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Jamie Lynn Spears

Jamie Lynn Marie Spears (born April 4, 1991) is an American actress and singer. She played Zoey Brooks on the Nickelodeon teen sitcom Zoey 101 (2005–2008) and its sequel film Zoey 102 (2023), and Noreen Fitzgibbons on the Netflix romantic drama series Sweet Magnolias (2020–present). She is the younger sister of singer Britney Spears.

Beginning in December 2007, Spears was subject to significant media attention and controversy after announcing her pregnancy at age 16. A misconception emerged that this canceled Zoey 101; however, the final season wrapped filming in September 2007 and premiered the following year. After giving birth in 2008, Spears stopped acting for five years.

Spears returned from her hiatus in 2013 and began her music career. She released her debut single "How Could I Want More" from her debut EP in country music, The Journey (2014). In 2015, Spears earned songwriting credits for Jana Kramer's certified Platinum single, "I Got the Boy", a song she wrote in 2011 during writing sessions in Nashville for her 2013 EP.

In 2023, due to the SAG-AFTRA strike, Spears began to appear on reality television shows, such as Dancing with the Stars, Special Forces: World's Toughest Test and I'm a Celebrity...Get Me Out of Here!.

Early life
Spears was born in McComb, Mississippi, and raised in Kentwood, Louisiana, to Jamie and Lynne Spears. Her older siblings are Bryan and Britney. Her parents divorced in 2002, but were in an on-and-off relationship after the divorce. She has described her home life as unstable, but remembers "feeling very loved". She attended Parklane Academy in McComb, where she was a cheerleader, and point guard of the basketball team. She was enrolled in dance and voice lessons, and participated in school plays and church performances. As a child, Spears and her family also spent time traveling and touring with Britney.

Spears studied with tutors while filming Zoey 101 in Los Angeles. She finished her education via online correspondence and received her GED in February 2008 through Tangipahoa Parish School System Adult Education Center.

2002–2008: Career beginnings and success with All That and Zoey 101
In February 2002, at age 10, Spears made her acting debut in the Paramount Pictures drama film Crossroads which starred her sister, who portrayed the central character Lucy Wagner. Spears made a cameo, playing the younger version of Britney's character. In April 2002, Spears walked the runway as a model for Kids R Us fashion show in New York City. Following Crossroads, Nickelodeon cast Spears as a regular performer in the sketch comedy series All That for the 2002 season; she appeared on the show through 2004. Spears portrayed various roles through the two seasons she appeared. Along with the cast of All That, Spears also made appearances on Snick On-Air Dare, Nickelodeon's version of Fear Factor. Deciding to pursue other roles on the network, Spears departed from the main cast on the tenth and final season of All That, and only appeared as a guest. At age 13, Spears was named as one of the celebrities on Teen People 's Young Hollywood/Hot List. In August 2004, Spears signed a development deal with Nickelodeon in which she would star as the protagonist in her own scripted television series that would air on the network. The series went on to be titled Zoey 101 and Spears portrayed the role of Zoey Brooks. The series centered on Zoey and her friends who attend a fictional boarding school, Pacific Coast Academy (PCA). Spears recorded the series theme song, titled "Follow Me", which was written by her sister Britney Spears. Filmed at Pepperdine University in Malibu, California, the series premiered on January 9, 2005, and was the network's highest-rated series premiere in over eight years. Spears won a Young Artist Award and a Nickelodeon Kids' Choice Awards for her performance. In September 2007, she won "The High School Sweetheart" award at Us Weekly's Hot Hollywood event.

Zoey 101 was positively received by critics and viewers. It was nominated for an Emmy for Outstanding Children's Programming, and the show's season 3 finale episode was the second most-watched show across all cable channels that week. The series ran for four seasons, with its series finale "PCA Confidential" airing on May 2, 2008. Nickelodeon issued a statement shortly after the announcement of Spears being pregnant at the age of 16, stating that the network respected Spears's decision to take responsibility and noting that its primary concern was for her well-being. Spears' co-star Seann Flynn praised her as hard-working and talented, telling Nylon: "One of the things that always stood out to me regarding her professionalism was how well she knew her lines. I don’t think she ever messed one up. She always brought her best work to the table."

In December 2007, Variety announced that Spears had signed on to guest-appear in the ABC comedy sitcom series Miss Guided, in which she played the role of the promiscuous high-school student Mandy Fener in the episode "Hot Sub" which aired on March 20, 2008. That same year, Spears voiced Goldilocks in the direct-to-DVD animation film Unstable Fables: Goldilocks & 3 Bears Show.

2011–2017: Country music debut
In 2011, Spears moved to Nashville and began working on a country music album with local music producers. On November 7, 2011, Spears held a small concert at The Rutledge in Nashville. On November 25, 2013, Spears released her first single "How Could I Want More". The song debuted at number 29 on Billboard's Hot Country Songs, and at number 8 on the Country Digital Songs chart, for the week ending December 7, 2013. Kevin Rutherford of Billboard described Spears' performance as "sweet and innocent" yet powerful, and suggested the song would fit right in at country radio. The song was named Taste of Country 's Top Critic's Pick.

Spears was featured on her older sister Britney Spears's eighth studio album Britney Jean on the song "Chillin' with You". On May 27, 2014, Spears released her debut EP The Journey. The EP peaked at number 5 on Billboard's Heatseekers Albums and at number 24 on Top Country Albums. That year, Spears made numerous appearances on country radio talk shows, made performances at fairs and festivals, and was an opening act for Alan Jackson. Spears co-wrote the Jana Kramer song "I Got the Boy" which was released as the second single from Thirty One in 2015.

In 2016, Spears released the documentary Jamie Lynn Spears: When the Lights Go Out, reflecting on the controversy of her pregnancy, showcasing her country music career after "realizing [her] true calling was music" as well as her personal life as a wife and mother. On March 15, 2016, Spears performed at the Grand Ole Opry in Nashville. On June 24, 2016, Spears released her second single "Sleepover". Spears was a presenter and a performer at Radio Disney Music Awards in April 2017.

2018–present: Return to acting, Zoey 102, memoir and reality television

In an episode that aired on July 13, 2018, Spears returned to Nickelodeon for the first time in over ten years as a contestant on the revived game show Double Dare with her daughter Maddie, competing against fellow All That alumnus Josh Server and his niece. She and Maddie won over $15,000 to donate to the New Orleans hospital that treated Maddie in 2017.

In July 2019, Netflix announced that Spears was cast as Noreen Fitzgibbons in the romance drama series Sweet Magnolias, based on the novel series by Sherryl Woods. In July 2020, the cast of Zoey 101 reunited in an episode of the eleventh season of All That. Spears played herself and her old All That character Thelma Stump. That year, Spears and Chantel Jeffries recorded a new version of the Zoey 101 theme song "Follow Me". The music video for the song featured appearances by numerous celebrities and Spears' elder daughter Maddie. On October 29, 2020, Spears performed "Follow Me" on Good Morning America.

In July 2021, Worthy Publishing, an imprint of Hachette Book Group announced Spears's plans to release a memoir, releasing a statement saying: "Jamie Lynn's book has been in development over the past 12 months and will allow the world to hear her inspiring story in her own words, for the first time." Her memoir titled Things I Should Have Said: Family, Fame, and Figuring it Out was released on January 18, 2022. Rolling Stone recommended the book but highlighted that Jamie Lynn ostensibly used her sister Britney Spears' highly publicized struggles and conservatorship to promote her new book. The book was not a financial success. On January 4, 2023, she began competing on Fox's survival television series Special Forces: World's Toughest Test, where contestants participate in special forces training challenges at a camp removed from society. She left the series during the competition's third episode, with Spears saying that she missed her children. On July 27, 2023, the sequel film Zoey 102 was released on Paramount+, with Spears reprising her role from Zoey 101 alongside many of the original cast members. Production took place in January 2023 in North Carolina, with Spears attached as executive producer. Billboard called it a "highly anticipated sequel", while Ready Steady Cut praised the film as a "a nostalgic and amusing watch, enjoyable for original fans and the series newcomers." One day after its release, the film was the third most watched on Paramount+ in the U.S. and the second most watched in Canada.

On September 12, 2023, Spears announced on Good Morning America that she would compete on season 32 of Dancing with the Stars, paired with Alan Bersten. She donated her weekly earnings to SAG-AFTRA and the WGA amid their ongoing strike. Spears was eliminated in the second week of the competition. On December 5, 2023, she returned for a group dance performance in the season 32 finale along with other eliminated contestants.

In November 2023, it was announced that Spears signed up as a contestant on the twenty-third season of British survival reality television show I'm a Celebrity...Get Me Out of Here!, which premiered on November 19, 2023. Ahead of entering the competition, Spears said: "There is literally every misconception you can have on a person about me so I don’t take any of it seriously. But I do think going on I’m a Celebrity will be a nice way for people to see the real me." Spears left the show after 11 days on November 29, 2023 for medical reasons. She was the second contestant to depart on medical grounds that season, as Grace Dent left days earlier. Spears placed 11th in the series. She described her decision to participate in the show by saying: "I think it is so far out of my comfort zone and this year with SAG being on strike, I have said ‘yes’ to a lot of unscripted projects. I am really enjoying throwing myself into these experiences." Fans of the show praised Spears for her determination in challenges, and her likable personality.

Public image
In the early years of her career, Spears had a generally wholesome image as a teen star. Her fame increased dramatically in the wake of her pregnancy announcement in December 2007, and during the following months, she became a frequent target of the paparazzi. In subsequent years, Spears was rarely seen in the public eye and focused on family, saying: "I used to be insecure at times, because I was always the youngest mom in the room. I quickly learned that being a good and competent mother has nothing to do with age."

Spears has been featured in several fashion blogs and magazines such as Nylon, Seventeen, Teen Vogue, CosmoGirl, Teen People, Teen, Girls' Life, Mizz and Nickelodeon Magazine. Spears has appeared in commercials and print ads throughout her career. She appeared in television commercials for Hasbro's eKara karaoke headset, Pepsi and Clorox bleach. In 2018, she appeared in an advertising campaign and social media ads for Kraft Foods with daughter Maddie.

Spears endorses beauty and wellness products on Instagram. She has partnered with FabFitFun subscription box, Smart Food popcorn and International Loops marketing agency.

Personal life
Spears attended First Baptist Church in Kentwood, Louisiana, but she is Catholic since her conversion in early 2018, as are her daughters and her mother Lynne.

In 2013, Spears was diagnosed with anxiety, depression, and obsessive-compulsive disorder. She underwent therapy and briefly used medication as well. "The anxiety was to the point where I couldn't function," she commented. "I'm very conscious not to lie to myself anymore because I think that my biggest issue my whole life was pretending to be perfect."

Teenage pregnancy
On December 20, 2007, in an interview with OK! magazine, Spears announced that she was pregnant by her then boyfriend, Casey Aldridge. She was 16 years old at the time. Aldridge, whom she had been dating for two years, was 18. Spears confirmed her engagement to Aldridge in March 2008. In May 2008, the couple moved to Liberty, Mississippi, and purchased a house, with Spears stating they would raise their child there. She gave birth to their daughter Maddie Briann Aldridge on June 19, 2008, in McComb, Mississippi. Spears and Aldridge ended their engagement in March 2009, and Spears moved out in December 2009. The couple reconciled in August 2010, but ended their relationship a second time in November 2010.

The announcement of Spears' pregnancy generated controversy, with criticism centering on the story purportedly glamorizing teenage pregnancy. Some teens were disappointed with the contrast between Spears' on-screen personality as a "good" girl and her real-life teen pregnancy. On June 4, 2008, while in Amite County, Mississippi, the Spears family filed a complaint against Edwin Merrino, a paparazzo who, they believed, was stalking the couple. Merrino denied the allegation. He was released later in the day after posting bond. After her daughter's birth, Spears disappeared from the public eye for five years, focusing on raising her child. In an interview with Nylon, Spears said: "So, I got me a little house. I put a big gate up around it, and I was like, 'I'm going to stay here, raise my baby, and figure this out because this is real life. I've put myself in this situation, I'm not condoning it or saying it's right, but these are the cards that I have to play.' And I tried to do the best that I could."

Marriage and family
She subsequently began a relationship with Jamie Watson, a businessman who owns the communications service Advanced Media Partners. After two years of on-and-off dating, they announced their engagement in March 2013. On March 14, 2014, she married Watson in New Orleans. On April 11, 2018, Spears gave birth to her second daughter.

ATV incident
On February 5, 2017, Spears' eight-year-old daughter Maddie almost drowned in a Polaris ATV accident, after accidentally driving into the pond on her stepfather and mother's Louisiana property, with the ATV flipping over upon impact. She was later airlifted to Children's Hospital of New Orleans, where she remained unconscious in a coma for two days. She woke up and was released from care on February 10. Police reported the incident as occurring before 3:00 p.m. on a Sunday afternoon, and the sheriff explained that she was steering "100 yards from her parents" when she took "a hard right to avoid running over a nearby drainage ditch". The police report stated, "The child was trapped and secured by the seatbelt and the ATV's safety netting. Within two minutes, Acadian Ambulance Services arrived and assisted in freeing the child from the cold waters."

In 2020, Spears explained the situation further. She elaborated that she and her husband Jamie Watson "[...] dove in and [...] were able to rescue her," then the first responder took her away. The firefighter was able to get a pulse, but her case looked "grim at the time", leading to her being airlifted to the hospital.

Feud with Britney Spears
In 2021, Jamie Lynn and her sister Britney were involved in a highly publicized feud via social media. Britney publicly accused Jamie Lynn of actively and knowingly being involved in her conservatorship, while not making any efforts to help her terminate it. Jamie Lynn denied the allegations and maintained that she was unaware of the details of the conservatorship. Britney threatened legal action against Jamie Lynn after the latter spoke extensively about Britney in both her book and the following promotional tour.

Published works

 * Spears, Jamie Lynn: Things I Should Have Said: Family, Fame, and Figuring it Out, Worthy Publishing, January 18, 2022.

title: Jamie Lynn Spears


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Isaac Barrow

Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was the first to calculate the tangents of the kappa curve. He is also notable for being the inaugural holder of the prestigious Lucasian Professorship of Mathematics, a post later held by his student, Isaac Newton.

Early life and education
Barrow was born in London. He was the son of Thomas Barrow, a linen draper by trade. In 1624, Thomas married Ann, daughter of William Buggin of North Cray, Kent and their son Isaac was born in 1630. It appears that Barrow was the only child of this union—certainly the only child to survive infancy. Ann died around 1634, and the widowed father sent the lad to his grandfather, Isaac, the Cambridgeshire J.P., who resided at Spinney Abbey. Within two years, however, Thomas remarried; the new wife was Katherine Oxinden, sister of Henry Oxinden of Maydekin, Kent. From this marriage, he had at least one daughter, Elizabeth (born 1641), and a son, Thomas, who apprenticed to Edward Miller, skinner, and won his release in 1647, emigrating to Barbados in 1680.

Early career
Isaac went to school first at Charterhouse (where he was so turbulent and pugnacious that his father was heard to pray that if it pleased God to take any of his children he could best spare Isaac), and subsequently to Felsted School, where he settled and learned under the brilliant puritan Headmaster Martin Holbeach who ten years previously had educated John Wallis. Having learnt Greek, Hebrew, Latin and logic at Felsted, in preparation for university studies, he continued his education at Trinity College, Cambridge; he enrolled there because of an offer of support from an unspecified member of the Walpole family, "an offer that was perhaps prompted by the Walpoles' sympathy for Barrow's adherence to the Royalist cause." His uncle and namesake Isaac Barrow, afterwards Bishop of St Asaph, was a Fellow of Peterhouse. He took to hard study, distinguishing himself in classics and mathematics; after taking his degree in 1648, he was elected to a fellowship in 1649. Barrow received an MA from Cambridge in 1652 as a student of James Duport; he then resided for a few years in college, and became candidate for the Greek Professorship at Cambridge, but in 1655 having refused to sign the Engagement to uphold the Commonwealth, he obtained travel grants to go abroad.

Travel
He spent the next four years travelling across France, Italy, Smyrna and Constantinople, and after many adventures returned to England in 1659. He was known for his courageousness. Particularly noted is the occasion of his having saved the ship he was upon, by the merits of his own prowess, from capture by pirates. He is described as "low in stature, lean, and of a pale complexion," slovenly in his dress, and having a committed and long-standing habit of tobacco use (an inveterate smoker). In respect to his courtly activities his aptitude to wit earned him favour with Charles II, and the respect of his fellow courtiers. In his writings one might find accordingly, a sustained and somewhat stately eloquence. He was an altogether impressive personage of the time, having lived a blameless life in which he exercised his conduct with due care and conscientiousness.

Work
On the Restoration in 1660, he was ordained and appointed to the Regius Professorship of Greek at the University of Cambridge. In 1662, he was made professor of geometry at Gresham College, and in 1663 was selected as the first occupier of the Lucasian chair at Cambridge. During his tenure of this chair he published two mathematical works of great learning and elegance, the first on geometry and the second on optics. In 1669 he resigned his professorship in favour of Isaac Newton. About this time, Barrow composed his Expositions of the Creed, The Lord's Prayer, Decalogue, and Sacraments. For the remainder of his life he devoted himself to the study of divinity. He was made a Doctor of Divinity by Royal mandate in 1670, and two years later Master of Trinity College (1672), where he founded the library, and held the post until his death.

His earliest work was a complete edition of the Elements of Euclid, which he issued in Latin in 1655, and in English in 1660; in 1657 he published an edition of the Data. His lectures, delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones Mathematicae; these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year, and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones Opticae et Geometricae. It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of the On Conic Sections of Apollonius of Perga, and of the extant works of Archimedes and Theodosius of Bithynia.

In the optical lectures many problems connected with the reflection and refraction of light are treated with ingenuity. The geometrical focus of a point seen by reflection or refraction is defined; and it is explained that the image of an object is the locus of the geometrical foci of every point on it. Barrow also worked out a few of the easier properties of thin lenses, and considerably simplified the Cartesian explanation of the rainbow.

Barrow was the first to find the integral of the secant function in closed form, thereby proving a conjecture that was well-known at the time.

Death and legacy
Barrow died unmarried in London at the early age of 46, and was buried at Westminster Abbey. John Aubrey, in the Brief Lives, attributes his death to an opium addiction acquired during his residence in Turkey.

Besides the works above mentioned, he wrote other important treatises on mathematics, but in literature his place is chiefly supported by his sermons, which are masterpieces of argumentative eloquence, while his Treatise on the Pope's Supremacy is regarded as one of the most perfect specimens of controversy in existence. Barrow's character as a man was in all respects worthy of his great talents, though he had a strong vein of eccentricity.

Calculating tangents
The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves, and this is sufficiently important to require a detailed notice, because it illustrates the way in which Barrow, Hudde and Sluze were working on the lines suggested by Fermat towards the methods of the differential calculus.

Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it were known; hence, if the length of the subtangent MT could be found (thus determining the point T), then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn, he got a small triangle PQR (which he called the differential triangle, because its sides QR and RP were the differences of the abscissae and ordinates of P and Q), so that K
 * TM : MP = QR : RP.

To find QR : RP he supposed that x, y were the co-ordinates of P, and x − e, y − a those of Q (Barrow actually used p for x and m for y, but this article uses the standard modern notation). Substituting the co-ordinates of Q in the equation of the curve, and neglecting the squares and higher powers of e and a as compared with their first powers, he obtained e : a. The ratio a/e was subsequently (in accordance with a suggestion made by Sluze) termed the angular coefficient of the tangent at the point.

Barrow applied this method to the curves
 * 1) x2 (x2 + y2) = r2y2, the kappa curve;
 * 2) x3 + y3 = r3;
 * 3) x3 + y3 = rxy, called la galande;
 * 4) y = (r − x) tan πx/2r, the quadratrix; and
 * 5) y = r tan πx/2r.

It will be sufficient here to take as an illustration the simpler case of the parabola y2 = px. Using the notation given above, we have for the point P, y2 = px; and for the point Q:
 * (y − a)2 = p(x − e).

Subtracting we get
 * 2ay − a2 = pe.

But, if a be an infinitesimal quantity, a2 must be infinitely smaller and therefore may be neglected when compared with the quantities 2ay and pe. Hence
 * 2ay = pe, that is, e : a = 2y : p.

Therefore,
 * TM : y = e : a = 2y : p.

Hence
 * TM = 2y2/p = 2x.

This is exactly the procedure of the differential calculus, except that there we have a rule by which we can get the ratio a/e or dy/dx directly without the labour of going through a calculation similar to the above for every separate case.

Publications

 * Epitome Fidei et Religionis Turcicae (1658)
 * "De Religione Turcica anno 1658" (poem)
 * Euclidis Elementorum (1659) [in Latin] Euclide's Elements (1660) [in English] translations of Euclid's Elements
 * Lectiones Opticae (1669)
 * Lectiones Geometricae (1670), translated as Geometrical Lectures (1735) by Edmund Stone, later translated as The Geometrical Lectures of Isaac Barrow (1916) by James M. Child
 * Apollonii Conica (1675) translation of Conics
 * Archimedis Opera (1675) translation of Archimedes’s works
 * Theodosii Sphaerica (1675) translation of Theodosius' Spherics
 * A Treatise on the Pope's Supremacy, to which is Added a Discourse Concerning the Unity of the Church (1680) (1859 edition)
 * Lectiones Mathematicae (1683) translated as The Usefulness of Mathematical Learning (1734) by John Kirkby
 * Of Contentment, Patience, and Resignation to the Will of God (1685)
 * The works of the learned Isaac Barrow, D.D. (1700) Vol. 1, Vol. 2–3
 * The Works of Dr. Isaac Barrow (1830), Vol. 1, Vol. 2, Vol. 3, Vol. 4, Vol. 5, Vol. 6, Vol. 7 [sermons and theological essays]

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Isaac Barrow


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Talk:Isaac Barrow

Untitled
An earlier version of this article was taken from the Rouse History of Mathematics, as marked up by Dr. David R. Wilkins, with permission: see article.

 * This article could do with some mathematical proof-reading for correctness.

Infobox
The infobox on this page has been removed. Please discuss for and against removal. To remind you what it looked like, here is a sample:

Please discuss to reach a consensus bunix 12:45, 5 August 2006 (UTC)

 * Infobox looks good idea to me. JackyR | Talk 15:32, 5 August 2006 (UTC)

Well, it has been two weeks since it was intially removed. This has now allowed adequate time for free discussion. So far the discussion has supported the box and there have been no arguments against posted here. Therefore I am now reinstating the box. In future, please can removers of large chunks of info always go to the discussion page first before removal, as per wiki policy. Immediate removal without discussion is only justified for vandalism and wiki violations. bunix 13:22, 11 August 2006 (UTC)

Discovery of FTC
Where the hell is Isaac Barrow's discovery of the fundamental theorem??? He's the father of calculus! —Preceding unsigned comment added by 204.69.190.75 (talk) 23:18, 17 November 2007 (UTC)

 * I just edited to adress this concern. While Barrow did discover the FTC, sometimes things are named for people who extended the scheme. For example, while Dirac discusses the path integral first, Feynman developed it to the point where it was useful. In this regard, Barrow is like Dirac and Newton like Feynman.Likebox (talk) 00:35, 16 February 2008 (UTC)

lectures on archimedes
The page states that Barrow "lectures for 1667 were published in the same year, and suggest the analysis by which Archimedes was led to his chief results." Does anyone have any additional information on this? Are the 1667 lectures online? Tkuvho (talk) 14:25, 22 July 2010 (UTC)

Allegations by Fountains of Bryn Mawr
Hi,

User:Fountains of Bryn Mawr disagrees that James Gregory had influenced Isaac Barrow, however I provided a very robust document that was a 1916 translation of his work, which clearly showed high praise for Gregory's work. It seems this user is just against establishing this relationship for some reason. It shouldn't be a surprise that Barrow had respect for gregory, and was influenced by his work, yet Bryn Mawr just can't let it go. I provided sources, and they try to claim a translated document from over 100 years ago, showing mutual respect, is somehow original research. It is not original research at all. Calling "the most learned man" is one of the highest compliments a scholar could pay another when using their work. I'm seeking moderation because the user may risk violating the 3RR, and I feel what I've provided is quite compelling. I would have to dig a little more, but I find a translated primary source document containing such a phrase as the most compelling piece of evidence. Secondary accounts often are less convincing. Here is some secondary analysis showing the impact of Gregory's work. Secondary analysis such as (page 4, footnote 11) also refers to the quote I've used, but builds on it slightly more by also mentioning Gottfried Leibniz' praise of Gregory's work. So we can see that the work of Gregory impressed both Leibniz and Barrow. To suggest such esteemed scholars (of almost 400 years ago) use phrases like "the most learned man" or "elegant theorem by Gregory" (quite the compliments) without being influenced by them (where such influence is in the form of using their work, at the very least) is unreasonable. I just do not know what the editor wants, and it feels like the evidence provided was sufficient. I will try to find more sources but I do not know how what is already provided isn't sufficient to constitute as evidence for influence. Here is a letter by a prominent publishing intermediary (between Gregory, Barrow, and Newton) John Collins addressed to Gregory While the letter itself is somewhat mundane (he met Sir Isaac, had dinner, Sir Isaac said Opticks was going to continue where Reverend Barrow left off, and that Collins needn't ask about Sir Isaac's publishing commitments because Reverend Barrow said the Lucasian chair had such requirements already), it establishes that, while communication was slow at the time, Collins was clearly acting as a facilitator. Thus, the work of Gregory clearly had an influence on Barrow given that the publisher was writing about their progress to the former. Trying to find more. Here is another secondary source, The Historical Development of the Fundamental Theorem of Calculus And Its Implication in Teaching: Apparently, Fermat reduced a problem of rectification by connecting tangents and the question of quadratures. Surprisingly, for all his deft use of infinitesimals in a variety of areas, he still failed to recognize this critical relation, denying himself the honored title of “true inventor of the calculus” (Boyer, 1959). The man first overtly aware of generality of the Fundamental Theorem of Calculus was James Gregory in 1668, exerting a significant influence on Isaac Barrow’s work. So I have to ask, why is Fountains of Bryn Mawr picking a fight over something that seems to be the consensus? Note: I am not talking about assigning priority/credit for the Fundamental Theorem of Calculus (which is clearly a bit more involved), but the influence of their investigations on eachother's work. Clearly assigning priority of the fundamental theorem is difficult because their collaboration (facilitated by Collins where appropriate) muddled the "who did what" part. Found probably the most definitive scholarly literature on this issue, however, again, I must emphasise that the focus of these scholars is much more than that: Barrow's heavy "borrowings" include his restatement of "Gregory's generalization of Wren's proof of a construction for the tangent at a general point on a cyclid arc," as well as his appropriation of Gregory's "rectifying transformation," the basis of the Scot's involutio and evolutio method. Even barrow's most famous achievement--the proof for the inverse nature of differentiation and integration--was now reinterpreted as a "neat amendment of Gregory's generalization of Neil's rectification method." - Mordechai Feingold

If you dig further into that cited passage, you will find references to DT Whiteside who argues this line of thinking. However, again, we are not discussing priority of the fundamental theorem, nor are we discussing originality. What we are discussing is whether it is reasonable to infer that Gregory's work influenced Barrow, and given the in-depth analyses of these scholars (admittedly focusing on originality and priority for the fundamental theorem of calculus), it is hard to argue against, at the very least, the idea that Gregory's work did influence Isaac Barrow. 174.3.155.181 (talk) 17:05, 5 June 2016 (UTC)

 * Seems to be a misunderstanding of Wikipedia somewhere. We do not "establishing .... relationship", that is a job for reliable secondary sources. These edits were reverted because they were interpretations of primary sources. Despite the opinion of the adding editor that this is a "very robust document", its not ours to judge, a reliable secondary source has to make that judgement. The guideline at Template:Infobox scientist on the influences/influenced parameter also states "Unless the scientist was clearly building on an earlier work, avoid adding influences that were only via study, as such influences are generally too many and hard to separate. Only list influences who are notable enough to warrant their own wiki article." Again, we don't judge that, there should be extensive secondary sources on this influences/influenced such as sections in several reliable text books on the subject, i.e. extensive coverage. Fountains of Bryn Mawr (talk) 20:49, 5 June 2016 (UTC)
 * Here's a published chapter of PhD Dissertation, which was communicated by Dt Whiteside:
 * In the summer of 1670 GREGORIE discovered a second, much more powerful method of tangents, which has to be viewed against the background of ISAAC BARROW'S 1670 Lectiones Geometricae. In July 1670 GREGORIE received from JOHN COLLINS a copy of BARROW'S book fresh from the printer, On September 5 he wrote back: "I have read over Mr Barrow's Lectures with much pleasure and attention, wherein I find him to have infinitely transcended all that ewer [sic] wrote before him. I have discovered from Barrow his method of drawing tangents together with some of my own, a general geometrical method, without calculation, of drawing tangents to all curves, and comprehending not only Barrow's particular methods, but also his general analytical method in the end of the 10th lecture. My method contains not above 12 propositions."(page 102) - James Gregory
 * Apart from the pieces of evidence produced above, other considerations support the reconstruction of GREGORIE'S techniques offered here. As we have seen, no calculus techniques are needed to explain GREGORIE'S series expansions. This is consistent with GREGORIE'S admiration for BARROW'S Lectiones Geometricae and with the content of the mathematical notes exchanged between GREGORIE and BARROW. (page 136)
 * If this reference above is not the type of "smoking gun" that demonstrates the collaborative relationship between Gregory and Barrow, where the influence of the former on the latter is on display, I don't know what is. 174.3.155.181 (talk) 17:05, 5 June 2016 (UTC)
 * with all due respect Fountains of Bryn Mawr, please spare me your superficially impartial response. On Gregory's page there are numerous (7) references around the statement that Gregory was acknowledged by Barrow to have produced the earliest rudimentary form of the fundamental theorem of calculus. This is no small acknowledgment because this theorem became central to Isaac Barrow's study. It is so central to his study that it's the ONLY entry under Barrow's "known for".
 * You then defer to the Wiki statement "Unless the scientist was clearly building on an earlier work, avoid adding influences that were only via study, as such influences are generally too many and hard to separate. Only list influences who are notable enough to warrant their own wiki article."
 * You do realise the "earlier work" is the Fundamental Theorem of Calculus, which Gregorie established, and Barrow continued, right? This invalidates your use of this quote because (I'm going to drop my claim into the first part of the quote you've used, just so we're clear that this statement doesn't apply): "Isaac Barrow was clearly building on an earlier work [of James Gregory's, involving the Fundamental theorem of Calculus]". This cannot be any more clear than it is already is... 174.3.155.181 (talk) 21:16, 5 June 2016 (UTC)
 * Sorry, you still have not filled in the primary claim "Isaac Barrow was primarily influenced by James Gregory" and "the person James Gregory primarily influenced was Isaac Barrow". Can you supply a reference directly supporting that? We are talking basic WP:V here re: "providing a citation to a reliable source that directly supports the contribution" Of the 7 references you note to 4 have nothing to do with the claim (they were there before the Barrow mention was added), two redundantly repeat the same claim (Barrow acknowledging Gregory's theorem), and one mentions Barrow and Gregory on the same page with many other thinkers. Cherry picking and repeating the same noted cross contributions and acknowledgments by Barrow and Gregory does not support the above statements, they are actually supported by some reliable secondary source making those statements. Fountains of Bryn Mawr (talk) 01:21, 6 June 2016 (UTC)
 * You do realise the "earlier work" is the Fundamental Theorem of Calculus, which Gregorie established, and Barrow continued, right? This invalidates your use of this quote because (I'm going to drop my claim into the first part of the quote you've used, just so we're clear that this statement doesn't apply): "Isaac Barrow was clearly building on an earlier work [of James Gregory's, involving the Fundamental theorem of Calculus]". This cannot be any more clear than it is already is... 174.3.155.181 (talk) 21:16, 5 June 2016 (UTC)
 * Sorry, you still have not filled in the primary claim "Isaac Barrow was primarily influenced by James Gregory" and "the person James Gregory primarily influenced was Isaac Barrow". Can you supply a reference directly supporting that? We are talking basic WP:V here re: "providing a citation to a reliable source that directly supports the contribution" Of the 7 references you note to 4 have nothing to do with the claim (they were there before the Barrow mention was added), two redundantly repeat the same claim (Barrow acknowledging Gregory's theorem), and one mentions Barrow and Gregory on the same page with many other thinkers. Cherry picking and repeating the same noted cross contributions and acknowledgments by Barrow and Gregory does not support the above statements, they are actually supported by some reliable secondary source making those statements. Fountains of Bryn Mawr (talk) 01:21, 6 June 2016 (UTC)

 * Sorry, you haven't done anything to refute my claims.
 * Bryn Mawr started this edit-war by first requiring a good source. I then produced such a source, which was a translated text from 1916 where Barrow pays high praise to Gregory and refers to his work. They reverted again, this time using the argument:
 * "The guideline at Template:Infobox scientist on the influences/influenced parameter also states 'Unless the scientist was clearly building on an earlier work, avoid adding influences that were only via study, as such influences are generally too many and hard to separate. Only list influences who are notable enough to warrant their own wiki article.' Again, we don't judge that, there should be extensive secondary sources on this influences/influenced such as sections in several reliable text books on the subject, i.e. extensive coverage. Fountains of Bryn Mawr (talk) 20:49, 5 June 2016 (UTC)"
 * Instead of using "the lede gives seven citations that support Barrow's acknowledgment of Gregory being first to publish a rudimentary form of the fundamental theorem of calculus" as my main argument to invalidate the need for secondary sources (again, because Barrow's work was primarily about the Fundamental theorem, and only Gregory preceded him in studying this specific concept), I entertained Bryn Mawr by procuring many secondary sources. Such effort on my behalf demonstrates a good faith effort to address the (invalid) concerns of Bryn Mawr, because again: Barrow's work on the fundamental theorem of calculus was building upon the work of Gregory, who he acknowledged was "first" (Barrow would be "second", by this measure).
 * I have called for additional people to comment, and notified User:TransporterMan about the failure to procure a productive discussion in which Bryn Mawr provides _some_ evidence that AT LEAST instills doubt in this claim. TransporterMan: can we please speed this up? I really think the references provided (especially the last few) are quite good, but given that the person who disagrees is representing a "prestigious" academic institution that was once home to Emmy Noether, I do not think this individual is going to "back down". They understand their institutional reputation is somewhat at stake here, especially with a "no name loser from Canada" (my words, not theirs) giving them a lesson on "it's not the institution one attends, but the work they produce".
 * They have not even read ANY of the content I've cited because, as I suggested before this entire exchange began, they are not going to accept anything other than "their way", which was to remove the justified insertion of Gregory as an influence on Barrow. It is unacceptable to expect the most productive contributors to this site to tolerate reversions that are borne out of bias instead of facts. It is the second time I've experienced "being right" (and providing the appropriate facts) and have had to continue a conversation that I felt was over.
 * I looked at their edit history, and in my opinion they haven't even demonstrated sufficient understanding of mathematics to truly comment on this issue. Being that it was incumbent upon me to make my case, I looked past it and did just that; for any observer to suggest that I did not "focus on the material" before "focusing on the user" would be disingenuous. The works I've provided (especially the published stuff) comprehensively study the originality/priority of the Fundamental Theorem, and in doing that, discuss the interactions between contributors such as Barrow, Newton and Gregory.
 * If you compare my edit history with Bryn Mawr, you will see that their edits are usually on the periphery. My edit commentaries may be occasionally excessive, but I stick to the facts and provide "meaty" additions in mathematics. Bryn Mawr user has not even provided any evidence they have an understanding of the issue being discussed, which I feel would help them understand why Barrow was influenced by Gregory.
 * It is unfair to expect those who invest the time and effort to improve this site, to have their voice erased because those without similar merit change the goalposts. I don't have time to read or edit pages of "richest people" or other meaningless things like Bryn Mawr. I'm here to stick to the facts, and when other users take bigger bites than that which they are capable of masticating, it is hard to "hold back". It is reminiscent of the edits on another page, where it was deemed that my arguments were correct but my edit was reverted because User:Floquenbeam wanted to exercise their authority. Where is Bryn Mawr's evidence to provide an alternative theory? Why aren't they being held accountable for prolonging of a discussion to which they've added little, but contested much? Do they feel that "I go to the school that Emmy Noether attended" is a sufficient basis to present arguments that not only change upon presentation of solid evidence, but are light on facts?
 * It is getting increasingly tiresome to have make a case to an editor whose opinion is not going to change, regardless of the facts provided. This is the second time I've experienced this, and I think it will be my last because the system is not conducive to correctness. Expecting someone to "eventually agree", in light of the facts provided, has proven futile twice now. Previously, a physicist User:Ymblanter was brave enough to agree with me and support my claim. This time I may not get any such support, yet provided a stronger case than the reverter. You are going to lose good editors on this site if you allow conduct like this to continue. It's not helpful to building an informative encyclopedia.174.3.155.181 (talk) 02:23, 6 June 2016 (UTC)

Votes

 * Support inclusion. Based on these sources, Gregory was clearly an important influence on the development of Barrow's mathematical thinking, indeed, perhaps the most important. In contrast, the other two people listed under "Influences", Gilles Personne de Roberval and Vincenzo Viviani, appear merely to have been professors he worked under; the article gives no evidence that they were particular influences. (WP does not normally list people as influences merely because they were professors under which a person had at some point studied.) Secondary sources are preferable, especially when the primary sources disagree or are inconclusive, but in this case that is not a problem. (Note: evidence for Barrow's appreciation of Gregory's work, on the other hand, is not sufficient to establish influence.) HGilbert (talk) 09:50, 6 June 2016 (UTC)
 * Oppose inclusion in influences/influenced but should be mentioned in the body. Despite the adding editor wanting to blow this up into an edit war and going personal immediately and continually on the talk page (somebody needs to read WP:TPG) there is clear merit to adding Barrow to the body. Mentioning in the lead is contrary to WP:LEAD, not covered in article. We have one noted instance of influence with no reference to whether this "influenced the scientist significantly", criteria at Template:Infobox scientist for influences/influenced inclusion. Fountains of Bryn Mawr (talk) 17:49, 6 June 2016 (UTC)
 * Abstain This is storm-in-teacup stuff. There seems to be general agreement on the plausibility of the influences; the disagreements are mainly about where and how it is OK to mention it (body-or-lede etc) I don't see anything wrong with its appearing in the table, except that it is not supported in context (yes, it is ref'd, but that is not enough to be of use ti the casual reader). If there is a case for influence one way or another, put it coherently and in context into the body. Create a special section on influence if you like. But in its current form this is just a waste of time and goodwill. JonRichfield (talk) 09:40, 10 June 2016 (UTC)
 * Comment The above discussion is practically unreadable, not helped by off-topic impugning of motives muddled layout and what seem to be unsigned posts. The disagreement seems to focus on whether the influence is sufficient to include in the info-box/lead. The argument for seems synth-y but it is too hard to find clearly what that argument is. I am not voting since it is just unreasonable to expect RfC commenters to wade through a sea of what is more argumentation than argument. Suggest you close and try again. Pincrete (talk) 09:24, 11 June 2016 (UTC)
 * Abstain This discussion is not of a sufficient standard to form any meaningful view. It should be closed with no consensus for change. Naturenet | Talk 17:42, 11 June 2016 (UTC)
 * Comment Like other commentators I'm struggling to parse exactly what the dispute here is over. In the future, please spend some time discussing your disagreements and seeking to clarify or narrow them before resorting to an RFC. However, some general principles: if we don't have a reliable secondary source that says "A was influenced by B", we should refrain from saying that "A was influenced by B". We generally shouldn't use primary sources; I think they can be allowed if their meaning is extremely obvious (e.g. if an author writes in the foreword to their book "I am greatly indebted to X whose work forms the basis of my own", it's okay to quote that)–but if there is any doubt about the correct interpretation of a primary source, we must defer to reliable secondary sources rather than seeking to solve those doubts ourselves. If we have a reliable secondary source which says this, we can cite it, but should refrain from making a stronger claim than the cited source actually supports; for example, we should not say "A was primarily influenced by B" if the source doesn't actually use the adverb primarily (or some of form of words with an obviously similar meaning.) SJK (talk) 04:53, 13 June 2016 (UTC)
 * Too long; didn't read. Anup   [Talk]  10:41, 18 June 2016 (UTC)
 * Comment - Try asking a civil concise question rather than posting an angry wall of text. Maybe I will read and parse later.  Not now.  Robert McClenon (talk) 15:01, 22 June 2016 (UTC)
 * Include the sourced connection, but not a claim of "influenced", without clear sources for that more specifically. It's original research to translate editorial praise into direct absorption of influence.  I have a great deal of respect for the music of Marvin Gaye, but he would not be any kind of influence on my industro-metal band, for example.   — SMcCandlish ☺ ☏ ¢ ≽ʌⱷ҅ᴥⱷʌ≼  23:30, 26 June 2016 (UTC)
 * Support Inclusion i put a decent amount of time into finding good sources, and to have someone close it before my vote is counted (whilst the opposition's is) feels disingenuous, especially since the opposition put no work into providing evidence countering the claims. — Preceding unsigned comment added by 174.3.155.181 (talk) 20:25, 24 July 2016 (UTC)

it is offensive, and consequently frustrating, that User:Fountains of Bryn Mawr, who has not shown any understanding, respect OR appreciation, of the *actually* RICH History of Calculus, continues to act as an authority on this matter when their pretentiousness is palpable with each piece of text.
 * Support Inclusion i put a decent amount of time into finding good sources, and to have someone close it before my vote is counted (whilst the opposition's is) feels disingenuous, especially since the opposition put no work into providing evidence countering the claims. — Preceding unsigned comment added by 174.3.155.181 (talk) 20:25, 24 July 2016 (UTC)
 * it was claimed User:S Marshall "gave very clear reasons for the edit in talk", which were Procedurally speaking, the only way I could decide to include that sentence in the infobox would be if I could find a clear consensus in favour, and I'm afraid it's not possible to find that consensus in the discussion below.
 * my edit prior to Mawr's reversion clearly stated that I had, per User:S Marshall's protocol, taken it to his talk page. however, given i was absent for the vote that he clearly stated motivated his action, i have reverted it until he & i have a discussion.
 * if he is unsatisfied with the sources i will try and find more to accommodate him, but let the process play out. i will have more confidence in his decision than your eager reversions. i think this is a completely acceptable course of action given my absence (for which i apologise) 174.3.155.181 (talk) 21:00, 24 July 2016 (UTC)

You asked for the RFC, you got an answer. Your opinion was well known and even if you inserted it again under "votes" it would make no difference --- "when there is no consensus among editors at RfC, the status quo ante applies". Rejecting community input, tendentious editing, ignoring WP:TPG, and other WP:DISRUPT is probably going to lead somewhere else fast. Fountains of Bryn Mawr (talk) 21:38, 24 July 2016 (UTC)

 * Please note this edit. For the time being, the correct place for further discussion of this issue is the administrators' noticeboard which is here.  The time allotted for the RfC has expired, and you are all respectfully advised not to make any further edits to it here.— S Marshall  T/C 17:05, 25 July 2016 (UTC)

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Death
Barrow died at the age 46. It would be worth mentioning what he died from — Preceding unsigned comment added by 73.50.99.110 (talk) 23:06, 14 January 2019 (UTC)

title: Talk:Isaac Barrow


text:
Jules Richard (mathematician)

Jules Richard (12 August 1862 – 14 October 1956) was a French mathematician who worked mainly in geometry but his name is most commonly associated with Richard's paradox.

Life and works
Richard was born in Blet, in the Cher département.

He taught at the lycées of Tours, Dijon and Châteauroux. He obtained his doctorate, at age of 39, from the Faculté des Sciences in Paris. His thesis of 126 pages concerns Fresnel's wave-surface. Richard worked mainly on the foundations of mathematics and geometry, relating to works by Hilbert, von Staudt and Méray.

In a more philosophical treatise about the nature of axioms of geometry Richard discusses and rejects the following basic principles:
 * 1) Geometry is founded on arbitrarily chosen axioms - there are infinitely many equally true geometries.
 * 2) Experience provides the axioms of geometry, the basis is experimental, the development deductive.
 * 3) The axioms of geometry are definitions (in contrast to (1)).
 * 4) Axioms are neither experimental nor arbitrary, they force themselves on us since without them experience is not possible.

The latter approach was essentially that proposed by Kant. Richard arrived at the result that the notion of identity of two objects and the invariability of an object are too vague and need to be specified more precisely. This should be done by axioms.

"Axioms are propositions, the task of which is to make precise the notion of identity of two objects pre-existing in our mind."

Further according to Richard, it is the aim of science to explain the material universe. And although non-Euclidean geometry had not found any applications (Albert Einstein finished his general theory of relativity only in 1915), Richard already stated clairvoyantly:

"One sees that having admitted the notion of angle, one is free to choose the notion of straight line in such a way that one or another of the three geometries is true."

Richard corresponded with Giuseppe Peano and Henri Poincaré. He became known to more than a small group of specialists by formulating his paradox which was extensively use by Poincaré to attack set theory whereupon the advocates of set theory had to refute these attacks.

He died in 1956 in Châteauroux, in the Indre département, at the age of 94.

Richard's paradox
The paradox was first stated in 1905 in a letter to Louis Olivier, director of the Revue générale des sciences pures et appliquées. It was published in 1905 in the article Les Principes des mathématiques et le problème des ensembles. The Principia Mathematica by Alfred North Whitehead and Bertrand Russell quote it together with six other paradoxes concerning the problem of self-reference. In one of the most important compendia of mathematical logic, compiled by Jean van Heijenoort, Richard's article is translated into English. The paradox can be interpreted as an application of Cantor's diagonal argument. It inspired Kurt Gödel and Alan Turing to their famous works. Kurt Gödel considered his incompleteness theorem as analogous to Richard's paradox which, in the original version runs as follows:

Let E be the set of real numbers that can be defined by a finite number of words. This set is denumerable. Let p be the nth decimal of the nth number of the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal, if p is not equal either to 8 or 9, and unity in the contrary case. This number N does not belong to the set E because it differs from any number of this set, namely from the nth number by the nth digit. But N has been defined by a finite number of words. It should therefore belong to the set E. That is a contradiction.

Richard never presented his paradox in another form, but meanwhile there exist several different versions, some of which being only very loosely connected to the original. For the sake of completeness they may be stated here.

Other versions of Richard's paradox
(A) The version given in Principia Mathematica by Whitehead and Russell is similar to Richard's original version, alas not quite as exact. Here only the digit 9 is replaced by the digit 0, such that identities like 1.000... = 0.999... can spoil the result.

(B) Berry's Paradox, first mentioned in the Principia Mathematica as fifth of seven paradoxes, is credited to Mr. G. G. Berry of the Bodleian Library. It uses the least integer not nameable in fewer than nineteen syllables; in fact, in English it denotes 111,777. But "the least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction

(C) Berry's Paradox with letters instead of syllables is often related to the set of all natural numbers which can be defined by less than 100 (or any other large number) letters. As the natural numbers are a well-ordered set there must be the least number which cannot be defined by less than 100 letters. But this number was just defined by 65 letters including spaces.

(D) König's Paradox was also published in 1905 by Julius König. All real numbers which can be defined by a finite number of words form a subset of the real numbers. If the real numbers can be well-ordered, then there must be a first real number (according to this order) which cannot be defined by a finite number of words. But the first real number which cannot be defined by a finite number of words has just been defined by a finite number of words.

(E) The smallest natural number without interesting properties acquires an interesting property by this very lack of any interesting properties.

(F) A loan of the Paradox of Grelling and Nelson. The number of all finite definitions is countable. In lexical order we obtain a sequence of definitions D1, D2, D3,  ... Now, it may happen that a definition defines its own number. This would be the case if D1 read "the smallest natural number". It may happen, that a definition does not describe its own number. This would be the case if D2 read "the smallest natural number". Also the sentence "this definition does not describe its number" is a finite definition. Let it be Dn. Is n described by Dn. If yes, then no, and if no, then yes. The dilemma is irresolvable. (This version is described in more detail in another article, Richard's paradox.)

Reactions to Richard's paradox
Georg Cantor wrote in a letter to David Hilbert:
 * "Infinite definitions" (i.e., definitions which cannot be done in finite time) are absurdities. If Königs statement was "correct", according to which all "finitely definable" real numbers form a collection of cardinal number $$\aleph_0$$, this would imply the countability of the whole continuum; but this is obviously wrong. The question is now what error the alleged proof of his wrong theorem is based upon. The error (which also appears in the note of a Mr. Richard in the last issue of the Acta mathematic, which Mr. Poincaré emphasizes in the last issue of the Revue de Métaphysique et de Morale) is, in my opinion, the following: It is assumed that the system {B} of notions B, which have to be used for the definition of individual numbers, is at most countably infinite. This assumption "must be in error" because otherwise we would have the wrong theorem: "the continuum of numbers has cardinality $$\aleph_0$$".

Here Cantor is in error. Today we know that there are uncountably many real numbers without the possibility of a finite definition.

Ernst Zermelo comments Richard's argument:
 * The notion "finitely definable" is not an absolute one but a relative one being always related to the "language" chosen. The conclusion according to which all finitely definable objects are countable is only valid in case that one and the same system of symbols is used; the question whether a single individual can be subject to a finite definition is void because to every thing an arbitrary name can be attached to.

Zermelo points to the reason why Richard's paradox fails. His last statement, however, is impossible to satisfy. A real number with infinitely many digits, which are not determined by some "rule", has an infinitely large contents of information. Such a number could only be identified by a short name if there were only one or few of them existing. If there exist uncountably many, as is the case, an identification is impossible.

Literature about the Richard's paradox

 * H. Meschkowski, W. Nilson: Georg Cantor - Briefe, Sphinhubyringer, Berlin 1991, p. 446.
 * W. Mückenheim: Die Mathematik des Unendlichen, Shaker, Aachen 2006.
 * A. N. Whitehead, B. Russell: Principia Mathematica I, Cambridge Univ. Press, Cambridge 1910, p. 64.
 * E. Zermelo: Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908) p. 107-128.

title: Jules Richard (mathematician)


text:
Richard's paradox

In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics.

Kurt Gödel specifically cites Richard's antinomy as a semantical analogue to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The paradox was also a motivation for the development of predicative mathematics.

Description
The original statement of the paradox, due to Richard (1905), is strongly related to Cantor's diagonal argument on the uncountability of the set of real numbers.

The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, "The real number the integer part of which is 17 and the nth decimal place of which is 0 if n is even and 1 if n is odd" defines the real number 17.1010101... = 1693/99, whereas the phrase "the capital of England" does not define a real number, nor the phrase "the smallest positive integer not definable in under sixty letters" (see Berry's paradox).

There is an infinite list of English phrases (such that each phrase is of finite length, but the list itself is of infinite length) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length lexicographically, so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1.

The preceding paragraph is an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers rn. However, r was constructed so that it cannot equal any of the rn (thus, r is an undefinable number). This is the paradoxical contradiction.

Analysis and relationship with metamathematics
Richard's paradox results in an untenable contradiction, which must be analyzed to find an error.

The proposed definition of the new real number r clearly includes a finite sequence of characters, and hence it seems at first to be a definition of a real number. However, the definition refers to definability-in-English itself. If it were possible to determine which English expressions actually do define a real number, and which do not, then the paradox would go through. Thus the resolution of Richard's paradox is that there is not any way to unambiguously determine exactly which English sentences are definitions of real numbers (see Good 1966). That is, there is not any way to describe in a finite number of words how to tell whether an arbitrary English expression is a definition of a real number. This is not surprising, as the ability to make this determination would also imply the ability to solve the halting problem and perform any other non-algorithmic calculation that can be described in English.

A similar phenomenon occurs in formalized theories that are able to refer to their own syntax, such as Zermelo–Fraenkel set theory (ZFC). Say that a formula φ(x) defines a real number if there is exactly one real number r such that φ(r) holds. Then it is not possible to define, by ZFC, the set of all (Gödel numbers of) formulas that define real numbers. For, if it were possible to define this set, it would be possible to diagonalize over it to produce a new definition of a real number, following the outline of Richard's paradox above. Note that the set of formulas that define real numbers may exist, as a set F; the limitation of ZFC is that there is not any formula that defines F without reference to other sets. This is related to Tarski's undefinability theorem.

The example of ZFC illustrates the importance of distinguishing the metamathematics of a formal system from the statements of the formal system itself. The property D(φ) that a formula φ of ZFC defines a unique real number is not itself expressible by ZFC, but must be considered as part of the metatheory used to formalize ZFC. From this viewpoint, Richard's paradox results from treating a construction of the metatheory (the enumeration of all statements in the original system that define real numbers) as if that construction could be performed in the original system.

Variation: Richardian numbers
A variation of the paradox uses integers instead of real numbers, while preserving the self-referential character of the original. Consider a language (such as English) in which the arithmetical properties of integers are defined. For example, "the first natural number" defines the property of being the first natural number, one; and "divisible by exactly two natural numbers" defines the property of being a prime number (It is clear that some properties cannot be defined explicitly, since every deductive system must start with some axioms. But for the purposes of this argument, it is assumed that phrases such as "an integer is the sum of two integers" are already understood). While the list of all such possible definitions is itself infinite, it is easily seen that each individual definition is composed of a finite number of words, and therefore also a finite number of characters. Since this is true, we can order the definitions, first by length and then lexicographically.

Now, we may map each definition to the set of natural numbers, such that the definition with the smallest number of characters and alphabetical order will correspond to the number 1, the next definition in the series will correspond to 2, and so on. Since each definition is associated with a unique integer, then it is possible that occasionally the integer assigned to a definition fits that definition. If, for example, the definition "not divisible by any integer other than 1 and itself" happened to be 43rd, then this would be true. Since 43 is itself not divisible by any integer other than 1 and itself, then the number of this definition has the property of the definition itself. However, this may not always be the case. If the definition: "divisible by 3" were assigned to the number 58, then the number of the definition does not have the property of the definition itself, since 58 is itself not divisible by 3. This latter example will be termed as having the property of being Richardian. Thus, if a number is Richardian, then the definition corresponding to that number is a property that the number itself does not have. (More formally, "x is Richardian" is equivalent to "x does not have the property designated by the defining expression with which x is correlated in the serially ordered set of definitions".) Thus in this example, 58 is Richardian, but 43 is not.

Now, since the property of being Richardian is itself a numerical property of integers, it belongs in the list of all definitions of properties. Therefore, the property of being Richardian is assigned some integer, n. For example, the definition "being Richardian" might be assigned to the number 92. Finally, the paradox becomes: Is 92 Richardian? Suppose 92 is Richardian. This is only possible if 92 does not have the property designated by the defining expression which it is correlated with. In other words, this means 92 is not Richardian, contradicting our assumption. However, if we suppose 92 is not Richardian, then it does have the defining property which it corresponds to. This, by definition, means that it is Richardian, again contrary to assumption. Thus, the statement "92 is Richardian" cannot consistently be designated as either true or false.

Relation to predicativism
Another opinion concerning Richard's paradox relates to mathematical predicativism. By this view, the real numbers are defined in stages, with each stage only making reference to previous stages and other things that have already been defined. From a predicative viewpoint it is not valid to quantify over all real numbers in the process of generating a new real number, because this is believed to result in a circularity problem in the definitions. Set theories such as ZFC are not based on this sort of predicative framework, and allow impredicative definitions.

Richard (1905) presented a solution to the paradox from the viewpoint of predicativism. Richard claimed that the flaw of the paradoxical construction was that the expression for the construction of the real number r does not actually define a real number unambiguously, because the statement refers to the construction of an infinite set of real numbers, of which r itself is a part. Thus, Richard says, the real number r will not be included as any rn, because the definition of r does not meet the criteria for being included in the sequence of definitions used to construct the sequence rn. Contemporary mathematicians agree that the definition of r is invalid, but for a different reason. They believe the definition of r is invalid because there is no well-defined notion of when an English phrase defines a real number, and so there is no unambiguous way to construct the sequence rn.

Although Richard's solution to the paradox did not gain favor with mathematicians, predicativism is an important part of the study of the foundations of mathematics. Predicativism was first studied in detail by Hermann Weyl in Das Kontinuum, wherein he showed that much of elementary real analysis can be conducted in a predicative manner starting with only the natural numbers. More recently, predicativism has been studied by Solomon Feferman, who has used proof theory to explore the relationship between predicative and impredicative systems.

title: Richard's paradox