The post The Important Theorem of Thomas Bayes appeared first on SciHi Blog.

]]>On April 17, 1761, English mathematician and Presbyterian minister **Thomas Bayes** passed away. He is best known as name giver of the Bayes’ theorem, of which he had developed a special case. It expresses (in the Bayesian interpretation) how a subjective degree of belief should rationally change to account for evidence, and finds application in in fields including science, engineering, economics (particularly microeconomics), game theory, medicine and law.

Thomas Bayes was born into a prominent family from Sheffield in 1701 and enrolled at the University of Edinburgh in 1719. He began studying logic and theology, assisting his father at the non-conformist chapel in London. He was ordained in 1727 and moved to Box Lane Chapel, Bovington, about 25 miles from London. In later years, Bayes became minister of the Mount Sion chapel.

We do know that in 1719 Bayes matriculated at the University of Edinburgh where he studied logic and theology. He had to choose a Scottish university if he was to obtain his education without going overseas since, at this time, Nonconformists were not allowed to matriculate at Oxford or Cambridge.[5] Even though Bayes is known to have published only two major works, they were quite influential. The first, published in 1731, was titled ‘*Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures*‘ was rather dedicated to the field of theology. Five years later however, Bayes published anonymously ‘*An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst*‘. This work was rather mathematical and in it, Bayes defended Isaac Newton‘s theories on calculus and its logical foundations.[6]

It is speculated that Bayes was elected as a Fellow of the Royal Society in 1742 on the strength of the Introduction to the Doctrine of Fluxions, as he is not known to have published any other mathematical works during his lifetime. In following years, Bayes’ interest in probability theories grew and his interesting ideas and findings were collected in his manuscripts and most of them only became known after his passing.

Unfortunately, Bayes’ most important and most influential work was published after his death. The ‘*Essay towards solving a Problem in the Doctrine of Chances*‘ was read to the Royal Society in 1763. The work contained a statement of a special case of probability now called Bayes’ theorem. The theorem can be seen as a “way of understanding how the probability that a theory is true is affected by a new piece of evidence”. Through the years, it has been helpful in a variety of scientific fields, and is often used to clarify the relationship between theory and evidence. Richard Price, who once published Bayes’ papers after his death believed that Bayes’ theorem could prove the existence of God. This technically combines Bayes’ research fields in mathematics and theology and Price’s claim opened up a wide field of discussion for future scientists and it still depicts a debate topic up to this day.

In probability theory and statistics, Bayes’ theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person’s age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person’s age. One of the many applications of Bayes’ theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayes’ theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics. After Bayes initial formulation, the theory was further developed by Pierre-Simon Laplace,[7] who first published the modern formulation in his 1812 “Théorie analytique des probabilités”. Sir Harold Jeffreys put Bayes’ algorithm and Laplace’s formulation on an axiomatic basis. Jeffreys wrote that Bayes’ theorem “*is to the theory of probability what the Pythagorean theorem is to geometry*“

At yovisto academic video search, you may enjoy a video lecture on Basic Probability Theory by Professor Dr Faber at Zurich.

**References and Further Reading:**

- [1] Bayes Theorem at Trinity University
- [2] Bayes Theorem at the Stanford Encyclopedia of Philosophy
- [3] International Society for Bayesian Analysis
- [4] Thomas Bayes at zbMATH
- [5] John J. O’Connor, Edmund F. Robertson: Thomas Bayes. In: MacTutor History of Mathematics archive
- [6] Standing on the Shoulders of Giants – Sir Isaac Newton, SciHi Blog
- [7] Pierre Simon de Laplace and his true love for Astronomy and Mathematics, SciHi Blog
- [8] Thomas Bayes at Wikidata

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]]>The post Joseph-Louis Lagrange and the Celestial Mechanics appeared first on SciHi Blog.

]]>On April 10, 1813, Italian mathematician and astronomer **Joseph-Louis Lagrange** passed away. Lagrange made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

Lagrange was born on January 25, 1736 as Giuseppe Ludovico Lagrangia in Turin, previously capital of the duchy of Savoy, but became the capital of the kingdom of Sardinia in 1720. His father was Giuseppe Francesco Lodovico Lagrangia, Treasurer of the Office of Public Works and Fortifications in Turin, but the family suffered considerable financial losses through speculation. Lagrange attended the Turin College, where he showed his first mathematical interest at the age of seventeen. His father wanted him to become a lawyer, but at school Lagrange eventually became more interested in mathematics. However, it was not until he was seventeen that he showed any taste for mathematics – his interest in the subject being first excited by a paper by Edmond Halley which he came across by accident. Within one year he taught himself all the knowledge of a fully qualified mathematician of his time.[4,5]

On 23 July 1754, at age 18, Lagrange published his first mathematical work which took the form of a letter written in Italian to Giulio Fagnano. Charles Emmanuel III appointed Lagrange to serve as the “*Sostituto del Maestro di Matematica*” (mathematics assistant professor) at the Royal Military Academy of the Theory and Practice of Artillery in 1755, where he taught courses in calculus and mechanics to support the Piedmontese army’s early adoption of the ballistics theories of Benjamin Robins and Leonhard Euler.[6] In that capacity, Lagrange was the first to teach calculus in an engineering school. According to Alessandro Papacino D’Antoni, the academy’s military commander and famous artillery theorist, Lagrange unfortunately proved to be a problematic professor with his oblivious teaching style, abstract reasoning, and impatience with artillery and fortification-engineering applications. Here he published his first scientific papers on differential equations and calculus of variations. In 1757 he was one of the founders of the Turin Academy, and most of his early writings are to be found in the five volumes of its transactions, usually known as the *Miscellanea Taurinensia*.

Lagrange is one of the founders of the calculus of variations. Starting in 1754, he worked on the problem of tautochrone, discovering a method of maximising and minimising functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to Leonhard Euler between 1754 and 1756 describing his results. He outlined his “δ-algorithm”, leading to the Euler–Lagrange equations of variational calculus and considerably simplifying Euler’s earlier analysis. Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and Maupertuis. Euler was very impressed with Lagrange’s results. It has been stated that “*with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus*“; however, this chivalric view has been disputed. Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773.

In 1756, Euler and Maupertuis, seeing his mathematical talent, tried to persuade him to come to Berlin, but Lagrange had no such intention and shyly refused the offer. The Académie des Sciences in Paris announced its prize competition for 1764 in 1762. The topic was on the libration of the Moon, that is the motion of the Moon which causes the face that it presents to the Earth to oscillate causing small changes in the position of the lunar features. Lagrange entered the competition, sending his entry to Paris in 1763 which arrived there not long before Lagrange himself. In 1765, Lagrange entered, later that year, for the Académie des Sciences prize of 1766 on the orbits of the moons of Jupiter.[1] The prize was again awarded to Lagrange, and he won the same distinction in 1772, 1774, and 1778.[2]

Finally, in 1765, d’Alembert interceded on Lagrange’s behalf with Frederick of Prussia and by letter, asked him to leave Turin for a considerably more prestigious position in Berlin.[7] Lagrange was finally persuaded and he spent the next twenty years in Prussia, where he produced not only the long series of papers published in the Berlin and Turin transactions, but also his monumental work, the *Mécanique analytique*. Here he dealt with problems of astronomy, but also with partial differential equations and questions of geometry and algebra. Lagrange was a favourite of the king, who used frequently to discourse to him on the advantages of perfect regularity of life. The lesson went home, and thenceforth Lagrange studied his mind and body as though they were machines, and found by experiment the exact amount of work which he was able to do without breaking down. Every night he set himself a definite task for the next day, and on completing any branch of a subject he wrote a short analysis to see what points in the demonstrations or in the subject-matter were capable of improvement. He always thought out the subject of his papers before he began to compose them, and usually wrote them straight off without a single erasure or correction.

Lagrange’s productivity in those years was prodigious: he published papers on the three-body problem, which concerns the evolution of three particles mutually attracted according to Sir Isaac Newton’s law of gravity; differential equations; prime number theory; the fundamentally important number-theoretic equation that has been identified (incorrectly by Euler) with John Pell’s name;[8] probability; mechanics; and the stability of the solar system. In his long paper “*Réflexions sur la résolution algébrique des équations*” (1770; “*Reflections on the Algebraic Resolution of Equations*”), he inaugurated a new period in algebra and inspired Évariste Galois to his group theory.[2]

In 1786, following Frederick’s death, Lagrange received invitations from states including Spain and Naples, and he accepted the offer of Louis XVI to move to Paris. In France he was received with every mark of distinction and special apartments in the Louvre were prepared for his reception, and he became a member of the French Academy of Sciences. At the beginning of his residence in Paris he was seized with an attack of melancholy, and even the printed copy of his *Mécanique* on which he had worked for a quarter of a century lay for more than two years unopened on his desk. Curiosity as to the results of the French revolution first stirred him out of his lethargy, a curiosity which soon turned to alarm as the revolution developed.

The Revolution, which began in 1789, pressed Lagrange into work on the committee to reform the metric system. When the great chemist Antoine-Laurent Lavoisier was guillotined, Lagrange commented, “*It took them only an instant to cut off that head, and a hundred years may not produce another like it.*”[2] Though Lagrange had been preparing to escape from France while there was yet time, he was never in any danger; different revolutionary governments (and at a later time, Napoleon) loaded him with honours and distinctions.

Lagrange was considerably involved in the process of making new standard units of measurement in the 1790s. He was offered the presidency of the Commission for the reform of weights and measures (la Commission des Poids et Mesures) when he was preparing to escape. And after Lavoisier’s death in 1794, it was largely owing to Lagrange’s influence that the final choice of the unit system of metre and kilogram was settled and the decimal subdivision was finally accepted by the commission of 1799. Lagrange was also one of the founding members of the Bureau des Longitudes in 1795. The École Polytechnique was founded on 11 March 1794 and opened in December 1794. Lagrange was its first professor of analysis, appointed for the opening in 1794. In 1795 the École Normale was founded with the aim of training school teachers. Lagrange taught courses on elementary mathematics there.

As for the solution of polynomial equations, Lagrange, in his *Reflexions sur la Theorie Algebriques des Equations* of 1770, tried to solve algebraically polynomial equations of degree five and higher starting with the procedure used by Cardano. He tried to generalize by considering permutations of the roots. However, he was unsuccessful with that and he was thus forced to abondon his quest. Nevertheless, his work did form the foundation on which all nineteenth-century work on the algebraic solutions of equations was based, especially that of Cauchy who was able later to take the theory of permutations to a deeper level.[3]

Finally, Lagrange’s commitment to the necessity of an algebraic foundation for the calculus led him to the major accomplishment of the *Fonctions Analytiques*, in which he studied functions by means of their power series expansions. He believed that every function could be expanded into a power series, where for Lagrange, a function was defined as follows:One names a function of one or several quantities any mathematical expression in which the quantities enter in any manner whatever, connected or not with other quantities which one regards as having given and constant values, whereas the quantities of the function may take any possible values.[3]

There is a nice anecdote about Lagrange’s later life. Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty of the parallel axiom. He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him which he had not observed. He muttered:

Il faut que j’y songe encore.

And he put the paper in his pocket.[3]

Joseph-Louis Lagrange, Senator, Count of the Empire, Grand Officer of the Legion of Honour, Grand Cross of the Imperial Order of the Reunion, Member of the Institute and the Bureau of Longitude, died in Paris on 10 April 1813.

References and Further Reading:

- [1] O’Connor, John J.; Robertson, Edmund F., “Joseph-Louis Lagrange“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Joseph-Louis Lagrange, comte de l’Empire, French mathematician, at Britannica Online
- [3] Joseph-Louis Lagrange biography at math.berkeley.edu
- [4] Edmond Halley and his famous Comet, SciHi Blog
- [5] Edmond Halley besides the Eponymous Comet, SciHi Blog
- [6] Read Euler, he is the Master of us all…, SciHi Blog
- [7] Jean Baptiste le Rond d’Alembert and the Great Encyclopedy, SciHi Blog
- [8] John Pell and the Obelus, SciHi Blog
- [9] Joseph-Louis Lagrange at zbMATH
- [10] Joseph-Louis Lagrange at the Mathematics Genealogy Project
- [11] Joseph-Louis Lagrange at Wikidata
- [12] Timeline for Joseph-Louis Lagrange, via Wikidata

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]]>The post Edmund Husserl’s Phenomenology appeared first on SciHi Blog.

]]>On April 8, 1859, German philosopher and mathematician** Edmund Gustav Albrecht Husserl **was born. He is best know as the founder of the 20th century philosophical school of phenomenology, where he broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic.

“First, anyone who seriously intends to become a philosopher must “once in his life” withdraw into himself and attempt, within himself, to overthrow and build anew all the sciences that, up to then, he has been accepting.”

Edmund Husserl, Cartesian Meditations (1931). Méditations cartésiennes

Edmund Husserl studied astronomy in Leipzig, but also attended several lectures in mathematics, physics and philosophy. Three years later, he continued his studies in Berlin and then completed his PhD in Vienna. Austria, and especially Franz Brentano, had a great influence on the scientist. Together, they studied logic and psychology, and Brentano suggested Husserl to finish his habilitation dissertation with Carl Stumpf in Halle.

When Husserl published his first monograph, the ‘*Philosophy of Arithmetic*‘, based on his dissertation, he combined his mathematical, psychological and philosophical knowledge in order to lay the foundations for arithmetic in psychology. However, it was Gottlob Frege, a founder of modern logic, who criticized Husserl’s work for its underlying psychologism. Therefore Husserl began developing the philosophical method of phenomenology.

At the very beginning of the 1900’s, he published his first work on the subject, ‘*Logical Investigations*‘ consisting of two volumes. In the work, Husserl attacked psychologism in general and explained “*descriptive-psychological*” and “*epistemological*” investigations. The work is credited with making continental philosophy possible and it depicted a significant foundation of phenomenology, going beyond systems like psychologism, formalism or realism.

In the following years, Edmund Husserl improved and modified his phenomenology methods and published his second major work in this field, ‘*Ideas*‘ in 1913 while working at the University of Göttingen. In Göttingen, Husserl presumably experienced his most productive years, developing the transcendental-phenomenological method, the phenomenological structure of time-consciousness and many more. Those were critically important for later publications, such as ‘*The Crisis of European Sciences and Transcendental Phenomenology*‘ and ‘*Experience and Judgement*‘.

Husserl criticized the logicians of his day for not focusing on the relation between subjective processes that give us objective knowledge of pure logic. All subjective activities of consciousness need an ideal correlate, and objective logic (constituted noematically) as it is constituted by consciousness needs a noetic correlate (the subjective activities of consciousness). He stated that logic has three strata, each further away from consciousness and psychology than those that precede it.

- The first stratum is what Husserl called a “
*morphology of meanings*” concerning a priori ways to relate judgments to make them meaningful. In this stratum we elaborate a “*pure grammar*” or a logical syntax, and he would call its rules “laws to prevent non-sense”, which would be similar to what logic calls today “*formation rules*“. Mathematics, as logic’s ontological correlate, also has a similar stratum, a “*morphology of formal-ontological categories*“. - The second stratum would be called by Husserl “
*logic of consequence*” or the “*logic of non-contradiction*” which explores all possible forms of true judgments. He includes here syllogistic classic logic, propositional logic and that of predicates. This is a semantic stratum, and the rules of this stratum would be the “*laws to avoid counter-sense*” or “*laws to prevent contradiction*“. They are very similar to today’s logic “*transformation rules*“. Mathematics also has a similar stratum which is based among others on pure theory of pluralities, and a pure theory of numbers. They provide a science of the conditions of possibility of any theory whatsoever. Husserl also talked about what he called “l*ogic of truth*” which consists of the formal laws of possible truth and its modalities, and precedes the third logical third stratum. - The third stratum is metalogical, what he called a “
*theory of all possible forms of theories*.” It explores all possible theories in an a priori fashion, rather than the possibility of theory in general. We could establish theories of possible relations between pure forms of theories, investigate these logical relations and the deductions from such general connection. The logician is free to see the extension of this deductive, theoretical sphere of pure logic.

In his last active working years, Edmund Husserl gave lectures across Europe on phenomenology. After his passing on April 27, 1938, over 40000 pages of manuscripts have been found and depicted the foundation for the first Husserl archive, established in 1939.

Husserl’s influence on European universities was enormous. He was invited to numerous talks and many institutions adapted his theories quickly. He was known as a fascinating teacher and scientist with Martin Heidegger as his most famous student.[5] He dedicated his work ‘Being and Time‘ to his former teacher even though the two scientists often had a difficult relationship due to Heidegger’s affiliation for the Nazi Party in Germany. Husserl influenced many younger scientists in concerns of their publications and methods. Jean-Paul Sartre also admired Husserl’s work, but rejected the transcendental theories and decided to rather follow Heideggers ontologies. Further admirers of Edmund Husserl were Kurt Gödel, Wilfrid Sellars, Jacques Derrida and many more.

At yovisto academic video search you may learn more about Edmund Husserl and Heidegger through a German lecture by Prof. Dr. Günther Figal at the University of Freiburg.

**References and Further Reading: **

- [1] Edmund Husserl Formal Ontology and Transcendental Logic
- [2] Edmund Husserl at Stanford Encyclopedia of Philosophy
- [3] Husserl Websites
- [4] Gottlob Frege and the Begriffsschrift, SciHi Blog
- [5] Martin Heidegger and the Question of Being, SciHi Blog
- [6] A Writer should not Allow Himself to be Turned into an Institution – Jean-Paul Sartre, SciHi Blog
- [7] Kurt Gödel and the Foundations of Mathematics, SciHi Blog
- [8] Edmund Husserl at Wikidata

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]]>The post The Short but Influential Life of Niels Henrik Abel appeared first on SciHi Blog.

]]>On April 6, 1829, Norwegian mathematician **Niels Henrik Abel** passed away. Abel is well known in mathematics for proving the impossibility of solving the quintic equation by radicals. In parallel to Évariste Galois – who also died very young – , he laid the foundations of group theory.[8]

Nils Henrik Abel was born into a family of educated people. His father, for instance earned himself a degree in theology and philosophy and taught his children at home for some years. At the age of 13, Nils Abel entered a Cathedral School and especially his mathematics teacher depicted a great influence on the talented boy. Even though, his brother Hans got better grades in general, their math teacher Bernt Michael Holmboe early detected Niels’ abilities in mathematics. He provided Niels Abel with advanced lessons after school and assigned him more and more difficult problems during classes.

Unfortunately, many issues in the Abel family followed in the next years. His father passed away and his mother dedicated her life to alcoholism, wherefore his brother went into a depression, quitting school and Niels himself began struggling in almost every subject but mathematics. Still, Holmboe supported his protégé and helped Abel with a scholarship wherefore he was able to attend the Royal Frederick University of Oslo, the oldest and largest university in Norway.

As it happened many times in history, at some point in teaching the student may become the master, and this was also the case with Abel. At the age of only 21, he was one of the most educated mathematicians in the country and since Holmboe had nothing left to teach him, Abel continued his research in libraries, studying the works of Newton, Euler, Lagrange, and Gauss.[9] About Gauss’ writing style, Abel once noted that ‘*He is like the fox, who effaces his tracks in the sand with his tail.*‘ However, in these years Abel also began working on what has made him so famous, the quintic equation in radicals. After working for a while on the problem that mathematicians have tried to solve for over 200 years, Abel instantly thought to have found its solution. His paper was checked and recalculated numerous times by experts and no one found any error. It was Abel himself, who discovered the mistakes in his method while trying to find specific examples.

After graduating, Abel began publishing several works, unfortunately only with modest success. Success in concerns of publications set in around 1824. Abel published his ‘*Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is proven*‘. Even though his contribution to the scientific community through this paper was very important, it was difficult to read. The scientist limited the paper to six pages in order to save printing money. In the following years, Niels Henrik Abel traveled though Europe, meeting many scientists and completing several publications with them.

“If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words,the most important parts of mathematics stand without a foundation” (Niels Henrik Abel)

Abel, the mathematical genius made several significant contributions to his field of study. He proved the binomial theorem for all numbers, extending the works of Leonard Euler. Next to Évariste Galois, it was Abel, who developed the group theory in order to complete his works on quintic equation. While in Paris, Abel contracted tuberculosis. At Christmas 1828, he traveled by sled to Froland to visit his fiancée. He became seriously ill on the journey; and, although a temporary improvement allowed the couple to enjoy the holiday together, he died relatively soon after on 6 April 1829, just two days before a letter arrived from August Crelle. Crelle had been searching for a new job for Abel in Berlin and had actually managed to have him appointed as a Professor at the University of Berlin. Crelle wrote to Abel to tell him, but the good news came too late. A statue of Abel stands in Oslo, and crater Abel on the Moon was named after him. In 2002, the Abel Prize was established in his memory.

At yovisto academic video search , you may learn more on Abel’s works through a short introduction to group theory.

**References and Further Reading**

- [1] Niels Henrik Abel at Fermat’s Last Theorem Blog
- [2] Abel Prize Website
- [3 ]Wolfram Research
- [4] O’Connor, John J.; Robertson, Edmund F., “Niels Henrik Abel“, MacTutor History of Mathematics archive, University of St Andrews.
- [5 ]Math2033, University of Arkansas
- [6] Niels Henrik Abel at NNDB
- [7] aggregated biographies and works related with Niels Henrik Abel
- [8] Only the Good Die Young – the Very Short Life of Évariste Galois, SciHi Blog, June 1, 2012.
- [9] Carl Friedrich Gauss – The Prince of Mathematicians, SciHi Blog, April 30, 2013.
- [10] Niels Henrik Abel at Wikidata
- [11] Niels Henrik Abel at zbMATH

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]]>The post Cogito Ergo Sum – The Philosophy of René Descartes appeared first on SciHi Blog.

]]>On March 31, 1596, French philosopher, mathematician, and writer **René Descartes **was born. The Cartesian coordinate system is named after him, allowing reference to a point in space as a set of numbers, and allowing algebraic equations to be expressed as geometric shapes in a two-dimensional coordinate system. He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial to the discovery of infinitesimal calculus and analysis. Descartes was also one of the key figures in the Scientific Revolution and has been described as an example of genius. He has been dubbed the ‘Father of Modern Philosophy’. His Meditations on First Philosophy continues to be a standard text at most university philosophy departments.

“Of all things, good sense is the most fairly distributed: everyone thinks he is so well supplied with it that even those who are the hardest to satisfy in every other respect never desire more of it than they already have.”, Rene Descartes, Discours de la Methode (1637)

René Descartes was born in the Touraine, France and attended the Jesuit College of La Fleche in 1606. At the school he learned Latin, Greek and studied the philosophies of Aristotele, Plato, the Stoics, and Cicero. Descartes also studied curiously mathematics, physics, and especially the works of Galileo Galilei.[4] Just like many of Descartes’ ancestors, he was supposed to become a lawyer, but never actually practiced law or anything like it after graduating in 1616. Instead, Descartes became a soldier as support to Protestant Prince Maurice for some years.

One of his first influences depicted Isaac Beeckman, a mathematician and natural philosopher, who met with Descartes while stationed at Breda. According to French scholar Adrien Baillet, on the night of 10–11 November 1619 (St. Martin’s Day), while stationed in Neuburg an der Donau, Descartes shut himself in a room with an “oven” to escape the cold. While within, he had three dreams and believed that a divine spirit revealed to him a new philosophy. Upon exiting, he had formulated analytical geometry and the idea of applying the mathematical method to philosophy. He concluded from these visions that the pursuit of science would prove to be, for him, the pursuit of true wisdom and a central part of his life’s work. Descartes also saw very clearly that all truths were linked with one another so that finding a fundamental truth and proceeding with logic would open the way to all science. Descartes discovered this basic truth quite soon: his famous “*I think, therefore I am*“.

In these years, Descartes discovered the technique of describing lines through mathematical equations, which led to the combination of both, algebra and geometry. Algebra and analysis evolved step by step after Descartes’ findings and the coordinate system of algebraic geometry came to be called “Cartesian coordinates” in honor to the scientist. Later on, Descartes enrolled at Leiden University, studying mathematics and astronomy and then became teacher at Utrecht University.

“Nothing comes out of nothing.”

Rene Descartes, Principia philosophiae, Part I, Article 49

In the 1620’s, René Descartes worked on a metaphysical piece on the existence of God, nature, and soul as well as tried to explain the set of parhelia in Rome. He combined both in the work *Treatise on the World*, which consisted of three parts. Only two of these, *The Treatise of Light* and the *Treastise of Man* survived. The two parts gave a good illustration of the universe as a system including all of its structures, operations, planet formations, light transmission, and the role of the human on Earth. However, Descartes abandoned his plans to publish the Treatise on the World after Galileo was condemned. He continued publishing works on philosophy, geometry, meterology and his most famous *Discours de la Méthode*, demonstrating four rules of thought. Further influential works followed after 1641, when Descartes published his *Mediations on First Philosophy* and his *Principles of Philosophy*.

The key points of *Discours de la Méthode* are:

- a theory of cognition that only accepts as correct what is verified as plausible by its own step-by-step analysis and logical reflection,
- an ethics according to which the individual must behave conscientiously and morally in the sense of proven social conventions,
- a metaphysics which (by logical proof) accepts the existence of a perfect Creator-God, but leaves little room for church-like institutions,
- a physics which regards nature as regulated by God-given but generally valid laws and makes its rational explanation and thus ultimately its control the task of man.

The philosophical method formulated in detail in the *Discours de la méthode* of Descartes is summarized in four rules (II. 7-10):

- Scepticism: Do not believe anything that is not so clearly recognized that it cannot be called into doubt.
- Analysis: Solving difficult problems in substeps.
- Construction: progressing from simple to difficult (inductive procedure: from concrete to abstract)
- Recursion: Always check whether the examination is complete.

During his lifetime, Descartes is now regarded as one of the first to write about the importance of reason in natural sciences rejecting any doubtable ideas. This was illustrated in his famous phrase ‘cogito ergo sum’ (*I think, therefore i am*) through which he concluded that doubting the existence of a person was already the prove of one’s presence. Descartes was also known for his dualism. He once wrote that a human body functioned like a machine with material properties and the mind, both interacting at the pineal gland. In other words, this means that the body is controlled by the mind and vise versa.

“In order to seek truth, it is necessary once in the course of our life, to doubt, as far as possible, of all things.”

Descartes, René, Principles of Philosophy (1644)

Through his works, René Descartes was able to set the foundations of the society’s emancipation from the Church, and shifting it from the medieval to the modern period. In mathematics, Descartes was able to lay the foundations for Leibniz and Newton to develop calculus and he discovered the law of reflection, achieving a critical contribution to the field of optics. One of Descartes’ most enduring legacies was his development of Cartesian or analytic geometry, which uses algebra to describe geometry. He “*invented the convention of representing unknowns in equations by x, y, and z, and knowns by a, b, and c*“. He also “*pioneered the standard notation*” that uses superscripts to show the powers or exponents. He was first to assign a fundamental place for algebra in our system of knowledge, using it as a method to automate or mechanize reasoning, particularly about abstract, unknown quantities.

René Descartes passed away on February 11, 1650 in Stockholm. In 1663, Pope Alexander VII set his works on the ‘Index of Prohibited Books’.

Dr. Richard Brown on Descartes’ Method of Doubt.

**References and Further Reading:**

- [1] René Descartes at Stanford
- [2] René Descartes at the Encyclopedia of Philosophy
- [3] René Descartes Website
- [4] The Galileo Affair, SciHi Blog, February 13, 2014.
- [5] Galileo Galilei and his Telescope, SciHi Blog, August 25, 2012.
- [6] Rene Descartes at Wikidata
- [7] Timeline for Rene Descartes, via Wikidata
- [8] Biography of Rene Descartes at at MacTutor’s History of mathematics

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]]>The post Pierre Simon de Laplace and his true love for Astronomy and Mathematics appeared first on SciHi Blog.

]]>On March 28, 1749, French mathematician and astronomer **Pierre Simon marquis de Laplace** was born, whose work was pivotal to the development of mathematical astronomy and statistics. One of his major achievements was the conclusion of the five-volume *Mécanique Céleste *(*Celestial Mechanics*) which translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems.

Pierre Simon Laplace, the son of a cider merchant was born in the Normandie, and grew up as a well educated child attending the local school at a Benedictine priory. In later years, he was sent to Caen in order to study theology and philosophy were he found out about his love to mathematics. Laplace quit his studies in 1768 applying to study mathematics under the most famous French mathematician of his time, Jean-Baptiste le Rond d’Alembert.[4] D’Alembert was immediately impressed by the young Laplace and ready to teach and support him.

Already three years later, Laplace taught geometry, trigonometry, analysis and statistics and published several works on game theory, probability theory and other difficult issues to improve his reputation. He was admitted to the Académie Française at only 24 years. Laplace was about to become one of France’s most influential scientists and had the honor to examine the future members of the royal artillery. One of his aspirants was the 16 year old Napoleon Bonaparte, who later offered Laplace a position as his minister of the interior.

During the French Revolution, Laplace was able to continue his research as far as possible and in 1792 he became a member of the Committee for Weights and Measures, which was later responsible for the introduction of the units metres and kilograms. But Laplace had to give up this office, because with the rule of Robespierre, participation in the revolution and hatred of the monarchy became a condition for the work. After Robespierre himself died by the guillotine on 28 July 1794, Laplace returned to Paris and became one of the two examiners for the École polytechnique in December of that year. In 1795 Laplace resumed his work on the Committee for Weights and Measures and became its chairman. In the same year, the academy was re-established with the umbrella organisation Institut de France. Laplace was a founding member and later also president of the institute. He also took over the management of the Paris Observatory and the research department.

After Laplace had voted in 1804 in the Senate for Napoleon’s appointment as emperor, he ennobled him to count in 1806. In the same year Laplace moved to Arcueil, a suburb of Paris, in the house next door to the chemist Claude-Louis Berthollet, with whom he founded the Société d’Arcueil. There, they conducted experiments with other, mostly young scientists. Among these scientists were Jean-Baptiste Biot and Alexander von Humboldt.[5] Through this work, however, he made enemies because he set up a clear research programme, which mainly included his own research priorities, and also carried this out mercilessly. Laplace lost further reputation because he continued to hold on to the particle nature of light, while wave theory was increasingly recognised by Augustine Jean Fresnel.[6]

Laplace created further opponents when he voted for Napoleon’s deposition in 1814 and immediately made himself available for the Bourbon restoration. King Louis XVIII, on the other hand, made Laplace a Pair of France in 1815 and a Marquis in 1817. In 1816 Laplace stopped working at the École polytechnique and became a member of the 40 Immortals of the Académie française.

However, Laplace’s scientific contributions are numerous. In the field of astronomy, he published a work titled *Traité de Mécanique Céleste*, a collected work of all scientific approaches after Newton. In the book series he also demonstrated the mathematical prove for the stability of planetary orbits – due to irregularities in the orbit curves, it was believed at the time that the solar system could collapse – and hypothesized about the possibility of black holes. This work was a great success and used and studied by every astronomer or those willing to be one. Laplace’s second major field of research was probability theory. For Laplace, it was a way out to achieve certain results despite a lack of knowledge. In his two-volume work *Théorie Analytique des Probabilités* (1812), Laplace gave a definition of probability and dealt with dependent and independent events, especially in connection with gambling. He also dealt in the book with the expected value, mortality and life expectancy. The work refuted the thesis that a strict mathematical treatment of probability was not possible. Laplace has always been more a physicist than a mathematician. Mathematics served him only as a means to an end. Today, however, the mathematical methods that Laplace developed and used are much more important than the actual work itself, like the Laplace operator, or the Laplace transform.

At yovisto academic video search, you may lean more about Laplace’s equation by watching Prof. Gilbert Strang’s video lecture at MIT.

**References and Further Reading:**

- [1] John J. O’Connor, Edmund F. Robertson: Pierre Simon de Laplace. In: MacTutor History of Mathematics archive
- [2] Laplace at Trinity College Dublin
- [3] Laplace Transform
- [4] Jean Baptiste le Rond d’Alembert and the Great Encyclopedy, SciHi Blog, November 16, 2012.
- [5] On the Road with Alexander von Humboldt, SciHi Blog, August 3, 2012.
- [6] Augustin-Jean Fresnel and the Wave Theory of Light, SciHi Blog, March 10, 2018.
- [7] Pierre Simon de Laplace at Wikidata
- [8] Pierre Simon de Laplace at zbMATH
- [9] Works of Pierre Simon de Laplace (digitized at Wiki Commons)
- [8] Timeline for Pierre Simon de Laplace via Wikidata

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]]>The post Emmy Noether and the Love for Mathematics appeared first on SciHi Blog.

]]>On April 23, 1882, German mathematician and physicist **Emmy Noether** was born, who is best known for her groundbreaking contributions to abstract algebra and theoretical physics. Albert Einstein called her the most important woman in the history of mathematics, as she revolutionized the theories of rings, fields, and algebras.

“My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously.”

Letter to Helmut Hasse (1931) as quoted in Auguste Dick, Emmy Noether, 1882-1935 (1981) Tr. H. I. Blocher, p. 61.

In 1900, Emmy Noether decided to enroll at the University of Erlangen, but as one of two women at the institution, she was only allowed to audit her classes instead of really participating in them. Noether also finished her graduation at a grammar school in Nuremberg three years later. While restrictions were hard on studying women in Erlangen, Noether attended lectures of famous scientists like Karl Schwarzschild [5] or David Hilbert [6] in Göttingen. After returning to Erlangen, she was allowed to finally study mathematics and taught at the universities’ mathematical institute without payment after her graduation.

David Hilbert had to put great effort into getting Emmy Noether into Göttingen University as privatdozent. Eventually she was allowed to teach at the university despite her sex, but still without any payment. Since the habilitation of women at Prussian universities was prohibited by a decree, the Department of Mathematics and Natural Sciences of the Faculty of Philosophy of the University of Göttingen submitted an official application to the Prussian minister on 26 November 1915, to make an exception for Emmy Noether. However, the application was rejected and Emmy Noether then had no choice but to announce her lectures under the name of Hilbert, whose assistant she acted as.

In these years, she proved the Noether theorem one of the most important contributions to the field of mathematics since the Pythagorean theorem, as many of her male colleagues noted. It proves a relationship between symmetries in physics and conservation principles. This basic result in the theory of relativity was praised by Einstein in a letter to Hilbert when he referred to Noether’s penetrating mathematical thinking. Noether enjoyed a great reputation and delivered her habilitation lecture in 1919 but was not given any salary for her work until 1924 when Noether was appointed a special teaching position in algebra.

A great part in the development of abstract algebra was achieved in the 20th century and Emmy Noether depicted a major influence on the topic with several papers and lectures. Beginning with the year 1920, Noether began publishing works on the ideal theory, defining left and right ideals as a ring followed by another publication, analyzing ascending chain conditions. Noether published *Abstrakter Aufbau der Idealtheorie in algebraischen Zahlkorpern* in 1924. In this paper she gave five conditions on a ring which allowed her to deduce that in such commutative rings every ideal is the unique product of prime ideals.[3]. After these works, Noether had many supporters in the scientific community and several mathematical terms were named after her. During the lectures she gave at university, Noether gave up regular lesson plans and rather used the time for intense discussions, bringing her research forward. Some students paid lots of respects to her methods, others were rather frustrated.

After a long friendship with the mathematician Pavel Alexandrov, Noether decided to continue her work at the Moscow State University in 1928 for some time. There she critically contributed to the development of Galois theory. Emmy Noether’s achievements are numerous. And even though she received several awards for her works she was still not promoted to being a full professor at the university, which caused much frustration along her colleagues who dearly respected her achievements and her personality.

Further recognition of her outstanding mathematical contributions came with invitations to address the International Congress of Mathematicians at Bologna in September 1928 and again at Zürich in September 1932. Her address to the 1932 Congress was entitled *Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie.* Much work on hypercomplex numbers and group representations was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these results and gave the first general representation theory of groups and algebras. In 1932 she also received, jointly with Emil Artin,[7] the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge.

In 1933, Emmy Noether received the same letter as many of her Jewish colleagues. She was expelled from her position at the University of Göttingen due to the new Law for the Restoration of the Professional Civil Service. However, she continued her lectures on class field theory secretly in her apartment until starting her job at the University of Oxford and later the Institute for Advanced Study in Princeton.

Emmy Noether passed away on April 14, 1935. At her memorial, many notable mathematicians and friend’s of Noether like Pavel Alexandrov, Bryn Mawr or Hermann Weyl paid their respects. Although she received little recognition in her lifetime considering the remarkable advances that she made, she has been honoured in many ways following her death.[3]

Emmy Noether is one of the founders of modern algebra. Her mathematical profiling developed in cooperation and discussion with Professor Paul Gordan from Erlangen, who also became her doctoral supervisor. He was often called the “King of the Invariants”. The theory of invariants occupied Emmy Noether until 1919 decidedly. In Göttingen, by then a world centre of mathematical research, she built up her own mathematical school. Noether is also ascribed a decisive role in the implementation of abstract algebraic methods in topology.

At yovisto academic video search you may enjoy a Google Tech Talk by Dr. Ransom Stephens on ‘Emmy Noether and The Fabric of Reality‘.

**References and Further Reading:**

- [1] Emmy Noether at the San Diego Supercomputer Center Website
- [2] Emmy Noether at the Agnes Scott College Website
- [3] O’Connor, John J.; Robertson, Edmund F., “Emmy Noether“, MacTutor History of Mathematics archive, University of St Andrews.
- [4] Noether’s Theorem at the University of California’s Website
- [5] Karl Schwarzschild and the Event Horizon, SciHi Blog, October 9, 2014.
- [6] David Hilbert’s 23 Problems, SciHi Blog, August 8, 2012.
- [7] Emil Artin and Algebraic Number Theory, SciHi Blog, March 3, 2017.
- [8] Emmy Noether at zbMATH
- [9] Emmy Noether at Mathematics Genealogy Project
- [10] Emmy Noether at Women in Mathematics
- [11] Emmy Noether at Wikidata
- [12] Timeline for Emmy Noether, via Wikidata

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]]>The post Georg Cantor and the Beauty of Infinity appeared first on SciHi Blog.

]]>On March 3, 1845, German mathematician **Georg Cantor**, creator of the set theory was born. **Set Theory** is considered the fundamental theory of mathematics. He also proved that the real numbers are “more numerous” than the natural numbers, which was quite shocking for his contemporaries that there should be different numbers of infinity.

“In mathematics the art of asking questions is more valuable than solving problems.”

Georg Cantor, Doctoral thesis (1867)

Georg Cantor was born in 1845 in the western merchant colony in Saint Petersburg, Russia. His father, Georg Waldemar Cantor, was a successful merchant, working as a wholesaling agent and later as a broker in the St Petersburg Stock Exchange. When his father became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated from the Realschule in Darmstadt with an outstanding report, which mentioned in particular his exceptional skills in mathematics, in particular trigonometry. In 1862, Cantor entered the University of Zürich to study mathematics. After receiving a substantial inheritance upon his father’s death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker [3], Karl Weierstrass [4] and Ernst Kummer [5]. He spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. In 1867, Cantor completed his dissertation in number theory with the title “*De aequationibus secundi gradus indeterminatis*“, under the supervision of Eduard Kummer and Karl Weierstrass [7], at the University of Berlin.

Cantor first taught at a girl’s school in Berlin and in 1868, he joined the Schellbach Seminar for mathematics teachers. During this time he worked on his habilitation and, immediately after being appointed to Halle in 1869, where he spent his entire career, he presented his thesis, again on number theory, and received his habilitation. In 1872 Cantor was promoted to Extraordinary Professor at Halle and 1879 became Full Professor. Cantor’s work between 1874 and 1884 is the origin of set theory. Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets, which are easy to understand, and “the infinite”, which was considered a topic for philosophical, rather than mathematical, discussion. Cantor succeeded in proving that there are many possible sizes for infinite sets – even infinitely many. Thereby, he established that set theory was anything else but trivial, and that it needed to be studied. In the course of this study, set theory has become a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects from all areas of mathematics in a single theory, and provides a standard set of axioms to prove or disprove them.

“A set is a Many that allows itself to be thought of as a One.”

Georg Cantor, as quoted in Infinity and the Mind (1995) by Rudy Rucker.

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor’s 1874 paper, “*Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen” (“On a Property of the Collection of All Real Algebraic Numbers”)*. This paper was the first to provide a rigorous proof that there was more than one kind of infinity. A first step towards Cantor’s set theory already was his 1873 proof that the rational numbers are countable, i.e. they may be placed in one-one correspondence with the natural numbers – and therefore their count is equal. However, when he tried to extend his proof to real numbers, he was facing difficulties. To decide whether the real numbers were countable proved much harder. But in December 1873 he succeeded with the counter argument and proved that the real numbers were not countable. Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous, i.e. having the same number of elements.

In 1891, he published a paper containing his elegant “diagonal argument” for the existence of an uncountable set. He applied the same idea to prove Cantor’s theorem: t*he cardinality of the power set of a set A is strictly larger than the cardinality of A*. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel‘s first incompleteness theorem.[9]

Also related with the problem of infinite sets is the famous continuum hypothesis – *There is no set whose cardinality is strictly between that of the integers and the real numbers* – introduced by Cantor in 1878. It was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris…but this is already another story [8].

“The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.”

Georg Cantor, as quoted in Infinity and the Mind (1995) by Rudy Rucker.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I. The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918.

David Hilbert described Cantor’s work as [1]:

…the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.

At yovisto academic video search you can learn more about the concept of infinity in the talk of Dennis Wildfogel on ‘How Big Is Infinity?’

**References and Further Reading:**

- [1] Georg Ferdinand Ludwig Philipp Cantor at The MacTutor History of Mathematics
- [2] God made the integers, all the rest is the work of man – Leopold Kronecker, SciHi Blog, December 7, 2014.
- [3] Karl Weierstrass – the Father of Modern Analysis, SciHi Blog, February 19, 2018.
- [4] Ernst Kummer and his Achievements in Mathematics, SciHi Blog, January 19, 2015.
- [5] Georg Cantor at Wikidata
- [6] Georg Cantor at zbMATH
- [7] Georg Cantor at Mathematics Genealogy Project
- [8] David Hilbert’s 23 Problems, SciHi Blog, August 8, 2012.
- [9] Kurt Gödel and the Foundations of Mathematics, SciHi Blog, April 28, 2012.
- [10] Timeline for Georg Cantor, via Wikidata

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]]>The post Karl Weierstrass – the Father of Modern Analysis appeared first on SciHi Blog.

]]>On February 19, 1897, German mathematician **Karl Theodor Wilhelm Weierstrass** passed away. Weierstrass often is cited as the “father of modern analysis“. He formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals.

“… it is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician.”

Karl Weierstrass in a letter to Sofia Kovalevskaya, August 27, 1883

Karl Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia, the son of Wilhelm Weierstrass, secretary to the mayor of Ostenfelde by the time of Karl’s birth and later a tax inspector, and Theodora Vonderforst. His interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn, where he regularly read *Crelle’s Journal* and also gave mathematical tuition to one of his brothers. However Weierstrass’s father wished him to study finance and so, after graduating from the Gymnasium in 1834, he entered the University of Bonn with a course planned out for him which included the study of law, finance and economics. Weierstrass was immediately in conflict with his original hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study, but continued private study in mathematics. Weierstrass did study mathematics on his own, however, reading Laplace’s *Mécanique céleste* [2] and then a work by Jacobi on elliptic functions [3].

Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration. After his decision, he spent one further semester at the University of Bonn, his eighth semester ending in 1838, and having failed to study the subjects he was enrolled for he simply left the University without taking the examinations. After that he studied mathematics at the Theological and Philosophical Academy of Münster and his father was able to obtain a place for him in a teacher training school in Münster, where he was later certified as a secondary school teacher.

In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg. Besides mathematics he also taught physics, botany, and even gymnastics. From around 1850 Weierstrass began to suffer from attacks of dizziness which were very severe and which ended after about an hour in violent sickness. Frequent attacks over a period of about 12 years made it difficult for him to work and it is thought that these problems may well have been caused by the mental conflicts he had suffered as a student, together with the stress of applying himself to mathematics in every free minute of his time while undertaking the demanding teaching job.[1]

In complete isolation from the mathematical world, Weierstrass worked intensively on his theory of Abel’s functions (direct generalizations of elliptical functions) and published in the journal of his school. However, it was not until an essay in *Crelle’s Journal* in 1854 *On the Theory of Abel’s Functions*, which was followed by a more detailed work in 1856, that attention was drawn. As a result, he received an honorary doctorate from the Albertus University of Königsberg in the same year, and the leading Berlin mathematicians Peter Gustav Lejeune Dirichlet [6] and Ernst Eduard Kummer [7] tried to move him to Berlin.

Since 1856 Weierstrass taught mathematics at the Königlichen Gewerbeinstitut (1879 integrated into the Technische Hochschule Berlin), but in the same year he became professor at the Friedrich-Wilhelms-Universität Berlin. There, a large school soon formed around him, whose characteristic feature was the introduction of “Swiss rigour” into the analysis. Even more so than through his publications, he worked through the numerous widely circulating transcripts of his lectures by his students, such as Wilhelm Killing or Adolf Hurwitz. He initially got along well with his Berlin colleague Leopold Kronecker [8], but in 1877 there was a disagreement due to his rejection of the set theory of Weierstrass’s pupil Georg Cantor [9].

Weierstrass, who never married, had a special relationship with his Russian pupil Sofia Kovalevskaya [10], whom he taught privately from 1870 onwards, since she was not admitted to the university as a woman. He exerted his influence so that she was able to obtain her doctorate in Göttingen in 1874 and to take up a position as a private lecturer in Stockholm in 1884. Until her death in 1891 he remained in constant correspondence with her.

Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions. Delta-epsilon proofs are first found in the works of Cauchy in the 1820s, who did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 *Cours d’analyse*, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass’s advisor, Christoph Gudermann in 1838. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.

Weierstrass also made significant advancements in the field of calculus of variations. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory which paved the way for the modern study of the calculus of variations. In Weierstrass’ 1863/64 course on *The general theory of analytic functions* he began to formulate his theory of the real numbers. In his 1863 lectures he proved that the complex numbers are the only commutative algebraic extension of the real numbers. In 1872 his emphasis on rigour led him to discover a function that, although continuous, had no derivative at any point. Analysts who depended heavily upon intuition for their discoveries were rather dismayed at this counter-intuitive function. Riemann had suggested in 1861 that such a function could be found, but his example failed to be non-differentiable at all points.

The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics. In 1883 he was elected a member of the Leopoldina. In 1892 he was elected to the National Academy of Sciences, in 1896 he was elected to the American Academy of Arts and Sciences. In 1887 Weierstrass was awarded the Cothenius Medal of Leopoldina.

Karl Weierstrass died on 19 February 1897 in Berlin from pneumonia at age 81.

**References and Further Reading:**

- [1] O’Connor, John J.; Robertson, Edmund F., “Karl Weierstrass“, MacTutor History of Mathematics archive, University of St Andrews.
- [2] Pierre Simon de Laplace and his true love for Astronomy and Mathematics, SciHi Blog, March 28, 2013.
- [3] Carl Jacobi and the Elliptic Functions, SciHi Blog, December 10, 2014.
- [4] Karl Weierstrass at zbMATH
- [5] Karl Weierstrass at Mathematics Genealogy Project
- [6] Lejeune Dirichlet and the Mathematical Function, SciHi Blog, February 13, 2015.
- [7] Ernst Kummer and his Achievements in Mathematics, SciHi Blog, January 29, 2015.
- [8] God made the integers, all the rest is the work of man – Leopold Kronecker, SciHi Blog, December 7, 2014.
- [9] Georg Cantor and the Set Theory, SciHi Blog, March 3, 2013.
- [10] Sofia Kovaleveskaya – Mathematician and Writer, SciHi Blog, January 15, 2016.
- [11] Karl Weierstrass at Wikidata
- [12] Timeline for Karl Weierstrass, via Wikidata

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]]>The post Lewis Carroll – Mathematician and Creator of the Wonderland appeared first on SciHi Blog.

]]>On January 27, 1832, British mathematician, photographer, and children’s book author** Lewis Carroll**, creator of the stories about ‘Alice in Wonderland’, was born. The English all round talented Carroll was first home schooled and confronted with challenging works like ‘*The Pilgrim´s Progress*‘ – a Christian allegory written by John Bunyan, regarded as one of the most significant works of religious English literature – from early years on. When being transferred to grammar school he could glance with his intellect as well as from the first day on when attending Oxford College. After graduating, the brilliant mathematician stayed at the Christ Church for teaching and studying. To his favored mathematical fields belonged matrix algebra, geometry, or logic. He was able to publish several mathematical books and developed Dodgson’s method (an improved voting method).

But despite his life as a mathematician, Carroll was enjoying the company of artists like Scottish poet George MacDonald or Preraphaelite painters Arthur Hughes and William Holman Hunt, influencing and motivating him to his own creative works he quickly became known for in his social circles.

From early years on, Carroll was fascinated by poetry and even published his own writings often with satirical and humorous content. His fame first grew when his romantic poem ‘*Solitude*‘ was published in 1856 and in the same year his inspirations for ‘*Alice´s Adentures in Wonderland*‘ began after meeting Henry Liddell at Christ Church. His family became a great part of Carroll’s life and especially his daughter, Alice Liddell evolved a good friendship with the author. On a trip with Liddell’s children, Carroll began inventing and writing down a story he would name ‘*Alice´s Adventures Under Ground*‘. Even though he denied it later on, it is assumed that Alice’s character in the story is leaned on Liddell’s daughter. However, his story was published in 1865 and caused instant success to the author. But Alice’s stories were not to stay his only creative success, the poem ‘*The Hunting of the Snark*‘ was able to increase Carroll’s fame and his photographies gained popularity as well. During his photographer career, he was able to make portraits of Alfred Lord Tennyson,[6] Michael Faraday,[7] Julia Margaret Cameron, and many more.

Lewis Carroll continued his creative works as well as his work at the Christ Church almost until his death in 1898, at age 65. His success remained after his passing and he is remembered by the lovers of his works not only for Alice’s stories, but also for poems like ‘*Jabberwocky*‘, one of the greatest nonsense poems ever written:

‘Twas brillig, and the slithy toves Did gyre and gimble in the wabe; All mimsy were the borogoves, And the mome raths outgrabe.”Beware the Jabberwock, my son! The jaws that bite, the claws that catch! Beware the Jubjub bird, and shun The frumious Bandersnatch!”He took his vorpal sword in hand: Long time the manxome foe he sought— So rested he by the Tumtum tree, And stood awhile in thought.And as in uffish thought he stood, The Jabberwock, with eyes of flame, Came whiffling through the tulgey wood, And burbled as it came! |
One, two! One, two! and through and through The vorpal blade went snicker-snack! He left it dead, and with its head He went galumphing back.”And hast thou slain the Jabberwock? Come to my arms, my beamish boy! O frabjous day! Callooh! Callay!” He chortled in his joy.’Twas brillig, and the slithy toves Did gyre and gimble in the wabe; All mimsy were the borogoves, And the mome raths outgrabe.— Lewis Carroll, 1871 |

As a mathematician, Dodgson was rather conservative but certainly thorough and careful.[11] Within the academic discipline of mathematics, Dodgson worked primarily in the fields of geometry, linear and matrix algebra, mathematical logic, and recreational mathematics, producing nearly a dozen books under his real name [9]. Dodgson also developed new ideas in linear algebra (e.g., the first printed proof of the Kronecker-Capelli theorem), probability, and the study of elections (e.g., Dodgson’s method) and committees; some of this work was not published until well after his death. His occupation as Mathematical Lecturer at Christ Church gave him some financial security. His mathematical work attracted renewed interest in the late 20th century. Martin Gardner‘s book on logic machines and diagrams and William Warren Bartley‘s posthumous publication of the second part of Carroll’s symbolic logic book have sparked a reevaluation of Carroll’s contributions to symbolic logic. Robbins’ and Rumsey’s investigation of Dodgson condensation, a method of evaluating determinants, led them to the Alternating Sign Matrix conjecture, now a theorem. The discovery in the 1990s of additional ciphers that Carroll had constructed, in addition to his “*Memoria Technica*“, showed that he had employed sophisticated mathematical ideas in their creation.

At yovisto you may enjoy the video lecture ‘Carroll in Numberland‘ at Oxford University.

**References and Further Reading:**

- [1] Carroll at Victorian Web
- [2] Lewis Carroll Web Exhibition
- [3] The Contrawise Association
- [4] Works by Lewis Carroll at Project Gutenberg
- [5] Lewis Carroll at the British Library
- [6] Victorian Poetry with Alfred Lord Tennyson, SciHi Blog, August 6, 2012
- [7] A Life of Discoveries – the great Michael Faraday, SciHi Blog, September 22, 2012.
- [8] Lewis Carroll at Wikidata
- [9] Charles Lutwidge Dodgson at zbMATH
- [10] Charles Lutwidge Dodgson at Mathematics Genealogy Project
- [11] “Charles Lutwidge Dodgson“. The MacTutor History of Mathematics archive. Retrieved 8 March 2011.
- [12] Timeline for Charles Lutwidge Dodgson, aka Lewis Carroll, via Wikidata

The post Lewis Carroll – Mathematician and Creator of the Wonderland appeared first on SciHi Blog.

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