Paul Erdős ( ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century.  pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics.

Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathematicians. He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; Time magazine called him "The Oddball's Oddball". He devoted his waking hours to mathematics, even into his later years—indeed, his death came at a mathematics conference in Warsaw. Erdős's prolific output with co-authors prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships.

Life
Paul Erdős was born on 26 March 1913, in Budapest, Austria-Hungary, the only surviving child of Anna (née Wilhelm) and Lajos Erdős (né Engländer). His two sisters, aged three and five, both died of scarlet fever a few days before he was born. His parents, both Jewish, were high school mathematics teachers. His fascination with mathematics developed early—he was often left home by himself because his father was held captive in Siberia as an Austro-Hungarian prisoner of war during 1914–1920, causing his mother to have to work long hours to support their household. His father had taught himself English while in captivity, but mispronounced many words. When Lajos later taught his son to speak English, Paul acquired his unique pronunciations, which he continued to use for the rest of his life.

He taught himself to read through mathematics texts that his parents left around in their home. By the age of four, given a person's age, he could calculate in his head how many seconds they had lived. Due to his sisters' deaths, he had a close relationship with his mother, with the two of them reportedly sharing the same bed until he left for college.

When he was 16, his father introduced him to two subjects that would become lifetime favourites—infinite series and set theory. In high school, Erdős became an ardent solver of the problems that appeared each month in KöMaL, the "Mathematical and Physical Journal for Secondary Schools".

Erdős began studying at the University of Budapest when he was 17 after winning a national examination. At the time, admission of Jews to Hungarian universities was severely restricted under the numerus clausus. By the time he was 20, he had found a proof for Chebyshev's theorem. In 1934, at the age of 21, he was awarded a doctorate in mathematics. Erdős's thesis advisor was Lipót Fejér, who was also the thesis advisor for John von Neumann, George Pólya, and Paul (Pál) Turán. He took up a post-doctoral fellowship at Manchester, as Jews in Hungary were suffering oppression under the authoritarian regime. While there he met Godfrey Harold Hardy and Stan Ulam.

Because he was Jewish, Erdős decided Hungary was dangerous and left the country, relocating to the United States in 1938. Many members of Erdős's family, including two of his aunts, two of his uncles, and his father, died in Budapest during World War II. His mother was the only one that survived. He was living in America and working at the Princeton Institute for Advanced Study at the time. However, his fellowship at Princeton only got extended by 6 months rather than the expected year due to Erdős not conforming to the standards of the place; they found him "uncouth and unconventional".

Described by his biographer, Paul Hoffman, as "probably the most eccentric mathematician in the world," Erdős spent most of his adult life living out of a suitcase. Except for some years in the 1950s, when he was not allowed to enter the United States based on the accusation that he was a Communist sympathizer, his life was a continuous series of going from one meeting or seminar to another. During his visits, Erdős expected his hosts to lodge him, feed him, and do his laundry, along with anything else he needed, as well as arrange for him to get to his next destination.

Ulam left his post at the University of Wisconsin–Madison in 1943 to work on the Manhattan Project in Los Alamos, New Mexico with other mathematicians and physicists. He invited Erdős to join the project, but the invitation was withdrawn when Erdős expressed a desire to return to Hungary after the war.

On 20 September 1996, at the age of 83, he had a heart attack and died while attending a conference in Warsaw. These circumstances were close to the way he wanted to die. He once said,

Erdős never married and had no children. He is buried next to his mother and father in grave 17A-6-29 in the Jewish Kozma Street Cemetery in Budapest. For his epitaph, he suggested "I've finally stopped getting dumber." (Hungarian: "Végre nem butulok tovább").

Erdős's name contains the Hungarian letter "ő" ("o" with double acute accent), but is often incorrectly written as Erdos or Erdös either "by mistake or out of typographical necessity".

Career
In 1934, Erdős moved to Manchester, England, to be a guest lecturer. In 1938, he accepted his first American position as a scholarship holder at the Institute for Advanced Study, Princeton, New Jersey, for the next ten years. Despite outstanding papers with Mark Kac and Aurel Wintner on probabilistic number theory, Pál Turán in approximation theory, and Witold Hurewicz on dimension theory, his fellowship was not continued, and
Erdos was forced to take positions as a wandering scholar at the UPenn, Notre Dame, Purdue, Stanford, and Syracuse. He would not stay long in one place, instead traveling among mathematical institutions until his death.

As a result of the Red Scare and McCarthyism, in 1954, the Immigration and Naturalization Service denied Erdős, a Hungarian citizen, a re-entry visa into the United States. Teaching at the University of Notre Dame at the time, Erdős could have chosen to remain in the country. Instead, he packed up and left, albeit requesting reconsideration from the U.S. Immigration Services at periodic intervals. At some point he moved to live in Israel, and was given a position for three months at the Hebrew University in Jerusalem, and then a "permanent visiting professor" position at the Technion.

Hungary at the time was under the Warsaw Pact with the Soviet Union. Although Hungary limited the freedom of its own citizens to enter and exit the country, in 1956 it gave Erdős the exclusive privilege of being allowed to enter and exit the country as he pleased.

In 1963, the U.S. Immigration Service granted Erdős a visa, and he resumed teaching at and traveling to American institutions. Ten years later, in 1973, the 60-year-old Erdős voluntarily left Hungary.

During the last decades of his life, Erdős received at least fifteen honorary doctorates. He became a member of the scientific academies of eight countries, including the U.S. National Academy of Sciences and the UK Royal Society. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1977. Shortly before his death, he renounced his honorary degree from the University of Waterloo over what he considered to be unfair treatment of colleague Adrian Bondy.

Mathematical work
Erdős was one of the most prolific publishers of papers in mathematical history, comparable only with Leonhard Euler; Erdős published more papers, mostly in collaboration with other mathematicians, while Euler published more pages, mostly by himself. Erdős wrote around 1,525 mathematical articles in his lifetime, mostly with co-authors. He strongly believed in and practiced mathematics as a social activity, having 511 different collaborators in his lifetime.

In his mathematical style, Erdős was much more of a "problem solver" than a "theory developer" (see "The Two Cultures of Mathematics" by Timothy Gowers for an in-depth discussion of the two styles, and why problem solvers are perhaps less appreciated). Joel Spencer states that "his place in the 20th-century mathematical pantheon is a matter of some controversy because he resolutely concentrated on particular theorems and conjectures throughout his illustrious career." Erdős never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the Wolf Prize, "for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over". In contrast, the works of the three winners after were recognized as "outstanding", "classic", and "profound", and the three before as "fundamental" or "seminal".

Of his contributions, the development of Ramsey theory and the application of the probabilistic method especially stand out. Extremal combinatorics owes to him a whole approach, derived in part from the tradition of analytic number theory. Erdős found a proof for Bertrand's postulate which proved to be far neater than Chebyshev's original one. He also discovered the first elementary proof for the prime number theorem, along with Atle Selberg. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg. Erdős also contributed to fields in which he had little real interest, such as topology, where he is credited as the first person to give an example of a totally disconnected topological space that is not zero-dimensional, the Erdős space.

Erdős's problems

Erdős had a reputation for posing new problems as well as solving existing ones – Ernst Strauss called him "the absolute monarch of problem posers".  Throughout his career, Erdős would offer payments for solutions to unresolved problems. These ranged from $25 for problems that he felt were just out of the reach of the current mathematical thinking (both his and others) up to $10,000 for problems that were both difficult to attack and mathematically significant.  Some of these problems have since been solved, including the most lucrative – Erdős's conjecture on prime gaps was solved in 2014, and the $10,000 paid.

There are thought to be at least a thousand remaining unsolved problems, though there is no official or comprehensive list. The offers remained active despite Erdős's death; Ronald Graham was the (informal) administrator of solutions, and a solver could receive either an original check signed by Erdős before his death (for memento only, cannot be cashed) or a cashable check from Graham.

Perhaps the most mathematically notable of these problems is the Erdős conjecture on arithmetic progressions:

If true, it would solve several other open problems in number theory (although one main implication of the conjecture, that the prime numbers contain arbitrarily long arithmetic progressions, has since been proved independently as the Green–Tao theorem). The payment for the solution of the problem is currently worth US$5,000.

The most familiar problem with an Erdős prize is likely the Collatz conjecture, also called the 3N + 1 problem. Erdős offered $500 for a solution.

Collaborators
Erdős' most frequent collaborators include Hungarian mathematicians András Sárközy (62 papers) and András Hajnal (56 papers), and American mathematician Ralph Faudree (50 papers). Other frequent collaborators were the following:

 Richard Schelp (42 papers)
 C. C. Rousseau (35 papers)
 Vera Sós (35 papers)
 Alfréd Rényi (32 papers)
 Pál Turán (30 papers)
 Endre Szemerédi (29 papers)
 Ron Graham (28 papers)
 Stefan Burr (27 papers)
 Carl Pomerance (23 papers)
 Joel Spencer (23 papers)
 János Pach (21 papers)
 Miklós Simonovits (21 papers)
 Ernst G. Straus (20 papers)
 Melvyn B. Nathanson (19 papers)
 Jean-Louis Nicolas (19 papers)
 Richard Rado (18 papers)
 Béla Bollobás (18 papers)
 Eric Charles Milner (15 papers)
 András Gyárfás (15 papers)
 John Selfridge (14 papers)
 Fan Chung (14 papers)
 Richard R. Hall (14 papers)
 George Piranian (14 papers)
 István Joó (12 papers)
 Zsolt Tuza (12 papers)
 A. R. Reddy (11 papers)
 Vojtěch Rödl (11 papers)
 Pál Révész (10 papers)
 Zoltán Füredi (10 papers)

For other co-authors of Erdős, see the list of people with Erdős number 1 in List of people by Erdős number.

Erdős number

Because of his prolific output, friends created the Erdős number as a tribute. An Erdős number describes a person's degree of separation from Erdős himself, based on their collaboration with him, or with another who has their own Erdős number. Erdős alone was assigned the Erdős number of 0 (for being himself), while his immediate collaborators could claim an Erdős number of 1, their collaborators have Erdős number at most 2, and so on. Approximately 200,000 mathematicians have an assigned Erdős number, and some have estimated that 90 percent of the world's active mathematicians have an Erdős number smaller than 8 (not surprising in light of the small-world phenomenon). Due to collaborations with mathematicians, many scientists in fields such as physics, engineering, biology, and economics also have Erdős numbers.

Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. For example, the roughly 268,000 mathematicians with a known Erdős number have a median value of 5. In contrast, the median Erdős number of Fields Medalists is 3. As of 2015, approximately 11,000 mathematicians have an Erdős number of 2 or less. Collaboration distances will necessarily increase over long time scales, as mathematicians with low Erdős numbers die and become unavailable for collaboration. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.

The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 2. Goffman published his observations about Erdős's prolific collaboration in a 1969 article titled "And what is your Erdős number?"

Jerry Grossman has written that it could be argued that Baseball Hall of Famer Hank Aaron can be considered to have an Erdős number of 1 because they both autographed the same baseball (for Carl Pomerance) when Emory University awarded them honorary degrees on the same day. Erdős numbers have also been proposed for an infant, a horse, and several actors.

Personality

Possessions meant little to Erdős; most of his belongings would fit in a suitcase, as dictated by his itinerant lifestyle. Awards and other earnings were generally donated to people in need and various worthy causes. He spent most of his life traveling between scientific conferences, universities and the homes of colleagues all over the world. He earned enough in stipends from universities as a guest lecturer, and from various mathematical awards, to fund his travels and basic needs; money left over he used to fund cash prizes for proofs of "Erdős's problems" (see above). He would typically show up at a colleague's doorstep and announce "my brain is open", staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the current collaborator about whom to visit next.

His colleague Alfréd Rényi said, "a mathematician is a machine for turning coffee into theorems", and Erdős drank copious quantities; this quotation is often attributed incorrectly to Erdős, but Erdős himself ascribed it to Rényi. After his mother's death in 1971 he started taking antidepressants and amphetamines, despite the concern of his friends, one of whom (Ron Graham) bet him $500 that he could not stop taking them for a month. Erdős won the bet, but complained that it impacted his performance: "You've showed me I'm not an addict. But I didn't get any work done. I'd get up in the morning and stare at a blank piece of paper. I'd have no ideas, just like an ordinary person. You've set mathematics back a month." After he won the bet, he promptly resumed his use of Ritalin and Benzedrine.

He had his own idiosyncratic vocabulary; although an agnostic atheist, he spoke of "The Book", a visualization of a book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, "You don't have to believe in God, but you should believe in The Book." He himself doubted the existence of God, whom he called the "Supreme Fascist" (SF). He accused SF of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, "This one's from The Book!" This later inspired a book titled Proofs from the Book.

Other idiosyncratic elements of Erdős's vocabulary include:
 Children were referred to as "epsilons" (because in mathematics, particularly calculus, an arbitrarily small positive quantity is commonly denoted by the Greek letter (ε)).
 Women were "bosses" who "captured" men as "slaves" by marrying them. Divorced men were "liberated".
 People who stopped doing mathematics had "died", while people who died had "left".
 Alcoholic drinks were "poison".
 Music (except classical music) was "noise".
 To be considered a hack was to be a "Newton".
 To give a mathematical lecture was "to preach".
Mathematical lectures themselves were "sermons".
 To give an oral exam to students was "to torture" them.

He gave nicknames to many countries, examples being: the U.S. was "samland" (after Uncle Sam) and the Soviet Union was "joedom" (after Joseph Stalin).  He claimed that Hindi was the best language because words for old age (bud̩d̩hā) and stupidity (buddhū) sounded almost the same.

Signature
Erdős signed his name "Paul Erdos P.G.O.M." When he became 60, he added "L.D.", at 65 "A.D.", at 70 "L.D." (again), and at 75 "C.D."
 P.G.O.M. represented "Poor Great Old Man"
 The first L.D. represented "Living Dead"
 A.D. represented "Archaeological Discovery"
 The second L.D. represented "Legally Dead"
 C.D. represented "Counts Dead"

Legacy

Books and films
Erdős is the subject of at least three books: two biographies (Hoffman's The Man Who Loved Only Numbers and Schechter's My Brain is Open, both published in 1998) and a 2013 children's picture book by Deborah Heiligman (The Boy Who Loved Math: The Improbable Life of Paul Erdős).

He is also the subject of George Csicsery's biographical documentary film N is a Number: A Portrait of Paul Erdős, made while he was still alive.

Astronomy
In 2021 the minor planet (asteroid) 405571 (temporarily designated 2005 QE87) was formally named "Erdőspál" to commemorate Erdős, with the citation describing him as "a Hungarian mathematician, much of whose work centered around discrete mathematics. His work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics." The naming was proposed by "K. Sárneczky, Z. Kuli" (Kuli being the asteroid's discoverer).

See also
  – including conjectures, numbers, prizes, and theorems

References

Sources

Further reading

External links

 Erdős's Google Scholar profile
 Searchable collection of (almost) all papers of Erdős

 Jerry Grossman at Oakland University. The Erdös Number Project
 The Man Who Loved Only Numbers – Royal Society public lecture by Paul Hoffman (video)
 Radiolab: Numbers, with a story on Paul Erdős
 Fan Chung, "Open problems of Paul Erdős in graph theory"

1913 births
1996 deaths
20th-century Hungarian mathematicians
Mental calculators
Hungarian agnostics
Hungarian atheists
Jewish atheists
Jewish agnostics
Hungarian Jews
Combinatorialists
Graph theorists
Set theorists
Number theorists
Network scientists
Probability theorists
Mathematicians from Austria-Hungary
Mathematicians from Budapest
Foreign Members of the Royal Society
Members of the Hungarian Academy of Sciences
Members of the Royal Netherlands Academy of Arts and Sciences
Eötvös Loránd University alumni
Institute for Advanced Study visiting scholars
Academics of the Victoria University of Manchester
Stanford University faculty
Syracuse University faculty
University of Pennsylvania faculty
Mathematicians at the University of Pennsylvania
University of Notre Dame faculty
Purdue University faculty
Princeton University faculty
Foreign associates of the National Academy of Sciences
Wolf Prize in Mathematics laureates

Paul Erdős


The Great Festival of Fujisaki Hachimangu Shrine is a festival of Fujisaki-hachimangu at Chūō-ku, Kumamoto every September, characterized by a parade of Shinto priests, followed by groups of followers who chase their horses shouting, "Boshita, Boshita", in earlier times; but now the parade followers, "Dookai Dookai", or other phrases. Recently 17,000 people participated in this festival.

Name
The formal name of this festival is 藤崎八旛宮秋季例大祭, or Fujisaki-hachimangu, autumn great festival. Previously it was called Boshita Matsuri in Japanese, though it is not used officially, because of a view that this comes from "Horoboshita," or Japan destroyed Korea, although Korea was not in the shouting phrase. However, there are other views, as described below in the section entitled Controversy.

The origin of Boshita is not known; decorated horses were not employed, and there was no horse chasing at the time of Katō Kiyomasa. A view states that it had been used since the First Sino-Japanese War. The use of Boshita was claimed at the time of world exhibition at Osaka in 1970, since Japanese of Korean origin felt miserable at the time of the festival. New shouting phrases have been up to the groups which participate; it was decided upon in August 1990.

Parade of Shinto priests
The climax of the autumn great festival of Fujisaki-hachimangu comes on the last day of the festival. It consists of the parade of portable shrines, followed by a parade of samurai re-enactors, and a parade of decorated horses, chased by many followers.

History
This festival originated in Ho-jo-e (the ritual for releasing living beings), a festival of letting wild animals go into the fields, a Buddhist custom, but which was introduced to the Shinto shrines and of which, Ho-jo-e of Iwashimizu Hachiman-gū, Kyoto is well known. The Kami of Fujisaki Hachimangu came from Iwashimizu Hachiman-gū. However, no trace of Ho-jo-e can be found in the present festival.

The Zuibyo, or parade of following samurai, certainly originated from the returning samurai of Katō Kiyomasa from Korea during the Japanese invasions of Korea (1592-1598). Katō Kiyomasa thanked the Kami of the Fujisaki-hachimangu for the safe return by leading the parade of his followers. This custom continued into the Hosokawa clan period, with three important samurai figures: head of followers, head of spears, parade head.

The decorated horses were originally those for Shinto priests (kannushi), twelve in number, and in the Hosokawa clan period, only upper-class samurai families presented horses. The distance from the shrine to the otabisho (destination of the parade) was short, and kannushi did not ride on the horses. Therefore, the decorations on the horses became gradually gala and greater. The decorations on the horses were originally from the symbols of sexual organs. There had been two schools, but only Ando school has been observed. After the Meiji Restoration, decorated horses were prepared by town people, and recently the decorated horses number about 60. In 2007, the horses numbered 67, and following people (seko) numbered 17,000, indicating that this is the biggest festival of Kumamoto city.

Number of horse chasing groups

The Festival
The festival continues for 5 days. On the first day, the head of Fujisakigu believers prays, and there are ceremonies of lion dance dress purification, musical instrument purification and purification of various instruments.

On the next day, there are a tea dedication ceremony and a haiku dedication ceremony. On the third day, Kenpei Sai (a divine ceremony), dedication of Japanese traditional fighting matches such as fencing, and dedication of traditional Japanese dancing. On the fourth day, purification and decorations of horses, dedication of flower arrangement and travelling portable shrines.

On the fifth day is the parade, headed by kannushi, starting at 6 a.m. (starting ceremony), three portable shrines, parades of followers, lion dances, portable shrines carried by children, and finally the groups of decorated horses. Decorated horse group people are dressed in uniforms of their own, dancing with folding fans, drums, trumpets, shouting "Dookai Dookai" (meaning "how about this?"),  chasing the decorated horse of their group. Some horses run violently and sometimes injure people nearby. These groups are from town groups, companies, and graduates of schools and other groups.

The order of the parade is determined by drawing lots. Exceptions are the top three groups which exist near the shrine; which must do preparations and cleanup after the festival.

Controversy

Korean language
The phrase "Boshita Boshita" originated in the Korean language.  A book entitled History of Kumamoto city, published in 1932, wrote that "Ehekoroboshita" was used as the shouting phrase which came from the Korean language. A Korean association in Kumamoto said that it might mean a great man (Toyotomi Hideyoshi) died.

Erotic origin
The Boshita Boshita was short of Boboshita (did sex) view. In a document by a kannushi of the shrine written in 1870, "Boboshita boboshita was the phrase of horse chasing". Another document written in 1865 "Boboshita Boboshita(did sex) was the phrase of horse chasing".

Japan destroyed Korea
The latter document added that it might be a mistake of Horoboshita. In a letter by Lafcadio Hearn, he wrote that he heard the shouting phrase Korea was destroyed many times, and in a newspaper during World War two that the Japanese people should understand that Japan destroyed Korea, apparently to whip up war sentiment.

1970 Osaka World Exposition
Kumamoto City was going to exhibit a Boshita parade at the 1970 Osaka World Exposition but criticism arose concerning the name of the Boshita festival, since it may mean Japan destroyed Korea. The Osaka World Exposition rejected the Boshita parade. Since then, the problem of the shouting phrase of Boshita had been under discussion.

1989 dispute
This problem was again under fire since the Yomiuri newspaper introduced the Boshita festival in a Korean language book in December 1989. In August, 1990, the Fujisaki shrine audit committee published that it would rate a minus point if a group shouted Boshita. This was reported substantially stopping the shouting of Boshita.

Present status
Many Kumamoto people use of the word Boshita Festival even today, but they do not use "Boshita Boshita" at horse chasing. Some miss the word Boshita even today.

See also
History of Kumamoto Prefecture

References
Higogaku Koza, Fujisaki Festival now and olden times Iwashita C. p 92-105, Kumamoto NichiNichi Shinbun, 2006,
Festivals and modern sociology in religion Tetsuro Ashida, Sekai Shisousha, , 2001, p 64-111

Footnotes

External links
Horse festival
horse festival
autumn festival of Fujisaki Hachiman
Fujisaki Hachimangu Homepage

Kumamoto
Tourist attractions in Kumamoto Prefecture
Shinto festivals
Cultural festivals in Japan
Religious festivals in Japan
Parades in Japan
Festivals established in the 16th century
Autumn events in Japan
1592 establishments in Asia


Vincenzo Grella (born 5 October 1979) is an Australian former footballer who played as a midfielder. Born in Dandenong to Italian immigrants, Antonio and Maria, Grella began his senior career in Australia before moving to Italy, where he spent over ten years, playing for Empoli, Ternana, Parma and Torino. He moved to Blackburn Rovers of the Premier League in 2008 where he spent four seasons, before returning to Australia in 2012 to play for Melbourne Heart.

He retired from professional football in January 2013, after a long-standing struggle with injuries. Grella represented the Australian national team on 46 occasions, and played at the 2006 and 2010 World Cups. His 2010 FIFA World Cup profile describes him as a possessing the "ability to mop up loose balls and halt opposition counter-attacks."

Club career

Italy
He had a ten-year stint in Italy's professional leagues after beginning his career in the NSL. He has been a regular for Australia since 2003, playing in his customary defensive midfield position, and was involved in both the 2006 World Cup and the 2007 Asian Cup.
Grella's friendship with Australian teammate Mark Bresciano is well known. Grella has described Bresciano as "even more than a brother"; they played together at the AIS, Carlton and Empoli, and were reunited in 2004 when Grella joined Parma, before Bresciano moved on to Palermo in 2006. Aged 19, Grella made his debut in Serie A with Empoli, in a match against Juventus at the Stadio Delle Alpi, where his first job was to mark Zinedine Zidane. (Empoli held Juventus to a 0–0 draw). Grella built up his reputation in Italy during his time with Empoli, to overcome initial doubts about his suitability to Serie A and earn the respect of the Italian media, coaches and fellow players. Grella was widely linked with a move to A.C. Milan in 2003, although the proposed transfer never eventuated, and he moved to Parma a year later. While at Parma, both Grella and Bresciano were excused from the 2005 Confederations Cup by the then-Australian manager Frank Farina to allow the pair to take part in a relegation playoff between Parma and Bologna–which Parma won, to avoid relegation.

Grella was named as captain of Parma for the 2006–07 season, and played a key role helping the team avoid relegation from Serie A for a second consecutive season. On 20 June 2007, Grella signed a three-year contract with Torino, one of several new signings made by The Bull. The move to Torino was a free transfer due to the expiry of his contract with Parma. During his one season in Turin, he was a regular member of the side, although his season was interrupted by injury on several occasions. He scored one league goal for Torino, a powerful volley from the edge of the penalty area, which cannoned in off the far post to draw level with Fiorentina. The goal was a candidate for goal of the month for January in Serie A. It was only the sixth league-goal of his career.  From July 2022 he assumes the position of vice president and CEO of the Catania SSD owned by Ross Pelligra Australian of Sicilian origins

Blackburn Rovers
On 26 August 2008, Grella joined English then Premier League outfit Blackburn Rovers on a four-year deal, for an undisclosed fee, rumoured to be around £4m (with the initial fee later revealed as €4.2m), having previously been linked to fellow Premiership club Fulham. He joined fellow Australian Brett Emerton at the club. He made his first appearance for the club in a 4–1 loss to West Ham United at Upton Park, where he played the first 45 minutes, picking up a yellow card and providing an assist with a header for the only Blackburn goal, before being substituted at half-time for Keith Andrews. A calf injury kept him out of the side for a month following his debut, and he returned to the side in a 1–1 draw with Middlesbrough at Ewood Park, playing the full game in the centre of midfield. This was his home debut, and he was later named man-of-the-match for his performance.

The following 3 years he spent at the club were a frustrating time for Grella, and Blackburn Rovers fans, with injuries disrupting his stay at Rovers. A succession of knee, groin and calf problems would ultimately rule him out for vast majority of his contract. After the 2010–11 season, Blackburn Rovers revealed their desire to transfer Grella. However, with his history of injuries, no offers were made, and he left Blackburn Rovers upon the expiry of his contract at the end of the 2011/12 Barclays Premier League season, a season in which he only made 2 appearances. Over the last 2 years of his contract, Grella was only able to make a total of 9 appearances due to injuries.

Melbourne City
On 16 October 2012, he signed with Australian A-League club Melbourne City on a one-year deal. Previously, Grella stated he announced his intention to return to Italy next season, rather go back to his home country. But in the unexpected event, Grella says former national teammate and now manager of Melbourne City, John Aloisi convinced Grella to play football in Australia. Two months after the move, continuously having injuries, Aloisi revealed he's making a comeback from injury and just returned to training.

Vince Grella made his A-League debut for the Melbourne City on Australia Day, 26 January 2013 against the Western Sydney Wanderers at the Parramatta Stadium as a 68th minute replacement for Jonatan Germano. He officially announced his retirement two days later due to a torn calf that he picked up in his one match for Melbourne City. In an interview with the Australian FourFourTwo Magazine, Grella spoke out his retirement and said:

International career
Grella was a regular member of the Socceroos; he represented the Australian Under-23s at the 2000 Summer Olympics and received his first senior cap when he came on as a second-half substitute in the international friendly against England in 2003, which Australia won 3–1 at West Ham's Upton Park stadium. In November 2005, Grella played in both legs of the World Cup play-off against Uruguay, where Australia qualified for the 2006 World Cup for the first time in 32 years. At the World Cup, Grella's displays in the heart of Australia's midfield saw him named in a preliminary squad for FIFA's team of the tournament, the only Australian to make this list.

On 13 November 2006, Grella was selected as captain for an international friendly in London against Ghana, making him the 51st player to captain the Australia football team. On 21 July 2007 in Hanoi, Vietnam, Grella was sent off in the 76th minute of Australia's 2007 Asian Cup quarter-final against Japan. Leaping for a header, Grella's forearm connected with Naohiro Takahara's head, who had come from behind Grella, sending the latter to the ground. The game was 1–1 at the time and remained that way until the end of extra time, with Japan eliminating Australia from the tournament in a penalty shoot-out. He has currently amassed 46 caps for Australia since making his debut back in February 2003.

Post-playing career
In July 2022, Grella moved back to Italy, after being named vice-president and CEO of Sicilian Serie D club Catania following the club's rebirth under an Australian ownership.

Career statistics

Club

International
Source:

Personal life
Grella and his wife, Barbara, have twin daughters, Victoria and Sophia, born in 2002. Grella first met Barbara while she was working in a shoe store in Empoli, Italy, which Grella frequently visited to buy shoes following their first meeting.

References

External links

 Blackburn profile
 OzFootball profile
 Article on SMH
 Article on The Age
 Migration Heritage Centre feature on Grella family

1979 births
Living people
Soccer players from Melbourne
Australian people of Italian descent
Sportspeople of Italian descent
Men's association football midfielders
Australian men's soccer players
Premier League players
Serie A players
Serie B players
Springvale White Eagles FC players
Blackburn Rovers F.C. players
Carlton SC players
Empoli FC players
Parma Calcio 1913 players
Ternana Calcio players
Torino FC players
National Soccer League (Australia) players
Australia men's international soccer players
Australian expatriate men's soccer players
Expatriate men's footballers in England
Expatriate men's footballers in Italy
Australian expatriate sportspeople in England
Australian expatriate sportspeople in Italy
Olympic soccer players for Australia
Footballers at the 2000 Summer Olympics
2004 OFC Nations Cup players
2006 FIFA World Cup players
2007 AFC Asian Cup players
2010 FIFA World Cup players
Australian Institute of Sport soccer players
A-League Men players
Melbourne City FC players
People from Dandenong, Victoria

Vince Grella


Brewster & Company was an American custom carriage and motor coachbuilder, James Brewster established the company in 1810; it operated for approximately 130 years. Brewster got its start in New Haven, Connecticut, and quickly gained a reputation for producing the best carriages in the country. In 1827, he set his shop at 52 Broad Street in New York City.

The earliest known automotive bodywork done by Brewster was on an electric vehicle in 1896, followed by a Delaunay-Belleville chassis with a gasoline engine in 1905. They eventually constructed bodywork on a variety of chassis, winning a special partnership with Rolls-Royce America Inc. in Springfield, Massachusetts

Brewster built a series of elegant and pricey cars at their Long Island City facility between 1915 and 1925. In 1929, the Great Depression started, and sales of high-end vehicles decreased. In 1934–35, they built and sold luxury bodies on 135 Ford V8 chassis, but bankruptcy proceedings began in mid-1935 and the last of Brewster's assets were sold by auction in 1937.

Carriage building

James Brewster (b.1788–d. 1866) of Connecticut apprenticed as a mechanic with carriage builder Colonel Charles Chapman from 1804–1809. He then worked as a journeyman for John Cook and married Cook's niece. Around the War of 1812, lightweight four-wheeled vehicles were introduced and Brewster made hundreds of 'Brewster Wagons'. In 1821 he purchased his employer's New Haven business, and his reputation as a carriage builder spread to the larger cities of eastern coastal US with exports to Cuba and South America.

In 1827, Brewster opened shop in New York City and purchased a local factory, travelling back and forth between New York and New Haven. Unable to keep up with the demand for his carriages, Brewster started purchasing vehicles from J. Cook & Sons as well as opened a new factory, which burned down in 1836. After the Panic of 1837 financial crisis, James Brewster sold his Connecticut and New York businesses to his business partners.

In 1838, he returned to the carriage business, and over the next few decades the Brewster name was involved in numerous carriage building companies with his sons James Benjamin and Henry, grandsons, and various partners in both New Haven and New York. His grandson William Brewster adopted the slogan, "Carriage Builder for the American Gentleman". Some of the many Brewster company names of the coachbuilding era include James Brewster & Son/Sons/Co., James B. Brewster & Co., J.B. Brewster & Co., Brewster & Collis, Brewster & Lawrence, and Brewster & Baldwin.

The Brewsters won many awards for their coachbuilding including the gold medal for best exhibit at the 1878 International Exposition in Paris and the Chevalier of the Legion of Honor presented by the president of France. More honors were received at the 1893 Columbian Exposition in Chicago.

Ignoring the emergent automobile industry, the firm's carriage trade failed in the early 1900s—their lawyers stating "the failure of the firm was due to the firm's inability to collect accounts, inability to dispose of high-grade carriages and vehicles in the current market without great sacrifice, and a gradual falling off of business due to the competition of the automobile industry."

Automobile bodies

In 1905, Brewster became importers for Delaunay-Belleville, one of many desirable French brands of the time. This was their first venture into automobile body building, beginning their history of providing coachwork for prestigious autos. In 1914, most Brewster coachwork sales were on Delaunay-Bellevilles and other French makes. In 1914 Brewster was carefully selected to be sales agents for Rolls-Royce Limited and they became the main body suppliers for Rolls-Royce in the US.

By 1925, Brewster's car had few sales, trading with Europe had resumed, and Rolls-Royce America Inc was expanding and gaining bargaining power against Brewster. Executives from Rolls-Royce of America and Brewster met and decided on the purchase of Brewster & Co. and their debt. Brewster had chassis fitted with temporary seats and protection and driven from Rolls-Royce's Massachusetts plant to the Brewster Building in Long Island City, New York for bodies. The Rolls-Royce showrooms offered 28 standardized body styles so as to deliver cars to customers quicker and for a lower price. Customers could purchase models directly from the showroom as well. Brewster gave English city names to the coachwork choices manufactured in America, to include the Derby Touring Sedan, the York Roadster, the Huntington Limousine, the Avon Sedan, Newmarket Convertible Sedan, and the French city of Trouville Town Car.

After Rolls-Royce America Inc. folded, from 1931 to 1934 Rolls-Royce Phantom II chassis were shipped directly from Britain to Brewster's large facility in Long Island City.

cars made by Brewster

Most bodywork were ordered to fit the imported chassis of the customers. Supplies were in danger once Europe entered the war in the summer of 1914. Following the 1915 shipwreck of the British liner Lusitania, Brewster started producing its own automobiles, a practice it kept up until 1925. They cost as much as "a Packard Twin Six limousine plus a fleet of five Model T Ford roadsters," while being smaller than their typical chassis for traversing Manhattan's streets. Brewster's personal cars had oval radiators and gleaming patent leather fenders, making them easy to identify. They were frequently equipped with Brewster's distinctive two-piece folding windscreen and were propelled by four-cylinder Knight engines with sleeve valves.

On a Ford V8 chassis, the 1934 Brewster Town Car was mounted.

When the Great Depression first began, there was strong sentiment against the wealthy and their archetypal Brewster-bodied Rolls-Royces and Brewster's bodies were not selling well. In 1934 sales chief J.S. Inskip, who had taken control of operations in the hope of saving Brewster, bought 135 Ford V8 roadster chassis for model year 1934 and designed a body for them easily identified by its swoopy fenders and a heart-shaped grille. Stylish and sold for US$3,500 ($ in  dollars ), it was a hit at the 1934 New York Auto Show. The bodies were worth more than the chassis. These cars were branded Brewster and sold at Rolls-Royce showrooms. Inskip marketed the cars to New York celebrities (see Notable Owners), with whom it became popular. Rolls-Royce Limited under English management strongly objected to coachwork being installed on Ford chassis, which lead to Rolls-Royce of America, Inc. changing their name to Springfield Manufacturing Corporation and renegotiating its contract with the British firm to continue importing British-made products into America. The Lincoln K series chassis installed with a V12 for 1934 did list the Brewster Non Collapsible Cabriolet with a 145" wheelbase but the coachwork choice did not continue for 1935.

Ford Brewster's initiative was originally profitable, but soon Brewster started to lose money, and the company's bondholders and directors asked that it be shut down. Dallas E. Winslow bought the company that was the subject of bankruptcy proceedings in July 1935.

Liquidation
The last of Brewster's possessions were auctioned off in public on August 18, 1937. Under the leadership of previous sales director John Inskip, the Rolls-Royce dealership and body shop operated on the same location as before. During World War II, the Brewster Aeronautical Corporation remained in business.

Fame

In 1956, Colonel Paul Downing penned the following for American Heritage Magazine: "It is unlikely, though, that America truly cemented its position in the world of genuinely stylish carriages prior to the company of Brewster & Company of Broome Street taking the initiative. The adage that a new design lacked value unless Brewster proved it became common knowledge in the industry."

Queens' Brewster & Co. building
Brewster maintained track of each customer's family crest and color. The Vanderbilts' was a shade of maroon, the Astors' was a shade of blue, J. P. Morgan's was a shade of dark green. New clients occasionally found it challenging to select a body color because of these exclusive choices.
Brewster created a top-secret oil-based finish that required far less upkeep than the varnishes of the day. Other businesses attempted to copy it but failed.
After conducting extensive investigation, Brewster created a windscreen with a four-pane design in response to drivers' complaints about blinding street lights at night. Even though it wasn't copyrighted, the "Brewster windshield" was widely imitated by body shops and production cars.
Brewster also produces the hulls for speedboats.
Children's pony carts and buses that can accommodate 20 or more passengers were also manufactured by Brewster.

Alumni
Brewster was the birthplace of many engineers and designers in the automotive industry. James Way initially worked at Brewster before becoming the designer and engineer of Pierce-cast-aluminum Arrow's bodies from 1904 to 1920. Edsel Ford, a Brewster native, convinced Henry Crecelius Sr., the head of Lincoln's coachbuilding section, to join the company. Brewster hired Raymond Dietrich as a draftsman beforee being fired for secret designing for other makes. Harry Lonschein founded Rollston after starting out at Brewster.

Notable Owners

Louis Comfort Tiffany - Brewster's second gasoline auto chassis, a Panhard et Levassor
John D. Rockefeller Sr. - His Crane-Simplex had two Brewster bodies, for summer and winter.
John D. Rockefeller Jr.
Frank Winfield Woolworth
Vincent Astor

Brewster V8 owners

Edsel Ford - Purchased the first Brewster-bodied Ford V8 available
Al Jolson
Cole Porter
Lily Pons
Gertrude Lawrence
Fred Waring
Victor Moore

Edsel Ford
The 1935 Ford grill was used to create 15 of the vehicles using a Ford V8 chassis. The first delivered example was purchased by Edsel Ford. A 1934 Brewster Town Cabriolet DeVille (chassis number 18-802233; engine number 49493; Brewster build number 9002), a "one off" custom with a lengthened 127-inch wheelbase, was the third Ford Brewster and the only one without the standard Brewster front end. Edsel requested a 1934 Ford grill, therefore that is what was installed. It is also the sole model produced with a banjo steering wheel, normal Ford dash rather than the Brewster dash, and 16-inch wheels rather than the standard 17-inch wheels. It was stored by Edsel Ford at a Ford dealer in New York so that he and his family may use it while in New York. In 1939, he had it retrofitted with a 239 Mercury flathead V8 engine that produced nearly 100 horsepower. It is one of Edsel Ford's few personal cars and still survives today in remarkable condition, unrestored. It was presented by RM Auctions at Automobiles of Amelia in 2008 where it sold for $198,000.

The Brewster-bodied Ford chassis Town Car with heart-shaped grill is the only classic Ford designated by the Classic Car Club of America.

References

External links

 Vanderbilt family Park Drag 1892

Coachbuilders of the United States
Defunct motor vehicle manufacturers of the United States
Cars powered by Knight engines
Luxury motor vehicle manufacturers
American companies established in 1810
Manufacturing companies established in 1810
1810 establishments in New York (state)
Defunct manufacturing companies based in New York City

Brewster & Co.


Corrida de carros esporte é um tipo de competição de automobilismo na qual participam carros esporte que tenham dois lugares e rodas cobertas. Eles podem ser baseados em carros de produção ou ser construídos especificamente para competição.

São muitas as variantes de modelos de carros esporte nesse tipo de competição. Os que têm obtido maior repercussão e popularidade são os do tipo GT (gran turismo), e seus subgrupos: GT1, GT2, GT3 e GT4.

Ver também
 Campeonato Mundial de Endurance da FIA
 Campeonato Mundial de Resistência
 Corrida de carros de turismo
 Automobilismo

Bibliografia
 Denis Jenkinson, "Automobile Year Book Of Sports Car Racing" (photographic history of sports car racing from the early 1950s to the 1970s)
 János Wimpffen, "Time and Two Seats" – 2 vols. Extensive history of World Championship sports car racing from 1952 to the late 1990s.
 János Wimpffen, "Open Roads And Front Engines" – a photographic companion to the above, covering the early 50s-early 60s.
 János Wimpffen, "Winged Sports Cars and Enduring Innovation" – a sequel to the above covering the early 60s-early 70s.
 János Wimpffen, "Spyders and Silhouettes" – a sequel to the above covering the early 70s-early 80s.
 John Wyer, "The Certain Sound" – memoirs of Aston Martin and Ford GT40 team manager
 Chris Nixon, "Racing With The David Brown Aston Martins", 2 vols.
 Anthony Pritchard, "Sports Racing Cars" – profiles of 25 sports racers through history.
 Brooklands Books, "Le Mans" – 5 volumes of contemporary race reports
 Brooklands Books, "Mille Miglia" – 2 volumes of contemporary race reports
 Brooklands Books, "Targa Florio" – 5 volumes of contemporary race reports
 Brooklands Books, "Carrera Panamericana" – 1 volume of contemporary race reports
 Ian Briggs, "Endurance Racing 1982–1991" – the Group C and IMSA GTP years, race by race.
 Michael Cotton, "Directory of World Sports Cars" – IMSA and GpC car histories outlined in detail.
 Andrew Whyte, "Jaguar: Sports Racing and Works Competition Cars" – 2 vols. Authoritative history of the marque.
 Ian Bamsey, ed. "Super Sports: The  Le Mans Cars" – technical summary of large-capacity coupés.
 Chris Nixon – "Sports Car Heaven" – Aston Martin vs Ferrari
 Karl Ludvigsen – "Quicksilver Century" – competition history of Mercedes-Benz
 Karl Ludvigsen – "Porsche: Excellence Was Expected" (3 vols) – extensive history of Porsche
 Vic Elford, "Reflections on a Golden Era of Motorsport" – covers Vic's rallying, single seater and mostly sports car career in depth.
 Norbert Singer, "24:16" – his role in Porsche's Le Mans wins
 John Horsman, "Racing In The Rain", an account of his engineering career with Aston Martin, John Wyer and Mirage.
 Curami/Vergnano, "'La Sport' e i suoi artigiani" – Italian domestic sports car competition from the 1930s–1960s and the 'specials' that competed in it.
 J. A. Martin & Ken Wells, "Prototypes: The History of the IMSA GTP Series" – team by team account of various racing teams and manufacturers that competed in the top flight IMSA series.
 Mike Fuller & J. A. Martin, "Inside IMSA's Legendary GTP Race Cars: The Prototype Experience", ISBN 0-7603-3069-7, Motorbooks International, 25 April 2008. Technical and historical overview of IMSA GTP racers

Ligações externas
 dailysportscar.com

Automóveis desportivos
Competições de automobilismo

Corrida com carros esportivos


Sir Douglas William Gretton Wass (15 April 1923 – 4 January 2017) was a British civil servant. He served as Permanent Secretary to HM Treasury from 1974 to 1983. Wass was Permanent Secretary to HM Treasury from 1974 to 1983 and served as joint head of the civil service in 1981.

He was appointed a Companion of the Order of the Bath (CB) in 1971, elevated to Knight Commander (KCB) in 1975 and further elevated to Knight Grand Cross (GCB) in 1980. In 1985, he was awarded an Honorary Degree (Doctor of Letters) by the University of Bath.

In 1983, the BBC invited Sir Douglas to present the annual Reith Lectures. In a series of six lectures titled Government and the Governed, he examined the role and responsibilities of government.

Wass died on 4 January 2017 in London at the age of 93.

References

1923 births
2017 deaths
British civil servants
People from Nottingham

Douglas Wass


The Imperial British East Africa Company (IBEAC) was a commercial association founded to develop African trade in the areas controlled by the British Empire. The company was incorporated in London on 18 April 1888 and granted a royal charter by Queen Victoria on 6 September 1888. It was led by William Mackinnon and built upon his company's trading activities in the region, with the encouragement of the British government through the granting of an imperial charter, although it remained unclear what that actually meant.

The IBEAC oversaw an area of about  along the eastern coast of Africa (from modern-day Somalia to modern-day Kenya), its centre being at about 39° East longitude and 0° latitude. Mombasa and its harbour were central to its operations, with an administrative office about  south in Shimoni. It granted immunity of prosecution to British subjects and allowed them the right to raise taxes, impose custom duties, administer justice, make treaties and otherwise act as the government of the area.

In 1893 the IBEAC transferred its administration rights of the territory to the British Government. The territory was then divided to form the Uganda Protectorate in 1894 and East Africa Protectorate (later Kenya) in 1895.

History
In the early 1880s, European powers rushed to obtain unclaimed territories within areas of interest in Africa. One of these areas, the Sultanate of Zanzibar and the interior of Eastern Africa, caught the attention of both Germany and Britain. Hoping to resolve this common interest in a peaceful manner, in 1886 Germany and Britain signed a treaty in which they agreed upon what lands they would exclusively pursue. Germany would lay claim to the coast of present-day Tanzania and Britain retained access to the area in which Kenya and Uganda lie.

At the same time, Britain was focusing its resources in other interests which included land acquired in Southern Africa. This left the British government reluctant in accepting full responsibility for this newly acknowledged region. In an effort to ease this potential burden, Britain considered allowing a commercial company the right to administer and develop the eastern territory. In 1888, Sir William Mackinnon and the Imperial British East Africa Company (IBEAC) were authorized to serve this purpose.

The IBEAC assumed responsibility for land stretching from the eastern coast of Kenya all the way to the northwestern shore of Lake Victoria.

Other than the expected work involved with governing the exportation and management of goods and agriculture, the main role of the IBEAC was to begin facilitating the construction of a railway connecting the east coast region of Mombasa to Lake Victoria.
The company employed James Macdonald assisted by John Wallace Pringle, both officers in the Royal Engineers, to undertake the survey in 1891–1892. The two reported favourably, noting that Kikuyuland would be suitable for European settlement.
However, the IBEAC lacked the funds needed to start the work.
The IBEAC started building the Mackinnon-Sclater road, a  ox cart track from Mombasa to Busia on the Uganda border, in 1890.

The company ordered a 110-ton general purpose steamship, the , to operate on Lake Victoria. She was built in Scotland in 1890 and delivered in kit form to Mombasa. However, the kit remained there in storage until 1895, presumably because the IBEAC did not succeed in starting to build the railway that would deliver the kit to the lake.

Brewing conflict between rival factions ultimately prevented the company from investing the necessary time and money into this venture. The four groups involved in Uganda, the Kabaka, French Catholics, Protestants, and the company, could not resolve their squabble amicably and with tensions continuing to rise, civil war broke out in January 1892.  With the aid of Frederick Lugard, the individual given the task of building the company a fortified location on Kampala Hill, the IBEAC earned itself a fruitless victory.

This conflict proved to be the company's final undoing. IBEAC was already struggling financially due to customs issues but the money spent funding this skirmish all but bankrupted it. This also made clear that the company would be unable to continue its poorly executed attempt at colonizing eastern Africa.

{{quote box|align=left|width=246px|quote=A miserable fraud and a disgrace to the English name.|source=– Sir Gerald Portal<ref>Perham, M, Lugard: The Years of Adventure, 1858-1899, p.435.</ref>}}

The British East Africa Company proved to be an ineffective attempt at allowing commercial businesses local administrative rights. Inevitably, in 1894, the British government declared a protectorate over Uganda effectively dissolving IBEAC and assuming full responsibility.

The main part of the Mackinnon-Sclater road was completed by the British government after the demise of the IBEAC. The British government eventually built the Uganda Railway to Kisumu on Lake Victoria, between 1896 and 1901. SS William Mackinnon reached Kisumu in kit form in 1898, was launched in 1900 and, like the railway, was completed and entered service in 1901.

See also
Colonial Heads of Kenya
German East Africa

References

Sources

Galbraith, John S, 1970, "Italy, the British East Africa Company, and the Benadir Coast, 1888–1893", The Journal of Modern History 42. 4, pages 549-563

Further reading
Galbraith, John S, 1972, Mackinnon and East Africa 1878–1895'', Cambridge, Cambridge University Press
The partition of East Africa (1856 – 1891)

British Empire
History of Kenya
East Africa
Companies established in 1888
1888 establishments in England
British companies established in 1888

Imperial British East Africa Company


Henry Capehart (March 18, 1825 – April 15, 1895) was a surgeon and officer in the U.S. Cavalry during the American Civil War. He received the Medal of Honor for saving the life of a drowning soldier while under fire at Greenbrier River, West Virginia, on May 22, 1864.

Early life
Born on March 18, 1825, near Johnstown, Pennsylvania, Capehart attended high school in Pittsburgh. He had a younger brother, Charles E. Capehart, whom he helped raise after their mother's early death. Upon graduating from Jefferson College (now known as Washington & Jefferson College), he moved to Waynesburg in 1847 to continue his medical education and then started a practice in Bridgeport, Ohio, after earning his license in 1849.

Military service
At the outbreak of the Civil War, Capehart left his medical practice in Bridgeport and volunteered for the Union Army. He was appointed regimental surgeon of the 1st West Virginia Cavalry on September 18, 1861. In the latter half of 1863, he participated in the battles of Gettysburg, Bristoe Station, and Mine Run. Upon the recommendations of General Judson Kilpatrick and others, Capehart was made colonel and succeeded Nathaniel P. Richmond, who resigned due to health issues, as commander of the regiment on February 22, 1864.

Beginning in May 1864, Capehart and the 1st West Virginia Cavalry took part in campaigns along the Shenandoah Valley. On May 22, while fording the Greenbrier River under Confederate fire, Private Watson Karr was swept off his horse and down the fast-moving stream. Capehart attempted to catch the soldier as he swept by, but was pulled off his horse as well. Both men were carried down the river and over a waterfall; Capehart then grabbed Karr and pulled him from the water. It was for this action that Capehart was awarded the Medal of Honor decades later, on February 12, 1895. His official citation reads simply: "Saved, under fire, the life of a drowning soldier."

Capehart was assigned command of a cavalry brigade in the Union Army of the Shenandoah and upon recommendation of General George Armstrong Custer was promoted to Brevet Brigadier General on March 13, 1865. Capehart's brigade was transferred to Custer's division, participating with it in the Appomattox Campaign.  Following the Confederate surrender at Appomattox, Capehart was promoted to Brevet Major General on June 17, 1865. He mustered out of service on July 8, 1865, at Wheeling, West Virginia.

He died in 1895 and was buried at Arlington National Cemetery, in Arlington, Virginia.

His brother, Major Charles E. Capehart was also awarded the Medal of Honor for gallantry in the Civil War.

See also

 List of American Civil War Medal of Honor recipients: A–F

References

External links

1825 births
1895 deaths
People from Johnstown, Pennsylvania
People of Pennsylvania in the American Civil War
Union Army colonels
Union Army generals
Washington & Jefferson College alumni
United States Army Medal of Honor recipients
Burials at Arlington National Cemetery
American Civil War recipients of the Medal of Honor
People from Bridgeport, Ohio

Henry Capehart


Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting's calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed.  Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK interpretation.

Several systems of semantics for intuitionistic logic have been studied. One of these semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras. Another semantics uses Kripke models. These, however, are technical means for studying Heyting's deductive system rather than formalizations of Brouwer's original informal semantic intuitions. Semantical systems claiming to capture such intuitions, due to offering meaningful concepts of "constructive truth" (rather than merely validity or provability), are Kurt Gödel's dialectica interpretation, Stephen Cole Kleene's realizability, Yurii Medvedev's logic of finite problems, or Giorgi Japaridze's computability logic.  Yet such semantics persistently induce logics properly stronger than Heyting's logic. Some authors have argued that this might be an indication of inadequacy of Heyting's calculus itself, deeming the latter incomplete as a constructive logic.

Mathematical constructivism

In the semantics of classical logic, propositional formulae are assigned truth values from the two-element set  ("true" and "false" respectively), regardless of whether we have direct evidence for either case. This is referred to as the 'law of excluded middle', because it excludes the possibility of any truth value besides 'true' or 'false'. In contrast, propositional formulae in intuitionistic logic are not assigned a definite truth value and are only considered "true" when we have direct evidence, hence proof. (We can also say, instead of the propositional formula being "true" due to direct evidence, that it is inhabited by a proof in the Curry–Howard sense.) Operations in intuitionistic logic therefore preserve justification, with respect to evidence and provability, rather than truth-valuation.

Intuitionistic logic is a commonly-used tool in developing approaches to constructivism in mathematics. The use of constructivist logics in general has been a controversial topic among mathematicians and philosophers (see, for example, the Brouwer–Hilbert controversy). A common objection to their use is the above-cited lack of two central rules of classical logic, the law of excluded middle and double negation elimination. David Hilbert considered them to be so important to the practice of mathematics that he wrote:"Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether."Intuitionistic logic has found practical use in mathematics despite the challenges presented by the inability to utilize these rules. One reason for this is that its restrictions produce proofs that have the existence property, making it also suitable for other forms of mathematical constructivism. Informally, this means that if there is a constructive proof that an object exists, that constructive proof may be used as an algorithm for generating an example of that object, a principle known as the Curry–Howard correspondence between proofs and algorithms. One reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants. These tools assist their users in the generation and verification of large-scale proofs, whose size usually precludes the usual human-based checking that goes into publishing and reviewing a mathematical proof. As such, the use of proof assistants (such as Agda or Coq) is enabling modern mathematicians and logicians to develop and prove extremely complex systems, beyond those that are feasible to create and check solely by hand. One example of a proof that was impossible to satisfactorily verify without formal verification is the famous proof of the four color theorem. This theorem stumped mathematicians for more than a hundred years, until a proof was developed that ruled out large classes of possible counterexamples, yet still left open enough possibilities that a computer program was needed to finish the proof. That proof was controversial for some time, but, later, it was verified using Coq.

Syntax

The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic.  However, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic (IPL) it is customary to use →, ∧, ∨, ⊥ as the basic connectives, treating ¬A as an abbreviation for . In intuitionistic first-order logic both quantifiers ∃, ∀ are needed.

Hilbert-style calculus
Intuitionistic logic can be defined using the following Hilbert-style calculus. This is similar to a way of axiomatizing classical propositional logic.

In propositional logic, the inference rule is modus ponens
 MP: from  and  infer
and the axioms are
 THEN-1:
 THEN-2:
 AND-1:
 AND-2:
 AND-3:
 OR-1:
 OR-2:
 OR-3:
 FALSE:
To make this a system of first-order predicate logic, the generalization rules
 -GEN: from  infer , if  is not free in
 -GEN: from  infer , if  is not free in
are added, along with the axioms
 PRED-1: , if the term  is free for substitution for the variable  in  (i.e., if no occurrence of any variable in  becomes bound in )
 PRED-2: , with the same restriction as for PRED-1

Negation
If one wishes to include a connective  for negation rather than consider it an abbreviation for , it is enough to add:
 NOT-1':
 NOT-2':

There are a number of alternatives available if one wishes to omit the connective  (false). For example, one may replace the three axioms FALSE, NOT-1', and NOT-2' with the two axioms
 NOT-1:
 NOT-2:
as at . Alternatives to NOT-1 are  or .

Equivalence
The connective  for equivalence may be treated as an abbreviation, with  standing for . Alternatively, one may add the axioms

 IFF-1:
 IFF-2:
 IFF-3:

IFF-1 and IFF-2 can, if desired, be combined into a single axiom  using conjunction.

Relation to classical logic
The system of classical logic is obtained by adding any one of the following axioms:

  (Law of the excluded middle. May also be formulated as .)
  (Double negation elimination)
  (Peirce's law)
  (Law of contraposition)

In general, one may take as the extra axiom any classical tautology that is not valid in the two-element Kripke frame  (in other words, that is not included in Smetanich's logic).

In 1932, Kurt Gödel defined a system of logics intermediate between classical and intuitionistic logic; Gödel logics are concomitantly known as intermediate logics.

Sequent calculus

Gerhard Gentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in a system that is sound and complete with respect to intuitionistic logic.  He called this system LJ.  In LK any number of formulas is allowed to appear on the conclusion side of a sequent; in contrast LJ allows at most one formula in this position.

Other derivatives of LK are limited to intuitionistic derivations but still allow multiple conclusions in a sequent.  LJ'  is one example.

Theorems
When explaining intuitionistic logic in terms of classical logic, it can be understood as a weakening thereof: It is more conservative in what it allows a reasoner to infer, while not permitting any new inferences that could not be made under classical logic. Each theorem of intuitionistic logic is a theorem in classical logic, but not conversely. Many tautologies in classical logic are not theorems in intuitionistic logicin particular, as said above, one of intuitionistic logic's chief aims is to not affirm the law of the excluded middle so as to vitiate the use of non-constructive proof by contradiction, which can be used to furnish existence claims without providing explicit examples of the objects that it proves exist.

Double negations
It does merely "not affirm" the law of the excluded middle because while it is not necessarily the case that the law is upheld in any context, no counterexample can be given either. Such a counterexample would be an inference (inferring the negation of the law for a certain proposition) disallowed under classical logic and thus is not allowed in a strict weakening like intuitionistic logic. Formally, it is a simple theorem that  for any two  propositions. By considering any  established to be false indeed shows that the double negation of the law  is retained as a tautology already in minimal logic. So the propositional calculus is always compatible with classical logic. When assuming the law implies a proposition, then by applying contraposition twice and using the double-negated excluded middle, one may prove double-negated variants of various strictly classical tautologies. The situation is more intricate for predicate logic formulas, when some quantified expressions are being negated.

Akin to the above, from modus ponens in the form  follows . The relation between them may always be used to obtain new formulas: A weakened premise makes for a strong implication, and vice versa. For example, note that if  holds, then so does , and even weaker than the latter are the four equivalent statements , ,  as well as . Propositions for which double-negation elimination is possible are also called stable. Intuitionistic logic proves stability only for restricted types of propositions, such as those of negated form. A formula for which excluded middle holds can be proven stable using the disjunctive syllogism, but the converse does not hold in general, unless the excluded middle statement is stable itself.

When  expresses a claim, then its double-negation  merely expresses the claim that a refutation of  would be inconsistent. Having proven such a mere double-negation also still aids in negating other statements through negation introduction . A double-negated existential statement does not denote existence of an entity with a property, but rather the absurdity of assumed non-existence of any such entity. Also all the principles in the next section involving quantifiers explain use of implications with hypothetical existence as premise.

Formula translation
Weakening statements by adding two negations before existential quantifiers (and atoms) is also the core step in the double-negation translation. It constitutes an embedding of classical first-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if its Gödel–Gentzen translation is provable intuitionistically. Therefore, intuitionistic logic can be seen as a means of extending classical logic with constructive semantics.

Non-interdefinability of operators
In classical propositional logic, it is possible to take one of conjunction, disjunction, or implication as primitive, and define the other two in terms of it together with negation, such as in Łukasiewicz's three axioms of propositional logic. It is even possible to define all four in terms of a sole sufficient operator such as the Peirce arrow (NOR) or Sheffer stroke (NAND). Similarly, in classical first-order logic, one of the quantifiers can be defined in terms of the other and negation.
These are fundamentally consequences of the law of bivalence, which makes all such connectives merely Boolean functions.
The law of bivalence is not required to hold in intuitionistic logic. As a result, none of the basic connectives can be dispensed with, and the above axioms are all necessary. So most of the classical identities between connectives and quantifiers are only theorems of intuitionistic logic in one direction. Some of the theorems go in both directions, i.e. are equivalences, as subsequently discussed.

Existential vs. universal quantification
Firstly, when  is not free in the proposition , then

When the domain of discourse is empty, then by the principle of explosion, an existential statement implies anything. When the domain contains at least one term, then assuming excluded middle for , the two sides become equivalent. Indeed, classically they are also equivalent to a disjunctive form discussed further below. Constructively, existence claims are however harder to come by.

If the domain of discourse is not empty and  is moreover independent of , such principles are equivalent to formulas in the propositional calculus. Here, the formula then just expresses the identity . This is the curried form of modus ponens , which in the special the case with  as a false proposition results in the law of non-contradiction principle . Considering a false proposition  for the original implication results in the important

In words: "If there exists an entity  that does not have the property , then the following is refuted: Each entity has the property ."

The quantifier formula with negations also immediately follows from the non-contradiction principle derived above, each instance of which itself already follows from the more particular . To derive a contradiction given , it suffices to establish its negation  (as opposed to the stronger ) and this makes proving double-negations valuable also. By the same token, the original quantifier formula in fact still holds with  weakened to . And so really

In words: "If there exists an entity  that does not have the property , then the following is refuted: For each entity, one is not able to prove that it does not have the property ".

Secondly,

where similar considerations apply. Here the existential part is always a hypothesis and this is an equivalence. Considering the special case again,

Lastly, by using the axiom THEN-1 once, THEN-2 can be reduced to its special case  and this conversion can then be used to obtain two further implications:

For simplicity, in the discussion here and below, the formulas are generally presented in weakened forms without double-negations. To name but one, a special case of the first formula here is . Note how also this formula indeed immediately implies the -direction of the equivalence bullet point listed above. The double-negations can also be placed in the antecedent.

More complicated variants hold. Incorporating the predicate  and currying, the following generalization also entails the relation between implication and conjunction in the predicate calculus, discussed below.

If the predicate  is decidedly false for all , then this equivalence is trivial. If  is decidedly true for all , the schema simply reduces to the previously stated equivalence. In the language of classes,  and , the special case of this equivalence with false  equates two characterizations of disjointness :

Disjunction vs. conjunction
There are finite variations of the quantifier formulas, with just two propositions:

And conversely,

And as mentioned, stronger forms of these formulas hold. For example, . This can easily be derived from the non-contradiction principle, as all of the formulas are about deriving absurd consequences from hypotheticals, and these statements do not have to be placed in a particular order. By the same reasoning, one may obtain the mixed form of the implication and its converse

But in particular, there is no distributivity principle deriving the claim  from . Negated propositions are comparably weak, in that the classically valid De Morgan's law, granting a disjunction from a single negative hypothetical, does not automatically hold constructively. See also the LLPO for a variant concerning two existentially closed decidable predicates. For an informal example of the constructive reading, consider the following: From conclusive evidence it not to be the case that both Alice and Bob showed up to their date, one cannot derive conclusive evidence, tied to either of the two persons, that this person did not show up. Instead, the intuitionistic propositional calculus and some of its extensions exhibit the disjunction property, implying the disjuncts individually would have to be derivable as well. When the fourth law is adopted, the weak excluded middle, i.e. the statement  for negated propositions, immediately follows.

Conjunction vs. implication
From the general equivalence also follows the following special case, expressing incompatibility of two predicates using two different connectives:

It may be understood as a special case of currying and uncurrying. Many more considerations regarding double-negations again apply. For example, as  is stronger than , from this equivalence the implication  follows. However, this cannot generally be reversed, as for true  the double-negation elimination principle would follow.

Finally, conversely,

For either of the propositions established to be true, this again reduces to the double-negation principle for the other, and such an implication cannot generally be reversed.

Due to the symmetry of the conjunction connective, the equivalence formula for the negated conjunction again implies the already established .

Disjunction vs. implication
Clearly  and classically this is an equivalence. Indeed, minimal logic already proves more general variants of this implication. In intuitionistic logic, one obtains variants involving , as follows.

Firstly, note that two different formulas for  mentioned above can be used to imply . The latter are forms of the disjunctive syllogism for negated propositions, . A strengthened form still holds in intuitionistic logic:

As in previous sections, the positions of  and  may be switched and  may be weakened to . These principles are provable from non-contradiction and explosion. So, for example, intuitionistically "Either  or " is a stronger propositional formula than "If not , then ", whereas these are classically interchangeable. The implication cannot generally be reversed, as that immediatenly implies excluded middle.

Considering  in place of  in the above form of the syllogism shows how excluded middle implies double-negation elimination. And in this way intuitionistic logic proves .

Of course the formulas established here may be combined to obtain yet more variations. For example, the disjunctive syllogism as presented generalizes to

If some term exists at all, the antecedent here even implies , which in turn itself also implies the conclusion here (this is again the very first formula mentioned in this section).

The bulk of the discussion in these sections applies just as well to just minimal logic. But as for the disjunctive syllogism with general , minimal logic can at most prove  where  denotes . The conclusion here can only be simplified to  using explosion.

Equivalences
The above lists also contain equivalences.
The equivalence involving a conjunction and a disjunction stems from  actually being stronger than . Both sides of the equivalence can be understood as conjunctions of independent implications. Above, absurdity  is used for . In functional interpretations, it corresponds to if-clause constructions.
So e.g. "Not ( or )" is equivalent to "Not , and also not ".

An equivalence itself is generally defined as, and then equivalent to, a conjunction of implications

With it, such connectives become in turn definable from it:

In turn,  and  are complete bases of intuitionistic connectives.

Functionally complete connectives
As shown by Alexander V. Kuznetsov, either of the following connectives – the first one ternary, the second one quinary – is by itself functionally complete: either one can serve the role of a sole sufficient operator for intuitionistic propositional logic, thus forming an analog of the Sheffer stroke from classical propositional logic:

Semantics
The semantics are rather more complicated than for the classical case. A model theory can be given by Heyting algebras or, equivalently, by Kripke semantics. Recently, a Tarski-like model theory was proved complete by Bob Constable, but with a different notion of completeness than classically.

Unproved statements in intuitionistic logic are not given an intermediate truth value (as is sometimes mistakenly asserted). One can prove that such statements have no third truth value, a result dating back to Glivenko in 1928. Instead they remain of unknown truth value, until they are either proved or disproved. Statements are disproved by deducing a contradiction from them.

A consequence of this point of view is that intuitionistic logic has no interpretation as a two-valued logic, nor even as a finite-valued logic, in the familiar sense. Although intuitionistic logic retains the trivial propositions  from classical logic, each proof of a propositional formula is considered a valid propositional value, thus by Heyting's notion of propositions-as-sets, propositional formulae are (potentially non-finite) sets of their proofs.

Heyting algebra semantics
In classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as the members of a Boolean algebra.  The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra.  Then we have the useful theorem that a formula is a valid proposition of classical logic if and only if its value is 1 for every valuation—that is, for any assignment of values to its variables.

A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from  a Boolean algebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case. A formula is valid in intuitionistic logic if and only if it receives the value of the top element for any valuation on any Heyting algebra.

It can be shown that to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elements are the open subsets of the real line R. In this algebra we have:

where int(X) is the interior of X and  X∁ its complement.

The last identity concerning A → B allows us to calculate the value of ¬A:

With these assignments, intuitionistically valid formulas are precisely those that are assigned the value of the entire line. For example, the formula ¬(A ∧ ¬A) is valid, because no matter what set X is chosen as the value of the formula A, the value of ¬(A ∧ ¬A) can be shown to be the entire line:

So the valuation of this formula is true, and indeed the formula is valid. But the law of the excluded middle, A ∨ ¬A, can be shown to be invalid by using a specific value of the set of positive real numbers for A:

The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in the top element, representing true, as the valuation of the formula, regardless of what values from the algebra are assigned to the variables of the formula. Conversely, for every invalid formula, there is an assignment of values to the variables that yields a valuation that differs from the top element. No finite Heyting algebra has the second of these two properties.

Kripke semantics

Building upon his work on semantics of modal logic, Saul Kripke created another semantics for intuitionistic logic, known as Kripke semantics or relational semantics.

Tarski-like semantics
It was discovered that Tarski-like semantics for intuitionistic logic were not possible to prove complete. However, Robert Constable has shown that a weaker notion of completeness still holds for intuitionistic logic under a Tarski-like model. In this notion of completeness we are concerned not with all of the statements that are true of every model, but with the statements that are true in the same way in every model. That is, a single proof that the model judges a formula to be true must be valid for every model. In this case, there is not only a proof of completeness, but one that is valid according to intuitionistic logic.

Metalogic

Admissible rules
In intuitionistic logic or a fixed theory using the logic, the situation can occur that an implication always hold metatheoretically, but not in the language. For example, in the pure propositional calculus, if  is provable, then so is . Another example is that  being provable always also means that so is . One says the system is closed under these implications as rules and they may be adopted.

Relation to other logics

Paraconsistent logic
Intuitionistic logic is related by duality to a paraconsistent logic known as Brazilian, anti-intuitionistic or dual-intuitionistic logic.
The subsystem of intuitionistic logic with the FALSE axiom removed is known as minimal logic and some differences have been elaborated on above.

Intermediate logics
Any finite Heyting algebra that is not equivalent to a Boolean algebra defines (semantically) an intermediate logic. On the other hand, validity of formulae in pure intuitionistic logic is not tied to any individual Heyting algebra but relates to any and all Heyting algebras at the same time.

Many-valued logic
Kurt Gödel's work involving many-valued logic showed in 1932 that intuitionistic logic is not a finite-valued logic.  (See the section titled Heyting algebra semantics above for an infinite-valued logic interpretation of intuitionistic logic.)

Modal logic
Any formula of the intuitionistic propositional logic (IPC) may be translated into the language of the normal modal logic S4 as follows:

and it has been demonstrated that the translated formula is valid in the propositional modal logic S4 if and only if the original formula is valid in IPC. The above set of formulae are called the Gödel–McKinsey–Tarski translation.
There is also an intuitionistic version of modal logic S4 called Constructive Modal Logic CS4.

Lambda calculus
There is an extended Curry–Howard isomorphism between IPC and simply-typed lambda calculus.

See also

 BHK interpretation
 Computability logic
 Constructive analysis
 Constructive proof
 Constructive set theory
 Curry–Howard correspondence
 Game semantics
 Harrop formula
 Heyting arithmetic
 Inhabited set
 Intermediate logics
 Intuitionistic type theory
 Kripke semantics
 Linear logic
 Paraconsistent logic
 Realizability
 Relevance theory
 Smooth infinitesimal analysis

Notes

References

 van Dalen, Dirk, 2001, "Intuitionistic Logic", in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell
 Morten H. Sørensen, Paweł Urzyczyn, 2006, Lectures on the Curry-Howard Isomorphism (chapter 2: "Intuitionistic Logic").  Studies in Logic and the Foundations of Mathematics vol. 149, Elsevier
W. A. Carnielli (with A. B. M. Brunner)."Anti-intuitionism and paraconsistency".  Journal of Applied Logic Volume 3, Issue 1, Mar 2005, pp. 161–184
Arend Heyting, 1930, "Die formalen Regeln der intuitionistischen Logik," in three parts, Sitzungsberichte der preussischen Akademie der Wissenschaften: 42–71, 158–169.

External links
 Stanford Encyclopedia of Philosophy: "Intuitionistic Logic" by Joan Moschovakis
 Intuitionistic Logic by Nick Bezhanishvili and Dick de Jongh (from the Institute for Logic, Language and Computation at the University of Amsterdam)
 Semantical Analysis of Intuitionistic Logic I by Saul A. Kripke from Harvard University, Cambridge, Mass., USA
 Intuitionistic Logic by Dirk van Dalen
 The discovery of E. W. Beth's semantics for intuitionistic logic by A. S. Troelstra and P. van Ulsen
 Expressing Database Queries with Intuitionistic Logic by Anthony J. Bonner. L. Thorne McCarty. Kumar Vadaparty. Rutgers University, Department of Computer Science
 Tableaux'method for intuitionistic logic through S4-translation tests the intuitionistic validity of propositional formulae; provided by the Laboratoire d'Informatique de Grenoble
The Oxford Handbook of Philosophy of Mathematics and Logic: "Intuitionism in Mathematics" by David Charles McCarty

Logic in computer science
Non-classical logic
Constructivism (mathematics)
Systems of formal logic
Intuitionism

Intuitionistic logic


Lenny the Wonder Dog is a 2005 American comedy film about the eponymous dog, Lenny, directed by Oren Goldman from a screenplay written by Michael Winslow, and filmed on locations in Florida.   The cast included many prominent actors and celebrities such as Craig Ferguson and Andy Richter as the voice of Lenny. The movie had television release in Europe and after video release received generally poor reviews.

The film is notable for being one of Oscar Isaac's earliest roles.

Plot
Doctor Island (Joe Morton) has invented a computer microchip he plans to use to train dogs instantly. However, his former colleague, Doctor Wagner (Craig Ferguson), has other plans for the microchip: he wants to implant copies in every child in order to brainwash them to come work for him after they turn 18, and be exceedingly loyal. In the fight between Doctor Island and Doctor Wagner's henchmen, Hanky (Andrew Shaifer), Panky (Jeff Chase), and Doctor Island get injured and his dog Lenny (Andy Richter) is injected with the computer microchip. On the run, Lenny finds his brainpower growing. And when he finds a young boy, Zach (Sammy Kahn), to be his friend and savior, Lenny discovers he can talk. Lenny (continually amazed by the new powers the embedded microchip keeps giving him), Zach (full of unrequited love for longtime friend Becky), and school newspaper reporter Becky (Stephanie Sherrin) (out to write the story of the century) team up with downtrodden police officer John Wyndham (Michael Winslow) to defeat Dr. Wagner and his henchmen. The showdown culminates at the school concert given by Pop Star Angie (Maria de los Angeles Monasterio), where Zach's inspiring speech and Lenny's audio powers combine to incriminate Doctor Wagner, who is arrested by the police chief, now appreciative of Officer Wyndham's talents for police-work and crime-solving.

Cast
 Sammy Kahn as Zach
 Joe Morton as Dr. Island
 Andy Richter as the voice of Lenny
 Craig Ferguson as Dr. Richard Wagner
 Kathy Kinney as Lisa Lathers
 Michael Winslow as John Wyndham
 Jeff Chase as Panky
 Evan Adams as Razel
 Paola Turbay as Mrs. Ripley
 Camila Banus as Rainbow
 Oscar Isaac as Detective Fartman
 Stephanie Sherrin as Becky
 Andrew Shaifer as Hanky
 "Boomer" as Lenny
 Rus Blackwell as Harley
 Jeffrey William Evans as Chief Greenwald
 Damaris Justamante as Mayor Jenkins
 Maria de los Angeles Monasterio as Angie
 P.C. Martinez as Bodyguard
 Graham Purdy as Jeremy
 Roberto Smith as Henry

Release
Filmed at locations in Florida under the working title of A Dog and his Boy, the film had its first television release December 5, 2005 in Hungary as Lenny, a Csodakutya, followed by screening in Germany as Lenny, der Wunderhund, in Finland as Lenny ihmekoira, and in France as Lenny le chien parlant. Later French DVD release was as Lenny Wonderdog.
DVD releases include by North by Northwest Entertainment in 2005, Third Millennium in 2006, Boulevard Entertainment in 2007, and Peace Arch Trinity in 2008.

Reception
Lenny the Wonder Dog has received generally negative reviews. Common Sense Media panned the film giving it one star and writing "a baffling plot, cheap special effects, the repetitious chase scenes, and overall silliness make this movie almost unwatchable for adults", adding "though kids might find the visual jokes and pratfalls hilarious, the film sets the quality bar at a very low level."  They also offered that Craig Ferguson's performance as the megalomaniacal Dr. Richard Wagner, "is so over-the-top it could become a career landmark."

The Dove Foundation criticized the language used in the film as non-family friendly, writing "Although this is an off-the-wall comedy that is geared toward the family, it cannot be family approved due to some language issues."

Florida real estate developer Jeff Sherrin "politely but abruptly shrugs off" any question about the film. His daughter, Stephanie Sherrin, appeared in this film as well as in Kids in America.

References

External links

  at the Wayback Machine

2000s adventure comedy films
2004 films
2005 films
American adventure comedy films
Films about dogs
2004 comedy films
2005 comedy films
2000s English-language films
2000s American films
2010s American films

Lenny the Wonder Dog