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5,000 | Barlog confirmed that "God of War" would not have microtransactions post-launch, a feature that had become prominent with other games and criticized. Barlog also confirmed there would not be any post-release DLC, like an expansion pack. He said he had pitched an idea for DLC, "but it was too ambitious". His idea was similar in scope to that of "" and "", large standalone expansions for "The Last of Us" (2013) and "" (2016), respectively. He said it would have been too big to be DLC, warranting its own standalone release. | https://en.wikipedia.org/wiki?curid=50810460 |
5,001 | Since launch, Santa Monica has supported the game with patch updates to address software bugs. As well, the developers have added new features along with these free updates. A Photo Mode was released as part of update patch 1.20 on May 9, 2018. It allows players to take customized in-game screenshots. Players can adjust the field of view, depth of view, filters, borders, the visibility of characters, and the ability to change the facial expressions of Kratos and Atreus. A New Game Plus mode was released as part of update patch 1.30 on August 20, 2018. To access the mode, players must have completed the game at any difficulty level. The mode itself can be played at any difficulty, but enemies are at a higher level with new maneuvers. All obtained items carry over to New Game Plus, and there are new resources to further upgrade gear, which also have new rarity levels. The option to skip cutscenes was also added. In November 2020, the PlayStation 5 (PS5) launched and is backwards compatible with PlayStation 4 games; these games see a performance boost when playing on the PS5 thanks to the power of the newer console. To further enhance the playing experience of "God of War" on the PS5, Santa Monica released an enhancement update on February 2, 2021, allowing the game to be played at full 60fps with checkerboard 4K resolution. | https://en.wikipedia.org/wiki?curid=50810460 |
5,002 | As part of Sony's larger efforts to port their first-party exclusive games to Windows (PC), Santa Monica Studio announced in October 2021 that "God of War" would be released for Windows on January 14, 2022. The port, handled by Jetpack Interactive with supervision by Santa Monica, includes additional graphic options support for Windows, including Nvidia's Deep Learning Super Sampling (DLSS) technology and ultra-widescreen support. This in turn marks the first main entry in the series to release on a non-PlayStation platform. According to Santa Monica's Matt DeWald, they had considered what options they could use to port their games to Windows, particularly as they used a non-standard game engine, and worked closely with Jetpack to determine the scope and technical issues associated with the port. | https://en.wikipedia.org/wiki?curid=50810460 |
5,003 | God of War: A Call from the Wilds is a text-based game playable through Facebook Messenger. To help further promote "God of War", Sony partnered with Facebook, Inc. to develop the play-by-web game, which released on February 1, 2018. Completing the game unlocks downloadable concept art. The short story follows Atreus on his first adventure in the Norse wilds. After archery training and learning runes with his mother, Atreus ventures into the wilderness after telepathically hearing the voice of a dying deer; he finds it covered in blood and stays with it during its final moments. A couple of draugrs appear and Atreus attempts to fight them but is injured. He is saved by his father, Kratos, who was out hunting. The two then battle a revenant before returning home. | https://en.wikipedia.org/wiki?curid=50810460 |
5,004 | God of War: Mímir's Vision is a smartphone companion app that was released on April 17, 2018, for Apple and Android devices. Using alternate reality, it provides some background for the Norse setting of "God of War". | https://en.wikipedia.org/wiki?curid=50810460 |
5,005 | Raising Kratos is a YouTube documentary of Santa Monica Studio's five-year process in making the game, showing the "herculean effort" that went into reviving the franchise. The documentary was announced on April 20, 2019, the one year anniversary of the game's launch, and was released the following month on May 10. | https://en.wikipedia.org/wiki?curid=50810460 |
5,006 | "The Art of God of War" is a book collecting various pieces of artworks created for the game during its development. It was written by Evan Shamoon and published by Dark Horse on April 24, 2018. | https://en.wikipedia.org/wiki?curid=50810460 |
5,007 | An official novelization of the game, written by Cory Barlog's father, James M. Barlog, was released on August 28, 2018, by Titan Books. An audiobook version is also available, narrated by Alastair Duncan, who voiced Mímir in the game. | https://en.wikipedia.org/wiki?curid=50810460 |
5,008 | The novel retells the events of the game, but unlike the series' previous two novels, this one closely follows the source material with a few notable exceptions. The game never revealed how or why Kratos ended up in ancient Norway, or how much time had passed since the ending of "God of War III", but the novel gives some indication. Kratos chose to leave ancient Greece to hide his identity and change who he was. At some point after leaving Greece, he battles some wolves and is saved by a cloaked female figure, presumably Faye. Later, during their journey, Kratos, Atreus, and Mímir see a mural with the wolves Sköll and Hati. This causes Kratos to have a flashback to that battle and makes him wonder if they dragged him to this new land and if so, why. There was also some retconning. At the end of "God of War III", Kratos had the Blades of Exile, but this novel says he had the Blades of Chaos after killing Zeus. It is also mentioned that he tried several times to get rid of the blades, but by fate they kept returning to him. (For example, he threw them off a cliff, but they washed up on shore near him.) Sometime after ending up in Norway, he decided to hide them under his house and never use them again. This moment was said to have occurred 50 years before the start of the current story. When Kratos does recover the Blades of Chaos, he hears Pandora's speech about hope from "God of War III". | https://en.wikipedia.org/wiki?curid=50810460 |
5,009 | In the game, Kratos sees one last image on the mural in Jötunheim. It seemingly shows Atreus holding Kratos's dead body, but in the novel, this mural is partially broken and does not show the corpse that Atreus is holding. Brok and Sindri also reveal why they made the Leviathan Axe for Faye. She had come to them as the last Guardian of Jötnar and needed a weapon to protect her people. The Huldra Brothers crafted the Leviathan Axe for her to be Mjölnir's equal. Mímir also mentioned that Faye, or rather Laufey the Just, thwarted many of the Æsir's plans, including freeing slaves, and Thor could never find her. Kratos's Guardian Shield is never mentioned, and Modi does not ambush them, resulting in Atreus falling ill. Atreus falls ill shortly after the first encounter when Kratos kills Magni. | https://en.wikipedia.org/wiki?curid=50810460 |
5,010 | "God of War: B is For Boy" is an "ABC storybook for adults" in which the story of the game is retold in an abridged format with illustrations. The title comes from Kratos referring to Atreus as "boy" for most of the game. It was written by Andrea Robinson, with the illustrations being provided by Romina Tempest. It released on September 1, 2020, by Insight Editions. | https://en.wikipedia.org/wiki?curid=50810460 |
5,011 | "God of War: Lore and Legends" is a tome that recreates Atreus' journal from the game. The book features new expanded lore that was written in collaboration with the writing team for the game. It was written by Rick Barba and published by Dark Horse on September 9, 2020. | https://en.wikipedia.org/wiki?curid=50810460 |
5,012 | "God of War" received "universal acclaim" according to review aggregator Metacritic, tying it with the original "God of War" for the highest score in the franchise. It has the fourth-highest score of all-time for a PlayStation 4 game, and the highest score for an original, non-remastered PlayStation 4 exclusive. It was the highest rated PlayStation 4 game of 2018 until the release of "Red Dead Redemption 2" in October, which pushed "God of War" to second. It is also tied with the Xbox One version of "Celeste" for the second-highest score of 2018, regardless of platform. "God of War" received particular praise for its art direction, graphics, combat system, music, story, use of Norse mythology, characters, and cinematic feeling. Many reviewers felt it had successfully revitalized the series without losing the core identity of its predecessors. | https://en.wikipedia.org/wiki?curid=50810460 |
5,013 | The story was well praised. Nick Plessas of "Electronic Gaming Monthly" ("EGM") said the story's most memorable moments were the interactions between Kratos and Atreus. He also noted, "there is often some comic relief to be found when Kratos's curtness and Atreus' charming naivety collide." He felt the presence of Atreus showed a side to Kratos not seen before, and that Kratos had evolved emotionally: "The rage and pain of his past is in constant conflict with his desire to spare his son from it, which comes across in even the most subtle actions and words, demonstrating the effort he is putting in." Plessas said Atreus' character was similarly complex. He commented it is easy for child characters "to succumb to a number of annoying child archetypes," but Atreus is more like a young man who is doing his best in an adult world. "Game Informer"s Joe Juba similarly praised the story, particularly the relationship between Kratos and Atreus: "The interactions of Kratos and Atreus range from adversarial to compassionate, and these exchanges have ample room to breathe and draw players in." Juba said that Kratos conveys more character than in any previous game. Peter Brown of "GameSpot" felt that although Kratos and Atreus were enjoyable, it was Mímir who stole the show. He also said that regardless of which character the player meets, the cast of "God of War" is "strong, convincing, and oddly enchanting." Writing for "Game Revolution", Jason Faulkner praised Santa Monica for creating a sequel that new players would be able to understand without having played any of the previous games, while at the same time providing story references to those past games that returning fans would appreciate. Speaking of the relationship between Kratos and Atreus, Faulkner wrote that, "Watching the two grow throughout their journey is incredibly rewarding," equating it to that of Ellie and Joel from "The Last of Us" or Lee and Clementine from Telltale Games' "The Walking Dead". | https://en.wikipedia.org/wiki?curid=50810460 |
5,014 | In terms of the game's combat system, Plessas said that unlike previous games, which often relied on the player to use many combos in a sequential fashion, this game is "more about individual moves strung together in response to the assortment of enemies being fought." Although that difference may be small, he said that the independent attacks of the axe "feature benefits and drawbacks players will need to understand and master to be as effective as possible." Furthermore, although the axe is "conceptually simple", it is "mechanically fascinating". It "succeeds as both a versatile means of dismembering foes and as a key element in puzzle solving." He felt the axe and all of its features was "distinctly rewarding to use" and that it had more versatility than all of the weapons in many other games. Juba said the Leviathan Axe is "a well-balanced and entertaining tool of destruction." He liked how it "emphasizes a more calculated style of combat; instead of zoomed-out, combo-driven encounters, Leviathan makes you a tactician." He also enjoyed how the combat system gradually unfolded through the course of the game; although seemingly restrictive at first, he noted players will be rapidly alternating between weapons and skills. While some reviewers greatly enjoyed the ability to call the Leviathan Axe back to Kratos's hand, Chris Carter of "Destructoid" felt it got old after a while. Atreus' implementation was praised. Plessas said Atreus is "surprisingly useful" and that he "lands in the perfect spot on the spectrum between independence and reliance." Faulkner noted that, "The interplay between Kratos ax, fists, and shield, and Atreus' bow makes for an impressive fighting system." Despite its different approach to combat, compared to the previous games, "GamesRadar+"s Leon Hurley felt the game was "every bit as brutally unflinching as previous games." | https://en.wikipedia.org/wiki?curid=50810460 |
5,015 | Writing for "Polygon", Chris Plante praised the camerawork as a technical marvel, noting the seamless nature of the game shifting from cinematic back to the gameplay. Juba said the decision to shift the camera closer to Kratos "[proved] immensely rewarding during big moments by giving [the player] an intimate view." Faulkner, however, claimed "it can be difficult to control the camera and keep a bead on the enemies you're fighting." In his review for "IGN", Jonathon Dornbush felt the intimacy of the camera makes all the emotions "more real and impactful." Speaking of the game's visuals, Faulkner said the game looks amazing, "and with 4K and HDR this game goes a step beyond what even games like "Horizon Zero Dawn" showed us was possible on this platform." Brown noted that ""God of War" is a technical and artistic showcase. It is without a doubt one of the best-looking console games ever released." Dan Ryckert of "Giant Bomb" claimed that games like "Uncharted: The Lost Legacy" and "Horizon Zero Dawn" "made great cases for a PS4 Pro and a 4K television, but "God of War"s visuals are a bigger selling point than anything I've seen on Sony's platform to date." | https://en.wikipedia.org/wiki?curid=50810460 |
5,016 | Despite the game's grandeur, Plessas felt that the boss fights "do not hit quite the same frequency as they did in the past few games." However, the few boss fights in the game "do the series proud". As to the vast world of "God of War", Faulkner said that, "The great thing about the exploration in "God of War" is that you can participate in it as little or as much as you want." He said an excellent design decision is that during main plot points, the game keeps the player on task, while in between, the player can explore, allowing "God of War" "to have the best of both worlds". Plessas noted that although the puzzles require thought, they were not "hair-pullingly" difficult as some were in previous games. Juba also found that the puzzles were not too challenging, saying they were fun. | https://en.wikipedia.org/wiki?curid=50810460 |
5,017 | Plessas felt that the RPG elements present in the game make this installment "unique" compared to previous entries. He said the game allows players to "specialize Kratos to meet the specific task at hand, or develop a build that best suits a preferred playstyle." Although this did not make the game easier, he felt it did make it more manageable. Juba noted that although this type of upgrading "may be less exciting" compared to previous games where Kratos just learns new moves, it still "provides a powerful incentive to explore." Ryckert was disappointed by this type of customization. He felt the presentation was "half-baked" and that some materials were confusing as there was little explanation given for their use. He did, however, say it was "cool" to see new armor on Kratos. | https://en.wikipedia.org/wiki?curid=50810460 |
5,018 | In terms of flaws, Plessas said that ""God of War" is so good that its most egregious failing is not letting fans play more of it", as New Game Plus was not an option at the time of the review. Juba said that ""God of War"s momentum rarely falters, and when it does, the inconvenience is brief." One example he gave was the map, saying that although players have freedom to explore, it can be difficult to track Kratos's position. He also felt the fast-travel system was "weirdly cumbersome" and that it opens up too late in the game. Although he enjoyed these features, Faulkner noted some players may dislike that "God of War" has a lack of player agency, and players have to explore the majority of the game on foot or by boat since the fast-travel feature is unlocked late in the game. Brown felt that if anything in "God of War" was a letdown, it was the final fight against Baldur: "He's great from a narrative standpoint, unraveling in a manner that changes your perspective, but it's the fight itself that leaves you wanting. There are plenty of big boss battles and tests of skill throughout the course of the game, yet this fight doesn't reach the same heights, and feels like it was played a little safe." Hurley said his only criticism was that, "You can occasionally find yourself unsure if you're doing something wrong, or don't have the right equipment yet." | https://en.wikipedia.org/wiki?curid=50810460 |
5,019 | During its release week in the United Kingdom, "God of War" became the fastest-selling entry in the franchise, selling 35% more physical copies than "God of War III". The game remained at the top of the all format sales chart for six consecutive weeks through April and May, setting a record for a PlayStation 4 exclusive having the most consecutive weeks at number one. It sold 46,091 units in its first week on sale in Japan, which placed it at number two on the sales chart. The game sold over 3.1 million units worldwide within three days of its release, making it the fastest-selling PlayStation 4 exclusive at the time. The game was the fastest-selling game of the month of its release and contributed to the PlayStation 4 being the best-selling console of that month. In total, the game sold over 5 million units in its first month, with 2.1 million in digital sales. By May 2019, the game had sold over 10 million units worldwide, making it the best-selling game in the series. By August 2021, total sales of the game had exceeded over 19.5 million units, making it the best-selling PlayStation 4 game. By March 2022, the PC version had sold 971,000 units, giving a total of over 20.4 million units sold. By November 2022, the game had sold 23 million units. | https://en.wikipedia.org/wiki?curid=50810460 |
5,020 | "God of War" won Game of the Year awards from several gaming publications, including British Academy Games Awards, "The Blade", CNET, "Destructoid", D.I.C.E. Awards, "Empire", "Entertainment Weekly", G1, The Game Awards, Game Developers Choice Awards, "Game Informer", "Game Revolution", "GamesRadar+", "IGN", Nerdist, New York Game Awards, "Polygon", "Push Square", "Slant Magazine", "Time" magazine, "Variety", and VideoGamer.com. The game was named among the best games of the 2010s by "Areajugones", "BuzzFeed", "GameSpew", "GamesRadar+", "Gaming Age", "GamingBolt", "The Hollywood Reporter", "IGN", Metacritic, "Slant Magazine", "Stuff", and "VG247". It was voted as the winner of IGN's Best Video Game of All Time. | https://en.wikipedia.org/wiki?curid=50810460 |
5,021 | The game was nominated for Game of the Show, Best PlayStation 4 Game, and Best Action Game at "IGN"s Best of E3 2016 Awards. It won the award for Game of the Year, Best PlayStation 4 Game, Best Action-Adventure Game, Best Art Direction, and Best Story at "IGN"s Best of 2018 Awards. It was a runner-up for Best Graphics, and was nominated for Best Music. | https://en.wikipedia.org/wiki?curid=50810460 |
5,022 | Following "God of War"s announcement in mid-2016, Barlog confirmed that it would not be Kratos' last game. He also said that future games could see the series tackling Egyptian or Mayan mythology, and although the 2018 installment focuses on Norse mythology, it alludes to the fact there are other mythologies co-existing in the world. Barlog also said he liked the idea of having different directors for each game, as had happened with the Greek era, and that although he might not direct another "God of War", he would still be at Santa Monica to work on future games. | https://en.wikipedia.org/wiki?curid=50810460 |
5,023 | A sequel, "God of War Ragnarök", was officially announced at the PlayStation 5 Showcase event in September 2020 and was originally scheduled to be released in 2021. However, the game was delayed, in part due to the COVID-19 pandemic, and released worldwide on November 9, 2022, for both the PlayStation 4 and PlayStation 5, marking the first cross-gen release in the franchise. Barlog stepped down as game director and became a producer while Eric Williams, who had worked on every previous game in the series, assumed the role of game director. Taking place three years after the 2018 installment, "Ragnarök" concluded the Norse era of the series. | https://en.wikipedia.org/wiki?curid=50810460 |
5,024 | On March 7, 2022, "Deadline" reported that a live action television (TV) series was said to be in negotiations at Amazon Prime Video by Mark Fergus and Hawk Ostby, creators of "The Expanse", and Rafe Judkins, the showrunner for "The Wheel of Time". During an investor briefing on May 26, 2022, Sony Interactive Entertainment president Jim Ryan confirmed that a "God of War" TV series was in development for Amazon Prime Video. The TV series was officially ordered on December 14, 2022. The adaptation is being produced by Sony Pictures Television and Amazon Studios in association with PlayStation Productions, and it will premiere on Prime Video in more than 240 countries and territories worldwide. It is being written by Fergus and Ostby, with Judkins serving as showrunner, who will all also be executive producers. Other executive producers include Santa Monica Studio's Creative Director Cory Barlog, PlayStation Productions' Asad Qizilbash and Carter Swan, Santa Monica Studio's Yumi Yang, and Vertigo Entertainment's Roy Lee. Santa Monica Studio's Jeff Ketcham will serve as a co-executive producer. It was also confirmed that the Prime Video series would adapt the Norse era, beginning with the events of the 2018 installment. | https://en.wikipedia.org/wiki?curid=50810460 |
5,025 | Google Scholar is a freely accessible web search engine that indexes the full text or metadata of scholarly literature across an array of publishing formats and disciplines. Released in beta in November 2004, the Google Scholar index includes peer-reviewed online academic journals and books, conference papers, theses and dissertations, preprints, abstracts, technical reports, and other scholarly literature, including court opinions and patents. | https://en.wikipedia.org/wiki?curid=1520204 |
5,026 | Google Scholar uses a web crawler, or web robot, to identify files for inclusion in the search results. For content to be indexed in Google Scholar, it must meet certain specified criteria. An earlier statistical estimate published in PLOS One using a mark and recapture method estimated approximately 80–90% coverage of all articles published in English with an estimate of 100 million. This estimate also determined how many documents were freely available on the internet. Google Scholar has been criticized for not vetting journals and for including predatory journals in its index. | https://en.wikipedia.org/wiki?curid=1520204 |
5,027 | The University of Michigan Library and other libraries whose collections Google scanned for Google Books and Google Scholar retained copies of the scans and have used them to create the HathiTrust Digital Library. | https://en.wikipedia.org/wiki?curid=1520204 |
5,028 | Google Scholar arose out of a discussion between Alex Verstak and Anurag Acharya, both of whom were then working on building Google's main web index. Their goal was to "make the world's problem solvers 10% more efficient" by allowing easier and more accurate access to scientific knowledge. This goal is reflected in the Google Scholar's advertising slogan "Stand on the shoulders of giants", which was taken from an idea attributed to Bernard of Chartres, , and is a nod to the scholars who have contributed to their fields over the centuries, providing the foundation for new intellectual achievements. One of the original sources for the texts in Google Scholar is the University of Michigan's print collection. | https://en.wikipedia.org/wiki?curid=1520204 |
5,029 | Scholar has gained a range of features over time. In 2006, a citation importing feature was implemented supporting bibliography managers, such as RefWorks, RefMan, EndNote, and BibTeX. In 2007, Acharya announced that Google Scholar had started a program to digitize and host journal articles in agreement with their publishers, an effort separate from Google Books, whose scans of older journals do not include the metadata required for identifying specific articles in specific issues. In 2011, Google removed Scholar from the toolbars on its search pages, making it both less easily accessible and less discoverable for users not already aware of its existence. Around this period, sites with similar features such as CiteSeer, Scirus, and Microsoft Windows Live Academic search were developed. Some of these are now defunct; in 2016, Microsoft launched a new competitor, Microsoft Academic. | https://en.wikipedia.org/wiki?curid=1520204 |
5,030 | A major enhancement was rolled out in 2012, with the possibility for individual scholars to create personal "Scholar Citations profiles". A feature introduced in November 2013 allows logged-in users to save search results into the "Google Scholar library", a personal collection which the user can search separately and organize by tags. Via the "metrics" button, it reveals the top journals in a field of interest, and the articles generating these journal's impact can also be accessed. A metrics feature now supports viewing the impact of whole fields of science, as well as academic journals. | https://en.wikipedia.org/wiki?curid=1520204 |
5,031 | Google Scholar allows users to search for digital or physical copies of articles, whether online or in libraries. It indexes "full-text journal articles, technical reports, preprints, theses, books, and other documents, including selected Web pages that are deemed to be 'scholarly.'" Because many of Google Scholar's search results link to commercial journal articles, most people will be able to access only an abstract and the citation details of an article, and have to pay a fee to access the entire article. The most relevant results for the searched keywords will be listed first, in order of the author's ranking, the number of references that are linked to it and their relevance to other scholarly literature, and the ranking of the publication that the journal appears in. | https://en.wikipedia.org/wiki?curid=1520204 |
5,032 | Using its "group of" feature, it shows the available links to journal articles. In the 2005 version, this feature provided a link to both subscription-access versions of an article and to free full-text versions of articles; for most of 2006, it provided links to only the publishers' versions. Since December 2006, it has provided links to both published versions and major open access repositories, including all those posted on individual faculty web pages and other unstructured sources identified by similarity. On the other hand, Google Scholar doesn't allow to filter explicitly between toll access and open access resources, a feature offered Unpaywall and the tools which embed its data, such as Web of Science, Scopus and Unpaywall Journals, used by libraries to calculate the real costs and value of their collections. | https://en.wikipedia.org/wiki?curid=1520204 |
5,033 | Through its "cited by" feature, Google Scholar provides access to abstracts of articles that have cited the article being viewed. It is this feature in particular that provides the citation indexing previously only found in CiteSeer, Scopus, and Web of Science. Google Scholar also provides links so that citations can be either copied in various formats or imported into user-chosen reference managers such as Zotero. | https://en.wikipedia.org/wiki?curid=1520204 |
5,034 | "Scholar Citations profiles" are public author profiles that are editable by authors themselves. Individuals, logging on through a Google account with a bona fide address usually linked to an academic institution, can now create their own page giving their fields of interest and citations. Google Scholar automatically calculates and displays the individual's total citation count, h-index, and i10-index. According to Google, "three-quarters of Scholar search results pages ... show links to the authors' public profiles" as of August 2014. | https://en.wikipedia.org/wiki?curid=1520204 |
5,035 | Through its "Related articles" feature, Google Scholar presents a list of closely related articles, ranked primarily by how similar these articles are to the original result, but also taking into account the relevance of each paper. | https://en.wikipedia.org/wiki?curid=1520204 |
5,036 | Google Scholar's legal database of US cases is extensive. Users can search and read published opinions of US state appellate and supreme court cases since 1950, US federal district, appellate, tax, and bankruptcy courts since 1923 and US Supreme Court cases since 1791. Google Scholar embeds clickable citation links within the case and the How Cited tab allows lawyers to research prior case law and the subsequent citations to the court decision. | https://en.wikipedia.org/wiki?curid=1520204 |
5,037 | While most academic databases and search engines allow users to select one factor (e.g. relevance, citation counts, or publication date) to rank results, Google Scholar ranks results with a combined ranking algorithm in a "way researchers do, weighing the full text of each article, the author, the publication in which the article appears, and how often the piece has been cited in other scholarly literature". Research has shown that Google Scholar puts high weight especially on citation counts, as well as words included in a document's title. In searches by author or year, the first search results are often highly cited articles, as the number of citations is highly determinant, whereas in keyword searches the number of citations is probably the factor with the most weight, but other factors also participate. | https://en.wikipedia.org/wiki?curid=1520204 |
5,038 | Some searchers found Google Scholar to be of comparable quality and utility to subscription-based databases when looking at citations of articles in some specific journals. The reviews recognize that its "cited by" feature in particular poses serious competition to Scopus and Web of Science. A study looking at the biomedical field found citation information in Google Scholar to be "sometimes inadequate, and less often updated". The coverage of Google Scholar may vary by discipline compared to other general databases. Google Scholar strives to include as many journals as possible, including predatory journals, which may lack academic rigor. Specialists on predatory journals say that these kinds of journals "have polluted the global scientific record with pseudo-science" and "that Google Scholar dutifully and perhaps blindly includes in its central index." | https://en.wikipedia.org/wiki?curid=1520204 |
5,039 | Google Scholar does not publish a list of journals crawled or publishers included, and the frequency of its updates is uncertain. Bibliometric evidence suggests Google Scholar's coverage of the sciences and social sciences is competitive with other academic databases; as of 2017, Scholar's coverage of the arts and humanities has not been investigated empirically and Scholar's utility for disciplines in these fields remains ambiguous. Especially early on, some publishers did not allow Scholar to crawl their journals. Elsevier journals have been included since mid-2007, when Elsevier began to make most of its ScienceDirect content available to Google Scholar and Google's web search. However, a 2014 study estimates that Google Scholar can find almost 90% (approximately 100 million) of all scholarly documents on the Web written in English. Large-scale longitudinal studies have found between 40 and 60 percent of scientific articles are available in full text via Google Scholar links. | https://en.wikipedia.org/wiki?curid=1520204 |
5,040 | Google Scholar puts high weight on citation counts in its ranking algorithm and therefore is being criticized for strengthening the Matthew effect; as highly cited papers appear in top positions they gain more citations while new papers hardly appear in top positions and therefore get less attention by the users of Google Scholar and hence fewer citations. Google Scholar effect is a phenomenon when some researchers pick and cite works appearing in the top results on Google Scholar regardless of their contribution to the citing publication because they automatically assume these works' credibility and believe that editors, reviewers, and readers expect to see these citations. Google Scholar has problems identifying publications on the arXiv preprint server correctly. Interpunctuation characters in titles produce wrong search results, and authors are assigned to wrong papers, which leads to erroneous additional search results. Some search results are even given without any comprehensible reason. | https://en.wikipedia.org/wiki?curid=1520204 |
5,041 | Google Scholar is vulnerable to spam. Researchers from the University of California, Berkeley and Otto-von-Guericke University Magdeburg demonstrated that citation counts on Google Scholar can be manipulated and complete non-sense articles created with SCIgen were indexed within Google Scholar. These researchers concluded that citation counts from Google Scholar should be used with care, especially when used to calculate performance metrics such as the h-index or impact factor, which is in itself a poor predictor of article quality. Google Scholar started computing an h-index in 2012 with the advent of individual Scholar pages. Several downstream packages like "Harzing's Publish or Perish" also use its data. The practicality of manipulating h-index calculators by spoofing Google Scholar was demonstrated in 2010 by Cyril Labbe from Joseph Fourier University, who managed to rank "Ike Antkare" ahead of Albert Einstein by means of a large set of SCIgen-produced documents citing each other (effectively an academic link farm). As of 2010, Google Scholar was not able to shepardize case law, as Lexis could. Unlike other indexes of academic work such as Scopus and Web of Science, Google Scholar does not maintain an Application Programming Interface that may be used to automate data retrieval. Use of web scrapers to obtain the contents of search results is also severely restricted by the implementation of CAPTCHAs. Google Scholar does not display or export Digital Object Identifiers (DOIs), a "de facto" standard implemented by all major academic publishers to uniquely identify and refer to individual pieces of academic work. | https://en.wikipedia.org/wiki?curid=1520204 |
5,042 | Search engine optimization (SEO) for traditional web search engines such as Google has been popular for many years. For several years, SEO has also been applied to academic search engines such as Google Scholar. SEO for academic articles is also called "academic search engine optimization" (ASEO) and defined as "the creation, publication, and modification of scholarly literature in a way that makes it easier for academic search engines to both crawl it and index it". ASEO has been adopted by several organizations, among them Elsevier, OpenScience, Mendeley, and SAGE Publishing, to optimize their articles' rankings in Google Scholar. ASEO has negatives. | https://en.wikipedia.org/wiki?curid=1520204 |
5,043 | Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. | https://en.wikipedia.org/wiki?curid=18831 |
5,044 | Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated with certain properties, called axioms. A "proof" consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered as true starting points of the theory under consideration. | https://en.wikipedia.org/wiki?curid=18831 |
5,045 | Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. The fundamental truths of mathematics are independent from any scientific experimentation, although mathematics is extensively used for modeling phenomena. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other mathematical areas are developed independently from any application (and are therefore called pure mathematics), but practical applications are often discovered later. A fitting example is the problem of integer factorization, which goes back to Euclid, but which had no practical application before its use in the RSA cryptosystem (for the security of computer networks). | https://en.wikipedia.org/wiki?curid=18831 |
5,046 | Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's "Elements". Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new areas of the subject. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. This gave rise to a dramatic increase in the number of mathematics areas and their fields of applications. This can be seen, for example, in the contemporary Mathematics Subject Classification, which lists more than 60 first-level areas of mathematics. | https://en.wikipedia.org/wiki?curid=18831 |
5,047 | The word "mathematics" comes from Ancient Greek "máthēma" (""), meaning "that which is learnt," "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. Its adjective is "mathēmatikós" (), meaning "related to learning" or "studious," which likewise further came to mean "mathematical." In particular, "mathēmatikḗ tékhnē" (; ) meant "the mathematical art." | https://en.wikipedia.org/wiki?curid=18831 |
5,048 | Similarly, one of the two main schools of thought in Pythagoreanism was known as the "mathēmatikoi" (μαθηματικοί)which at the time meant "learners" rather than "mathematicians" in the modern sense. The Pythagoreans were likely the first to constrain the use of the word to just the study of arithmetic and geometry. By the time of Aristotle (384–322 BC) this meaning was fully established. | https://en.wikipedia.org/wiki?curid=18831 |
5,049 | In Latin, and in English until around 1700, the term "mathematics" more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of "mathematici", meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians. | https://en.wikipedia.org/wiki?curid=18831 |
5,050 | The apparent plural form in English goes back to the Latin neuter plural (Cicero), based on the Greek plural "ta mathēmatiká" () and means roughly "all things mathematical", although it is plausible that English borrowed only the adjective "mathematic(al)" and formed the noun "mathematics" anew, after the pattern of "physics" and "metaphysics", which were inherited from Greek. In English, the noun "mathematics" takes a singular verb. It is often shortened to "maths" or, in North America, "math". | https://en.wikipedia.org/wiki?curid=18831 |
5,051 | Before the Renaissance, mathematics was divided into two main areas: arithmeticregarding the manipulation of numbers, and geometryregarding the study of shapes. Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics. | https://en.wikipedia.org/wiki?curid=18831 |
5,052 | During the Renaissance, two more areas appeared. Mathematical notation led to algebra, which, roughly speaking, consists of the study and the manipulation of formulas. Calculus, consisting of the two subfields "differential calculus" and "integral calculus", is the study of continuous functions, which model the typically nonlinear relationships between varying quantities, as represented by variables. This division into four main areasarithmetic, geometry, algebra, calculusendured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were then studied by mathematicians, but now are considered as belonging to physics. The subject of combinatorics has been studied for much of recorded history, yet it did not become a separate branch of mathematics until the seventeenth century. | https://en.wikipedia.org/wiki?curid=18831 |
5,053 | At the end of the 19th century, the foundational crisis in mathematics and the resulting systematization of the axiomatic method led to an explosion of new areas of mathematics. The 2020 Mathematics Subject Classification contains no less than first-level areas. Some of these areas correspond to the older division, as is true regarding number theory (the modern name for higher arithmetic) and geometry. Several other first-level areas have "geometry" in their names or are otherwise commonly considered part of geometry. Algebra and calculus do not appear as first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century or had not previously been considered as mathematics, such as mathematical logic and foundations. | https://en.wikipedia.org/wiki?curid=18831 |
5,054 | Number theory began with the manipulation of numbers, that is, natural numbers formula_1 and later expanded to integers formula_2 and rational numbers formula_3 Formerly, number theory was called arithmetic, but nowadays this term is mostly used for numerical calculations. The origin of number theory dates back to ancient Babylon and probably China. Two prominent early number theorists were Euclid and Diophantus. The modern study of number theory in its abstract form is largely attributed to Pierre de Fermat and Leonhard Euler. The field came to full fruition with the contributions of Adrien-Marie Legendre and Carl Friedrich Gauss. | https://en.wikipedia.org/wiki?curid=18831 |
5,055 | Many easily-stated number problems have solutions that require sophisticated methods from across mathematics. One prominent example is Fermat's last theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic geometry, category theory, and homological algebra. Another example is Goldbach's conjecture, which asserts that every even integer greater than 2 is the sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven to this day despite considerable effort. | https://en.wikipedia.org/wiki?curid=18831 |
5,056 | Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers (method oriented), diophantine equations, and transcendence theory (problem oriented). | https://en.wikipedia.org/wiki?curid=18831 |
5,057 | Geometry is one of the oldest branches of mathematics. It started with empirical recipes concerning shapes, such as lines, angles and circles, which were developed mainly for the needs of surveying and architecture, but has since blossomed out into many other subfields. | https://en.wikipedia.org/wiki?curid=18831 |
5,058 | A fundamental innovation was the introduction of the concept of proofs by ancient Greeks, with the requirement that every assertion must be "proved". For example, it is not sufficient to verify by measurement that, say, two lengths are equal; their equality must be proven via reasoning from previously accepted results (theorems) and a few basic statements. The basic statements are not subject to proof because they are self-evident (postulates), or they are a part of the definition of the subject of study (axioms). This principle, which is foundational for all mathematics, was first elaborated for geometry, and was systematized by Euclid around 300 BC in his book "Elements". | https://en.wikipedia.org/wiki?curid=18831 |
5,059 | The resulting Euclidean geometry is the study of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane (plane geometry) and the (three-dimensional) Euclidean space. | https://en.wikipedia.org/wiki?curid=18831 |
5,060 | Euclidean geometry was developed without change of methods or scope until the 17th century, when René Descartes introduced what is now called Cartesian coordinates. This was a major change of paradigm, since instead of defining real numbers as lengths of line segments (see number line), it allowed the representation of points using their "coordinates" (which are numbers). This allows one to use algebra (and later, calculus) to solve geometrical problems. This split geometry into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically. | https://en.wikipedia.org/wiki?curid=18831 |
5,061 | Analytic geometry allows the study of curves that are not related to circles and lines. Such curves can be defined as graph of functions (whose study led to differential geometry). They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions. | https://en.wikipedia.org/wiki?curid=18831 |
5,062 | In the 19th century, mathematicians discovered non-Euclidean geometries, which do not follow the parallel postulate. By questioning the truth of that postulate, this discovery has been viewed as joining Russel's paradox in revealing the foundational crisis of mathematics.This aspect of the crisis was solved by systematizing the axiomatic method, and adopting that the truth of the chosen axioms is not a mathematical problem. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are invariant under specific transformations of the space. | https://en.wikipedia.org/wiki?curid=18831 |
5,063 | Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. The first one solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. The second one introduced systematic methods for transforming equations (such as moving a term from a side of an equation into the other side). The term "algebra" is derived from the Arabic word "al-jabr" meaning "the reunion of broken parts" that he used for naming one of these methods in the title of his main treatise. | https://en.wikipedia.org/wiki?curid=18831 |
5,064 | Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. This allows mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas. | https://en.wikipedia.org/wiki?curid=18831 |
5,065 | Until the 19th century, algebra consisted mainly of the study of linear equations (presently "linear algebra"), and polynomial equations in a single unknown, which were called "algebraic equations" (a term that is still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. Due to this change, the scope of algebra grew to include the study of algebraic structures. This object of algebra was called "modern algebra" or abstract algebra, as established by the influence and works of Emmy Noether. (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.) | https://en.wikipedia.org/wiki?curid=18831 |
5,066 | Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include: | https://en.wikipedia.org/wiki?curid=18831 |
5,067 | The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology. | https://en.wikipedia.org/wiki?curid=18831 |
5,068 | Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, "calculus" refers mainly to the elementary part of this theory, and "analysis" is commonly used for advanced parts. | https://en.wikipedia.org/wiki?curid=18831 |
5,069 | Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include: | https://en.wikipedia.org/wiki?curid=18831 |
5,070 | Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithmsespecially their implementation and computational complexityplay a major role in discrete mathematics. | https://en.wikipedia.org/wiki?curid=18831 |
5,071 | The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems. | https://en.wikipedia.org/wiki?curid=18831 |
5,072 | The two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century. Before this period, sets were not considered to be mathematical objects, and logic, although used for mathematical proofs, belonged to philosophy and was not specifically studied by mathematicians. | https://en.wikipedia.org/wiki?curid=18831 |
5,073 | Before Cantor's study of infinite sets, mathematicians were reluctant to consider actually infinite collections, and considered infinity to be the result of endless enumeration. Cantor's work offended many mathematicians not only by considering actually infinite sets but by showing that this implies different sizes of infinity, per Cantor's diagonal argument. This led to the controversy over Cantor's set theory. | https://en.wikipedia.org/wiki?curid=18831 |
5,074 | In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a point is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc. | https://en.wikipedia.org/wiki?curid=18831 |
5,075 | This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, each mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number has a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. This mathematical abstraction from reality is embodied in the modern philosophy of formalism, as founded by David Hilbert around 1910. | https://en.wikipedia.org/wiki?curid=18831 |
5,076 | The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinionsometimes called "intuition"to guide their study and proofs. The approach allows considering "logics" (that is, sets of allowed deducing rules), theorems, proofs, etc. as mathematical objects, and to prove theorems about them. For example, Gödel's incompleteness theorems assert, roughly speaking that, in every consistent formal system that contains the natural numbers, there are theorems that are true (that is provable in a stronger system), but not provable inside the system. This approach to the foundations of mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle. | https://en.wikipedia.org/wiki?curid=18831 |
5,077 | These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory (modeling some logical theories inside other theories), proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in turn to the expansion of these logical theories. | https://en.wikipedia.org/wiki?curid=18831 |
5,078 | The field of statistics is a mathematical application that is employed for the collection and processing of data samples, using procedures based on mathematical methods especially probability theory. Statisticians generate data with random sampling or randomized experiments. The design of a statistical sample or experiment determines the analytical methods that will be used. Analysis of data from observational studies is done using statistical models and the theory of inference, using model selection and estimation. The models and consequential predictions should then be tested against new data. | https://en.wikipedia.org/wiki?curid=18831 |
5,079 | Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints. For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics. | https://en.wikipedia.org/wiki?curid=18831 |
5,080 | Computational mathematics is the study of mathematical problems that are typically too large for human, numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory; numerical analysis broadly includes the study of approximation and discretization with special focus on rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic-matrix-and-graph theory. Other areas of computational mathematics include computer algebra and symbolic computation. | https://en.wikipedia.org/wiki?curid=18831 |
5,081 | The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals, was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like timedays, seasons, or years. | https://en.wikipedia.org/wiki?curid=18831 |
5,082 | Evidence for more complex mathematics does not appear until around 3000 , when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system which is still in use today for measuring angles and time. | https://en.wikipedia.org/wiki?curid=18831 |
5,083 | In the 6th century BC, Greek mathematics began to emerge as a distinct discipline and some Ancient Greeks such as the Pythagoreans appeared to have considered it a subject in its own right. Around 300 BC, Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, "Elements", is widely considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse. He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), trigonometry (Hipparchus of Nicaea, 2nd century BC), and the beginnings of algebra (Diophantus, 3rd century AD). | https://en.wikipedia.org/wiki?curid=18831 |
5,084 | The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, and an early form of infinite series. | https://en.wikipedia.org/wiki?curid=18831 |
5,085 | During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. The Greek and Arabic mathematical texts were in turn translated to Latin during the Middle Ages and made available in Europe. | https://en.wikipedia.org/wiki?curid=18831 |
5,086 | During the early modern period, mathematics began to develop at an accelerating pace in Western Europe, with innovations that revolutionized mathematics, such as the introduction of variables and symbolic notation by François Viète (1540–1603), the introduction of coordinates by René Descartes (1596–1650) for reducing geometry to algebra, and the development of calculus by Isaac Newton (1642–1726/27) and Gottfried Leibniz (1646–1716) in the 17th century. Leonhard Euler (1707–1783), the most notable mathematician of the 18th century, unified these innovations into a single corpus with a standardized terminology, and completed them with the discovery and the proof of numerous theorems. Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic systemif powerful enough to describe arithmeticwill contain true propositions that cannot be proved. | https://en.wikipedia.org/wiki?curid=18831 |
5,087 | Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made to this very day. According to Mikhail B. Sevryuk, in the January 2006 issue of the "Bulletin of the American Mathematical Society", "The number of papers and books included in the "Mathematical Reviews" database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." | https://en.wikipedia.org/wiki?curid=18831 |
5,088 | Mathematical notation is widely used in science and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. This notation consists of symbols used for representing operations, unspecified numbers, relations and any other mathematical objects, and then assembling them into expressions and formulas. More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally Latin or Greek letters, and often include subscripts. Operation and relations are generally represented by specific symbols or glyphs, such as (plus), (multiplication), formula_4 (integral), (equal), and (less than). All these symbols are generally grouped according to specific rules to form expressions and formulas. Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of noun phrases and formulas play the role of clauses. | https://en.wikipedia.org/wiki?curid=18831 |
5,089 | Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corallary. | https://en.wikipedia.org/wiki?curid=18831 |
5,090 | Numerous technical terms used in mathematics are neologisms, such as "polynomial" and "homeomorphism". Other technical terms are words of the common language that are used in an accurate meaning that may differs slightly from their common meaning. For example, in mathematics, "or" means "one, the other or both", while, in common language, it is either amiguous or means "one or the other but not both" (in mathematics, the latter is called "exclusive or"). Finally, many mathematical terms are common words that are used with a completely different meaning. This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every free module is flat" and "a field is always a ring". | https://en.wikipedia.org/wiki?curid=18831 |
5,091 | Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model. | https://en.wikipedia.org/wiki?curid=18831 |
5,092 | There is still a philosophical debate whether mathematics is a science. However, in practice, mathematicians are typically grouped with scientists, and mathematics shares much in common with the physical sciences. Like them, it is falsifiable, which means in mathematics that, if a result or a theory is wrong, this can be proved by providing a counterexample. Similarly as in science, theories and results (theorems) are often obtained from experimentation. In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support). For example, when asked how he came about his theorems, Gauss once replied "durch planmässiges Tattonieren" (through systematic experimentation). However, some authors emphasize that mathematics differs from the modern notion of science by not on empirical evidence. | https://en.wikipedia.org/wiki?curid=18831 |
5,093 | Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics. For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece. | https://en.wikipedia.org/wiki?curid=18831 |
5,094 | In the 19th century, mathematicians such as Karl Weierstrass and Richard Dedekind increasingly focused their research on internal problems, that is, "pure mathematics". This led to split mathematics into "pure mathematics" and "applied mathematics", the latter being often considered as having a lower value among mathematical purists. However, the lines between the two are frequently blurred. | https://en.wikipedia.org/wiki?curid=18831 |
5,095 | The aftermath of World War II led to a surge in the development of applied mathematics in the US and elsewhere. Many of the theories developed for applications were found interesting from the point of view of pure mathematics, and many results of pure mathematics were shown to have applications outside mathematics; in turn, the study of these applications may give new insights on the "pure theory". | https://en.wikipedia.org/wiki?curid=18831 |
5,096 | An example of the first case is the theory of distributions, introduced by Laurent Schwartz for validating computations done in quantum mechanics, which became immediately an important tool of (pure) mathematical analysis. An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. For getting an algorithm that can be implemented and can solve systems of polynomial equations and inequalities, George Collins introduced the cylindrical algebraic decomposition that became a fundamental tool in real algebraic geometry. | https://en.wikipedia.org/wiki?curid=18831 |
5,097 | In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas. The Mathematics Subject Classification has a section for "general applied mathematics" but does not mention "pure mathematics". However, these terms are still used in names of some university departments, such as at the Faculty of Mathematics at the University of Cambridge. | https://en.wikipedia.org/wiki?curid=18831 |
5,098 | The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner. It is the fact that many mathematical theories, even the "purest" have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. | https://en.wikipedia.org/wiki?curid=18831 |
5,099 | A notable example is the prime factorization of natural numbers that was discovered more than 2,000 years before its common use for secure internet communications through the RSA cryptosystem. A second historical example is the theory of ellipses. They were studied by the ancient Greek mathematicians as conic sections (that is, intersections of cones with planes). It is almost 2,000 years later that Johannes Kepler discovered that the trajectories of the planets are ellipses. | https://en.wikipedia.org/wiki?curid=18831 |
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