Trinity College Dublin
Trinity College the College of the Holy and Undivided Trinity of Queen Elizabeth near Dublin, is the sole constituent college of the University of Dublin, a research university located in Dublin, Ireland. The college was founded in 1592 by Queen Elizabeth I as the "mother" of a new university, modelled after the collegiate universities of Oxford and Cambridge, but unlike these other ancient universities, only one college was established; the college is incorporated by "the Provost, Foundation Scholars and other members of the Board" as outlined by its founding charter. It is one of the seven ancient universities of Britain and Ireland, as well as Ireland's oldest surviving university. Trinity College is considered the most prestigious university in Ireland and amongst the most elite in Europe, principally due to its extensive history, reputation for social elitism and unique relationship with both the University of Oxford and the University of Cambridge. In accordance with the formula of ad eundem gradum, a form of recognition that exists among the three universities, a graduate of Oxford, Cambridge, or Dublin can be conferred with the equivalent degree at either of the other two universities without further examination.
Trinity College, Dublin is a sister college to St John's College and Oriel College, Oxford. Trinity was established outside the city walls of Dublin in the buildings of the outlawed Catholic Augustinian Priory of All Hallows. Trinity College was set up in part to consolidate the rule of the Tudor monarchy in Ireland, as a result was the university of the Protestant Ascendancy for much of its history. While Catholics were admitted from 1793 certain restrictions on membership of the college remained as professorships and scholarships were reserved for Protestants; these restrictions were lifted by Act of Parliament in 1873. However, from 1871 to 1970, the Catholic Church in Ireland in turn forbade its adherents from attending Trinity College without permission. Women were first admitted to the college as full members in January 1904. Trinity College is now surrounded by central Dublin and is located on College Green, opposite the historic Irish Houses of Parliament; the college proper occupies 190,000 m2, with many of its buildings ranged around large quadrangles and two playing fields.
Academically, it is divided into three faculties comprising 25 schools, offering degree and diploma courses at both undergraduate and postgraduate levels. The Library of Trinity College is a legal deposit library for Ireland and Great Britain, containing over 6.2 million printed volumes and significant quantities of manuscripts, including the Book of Kells. The first University of Dublin was created by the Pope in 1311, had a Chancellor and students over many years, before coming to an end at the Reformation. Following this, some debate about a new university at St. Patrick's Cathedral, in 1592 a small group of Dublin citizens obtained a charter by way of letters patent from Queen Elizabeth incorporating Trinity College at the former site of All Hallows monastery, to the south east of the city walls, provided by the Corporation of Dublin; the first provost of the college was the Archbishop of Dublin, Adam Loftus, he was provided with two initial Fellows, James Hamilton and James Fullerton.
Two years after foundation, a few Fellows and students began to work in the new college, which lay around one small square. During the following fifty years the community increased the endowments, including considerable landed estates, were secured, new fellowships were founded, the books which formed the foundation of the great library were acquired, a curriculum was devised and statutes were framed; the founding Letters Patent were amended by succeeding monarchs on a number of occasions, such as by James I in 1613 and most notably in 1637 by Charles I and supplemented as late as the reign of Queen Victoria. During the eighteenth century Trinity College was seen as the university of the Protestant Ascendancy. Parliament, meeting on the other side of College Green, made generous grants for building; the first building of this period was the Old Library building, begun in 1712, followed by the Printing House and the Dining Hall. During the second half of the century Parliament Square emerged.
The great building drive was completed in the early nineteenth century by Botany Bay, the square which derives its name in part from the herb garden it once contained. Following early steps in Catholic Emancipation, Catholics were first allowed to apply for admission in 1793, prior to the equivalent change at the University of Cambridge and the University of Oxford. Certain disabilities remained. In December 1845 Denis Caulfield Heron was the subject of a hearing at Trinity College. Heron had been examined and, on merit, declared a scholar of the college but had not been allowed to take up his place due to his Catholic religion. Heron appealed to the Courts which issued a writ of mandamus requiring the case to be adjudicated by the Archbishop of Dublin and the Primate of Ireland; the decision of Richard Whately and John George de la Poer Beresf
National University of Ireland
The National University of Ireland is a federal university system of constituent universities and recognised colleges set up under the Irish Universities Act, 1908, amended by the Universities Act, 1997. The constituent universities are for all essential purposes independent universities, except that the degrees and diplomas are those of the National University of Ireland with its seat in Dublin. In post-nominals, the abbreviation NUI is used for degrees from all the constituent universities of the National University of Ireland. Queen's Colleges at Belfast and Galway were established in 1845. In 1849 teaching commenced and a year they were united under the Queen's University of Ireland; the Catholic University of Ireland was created as an independent university on 3 November 1854 for the education of Catholics. This university did not offer recognised degrees. In 1880 the Royal University of Ireland took over the degree awarding functions of the two former universities and offered recognised degrees to the graduates of the new University College Dublin and St Patrick's College, Maynooth awarded under the Catholic University.
The Catholic University became University College Dublin in 1882 under the direction of the Jesuits. In the 1890s its students achieved more distinctions than their counterparts in Belfast and Galway, established as secular institutions; the 1908 reforms created the National University of Ireland and a separate Queen's University of Belfast. The Royal University was dissolved in 1909, in 1910 Maynooth became a recognised college of the NUI; the National University, unlike the Royal University, did not award degrees for part-time or external students. To the Royal University, the National University was still banned from awarding degrees in Theology. In 1975 the teacher training colleges of Carysfort College, Blackrock, St. Patrick's College Drumcondra and Mary Immaculate College, Limerick became recognised colleges of the NUI. During 1976 and 1977 Thomond College of Education, Limerick was a recognized college of the NUI, also. In 1978 St. Angela's College, Sligo became affiliated to the NUI. In 1996 the National College of Art and Design became a recognised college of the NUI.
The 1997 reforms restructured the National University of Ireland, an additional university at Maynooth was created from certain faculties of the previous recognised college, St Patrick's College, Maynooth. These reforms removed the prohibition on theology, imposed on the National University and its predecessors. Since 1918 the university's graduates have formed a constituency in parliamentary elections. In 1918 it was formed as a constituency for the UK House of Commons. After the first election Eoin MacNeill sat in the first Dáil; the NUI graduates elected four TDs from 1921 until 1934 when the university constituencies were abolished by Fianna Fáil. Under the Constitution of Ireland, adopted in 1937, the graduates of the university elect three members of Seanad Éireann. All graduates that are Irish citizens are entitled to vote if on the university's register of electors. An honorary degree does not give the entitlement to vote; the election is conducted by postal vote. The most recent election was in 2016, for the 25th Seanad, returned three independents: Alice Mary Higgins, Michael McDowell and Rónán Mullen.
The governing body of the NUI is styled the Senate under its 1908 charter. Members are called "Members of the Senate" rather than Senators; the NUI Senate meets in the Phelan Room, called after Edward J. Phelan, who funded its refurbishment; the Universities Act 1997 increased the size of the Senate and devolved power from it to the constituent universities. The NUI's Convocation comprises the Chancellor, the Vice-Chancellor, the Members of Senate, the Professors and Lecturers, the Graduates of the University. Eight Members of the NUI Senate are elected for terms of five years; the chancellor is the notional head of the university, constituent universities and recognised colleges have their own heads, which exercise most powers in practice. When the university was established in 1908 by Royal Charter, the first chancellor was appointed; the chancellor is elected by graduates and staff. William Joseph Walsh Éamon de Valera T. K. Whitaker Garret FitzGerald Maurice Manning Within the university there is a common faculty structure in operation in the constituent universities.
These ten faculties are: Agriculture. The constituent universities are: University College Cork University College Dublin NUI Galway Maynooth UniversityThe recognised college is: Royal College of Surgeons in Ireland++++++ Since April 2009 the Senate of the NUI decided that medical graduates of RCSI Bahrain will be eligible to receive the NUI degrees of MB BCh BAO. Former recognised colleges, now colleges of constituent universities, are: Institute of Public Administration †† National College of Art and Design †† Shannon College of Hotel Management † St. Angela's College, Sligo †† St Angela's College and Shannon College of Hotel Management are each "A Colleg
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is one of the oldest learned societies in the United States. Founded in 1780, the Academy is dedicated to honoring excellence and leadership, working across disciplines and divides, advancing the common good. Membership in the academy is achieved through a thorough petition and election process and has been considered a high honor of scholarly and societal merit since the academy was founded during the American Revolution by John Adams, John Hancock, James Bowdoin, others of their contemporaries who contributed prominently to the establishment of the new nation, its government, the United States Constitution. Today the Academy is charged with a dual function: to elect to membership the finest minds and most influential leaders, drawn from science, business, public affairs, the arts, from each generation, to conduct policy studies in response to the needs of society. Major Academy projects now have focused on higher education and research and cultural studies and technological advances, politics and the environment, the welfare of children.
Dædalus, the Academy's quarterly journal, is regarded as one of the world's leading intellectual journals. The Academy carries out nonpartisan policy research by bringing together scientists, artists, business leaders, other experts to make multidisciplinary analyses of complex social and intellectual topics; the Academy's current areas of work are Arts & Humanities, Democracy & Justice, Energy & Environment, Global Affairs, Science & Technology. David W. Oxtoby began his term as the organization’s President in January 2019. A chemist by training, he served as President of Pomona College from 2003 to 2017, he was elected a member of the American Academy in 2012. The Academy is headquartered in Massachusetts; the Academy was established by the Massachusetts legislature on May 4, 1780. Its purpose, as described in its charter, is "to cultivate every art and science which may tend to advance the interest, honor and happiness of a free and virtuous people." The sixty-two incorporating fellows represented varying interests and high standing in the political and commercial sectors of the state.
The first class of new members, chosen by the Academy in 1781, included Benjamin Franklin and George Washington as well as several international honorary members. The initial volume of Academy Memoirs appeared in 1785, the Proceedings followed in 1846. In the 1950s, the Academy launched its journal Daedalus, reflecting its commitment to a broader intellectual and socially-oriented program. Since the second half of the twentieth century, independent research has become a central focus of the Academy. In the late 1950s, arms control emerged as one of its signature concerns; the Academy served as the catalyst in establishing the National Humanities Center in North Carolina. In the late 1990s, the Academy developed a new strategic plan, focusing on four major areas: science and global security. In 2002, the Academy established a visiting scholars program in association with Harvard University. More than 75 academic institutions from across the country have become Affiliates of the Academy to support this program and other Academy initiatives.
The Academy has sponsored a number of awards and prizes, now numbering 11, throughout its history and has offered opportunities for fellowships and visiting scholars at the Academy. Charter members of the Academy are John Adams, Samuel Adams, John Bacon, James Bowdoin, Charles Chauncy, John Clarke, David Cobb, Samuel Cooper, Nathan Cushing, Thomas Cushing, William Cushing, Tristram Dalton, Francis Dana, Samuel Deane, Perez Fobes, Caleb Gannett, Henry Gardner, Benjamin Guild, John Hancock, Joseph Hawley, Edward Augustus Holyoke, Ebenezer Hunt, Jonathan Jackson, Charles Jarvis, Samuel Langdon, Levi Lincoln, Daniel Little, Elijah Lothrup, John Lowell, Samuel Mather, Samuel Moody, Andrew Oliver, Joseph Orne, Theodore Parsons, George Partridge, Robert Treat Paine, Phillips Payson, Samuel Phillips, John Pickering, Oliver Prescott, Zedekiah Sanger, Nathaniel Peaslee Sargeant, Micajah Sawyer, Theodore Sedgwick, William Sever, David Sewall, Stephen Sewall, John Sprague, Ebenezer Storer, Caleb Strong, James Sullivan, John Bernard Sweat, Nathaniel Tracy, Cotton Tufts, James Warren, Samuel West, Edward Wigglesworth, Joseph Willard, Abraham Williams, Nehemiah Williams, Samuel Williams, James Winthrop.
From the beginning, the membership and elected by peers, has included not only scientists and scholars, but writers and artists as well as representatives from the full range of professions and public life. Throughout the Academy's history, 10,000 fellows have been elected, including such notables as John Adams, Thomas Jefferson, John James Audubon, Joseph Henry, Washington Irving, Josiah Willard Gibbs, Augustus Saint-Gaudens, J. Robert Oppenheimer, Willa Cather, T. S. Eliot, Edward R. Murrow, Jonas Salk, Eudora Welty, Duke Ellington. International honorary members have included Jose Antonio Pantoja Hernandez, Leonhard Euler, Marquis de Lafayette, Alexander von Humboldt, Leopold von Ranke, Charles Darwin, Otto Hahn, Jawaharlal Nehru, Pablo Picasso, Liu Kuo-Sung, Lucian Michael Freud, Galina Ulanova, Werner Heisenberg, Alec Guinness and Sebastião Salgado. Astronomer Maria Mitchell was the first woman elected to the Academy, in 1848; the current membership encompasses over 5,700 members based across the United States and around the world.
Academy members include more than 60 Pulitzer Prize winners. The current membership is divided into five classes and twen
County Cork is a county in Ireland. It is the largest and southernmost county of Ireland, situated in the province of Munster and named after the city of Cork, Ireland's second-largest city; the Cork County Council is the local authority for the county. Its largest market towns are Mallow, Macroom and Skibbereen. In 2016, the county's population was 542,868. Notable Corkonians include Michael Collins, Jack Lynch, Sonia O'Sullivan. Cork borders four other counties; the county contains the Golden Vale pastureland and stretches from Kanturk in the north to Allihies in the south. The south-west region, including West Cork, is one of Ireland's main tourist destinations, known for its rugged coast, megalithic monuments, as the starting point for the Wild Atlantic Way; the county is known as the "Rebel county", a name given to them by King Henry VII of England for its support of a man claiming to be Richard, Duke of York in a futile attempt at a rebellion. The main third-level educator is University College Cork, founded in 1845, with a current undergraduate population around 15,000.
Significant local industry and employers include technology company Dell EMC, the European headquarters of Apple, Dairygold, which own milk-processing factories in Mitchelstown and Mallow. Two local authorities have remits which collectively encompass the geographic area of the county and city of Cork; the county, excluding Cork city, is administered by Cork County Council, while the city is administered separately by Cork City Council. Both city and county are part of the South-West Region. For standardized European statistical purposes, both Cork County Council and Cork City Council rank as first-level local administrative units of the NUTS 3 South-West Region. Thirty-four such LAU 1 entities are in the Republic of Ireland. For elections to Dáil Éireann, the county is divided into five constituencies—Cork East, Cork North-Central, Cork North-West, Cork South-Central and Cork South-West. Together they return 18 deputies to the Dáil; the county is part of the South constituency for the purposes of European elections.
For purposes other than local government, such as the formation of sporting teams, the term "County Cork" is taken to include both city and county. County Cork is located in the province of Munster, bordering Kerry to the west, Limerick to the north, Tipperary to the north-east and Waterford to the east, it is the largest county in Ireland by land area, the largest of Munster's six counties by population and area. At the last census in 2016, Cork city stood at 125,657; the population of the entire county is 542,868 making it the state's second-most populous county and the third-most populous county on the island of Ireland. The remit of Cork County Council includes some suburbs of the city not within the area of Cork City Council. Twenty-four historic baronies are in the county—the most of any county in Ireland. While baronies continue to be defined units, they are no longer used for many administrative purposes, their official status is illustrated by Placenames Orders made since 2003, where official Irish names of baronies are listed.
The county has 253 civil parishes. Townlands are the smallest defined geographical divisions in Ireland, with about 5447 townlands in the county; the county's mountain rose during a period mountain formation some 374-360 million years ago and include the Slieve Miskish and Caha Mountains on the Beara Peninsula, the Ballyhoura Mountains on the border with Limerick and the Shehy Mountains which contain Knockboy, the highest point in Cork. The Shehy Mountains are on the border with Kerry and may be accessed from the area known as Priests Leap, near the village of Coomhola; the Galtee Mountains are located across parts of Tipperary and Cork and are Ireland's highest inland mountain range. The upland areas of the Ballyhoura, Boggeragh and Mullaghareirk Mountain ranges add to the range of habitats found in the county. Important habitats in the uplands include blanket bog, glacial lakes, upland grasslands. Cork has the 13th-highest county peak in Ireland. Three rivers, the Bandon and Lee, their valleys dominate central Cork.
Habitats of the valleys and floodplains include woodlands, marshes and species-rich limestone grasslands. The River Bandon flows through several towns, including Dunmanway to the west of the town of Bandon before draining into Kinsale Harbour on the south coast. Cork's sea loughs include Lough Hyne and Lough Mahon, the county has many small lakes. An area has formed where the River Lee breaks into a network of channels weaving through a series of wooded islands. About 85 hectares of swamp are around Cork's wooded area; the Environmental Protection Agency carried out a survey of surface waters in County Cork between 1995 and 1997, which identified 125 rivers and 32 lakes covered by the regulations. Cork has a flat landscape with many beaches and sea cliffs along its coast; the southwest of Ireland is known for its peninsulas and some in Cork include the Beara Peninsula, Sheep's Head, Mizen Head, Brow Head. Brow Head is the most southerly point of mainland Ireland. There are many islands off the coast in particular, off West Cork.
Carbery's Hundred Isles are the islands around Long Island Roaringwater Bay. Fastnet Rock lies in the Atlantic Ocean 11.3 km south of mainland Ireland, making it the most southerly point of Ireland. Many notable islands lie off Cork, including Bere, Great and Cape Clear. Cork has 1,094 km of coastline, the second-longest coastline of any county after Mayo
In a navy, a rate, rating or bluejacket is a junior enlisted member of that navy, not a warrant officer or commissioned officer. Depending on the country and navy that uses it, the exact term and the range of ranks that it refers to may vary. In the Royal Navy and other navies in the Commonwealth and rating are interchangeably used to refer to an enlisted member of the navy, ranked below warrant officers and commissioned officers but may include petty officers and chief petty officers; the term comes from the general nautical usage of rating to refer to a seaman's class or grade as recorded in the ship's books. The system of conferring authority on sailors in the Royal Navy evolved through the recognition of competence: landsman, ordinary seaman, able seaman, through to the appointment of authority as a petty officer; the general structure now used breaks down into four major groupings: Able rate Leading rate Petty officer Chief petty officer In the United States Navy, the term bluejacket is used instead to refer enlisted sailors that rank below a chief petty officer.
Bluejacket derives itself from an item of clothing, worn by junior enlisted sailors before 1886. It is used when the sailors are deployed ashore as infantry. Royal Navy ratings rank insignia Rating system of the Royal Navy List of United States Navy ratings List of United States Navy enlisted rates List of United States Coast Guard ratings The Bluejacket's Manual Baker, Ernest A; the New English Dictionary, Odhams Press, London, 1932. Cutler, Thomas J; the Blue Jacket's Manual Centennial Edition, Naval Institute Press, Maryland, 2002. ISBN 9781557502087
The British Academy
The British Academy is the United Kingdom's national academy for the humanities and the social sciences. It received its Royal Charter in the same year, it is now a fellowship of more than 1,000 leading scholars spanning all disciplines across the humanities and social sciences and a funding body for research projects across the United Kingdom. The academy is a self-governing and independent registered charity, based at 10–11 Carlton House Terrace in London; the British Academy is funded with an annual grant from the Department for Business and Skills. In 2014/15 the British Academy's total income was £33,100,000, including £27,000,000 from BIS. £32,900,000 was distributed during the year in research grants and charitable activities. The academy states that it has five fundamental purposes: To speak up for the humanities and the social sciences To invest in the best researchers and research To inform and enrich debate around society’s greatest questions To ensure sustained international engagement and collaboration To make the most of the Academy’s assets to secure the Academy for the future.
The creation of a "British Academy for the Promotion of Historical and Philological Studies" was first proposed in 1899 in order that Britain could be represented at meetings of European and American academies. The organisation, which has since become "the British Academy", was initiated as an unincorporated society on 17 December 1901, received its Royal Charter from King Edward VII on 8 August 1902. Since many of Britain's most distinguished scholars in the humanities and social sciences have been involved in the life of the academy, including John Maynard Keynes, Isaiah Berlin, C. S. Lewis and Henry Moore; until 1927–28 the academy had no premises. It moved to some rooms in No. 6 Burlington Gardens. In 1968 it moved the short distance to Burlington House, it subsequently moved to headquarters near Regent's Park. In 1998 the Academy moved to its present headquarters in Carlton House Terrace. Overlooking St James's Park, the terrace was built in the 1820s and 1830s. Number 10 was the London residence of the Ridley family and number 11 was from 1856 to 1875 the home of Prime Minister William Gladstone.
In March 2010, the academy embarked on a £2.75m project to renovate and restore the public rooms in No. 11, following the departure of former tenant the Foreign Press Association, link the two buildings together. The work was completed in January 2011 and the new spaces include a new 150-seat Wolfson Auditorium are available for public hire; the history and achievements of the academy have been recorded in works by two of its secretaries. Sir Frederic Kenyon's volume of 37 pages covers the years up to 1951. Election as a Fellow of the British Academy recognises high scholarly distinction in the humanities or social sciences, evidenced by published work. Fellows may use the letters FBA after their names. Fellows are elected into one of the following disciplinary sections: HumanitiesClassical Antiquity Theology and Religious Studies African and Oriental Studies Linguistics and Philology Early Modern Languages and Literatures Modern Languages and other Media Archaeology Medieval Studies Early Modern History to c1800 Modern History from c1800 History of Art and Music Philosophy Culture and PerformanceSocial SciencesLaw Economics and Economic History Anthropology and Geography Sociology and Social Statistics Political Studies: Political Theory and International Relations Psychology Management and Business StudiesThere is an Education'ginger group'.
The British Academy channels substantial public funding into support for individuals and organisations pursuing humanities and social sciences research and scholarship in the UK and overseas. These funding schemes are designed to aid scholars at different stages of their academic career and include postdoctoral fellowships, Wolfson Research Professorships, Leverhulme Senior Research Fellowships, small research grants and British Academy Research Projects. In addition to its main public funds supported by the Department for Business and Skills, the academy draws on private funds arising from gifts, contributions made by fellows and grants from research foundations to support a further range of research activities. In 2014/15, the academy received around £30m to support research and researchers across the humanities and social sciences. Funds available to the academy were invested in the following main areas: research career development; the demand and quality of applications submitted for academy funding remains high.
This year the academy received around 3,600 applications and made 588 awards to scholars based in around 100 different universities across the UK – a success rate of 16%. In order to promote the interests of UK research and learning around the world, the Academy works to create frameworks to support international networking and collaboration and develop the role of humanities and social sciences research in tackling global challenges, it draws on expertise from a wide range of sources from within the fellowship and on specialist advice from its seven Area Panels for Africa, the Middle East, South Asia, Latin America/Caribbean. The Academy funds and coordinates a network of overseas institutes which provide local expertise, logistical support and a working base for UK scholars; these include research institutes in Amman, Athens, Nairobi and Tehran, as well as UK-based specialist learned societies which run strategic research programmes in o
In mathematics, a theorem is a statement, proven on the basis of established statements, such as other theorems, accepted statements, such as axioms. A theorem is a logical consequence of the axioms; the proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, experimental. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. In light of the interpretation of proof as justification of truth, the conclusion is viewed as a necessary consequence of the hypotheses, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.
Although they can be written in a symbolic form, for example, within the propositional calculus, theorems are expressed in a natural language such as English. The same is true of proofs, which are expressed as logically organized and worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, from which a formal symbolic proof can in principle be constructed; such arguments are easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but explains in some way why it is true. In some cases, a picture alone may be sufficient to prove a theorem; because theorems lie at the core of mathematics, they are central to its aesthetics. Theorems are described as being "trivial", or "difficult", or "deep", or "beautiful"; these subjective judgments vary not only from person to person, but with time: for example, as a proof is simplified or better understood, a theorem, once difficult may become trivial.
On the other hand, a deep theorem may be stated but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a well-known example of such a theorem. Logically, many theorems are of the form of an indicative conditional: if A B; such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion; the theorem "If n is an natural number n/2 is a natural number" is a typical example in which the hypothesis is "n is an natural number" and the conclusion is "n/2 is a natural number". To be proved, a theorem must be expressible as a formal statement. Theorems are expressed in natural language rather than in a symbolic form, with the intention that the reader can produce a formal statement from the informal one, it is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses are called axioms or postulates.
The field of mathematics known as proof theory studies formal languages and the structure of proofs. Some theorems are "trivial", in the sense that they follow from definitions and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas. Other theorems have a known proof that cannot be written down; the most prominent examples are the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search, verified by a computer program. Many mathematicians did not accept this form of proof, but it has become more accepted.
The mathematician Doron Zeilberger has gone so far as to claim that these are the only nontrivial results that mathematicians have proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. To establish a mathematical statement as a theorem, a proof is required, that is, a line of reasoning from axioms in the system to the given statement must be demonstrated. However, the proof is considered as separate from the theorem statement. Although more than one proof may be known for a single theorem, only one proof is required to establish the status of a statement as a theorem; the Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs. Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved.