Mathematical proof

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.

Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Mathematics

Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),structure,space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.

Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.

Mathematics (producer)

Mathematics, also known as Allah Mathematics, (born: Ronald Maurice Bean) is a hip hop producer and DJ for the Wu-Tang Clan and its solo and affiliate projects.

Biography

Born and raised in Jamaica, Queens, New York, Mathematics was introduced to hip hop by his brother who used to bring home recordings of the genre's pioneers like Grandmaster Flash & The Furious Five, Treacherous Three and Cold Crush Brothers. He began his career in 1987 DJing block parties and park jams in Baisley Projects, going by the name Supreme Cut Master. In 1988, he became the full-time DJ for experienced rapper Victor C, doing countless shows in clubs and colleges in New York City.

In 1990, Mathematics linked up with GZA/Genius; he soon became one of the Wu-Tang Clan's founding members, but at the time GZA was struggling to build a career on the Cold Chillin' label. This partnership earned Mathematics a spot on his first official tour, The Cold Chillin Blizzard Tour (with popular acts such as Biz Markie, Big Daddy Kane, Kool G. Rap & DJ Polo and Marley Marl).

Mathematics (song)

"Mathematics" is a b-side single from Mos Def's solo debut album, Black on Both Sides. It contains lyrics about various social issues and asks the listener to add them up and come to conclusions about them. Many references to numbers are found in this song and at times, Mos Def rhymes statistics in numerical order. The song highlights the differences between the White and African-American citizens of the US and uses the lyrics "Do your math..." (from Erykah Badu's "On & On") telling young African-American's to 'do their maths' so they can avoid being part of the numerous degrading statistics he raps about in the opening and third verses of the song. The song is produced by DJ Premier whose famous scratch samples make up the song's bridge. Premier has called it one of his favorite beats.

It can be found on the soundtrack of Madden NFL 2002. The song is also played briefly in the CSI: Crime Scene Investigation episode "Crate 'n' Burial".

Samples

The bridge of "Mathematics" contains DJ Premier's signature scratched vocals from various hip hop songs. The lyrics of those samples as well as information about their origin can be found below:

Colour Coding

Colour Coding was an Australian indie pop band officially formed in 2011 and based in Sydney. The band was formed by Chris Holland and Tim Commandeur, both members of the group Operator Please; after Operator Please's second album, cousins Holland and Commandeur began writing as Colour Coding. The band released its first EP Proof in March 2012 with the lead single "Perfect" uploaded as a free track online in November 2011.

Early life

At ages 12 and 13, cousins Chris and Tim recorded an extended play in Tim's father's mark studio. In 2005, Tim joined the pop group Operator Please with several of his Gold Coast schoolmates, including founder and lead vocalist Amandah Wilkinson. In 2008, Chris joined the band as keyboardist.

This early experience helped the boys realize their dream of forming their own band, Colour Coding.

2011–present: Proof

In September 2011, Hoppy Studios (Sydney) produced their first single, "Perfect". It was a free-to-download online release track. The track was then added to the fourth digital release by New York label Cosine Records in the US.

Die proof (philately)

In philately a Die Proof is a printed image pulled directly from the master die for an engraved stamp.

As a stamp is engraved it is necessary to check progress and a series of proofs are printed or 'pulled' from the die. These are known as progressive or contemporary die proofs.

Progressive proofs also form part of the design and approval process for a stamp. Any changes made during this process turn the proofs into essays which may be identified retrospectively because they differ from the issued stamp. Die proofs for engraved stamps are normally printed under great pressure onto oversized card and, as they are printed from the master die, they are normally of high quality. By contrast they may also be printed on India Paper, a strong, thin, opaque paper. The proofs are not necessarily in the same colour(s) as the issued stamp. A proof on India or other paper mounted on a die-sunk card is known as a hybrid proof.

Once the die is completed, it is transferred multiple times to the plate from which the stamps are printed.

Proof (2005 film)

Proof is a 2005 American drama film directed by John Madden and starring Gwyneth Paltrow, Anthony Hopkins, Jake Gyllenhaal, and Hope Davis. It was written by Rebecca Miller, based on David Auburn's Pulitzer Prize-winning play of the same title.

Plot

The film opens with the 27-year-old protagonist, young mathematician Catherine (Gwyneth Paltrow) talking to her father Robert (Anthony Hopkins) after he startles her watching TV in the middle of the night. He gives her a bottle of champagne for her birthday, and they chat for a while about the nature of insanity, ending with the revelation that Robert died last week and his funeral is tomorrow.

Awakened from this dream, Catherine realizes that Hal (Jake Gyllenhaal), a former graduate student of Robert's, is still upstairs, reading through Robert's books. Hal believes that somewhere in the gibberish there might be additional products of his mathematical genius. On his way out, he comments on the vast amount of work Robert did. Catherine, suspicious, searches his backpack but finds nothing, but when she picks up Hal's coat to give to him, a notebook falls out. He explains that he wanted to give the notebook as a birthday present because it "had something written in it about her, not math, her". Hal is forced to leave, giving the notebook as intended, when Catherine calls the police.