the Hebrew Funnel.

In addition to his teaching Hebrew, he was involved in astronomy. In 1623 he invented an *astroscopium*, a cone made of paper inside which the starry sky was depicted. Among other things, he developed a theory of the moon’s orbit, which provided the most accurate ephemeris of his time. He was the first to determine meteor orbits from simultaneous observations from different locations. While toiling over the many tedious calculations necessary in astronomy work, Schickard’s thoughts turned to the notion of mechanically performing mathematical calculations. Although the discovery of logarithms and logarithmic tables by John Napier (1550-1617) several years earlier had greatly simplified the process of multiplication and division, Schickard sought to develop a calculating machine to completely automate these functions.

In 1623, Schickard finally succeeded in building a mechanical device which could perform additions and subtractions. On September 20, 1623, he wrote in a letter to Johannes Kepler as follows:

“What you have done by calculation I have just tried to do by way of mechanics. I have conceived a machine consisting of eleven complete and six incomplete sprocket wheels; it calculates instantaneously and automatically from given numbers, as it adds, subtracts, multiplies and divides. You would enjoy to see how the machine accumulates and transports spontaneously a ten or a hundred to the left and, vice-versa, how it does the opposite if it is subtracting”

Schickard’s letter also mentions that the first machine to be built by a professional, a clockmaker named Johann Pfister, was destroyed in a fire while still incomplete. Schickard abandoned his project soon after. However, in the 1950s, scholars who were collecting the works of Kepler found, tucked into a book, Schickard’s original drawings of his device. This made it possible for Professor Bruno Baron von Freytag Loringhoff of the University of Tübingen to reconstruct Schickard’s calculator. Even though Wilhelm Schickard designed his mechanical calculator twenty years earlier, Pascal is still the inventor of the mechanical calculator because the drawings of Schickard’s calculating clock described a machine that was neither complete nor fully usable.

From 1624 Schickard began on his travels through Württemberg as a school supervisor for the Latin schools to measure the country again. So that others could support him, he wrote an instruction in 1629 on how to make artificial land tables. He used the method of geodetic triangulation, which Willebrord Snell had invented a few years earlier. In 1631 the astronomy professor Michael Mästlin died and Schickard was appointed as his successor. He gave the astronomical lectures from now on. One of his most important works concerned the theory of lunar motion. In 1631 he published the *Ephemeris Lunaris* to calculate the moon’s orbit, with which one could graphically determine the moon’s position in the sky at any time. He was a convinced supporter of the heliocentric system and invented the first hand planetarium for his depiction, which is depicted on his portrait of 1631.

After the Battle of Nördlingen in 1634, the imperial troops occupied Tübingen and the plague came with them. In the autumn of 1634 Schickard’s mother died of mistreatment by soldiers, then his wife and three daughters died of the plague, leaving only his nine-year-old son. Schickard, who himself fell ill with the plague at the turn of the year and recovered, managed to come to terms with the occupying forces. On behalf of Count Gronsfeld, who was interested in his mathematical and even more in his geodetic work, he carried out surveys in the Stuttgart-Herrenberg-Tübingen area and in the Sinzheim-Bruchsal-Pforzheim area from February to July 1635. In mid-October he fell ill again, died on 23 October 1635 and was buried the following day.

Learn more about Schickards calculating machine in an animated video explaining its mechanic design:

**References and Further Reading:**

- [1] John Napier and his Napier Bones, SciHi Blog
- [2] It is not Certain that Everything is Uncertain – Blaise Pascal’s Thoughts, SciHi Blog, 19.06.2012
- [3] And Kepler Has His Own Opera – Kepler’s 3rd Planetary Law, SciHi Blog, 15.05.2012
- [4] O’Connor, John J.; Robertson, Edmund F., “Wilhelm Schickard“, MacTutor History of Mathematics archive, University of St Andrews.
- [5] Wilhelm Schickard at the University of Tübingen
- [6] Wilhelm Schickard’s Calculating Clock
- [7] Wilhelm Schickard at Wikidata

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]]>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.

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]]>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

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]]>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

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