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Distributive property

In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality $$x \cdot (y + z) = x \cdot y + x \cdot z$$ is always true in elementary algebra. For example, in elementary arithmetic, one has $$2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3).$$ Therefore, one would say that multiplication distributes over addition.

This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted $$\,\land\,$$) and the logical or (denoted $$\,\lor\,$$) distributes over the other.

Definition
Given a set $$S$$ and two binary operators $$\,*\,$$ and $$\,+\,$$ on $$S,$$ $$x * (y + z) = (x * y) + (x * z);$$ $$(y + z) * x = (y * x) + (z * x);$$
 * the operation $$\,*\,$$ is over (or with respect to) $$\,+\,$$ if, given any elements $$x, y, \text{ and } z$$ of $$S,$$
 * the operation $$\,*\,$$ is over $$\,+\,$$ if, given any elements $$x, y, \text{ and } z$$ of $$S,$$
 * and the operation $$\,*\,$$ is over $$\,+\,$$ if it is left- and right-distributive.

When $$\,*\,$$ is commutative, the three conditions above are logically equivalent.

Meaning
The operators used for examples in this section are those of the usual addition $$\,+\,$$ and multiplication $$\,\cdot.\,$$

If the operation denoted $$\cdot$$ is not commutative, there is a distinction between left-distributivity and right-distributivity:

$$a \cdot \left( b \pm c \right) = a \cdot b \pm a \cdot c \qquad \text{ (left-distributive) }$$ $$(a \pm b) \cdot c = a \cdot c \pm b \cdot c \qquad \text{ (right-distributive) }.$$

In either case, the distributive property can be described in words as:

To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).

If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa, and one talks simply of.

One example of an operation that is "only" right-distributive is division, which is not commutative: $$(a \pm b) \div c = a \div c \pm b \div c.$$ In this case, left-distributivity does not apply: $$a \div(b \pm c) \neq a \div b \pm a \div c$$

The distributive laws are among the axioms for rings (like the ring of integers) and fields (like the field of rational numbers). Here multiplication is distributive over addition, but addition is not distributive over multiplication. Examples of structures with two operations that are each distributive over the other are Boolean algebras such as the algebra of sets or the switching algebra.

Multiplying sums can be put into words as follows: When a sum is multiplied by a sum, multiply each summand of a sum with each summand of the other sum (keeping track of signs) then add up all of the resulting products.

Real numbers
In the following examples, the use of the distributive law on the set of real numbers $$\R$$ is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a field, which ensures the validity of the distributive law.

First example (mental and written multiplication):During mental arithmetic, distributivity is often used unconsciously: $6 \cdot 16 = 6 \cdot (10 + 6) = 6\cdot 10 + 6 \cdot 6 = 60 + 36 = 96$

Thus, to calculate $6 \cdot 16$ in one's head, one first multiplies $6 \cdot 10$ and $6 \cdot 6$ and add the intermediate results. Written multiplication is also based on the distributive law. Second example (with variables):$3 a^2 b \cdot (4 a - 5 b) = 3 a^2 b \cdot 4a - 3 a^2 b \cdot 5 b = 12 a^3 b - 15 a^2 b^2$ Third example (with two sums):$\begin{align} (a + b) \cdot (a - b) & = a \cdot (a - b) + b \cdot (a - b) = a^2 - ab + ba - b^2 = a^2 - b^2 \\ & = (a + b) \cdot a - (a + b) \cdot b = a^2 + ba - ab - b^2 = a^2 - b^2 \\ \end{align}$

Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out. Fourth example:Here the distributive law is applied the other way around compared to the previous examples. Consider $12 a^3 b^2 - 30 a^4 b c + 18 a^2 b^3 c^2 \,.$

Since the factor $6 a^2 b$ occurs in all summands, it can be factored out. That is, due to the distributive law one obtains $12 a^3 b^2 - 30 a^4 b c + 18 a^2 b^3 c^2 = 6 a^2 b \left(2 a b - 5 a^2 c + 3 b^2 c^2\right).$

Matrices
The distributive law is valid for matrix multiplication. More precisely, $$(A + B) \cdot C = A \cdot C + B \cdot C$$ for all $$l \times m$$-matrices $$A, B$$ and $$m \times n$$-matrices $$C,$$ as well as $$A \cdot (B + C) = A \cdot B + A \cdot C$$ for all $$l \times m$$-matrices $$A$$ and $$m \times n$$-matrices $$B, C.$$ Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.

Other examples

 * Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.
 * The cross product is left- and right-distributive over vector addition, though not commutative.
 * The union of sets is distributive over intersection, and intersection is distributive over union.
 * Logical disjunction ("or") is distributive over logical conjunction ("and"), and vice versa.
 * For real numbers (and for any totally ordered set), the maximum operation is distributive over the minimum operation, and vice versa: $$\max(a, \min(b, c)) = \min(\max(a, b), \max(a, c)) \quad \text{ and } \quad \min(a, \max(b, c)) = \max(\min(a, b), \min(a, c)).$$
 * For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: $$\gcd(a, \operatorname{lcm}(b, c)) = \operatorname{lcm}(\gcd(a, b), \gcd(a, c)) \quad \text{ and } \quad \operatorname{lcm}(a, \gcd(b, c)) = \gcd(\operatorname{lcm}(a, b), \operatorname{lcm}(a, c)).$$
 * For real numbers, addition distributes over the maximum operation, and also over the minimum operation: $$a + \max(b, c) = \max(a + b, a + c) \quad \text{ and } \quad a + \min(b, c) = \min(a + b, a + c).$$
 * For binomial multiplication, distribution is sometimes referred to as the FOIL Method (First terms $$a c,$$ Outer $$a d,$$ Inner $$b c,$$ and Last $$b d$$) such as: $$(a + b) \cdot (c + d) = a c + a d + b c + b d.$$
 * In all semirings, including the complex numbers, the quaternions, polynomials, and matrices, multiplication distributes over addition: $$u (v + w) = u v + u w, (u + v)w = u w + v w.$$
 * In all algebras over a field, including the octonions and other non-associative algebras, multiplication distributes over addition.

Rule of replacement
In standard truth-functional propositional logic,  in logical proofs uses two valid rules of replacement to expand individual occurrences of certain logical connectives, within some formula, into separate applications of those connectives across subformulas of the given formula. The rules are $$(P \land (Q \lor R)) \Leftrightarrow ((P \land Q) \lor (P \land R)) \qquad \text{ and } \qquad (P \lor (Q \land R)) \Leftrightarrow ((P \lor Q) \land (P \lor R))$$ where "$$\Leftrightarrow$$", also written $$\,\equiv,\,$$ is a metalogical symbol representing "can be replaced in a proof with" or "is logically equivalent to".

Truth functional connectives
is a property of some logical connectives of truth-functional propositional logic. The following logical equivalences demonstrate that distributivity is a property of particular connectives. The following are truth-functional tautologies. $$\begin{alignat}{13} &(P &&\;\land &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\lor (P \land R)) && \quad\text{ Distribution of } && \text{ conjunction } && \text{ over } && \text{ disjunction } \\ &(P &&\;\lor &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\land (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ conjunction } \\ &(P &&\;\land &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \land Q) &&\;\land (P \land R)) && \quad\text{ Distribution of } && \text{ conjunction } && \text{ over } && \text{ conjunction } \\ &(P &&\;\lor &&(Q \lor R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\;\lor (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ disjunction } \\ &(P &&\to &&(Q \to R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\to (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{ } && \text{  } \\ &(P &&\to &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\leftrightarrow (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{ over } && \text{ equivalence } \\ &(P &&\to &&(Q \land R)) &&\;\Leftrightarrow\;&& ((P \to Q) &&\;\land (P \to R)) && \quad\text{ Distribution of } && \text{ implication } && \text{ over } && \text{ conjunction } \\ &(P &&\;\lor &&(Q \leftrightarrow R)) &&\;\Leftrightarrow\;&& ((P \lor Q) &&\leftrightarrow (P \lor R)) && \quad\text{ Distribution of } && \text{ disjunction } && \text{ over } && \text{ equivalence } \\ \end{alignat}$$

$$\begin{alignat}{13} &((P \land Q) &&\;\lor (R \land S)) &&\;\Leftrightarrow\;&& (((P \lor R) \land (P \lor S)) &&\;\land ((Q \lor R) \land (Q \lor S))) && \\ &((P \lor Q) &&\;\land (R \lor S)) &&\;\Leftrightarrow\;&& (((P \land R) \lor (P \land S)) &&\;\lor ((Q \land R) \lor (Q \land S))) && \\ \end{alignat}$$
 * Double distribution:

Distributivity and rounding
In approximate arithmetic, such as floating-point arithmetic, the distributive property of multiplication (and division) over addition may fail because of the limitations of arithmetic precision. For example, the identity $$1/3 + 1/3 + 1/3 = (1 + 1 + 1) / 3$$ fails in decimal arithmetic, regardless of the number of significant digits. Methods such as banker's rounding may help in some cases, as may increasing the precision used, but ultimately some calculation errors are inevitable.

In rings and other structures
Distributivity is most commonly found in semirings, notably the particular cases of rings and distributive lattices.

A semiring has two binary operations, commonly denoted $$\,+\,$$ and $$\,*,$$ and requires that $$\,*\,$$ must distribute over $$\,+.$$

A ring is a semiring with additive inverses.

A lattice is another kind of algebraic structure with two binary operations, $$\,\land \text{ and } \lor.$$ If either of these operations distributes over the other (say $$\,\land\,$$ distributes over $$\,\lor$$), then the reverse also holds ($$\,\lor\,$$ distributes over $$\,\land\,$$), and the lattice is called distributive. See also.

A Boolean algebra can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra.

Similar structures without distributive laws are near-rings and near-fields instead of rings and division rings. The operations are usually defined to be distributive on the right but not on the left.

Generalizations
In several mathematical areas, generalized distributivity laws are considered. This may involve the weakening of the above conditions or the extension to infinitary operations. Especially in order theory one finds numerous important variants of distributivity, some of which include infinitary operations, such as the infinite distributive law; others being defined in the presence of only binary operation, such as the according definitions and their relations are given in the article distributivity (order theory). This also includes the notion of a completely distributive lattice.

In the presence of an ordering relation, one can also weaken the above equalities by replacing $$\,=\,$$ by either $$\,\leq\,$$ or $$\,\geq.$$ Naturally, this will lead to meaningful concepts only in some situations. An application of this principle is the notion of sub-distributivity as explained in the article on interval arithmetic.

In category theory, if $$(S, \mu, \nu)$$ and $$\left(S^{\prime}, \mu^{\prime}, \nu^{\prime}\right)$$ are monads on a category $$C,$$ a distributive law $$S. S^{\prime} \to S^{\prime}. S$$ is a natural transformation $$\lambda : S. S^{\prime} \to S^{\prime}. S$$ such that $$\left(S^{\prime}, \lambda\right)$$ is a lax map of monads $$S \to S$$ and $$(S, \lambda)$$ is a colax map of monads $$S^{\prime} \to S^{\prime}.$$ This is exactly the data needed to define a monad structure on $$S^{\prime}. S$$: the multiplication map is $$S^{\prime} \mu. \mu^{\prime} S^2. S^{\prime} \lambda S$$ and the unit map is $$\eta^{\prime} S. \eta.$$ See: distributive law between monads.

A generalized distributive law has also been proposed in the area of information theory.

Antidistributivity
The ubiquitous identity that relates inverses to the binary operation in any group, namely $$(x y)^{-1} = y^{-1} x^{-1},$$ which is taken as an axiom in the more general context of a semigroup with involution, has sometimes been called an antidistributive property (of inversion as a unary operation).

In the context of a near-ring, which removes the commutativity of the additively written group and assumes only one-sided distributivity, one can speak of (two-sided) distributive elements but also of antidistributive elements. The latter reverse the order of (the non-commutative) addition; assuming a left-nearring (i.e. one which all elements distribute when multiplied on the left), then an antidistributive element $$a$$ reverses the order of addition when multiplied to the right: $$(x + y) a = y a + x a.$$

In the study of propositional logic and Boolean algebra, the term antidistributive law is sometimes used to denote the interchange between conjunction and disjunction when implication factors over them: $$(a \lor b) \Rightarrow c \equiv (a \Rightarrow c) \land (b \Rightarrow c)$$ $$(a \land b) \Rightarrow c \equiv (a \Rightarrow c) \lor (b \Rightarrow c).$$

These two tautologies are a direct consequence of the duality in De Morgan's laws.

title: Distributive property


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Modern flat Earth beliefs

Pseudoscientific beliefs in a flat Earth are promoted by a number of organizations and individuals. The claims of modern flat Earth proponents are not based on scientific knowledge and are contrary to over two millennia of scientific consensus based on multiple confirming lines of evidence that Earth is spherical. Flat Earth beliefs are classified by experts in philosophy and physics as a form of science denial.

Flat Earth groups of the modern era date from the middle of the 20th century; some adherents are serious and some are not. Those who are serious are often motivated by religion or conspiracy theories. Through the use of social media, flat Earth theories have been increasingly espoused and promoted by individuals unaffiliated with larger groups. Many believers make use of social media to spread their views.

Background
Many ancient cultures may have believed that the Earth, Sun, Moon, stars, and planets were flat as they can sometimes appear so on a local level. The Greeks deduced that they were instead a sphere, and that became the scientific consensus.

19th and early 20th centuries
Contrary to the popular belief that the Earth was generally believed to be flat until a few hundred years ago, Earth's sphericity has been widely accepted in the Western world (and universally by scholars) since at least the Hellenistic period (323 BCE–31 BCE). It was not until the 19th century that the Flat Earth concept had a resurgence.

Modern flat Earth belief originated with the English writer Samuel Rowbotham (1816–1884). Based on conclusions derived from his 1838 Bedford Level experiment, Rowbotham published the 1849 pamphlet titled Zetetic Astronomy, writing under the pseudonym "Parallax". He later expanded this into the book Earth Not a Globe, proposing the Earth is a flat disc centred at the North Pole and bounded along its southern edge by a wall of ice, Antarctica. Rowbotham further held that the Sun and Moon were 3000 mi above Earth and that the "cosmos" was 3100 mi above the Earth. He also published a leaflet titled The Inconsistency of Modern Astronomy and its Opposition to the Scriptures, which argued that the "Bible, alongside our senses, supported the idea that the Earth was flat and immovable and this essential truth should not be set aside for a system based solely on human conjecture".

Rowbotham and followers like William Carpenter gained attention by successful use of pseudoscience in public debates with leading scientists such as Alfred Russel Wallace. Rowbotham created a Zetetic Society in England and New York, shipping over a thousand copies of Zetetic Astronomy to the New York branch. Wallace repeated the Bedford Level experiment in 1870, correcting for atmospheric refraction and showing a spherical Earth.

In 1877, John Hampden produced a book A New Manual of Biblical Cosmography. Rowbotham also produced studies that purported to show that the effects of ships disappearing below the horizon could be explained by the laws of perspective in relation to the human eye.

After Rowbotham's death, Lady Elizabeth Blount established the Universal Zetetic Society in 1893, whose objective was "the propagation of knowledge related to Natural Cosmogony in confirmation of the Holy Scriptures, based on practical scientific investigation". The society published a magazine, The Earth Not a Globe Review, which sold for twopence and remained active well into the early 20th century. A flat Earth journal, Earth: a Monthly Magazine of Sense and Science, was published between 1901 and 1904, edited by Lady Blount. She held that the Bible was the unquestionable authority on the natural world and argued that one could not be a Christian and believe the Earth is a globe. Well-known members included E. W. Bullinger of the Trinitarian Bible Society, Edward Haughton, senior moderator in natural science in Trinity College Dublin and an archbishop. She repeated Rowbotham's experiments, generating some counter-experiments, but interest declined after the First World War. The Universal Zetetic Society "was revived under different names over the years—in 1956, 1972, and 2004". The movement gave rise to several books that argued for a flat, stationary Earth, including Terra Firma by David Wardlaw Scott.

Other notable flat Earthers from this time period include:
 * William Carpenter, a printer originally from Greenwich, was a supporter of Rowbotham. Carpenter published Theoretical Astronomy Examined and Exposed – Proving the Earth not a Globe in eight parts from 1864 under the name Common Sense. He later emigrated to Baltimore, where he published One Hundred Proofs the Earth is Not a Globe in 1885. He wrote: "There are rivers that flow for hundreds of miles towards the level of the sea without falling more than a few feet – notably, the Nile, which, in a thousand miles, falls but a foot. A level expanse of this extent is quite incompatible with the idea of the Earth's convexity. It is, therefore, a reasonable proof that Earth is not a globe", as well as: "If the Earth were a globe, a small model globe would be the very best – because the truest – thing for the navigator to take to sea with him. But such a thing as that is not known: with such a toy as a guide, the mariner would wreck his ship, of a certainty! This is a proof that Earth is not a globe."
 * John Jasper, an American slave turned prolific preacher, and friend of Carpenter's, echoed his friend's sentiments in his most famous sermon "The Sun do move", preached over 250 times, always by invitation. In a written account of his sermon, published in The Richmond Whig of March 19, 1878, Jasper says he would frequently cite the verse "I saw four angels standing on the four corners of the earth" and follow up by arguing: "So we are living on a four-cornered earth; then, my friends, will you tell me how in the name of God can an earth with four corners be round!" In the same article he argued: "if the earth is like others say, who hold a different theory, peopled on the other side, those people would be obliged to walk on the ground with their feet upward like flies on the ceiling of a room".
 * In Brockport, New York, in 1887, M. C. Flanders argued the case of a flat Earth for three nights against two scientific gentlemen defending sphericity. Five townsmen chosen as judges voted unanimously for a flat Earth at the end. The case was reported in the Brockport Democrat.
 * Joseph W. Holden of Maine, a former justice of the peace, gave numerous lectures in New England and lectured on flat-Earth theory at the Columbian Exposition in Chicago. His fame stretched to North Carolina, where the Statesville Semi-weekly Landmark recorded at his death in 1900: "We hold to the doctrine that the Earth is flat ourselves and we regret exceedingly to learn that one of our members is dead."
 * In 1898, during his solo circumnavigation of the world, Joshua Slocum encountered a group of flat-Earthers in Durban, South Africa. Three Boers, one of them a clergyman, presented Slocum with a pamphlet in which they set out to prove that the world was flat. Paul Kruger, President of the Transvaal Republic, advanced the same view: "You don't mean round the world, it is impossible! You mean in the world. Impossible!"
 * From 1915 to 1942 Wilbur Glenn Voliva, who in 1906 took over the Christian Catholic Church, a Pentecostal sect that established a utopian community in Zion, Illinois, preached flat Earth doctrine. He used a photograph of a 12 mi stretch of the shoreline at Lake Winnebago, Wisconsin, taken 3 ft above the waterline to prove his point. When the airship Italia disappeared on an expedition to the North Pole in 1928, he warned the world's press that it had sailed over the edge of the world. He offered a $5000 award ($98,590 in 2022 terms) for proving that the Earth is not flat, under his own conditions. Teaching a globular Earth was banned in the Zion schools, and the message was transmitted on his WCBD radio station.
 * Along with those who followed him, Frank Cherry (died 1963), the founder of the Black Hebrew Israelite religion, taught the existence of a flat Earth "surrounded by three layers of heaven."

International Flat Earth Research Society
In 1956, Samuel Shenton created the International Flat Earth Research Society, better known as the "Flat Earth Society", as a successor to the Universal Zetetic Society, running it as "organising secretary" from his home in Dover, England. Given Shenton's interest in alternative science and technology, the emphasis on religious arguments was less than in the predecessor society. This was just before the Soviet Union launched the first artificial satellite, Sputnik; he responded: "Would sailing round the Isle of Wight prove that it were spherical? It is just the same for those satellites."

His primary aim was to reach children before they were convinced about a spherical Earth. Despite plenty of publicity, the space race eroded Shenton's support in Britain until 1967, when he started to gain attention during the Apollo program. When satellite images showed Earth as a sphere, Shenton remarked: "It's easy to see how a photograph like that could fool the untrained eye". Later asked about similar photographs taken by astronauts, he attributed curvature to the use of wide-angle lens, adding, "It's a deception of the public and it isn't right".

In 1969, Shenton persuaded Ellis Hillman, a Polytechnic of East London lecturer, to become president of the Flat Earth Society after attempts to convince Eden Thomas, a former chairman, to take on the role; but there is little evidence of any activity on his part until after Shenton's death, when he added most of Shenton's library to the archives of the Science Fiction Foundation he helped to establish.

Shenton died in 1971. Charles K. Johnson, a correspondent from California, inherited part of Shenton's library from Shenton's wife; he incorporated and became president of the International Flat Earth Research Society of America and Covenant People's Church in California. Over the next three decades, under his leadership, the Flat Earth Society grew to a reported 3,500 members.

Johnson spent years examining the studies of flat- and round-Earth theories and proposed evidence of a conspiracy against flat Earth: "The idea of a spinning globe is only a conspiracy of error that Moses, Columbus, and FDR all fought..." His article was published in the magazine Science Digest in 1980. It goes on to state: "If it is a sphere, the surface of a large body of water must be curved. The Johnsons have checked the surfaces of Lake Tahoe and the Salton Sea without detecting any curvature."

Johnson issued many publications and handled all membership applications. The most famous publication was Flat Earth News, a quarterly, four-page tabloid. Johnson paid for these publications through annual member dues costing US$6 to US$10 over the course of his leadership. Johnson cited the Bible for his beliefs, and he saw scientists as pulling a hoax which would replace religion with science.

The Flat Earth Society's most recent planet model is that humanity lives on a disc, with the North Pole at its centre and a 150 ft wall of ice, Antarctica, at the outer edge. The resulting map resembles the symbol of the United Nations, which Johnson used as evidence for his position. In this model, the Sun and Moon are each 32 mi in diameter.

The Flat Earth Society recruited members by speaking against the US government and all its agencies, particularly NASA. Much of the society's literature in its early days focused on interpreting the Bible to mean that the Earth is flat, although they did try to offer scientific explanations and evidence.

Criticism
Eugenie Scott called the group an example of "extreme Biblical-literalist theology: The earth is flat because the Bible says it is flat, regardless of what science tells us".

According to some flat Earthers, the Flat Earth Society is a government-controlled organization whose true purpose is to make ridiculous claims about flat Earth and therefore discredit the flat Earth movement.

Decline and relaunch
According to Charles K. Johnson, the membership of the group rose to 3,500 under his leadership but began to decline after a fire at his house in 1997 which destroyed all of the records and contacts of the society's members. Johnson's wife, who helped manage the membership database, died shortly thereafter. Johnson himself died on 19 March 2001.

In 2004, Daniel Shenton (not related to Samuel) resurrected the Flat Earth Society, basing it around a web-based discussion forum. In the late 1990s, Thomas Dolby's 1984 album The Flat Earth inspired Shenton to look into the Flat Earth Society, and he came to believe in its ideas. He believes that no one has provided proof that the world is not flat. This eventually led to the official relaunch of the society in October 2009, and the creation of a new website, featuring a public collection of flat Earth literature and a wiki. Moreover, the society began accepting new members for the first time since 2001. Dolby accepted Shenton's offer of membership number 00001, although he does not believe the Earth is flat. , over 500 people had become members.

In 2013, part of this society broke away to form a new web-based group also featuring a forum and wiki.

Canada
Flat Earth Society of Canada was established on 8 November 1970 by philosopher Leo Ferrari, writer Raymond Fraser and poet Alden Nowlan; and was active until 1984. Its archives are held at the University of New Brunswick.

Calling themselves "planoterrestrialists", their aims were quite different from other flat Earth societies. They claimed a prevailing problem of the new technological age was the willingness of people to accept theories "on blind faith and to reject the evidence of their own senses." The parodic intention of the Society appeared in the writings of Ferrari, as he attributed everything from gender to racial inequality on the globularist and the spherical Earth model. Ferrari even claimed to have nearly fallen off "the Edge" of the Earth at Brimstone Head on Fogo Island.

Ferrari was interviewed as an "expert" in the 1990 flat Earth mockumentary In Search of the Edge by Pancake Productions (a reference to the expression "as flat as a pancake"). In the accompanying study guide, Ferrari is outed as a "globularist", a nonce word for someone who believes the Earth is spherical. The real intent of the film, which was part-funded by the Ontario Arts Council and National Film Board of Canada, was to promote schoolchildren's critical thinking and media literacy by "[attempting] to prove in convincing fashion, something everyone knew to be false."

Relaunch
Multi-media artist Kay Burns re-created the Flat Earth Society of Canada as an art project with her alter ego Iris Taylor as its president. Burns created an installation entitled the Museum of the Flat Earth, which included some artefacts from the 1970 group. It was exhibited in 2016 at the Flat Earth Outpost Café in Shoal Bay, Newfoundland.

Italy
In Italy there are no centralised societies on flat Earth. However, since the 2010s, small groups of conspiracy theorists, who carry out meetings, started to emerge and to spread flat Earth theories. Among these are Calogero Greco, Albino Galuppini and Agostino Favari, who in 2018–2019 organised several meetings in Palermo, Sicily, with an entry price of €20.

Among their claims, some include:
 * NASA is similar to Disneyland and that astronauts are actors.
 * The April 2019 supermassive black hole photo at the core of the supergiant elliptical galaxy Messier 87 is fake.
 * The proof Earth is flat lies in a filled bottle where, if placed horizontally, water never curves.

In addition to these, it is their common belief that the United States has a plan to create in Europe a new America open to everyone, where the only value is consumerism and that George Soros commands a satanic globalist conspiracy. They reject the past existence of dinosaurs, the Darwinian theory of evolution, and the authority of the scientific community, claiming scientists are Freemasons.

Former leader of the Five Star Movement political party Beppe Grillo showed interest in the group, admitting to admiring their free speech spirit and to wanting to participate at the May 2019 conference. However, Grillo did not appear.

Internet-era resurgence
In November 2017 "more than five hundred people ... paid as much as $249 each to attend "the first-ever Flat Earth Conference", in a suburb of Raleigh, North Carolina, U.S.  According to a 2018 YouGov opinion poll, "just 66% of millennials firmly believe" that the earth is round, with celebrities (rapper B.o.B., basketball players Kyrie Irving, Wilson Chandler, Draymond Green) advocating for flatness.

Sociological explanations for counterfactual beliefs
In the Information Age, the availability of communications technology and social media like YouTube, Facebook and Twitter have made it easy for individuals, famous or not, to spread disinformation and attract others to erroneous ideas. One of the topics that has flourished in this environment is that of the flat Earth. These sites have made it easier for like-minded theorists to connect with one another and mutually reinforce their beliefs. Social media has had a "levelling effect", in that experts have less sway in the public mind than they used to.

YouTube had faced criticism for allowing the spread of misinformation and conspiracy theories through its platform. In 2019, YouTube stated that it was making changes in its software to reduce the distribution of videos based on conspiracy theories including flat Earth.

In the documentary Behind the Curve (2018) (which follows prominent modern flat-Earthers including Mark Sargent and Patricia Steere, as well as astrophysicists and psychologists who attempt to explain the growing fad), professor of psychiatry Joe Pierre offers as explanations: the Dunning-Kruger effect (the phenomenon whereby ignorance in a given field makes people unable to recognize their own ignorance or lack of ability in that field); misunderstandings of simple observation; pseudoscientific practices which fail to separate reliable from unreliable conclusions; and a progressive divergence from reality that starts with a belief that conventional information sources and the government cannot be trusted.

Out of the necessity to explain photographs of the earth in space, the observations of astronauts, why all major institutions such as governments, media outlets, schools, scientists, and airlines assert that the world is a sphere, etc., modern flat-Earthers very commonly embrace some form of conspiracy theory. As Darryle Marble, a speaker at the Flat Earth Conference, told his audience, after watching hours of YouTube conspiracy videos on Sandy Hook, 9/11, false flags, the Bilderbergers, Rothschilds, Illuminati – "Each thing started to make that much more sense. I was already primed to receive the whole flat-Earth idea, because we had already come to the conclusion that we were being deceived about so many other things. So of course they would lie to us about this."

Conspiracy belief is often intertwined with conservative Christian belief. According to internet influencer Rob Skiba, "the ultimate motivation" of the (alleged) conspiracy of a round earth in space, "many of us have come to believe, is hiding God." Reading the Bible, "when you break down the text of what it represents, there's no way you can get a spinning heliocentric globe out of anything in the Bible." (According to author Alan Burdick, "in style and substance, the flat-Earth movement is a close cousin of creationism.")

They tend to not trust observations they have not made themselves, and often distrust or disagree with each other. Patricia Steere admitted in Behind the Curve that she wouldn't believe an event like the Boston Marathon bombing was real unless she had gotten her own leg blown off. Flat Earth believers in the documentary also professed belief in conspiracy theories about vaccines, genetically modified organisms, chemtrails, 9/11, and transgender people; some said dinosaurs and evolution were also fake, and that heliocentrism is a form of Sun worship.

The scientific experts in Behind the Curve pointed to confirmation bias as a way to maintain a counterfactual belief, by cherry-picking only supporting evidence, and dismissing any disconfirming evidence as part of the purported global conspiracy.

Some flat Earth believers, such as authors Zen Garcia and Edward Hendrie, cite the Christian Bible as evidence. Some critics of the flat Earth idea, such as astronomer Danny R. Faulkner, are young Earth creationists and attempt to explain away the Bible's supposed flat Earth language.

On 3 May 2018, Steven Novella analysed the modern belief in a flat Earth, and concluded that, despite what most people think about the subject, the believers are being sincere in their belief that the Earth is flat, and are not "just saying that to wind us up". He stated that: "In the end that is the core malfunction of the flat-earthers, and the modern populist rejection of expertise in general. It is a horrifically simplistic view of the world that ignores (partly out of ignorance, and partly out of motivated reasoning) to [sic] real complexities of our civilisation. It is ultimately lazy, childish, and self-indulgent, resulting in a profound level of ignorance drowning in motivated reasoning."

The British sceptical activist Michael Marshall attended the annual Flat Earth UK Convention on 27–29 April 2018 and noted disagreement on several views among believers in a flat Earth. To Marshall, one of the most telling moments at the convention was the "Flat Earth Addiction" test that was based on a checklist used to determine whether someone is in a cult, without the convention attendees realising the possibility of themselves being in a cult.

Beliefs
Based on the speakers at the 2018 UK's Flat Earth UK Convention, believers in a flat Earth vary widely in their views. While most agree upon a disc-shaped Earth, some are convinced the Earth is diamond-shaped. Furthermore, while most believers do not believe in outer space and none believe humans have ever travelled there, they vary widely in their views of the universe. (Flat Earth International Conferences, organized by Robbie Davidson, are unaffiliated with the Flat Earth Society. According to Davidson, the "Earth is ... a stationary plane, with the sun, moon, and stars inside a dome", while the Flat Earth Society promotes a model in which Earth is "a disk flying through space", and which Davidson finds "incredibly ridiculous".)

Filmmakers of Behind the Curve attended another flat Earth conference at which a substantial number of people believed the Earth was an infinite plane, potentially with more continents beyond the purported circular ice wall of Antarctica. Members of the Flat Earth Society and other flat-Earthers claim that NASA and other government agencies conspire to fabricate evidence that the Earth is spherical. According to the most widely spread version of current flat-Earth theory, NASA is guarding the Antarctic ice wall that surrounds Earth. Flat-Earthers argue that NASA manipulates and fabricates its satellite images, based on observations that the color of the oceans changes from image to image and that continents seem to be in different places. The publicly perpetuated image is kept up through a large-scale practice of "compartmentalization", according to which only a select number of individuals have knowledge about the truth.

Research by Carlos Diaz Ruiz and Tomas Nilsson on the arguments that flat Earthers wield, shows three factions, each one subscribing to its own set of beliefs. The first faction subscribes to a faith-based conflict in which atheists use science to suppress the Christian faith. Their argument is that atheists use pseudo-science – evolution, Big Bang, and the round Earth – to make people believe that God is an abstract idea, not real. Instead, their arguments use the Scripture – word-by-word – to support an argument that enables God to really exist. This faction frames flat-Earth arguments as revelatory. (For example, a literal interpretation of Revelation 7 -- "I saw four angels standing at the four corners of the earth ..."—indicates the earth must have four corners.)

The second faction believes in an overarching conspiracy for knowledge suppression. Building upon the premise that knowledge is power, the flat Earth conspiracy argues that a shadowy group of "elites" control knowledge to remain in power. In their view, lying about the fundamental nature of the Earth primes the population to believe a host of other conspiracies. This faction frames flat-Earth arguments as liberatory.

The third faction believes that knowledge is personal and experiential. They are dismissive of knowledge that comes from authoritative sources, especially book knowledge. This faction would like to find out themselves whether the Earth truly is round or flat. Because they distrust book knowledge and mathematical proof, this faction believes that the Earth is flat because their observations and lived experiences make it appear as if we live on a flat surface. This faction frames flat-Earth arguments as experimental.

Fellow flat earthers are not exempt from distrust and belief that they may be in cahoots with round earthers. In Behind the Curve, conference attendees were warned against attending by Math Powerland a.k.a. Matt Boylan, who posted videos alleging others were working for the CIA or Warner Brothers. At the 2017 Flat Earth conference: "several speakers made reference to 'shills' within the community, people purporting to espouse the theory but who in fact belong to some deep-state counterintelligence program aimed at making the movement seem laughable. In 2016, [Eric] Dubay, of the '200 Proofs' video, called out Mark Sargent, Jeran Campanella, and other figures as 'suspected controlled opposition shills,' and last year in a radio interview he called the November conference a 'shill-fest.' Even the flat-Earth bureaucracy is suspect. At the end of the conference's second day, a panelist mentioned a plan to set up a nonprofit to carry on the work. This brought a rebuke from a woman in the audience. 'You had me up until I heard the gentleman say, 'The reason we had to scramble to get the 501(c)(3),' ' she said. 'In my research, I found out that's a Luciferian contract.'"

Social and experimental activities of skeptics and believers
Organisations sceptical of fringe beliefs have occasionally performed tests to demonstrate the local curvature of the Earth. One of these, conducted by members of the Independent Investigations Group of the Center for Inquiry, at the Salton Sea on 10 June 2018 was attended also by supporters of a flat Earth, and the encounter between the two groups was recorded by the National Geographic Explorer. This experiment successfully demonstrated the curvature of the Earth via the disappearance over distance of boat-based and shore-based targets. IIG founder Jim Underdown reported that the flat Earth supporters in attendance immediately rejected the results, denying the validity of the demonstration after the fact, and the discussion degenerated into tangents about Moon landing conspiracy theories and alleged NASA cover-ups.

The 2018 documentary Behind the Curve followed two groups of American flat Earth believers who were attempting to gather first-hand empirical proof for that belief. One group from the YouTube show GlobeBusters used a ring laser gyroscope in an attempt to show the Earth was not rotating. Instead, they detected the actual 15-degree-per-hour rotation of the Earth, a measurement they dismissed as corrupted by the device somehow picking up the rotation of the "firmament". Another group used lasers in an attempt to show a several-mile stretch of water is perfectly flat by measuring the distance between the water level and the laser beam along three vertical posts. They were unable to align the beam as they expected to because the surface of the still water was in fact bent by several feet over the distance measured; the experiment was dismissed as inconclusive.

Behind the Curve illustrated how flat Earth believers rely on poorly-verified claims. Mark Sargent claimed to have watched flightaware.com for a very long time to check if any flights traveled between continents in the Southern Hemisphere, which in his disc model would be much further apart than they are on the globe. He stated that he saw no such flights, and took this as evidence for the disc model. Caltech astrophysicist Hannalore Gerling-Dunsmore went to the site and immediately found flights that contradicted Sargent's claims.

The solar eclipse of 21 August 2017 gave rise to numerous YouTube videos purporting to show how the details of the eclipse prove the Earth is flat.

In 2017, "the Tunisian and Arab scientific and educational world" had a scandal when a Ph.D. student at the University of Sfax in Tunisia submitted a Ph.D. dissertation "declaring Earth to be flat, unmoving, young (only 13,500 years of age), and the centre of the universe". In 2018, astronomer Yaël Nazé analyzed the controversy over the dissertation. The dissertation, which had not been approved by the committee overseeing environmental studies theses, had been made public and denounced in 2017 by Hafedh Ateb, a founder of the Tunisian Astronomical Society, on his Facebook page.

In March 2019, social media personality Logan Paul released a satirical documentary film about the flat Earth called FLAT EARTH: To The Edge And Back.

The Flat Earth Society has a Twitter account, @FlatEarthOrg. This account shares information about their group and promotes flat Earth ideologies.

Mike Hughes
Mike Hughes, a daredevil/stuntman, planned to use a homebuilt crewed rocket to reach outer space. In a practice flight on 22 February 2020, the early deployment and separation of the return parachute allowed his rocket to fall unimpeded from an altitude of several hundred feet, killing him instantly.

After Hughes' death, his public relations representative Darren Shuster stated that Hughes "didn't believe in flat Earth" and that it was "a PR stunt" to get publicity, while Michael Linn, who worked on the documentary Rocketman: Mad Mike's Mission to Prove the Flat-Earth, said that Hughes' belief appeared genuine.

Social consequences and responses
Behind the Curve's filmmakers spoke with several people who said that as a result of their flat Earth beliefs they had lost romantic partners and no longer spoke to many friends and family. One said he was tired of being told he was an idiot. The Facebook group Flat Earth Match is a dating site used by some to find romantic partners who share these beliefs. Experts pointed out that after social ties to people outside the flat Earth community are lost, one consequence of abandoning the flat Earth belief would be loss of all remaining relationships.

Caltech physicist Spiros Michaelakis stated that instead of denigrating flat Earthers, scientists should do a better job of teaching scientific facts. Various scientific and medical experts in the documentary supported improving scientific literacy and avoiding marginalization of flat Earthers. They pointed out that people who distrust all of science, including truths about vaccines, evolution, and climate change, would make poorly informed-decisions, and that people who do not exercise the skill of critical thinking can be easily manipulated. They also pointed out that some believers were motivated to spread false ideas, and that because they are unconstrained by facts they can mutate and become less harmless than a mere belief about the shape of the Earth.

Prevalence
In 2020, it was reported that based on polling by Datafolha, 7% of Brazilians believed in a flat Earth. A 2018 YouGov poll found that around 4% of the population of the United States believed in flat Earth, while the POLES 2021 Survey found around 10% of the United States population believed that the Earth is flat. A 2019 YouGov survey found that around 3% of British people supported flat Earth.

The term "flat-Earther"
The term flat-Earth-man, used in a derogatory sense to mean anyone who holds ridiculously antiquated or impossible views, predates the more compact flat-earther. It was recorded in 1908: "Fewer votes than one would have thought possible for any human candidate, were he even a flat-earth-man." According to the Oxford English Dictionary, the first use of the term flat-Earther was in 1934 in Punch magazine: "Without being a bigoted flat-earther, [Mercator] perceived the nuisance ... of fiddling about with globes ... in order to discover the South Seas."

Scientific sources

 * Raymond Fraser (2007). When The Earth Was Flat: Remembering Leonard Cohen, Alden Nowlan, the Flat Earth Society, the King James monarchy hoax, the Montreal Story Tellers and other curious matters. Black Moss Press, ISBN 978-0-88753-439-3
 * Christine Garwood (2007) Flat Earth: The History of an Infamous Idea, Pan Books, ISBN 1-4050-4702-X

Flat Earth believers

 * Arguments based on Christian Bible and related writings.
 * Christian basis.

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

title: Modern flat Earth beliefs


text:
Talk:Belief

Use of the word tenanted
"tenanted belief systems are difficult for the tenants to completely revise or reject."I am not familiar with this use of tenanted, but when I search the phrase tenanted belief systems, I see repeated paraphrases of this Wikipedia article. Can a less novel term be substituted? / edg ☺ ☭ 00:43, 27 November 2017 (UTC)

Assumption of empirical epistemology?
The first sentence of the article, "Belief is the state of mind in which a person thinks something to be the case with or without there being empirical evidence to prove that something is the case with factual certainty." seems to assume a certain, empirical epistemology. This strikes me as biased. — Preceding unsigned comment added by 2600:1700:9580:3FF0:4D00:4183:E29:E7F9 (talk) 14:36, 21 October 2018 (UTC)

Influence by emotions
Surprised to see article not even mention how emotions influence one's beliefs. Here's one reference. I'll take a stab at writing it one of these days. BecomeFree (talk) 12:13, 27 January 2020 (UTC)

The emotions and beliefs section is an important section to have in this article, but it seems to me that it would be hard to write this section from a neutral standpoint. Brandon436 (talk) 21:20, 2 October 2021 (UTC)

Applying a Greek, Western idea (belief) to every culture in the world?
This article says that belief is a Greek philosophical idea, but then goes on through the entire article, acting that every non-Western people in the world, have something that identifies with the Greek idea of a belief. This is completely untrue. There is dogma in Christianity, and Iman in Islam (Islam was influenced by Greek philosophy), but Judaism didn't have any declared "beliefs" before an Arabic speaking Jewish philosopher in the Middle Ages created a list of what he thought would be them. Buddhism wouldn't say it has beliefs - it would say it used reason to figure out things. Any other religion or culture wouldn't say they "believed", they would say they "know". For example, Hinduism would say they know, Zoroastrians would say they know. Chinese or Japanese would say they know. Tribal religions or cultures would say they know. Belief is a Greco-Roman construct that only has meaning for westerners, but the article doesn't address that at all.Jimhoward72 (talk) 19:07, 3 May 2020 (UTC)

You could try to address the issue. "Belief" in Greek thought long predates Christianity. See dogma. The corresponding term in Buddhism is View (Buddhism). How do you know "belief" only has meaning for Westerners? Teishin (talk) 20:09, 3 May 2020 (UTC)

Justified true belief
Since any belief is either justified or justified by other beliefs, how can JTB generate knowledge, since basic beliefs are unjustifiable? -- Hey mid  (contribs) 12:06, 20 April 2021 (UTC)

 * There are different theories of justification. According to internalist evidentialism, a belief is justified if the believer is in possession of sufficient evidence for it. If this evidence comes in the form of other beliefs, it is inferentially justified. If the evidence comes from perception, for example, it's non-inferentially justified. Non-inferentially justified beliefs are often referred to as basic beliefs. So basic beliefs are justifiable, just not by other beliefs. Phlsph7 (talk) 16:02, 20 April 2021 (UTC)

Dispositionalism and the secret pie
The "Dispositionalism" sub-section states: "For example, one may decide not to affirm that there is a pie in the pantry when asked, because one wants to keep it secret." This seems not to be about a disposition to believe but to be about a disposition to declare a belief to which one is disposed - and hence not relevant here. What if anything like this arises from the sources? If nothing, then remove? Errantius (talk) 23:58, 6 August 2021 (UTC)
 * Thanks for your extensive copy editing. Dispositionalism is about dispositions to act, including to perform speech acts: Dispositionalism ... defines beliefs only concerning their role as causes of behavior or as dispositions to behave in a certain way. Otherwise it would be circular: defining beliefs as dispositions to believe. So the example should be fine. I think this example is in one of the articles cited but I would have to check to make sure. Phlsph7 (talk) 05:58, 7 August 2021 (UTC)

‘Emotion and beliefs’ and ‘Delusion’
These topics seem to be related, especially regarding the discussion in ‘Emotion and beliefs’ on emotion and cognition acting together to produce belief (an interaction that can be similarly attributed to delusion.) It may be best to consolidate these two sections into one for greater cohesion. Eid1803 (talk) 03:12, 26 September 2023 (UTC)
 * The two sections are under developed, but "Emotion and beliefs" would about maintaining belief regardless of feelings of doubt, peer pressure, etc. "Delusion" is about a break from reality, I do not see them as the same at all. Richard-of-Earth (talk) 01:29, 2 October 2023 (UTC)
 * I see, thank you for the clarification. Eid1803 (talk) 20:28, 2 October 2023 (UTC)

Wiki Education assignment: Adult Development Fall 2023
— Assignment last updated by Simreestolk0 (talk) 20:18, 3 November 2023 (UTC)

Human Senses and Belief
The human senses are based on recognition, we recognize by sight, by sound, by taste, by smell, by touch and by feel. We use feel to recognize energy that makes us feel good, the effect of love that comes from God. Belief data is not transferable to all people while recognialbe data is.Question46 (talk) 17:14, 26 December 2023 (UTC)
 * Sooooo, what change did you want to make to the article and what citations do you have to support it? If you are just chatting about the subject, this is not a forum. See WP:NOTAFORUM. Richard-of-Earth (talk) 02:29, 31 December 2023 (UTC)

Friesian.com
Friesian.com has been determined to not be a reliable source, but it is used in the section on epistemology to support a description of Socrates/Plato's understanding of knowledge. There must be better sources for this. Can one be substituted? 2601:601:1B80:820:9FF:92D0:6C29:7C52 (talk) 15:13, 9 February 2024 (UTC)

 * Thanks for pointing this out, I replaced the unreliable source with a citation-needed tag. Phlsph7 (talk) 09:22, 10 February 2024 (UTC)

title: Talk:Belief


text:
Talk:The Star-Spangled Banner

Absent elaboration by Francis Scott Key prior to his death in 1843...
,,, This phrase makes no sense. Does it mean 'In the absence of any explanation by Francis Scott Key...'? 2001:8003:3020:1C00:188C:8B35:298B:3B7B (talk) 22:52, 7 September 2023 (UTC)

The Tabernacle Choir at Temple Square
Is pretty decent.  scope_creep Talk  15:54, 27 October 2023 (UTC)

Edit request
Could "1990's" in the lede be replaced with "1990s" (i.e. remove the apostrophe), per MOS:DECADE ("Decades are written in the format the 1980s, with no apostrophe.")

Before: It has become controversial in the United States since the 1990's due...

After: It has become controversial in the United States since the 1990s due...

Thank you!

51.148.251.110 (talk) 13:49, 28 May 2024 (UTC)

✅ Thwaluigi (talk) 15:44, 28 May 2024 (UTC)

Semi-protected edit request on 1 June 2024
I would like to edit the sheet music image (not the one on the top of the page, the one near the bottom). The three beats are somewhat incorrect, and I also wish to make it more professional while allowing it to be easy on that page. Samvielsaa (talk) 20:51, 1 June 2024 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate.  JTP (talk • contribs) 21:19, 1 June 2024 (UTC)
 * "The three beats" -- which three beats? I'll agree, the text doesn't look like it's been set by a decent music program -- the words are run together badly. --jpgordon𝄢𝄆𝄐𝄇 21:27, 1 June 2024 (UTC)

Semi-protected edit request on 1 June 2024 (2)
In the "Lyrics" page, there is an incorrect sheet music piece that has the 3 beats p/ bar not lining up with the actual music on the piece. There is also an absence of a 120 p/ quarter note at the top, the absence of a title, supertitle, lyricist, composer, and subtitle as well as a oversimplified piece altogether. I have a png (and a jpg copy) score that fix these problems, while maintaining it is playable to the same easy level the previous score could be played at. Samvielsaa (talk) 23:10, 1 June 2024 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. M.Bitton (talk) 00:10, 2 June 2024 (UTC)
 * That would be pointless. The entire image should be replaced with something more correct and representative; the current one is a quite crappy rendition., please upload your score image (or provide a link to it); I'll take a look at it and evaluate it for this article. --jpgordon𝄢𝄆𝄐𝄇 03:47, 2 June 2024 (UTC)
 * it isn't letting me upload the file, saying the page is protected. When i tried to upload last time it said it was not known if it was your own work and it would not be uploaded to wikimedia commons
 * please respond Samvielsaa (talk) 16:43, 8 June 2024 (UTC)
 * nevermind, i have the file, i made it on wikimedia commons, here:https://commons.wikimedia.org/wiki/File:The_Star_Spangled_Banner_Score,_With_Lyrics.png Samvielsaa (talk) 18:20, 8 June 2024 (UTC)
 * @Jpgordon i have the file right here, i uploaded it to wikimedia commons, and its my own work.
 * https://commons.wikimedia.org/wiki/File:The_Star_Spangled_Banner_Score,_With_Lyrics.png Samvielsaa (talk) 18:23, 8 June 2024 (UTC)
 * I see. I'm sorry, but that rendition is very incorrect. Every measure is off by one beat, and the accompaniment is quite problematic. --jpgordon𝄢𝄆𝄐𝄇 18:36, 8 June 2024 (UTC)
 * I am using a software called Noteflight, @Jpgordon, I will match the base clef with the treble and try to bring the note beats closer together, if that's what is needed Samvielsaa (talk) 18:59, 8 June 2024 (UTC)
 * @Jpgordon I'm also confused, the recording playback on the software is going through it perfectly.
 * If you have a discord, I can send you the audio Samvielsaa (talk) 19:05, 8 June 2024 (UTC)
 * The audio is irrelevant; the chart you've provided is wrong. Compare it with the version you wish replaced. Every phrase in the song starts on the upbeat (3-count) of a bar. You have every phrase in the song starting on the downbeat (1-count) of the bar. --jpgordon𝄢𝄆𝄐𝄇 23:36, 8 June 2024 (UTC)
 * And it only needs treble clef for the melody, you can and should omit the bass (it's not part of the song.) --jpgordon𝄢𝄆𝄐𝄇 23:40, 8 June 2024 (UTC)

title: Talk:The Star-Spangled Banner


text:
Brute-force search

In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically checking all possible candidates for whether or not each candidate satisfies the problem's statement.

A brute-force algorithm that finds the divisors of a natural number n would enumerate all integers from 1 to n, and check whether each of them divides n without remainder. A brute-force approach for the eight queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard and for each arrangement, check whether each (queen) piece can attack any other.

While a brute-force search is simple to implement and will always find a solution if it exists, implementation costs are proportional to the number of candidate solutions – which in many practical problems tends to grow very quickly as the size of the problem increases (§Combinatorial explosion). Therefore, brute-force search is typically used when the problem size is limited, or when there are problem-specific heuristics that can be used to reduce the set of candidate solutions to a manageable size. The method is also used when the simplicity of implementation is more important than processing speed.

This is the case, for example, in critical applications where any errors in the algorithm would have very serious consequences or when using a computer to prove a mathematical theorem. Brute-force search is also useful as a baseline method when benchmarking other algorithms or metaheuristics. Indeed, brute-force search can be viewed as the simplest metaheuristic. Brute force search should not be confused with backtracking, where large sets of solutions can be discarded without being explicitly enumerated (as in the textbook computer solution to the eight queens problem above). The brute-force method for finding an item in a table – namely, check all entries of the latter, sequentially – is called linear search.

Basic algorithm
In order candidate for P after the current one c. The next procedure must also tell when there are no more candidates for the instance P, after the current one c. A convenient way to do that is to return a "null candidate", some conventional data value Λ that is distinct from any real candidate. Likewise the first procedure should return Λ if there are no candidates at all for the instance P. The brute-force method is then expressed by the algorithm c ← first(P) while c ≠ Λ do if valid(P,c) then output(P, c) c ← next(P, c) end while For example, when looking for the divisors of an integer n, the instance data P is the number n. The call first(n) should return the integer 1 if n ≥ 1, or Λ otherwise; the call next(n,c) should return c + 1 if c < n, and Λ otherwise; and valid(n,c) should return true if and only if c is a divisor of n. (In fact, if we choose Λ to be n + 1, the tests n ≥ 1 and c < n are unnecessary.)The brute-force search algorithm above will call output for every candidate that is a solution to the given instance P. The algorithm is easily modified to stop after finding the first solution, or a specified number of solutions; or after testing a specified number of candidates, or after spending a given amount of CPU time.
 * 1) valid (P, c): check whether candidate c is a solution for P.
 * 2) output (P, c): use the solution c of P as appropriate to the application.

Combinatorial explosion
The main disadvantage of the brute-force method is that, for many real-world problems, the number of natural candidates is prohibitively large. For instance, if we look for the divisors of a number as described above, the number of candidates tested will be the given number n. So if n has sixteen decimal digits, say, the search will require executing at least 1015 computer instructions, which will take several days on a typical PC. If n is a random 64-bit natural number, which has about 19 decimal digits on the average, the search will take about 10 years. This steep growth in the number of candidates, as the size of the data increases, occurs in all sorts of problems. For instance, if we are seeking a particular rearrangement of 10 letters, then we have 10! = 3,628,800 candidates to consider, which a typical PC can generate and test in less than one second. However, adding one more letter – which is only a 10% increase in the data size – will multiply the number of candidates by 11, a 1000% increase. For 20 letters, the number of candidates is 20!, which is about 2.4×1018 or 2.4 quintillion; and the search will take about 10 years. This unwelcome phenomenon is commonly called the combinatorial explosion, or the curse of dimensionality.

One example of a case where combinatorial complexity leads to solvability limit is in solving chess. Chess is not a solved game. In 2005, all chess game endings with six pieces or less were solved, showing the result of each position if played perfectly. It took ten more years to complete the tablebase with one more chess piece added, thus completing a 7-piece tablebase. Adding one more piece to a chess ending (thus making an 8-piece tablebase) is considered intractable due to the added combinatorial complexity.

Speeding up brute-force searches
One way to speed up a brute-force algorithm is to reduce the search space, that is, the set of candidate solutions, by using heuristics specific to the problem class. For example, in the eight queens problem the challenge is to place eight queens on a standard chessboard so that no queen attacks any other. Since each queen can be placed in any of the 64 squares, in principle there are 648 = 281,474,976,710,656 possibilities to consider. However, because the queens are all alike, and that no two queens can be placed on the same square, the candidates are all possible ways of choosing of a set of 8 squares from the set all 64 squares; which means 64 choose 8 = 64!/(56!*8!) = 4,426,165,368 candidate solutions – about 1/60,000 of the previous estimate. Further, no arrangement with two queens on the same row or the same column can be a solution. Therefore, we can further restrict the set of candidates to those arrangements. As this example shows, a little bit of analysis will often lead to dramatic reductions in the number of candidate solutions, and may turn an intractable problem into a trivial one.

In some cases, the analysis may reduce the candidates to the set of all valid solutions; that is, it may yield an algorithm that directly enumerates all the desired solutions (or finds one solution, as appropriate), without wasting time with tests and the generation of invalid candidates. For example, for the problem "find all integers between 1 and 1,000,000 that are evenly divisible by 417" a naive brute-force solution would generate all integers in the range, testing each of them for divisibility. However, that problem can be solved much more efficiently by starting with 417 and repeatedly adding 417 until the number exceeds 1,000,000 – which takes only 2398 (= 1,000,000 ÷ 417) steps, and no tests.

Reordering the search space
In applications that require only one solution, rather than all solutions, the expected running time of a brute force search will often depend on the order in which the candidates are tested. As a general rule, one should test the most promising candidates first. For example, when searching for a proper divisor of a random number n, it is better to enumerate the candidate divisors in increasing order, from 2 to n − 1, than the other way around – because the probability that n is divisible by c is 1/c. Moreover, the probability of a candidate being valid is often affected by the previous failed trials. For example, consider the problem of finding a 1 bit in a given 1000-bit string P. In this case, the candidate solutions are the indices 1 to 1000, and a candidate c is valid if P[c] = 1. Now, suppose that the first bit of P is equally likely to be 0 or 1, but each bit thereafter is equal to the previous one with 90% probability. If the candidates are enumerated in increasing order, 1 to 1000, the number t of candidates examined before success will be about 6, on the average. On the other hand, if the candidates are enumerated in the order 1,11,21,31...991,2,12,22,32 etc., the expected value of t will be only a little more than 2.More generally, the search space should be enumerated in such a way that the next candidate is most likely to be valid, given that the previous trials were not. So if the valid solutions are likely to be "clustered" in some sense, then each new candidate should be as far as possible from the previous ones, in that same sense. The converse holds, of course, if the solutions are likely to be spread out more uniformly than expected by chance.

Alternatives to brute-force search
There are many other search methods, or metaheuristics, which are designed to take advantage of various kinds of partial knowledge one may have about the solution. Heuristics can also be used to make an early cutoff of parts of the search. One example of this is the minimax principle for searching game trees, that eliminates many subtrees at an early stage in the search. In certain fields, such as language parsing, techniques such as chart parsing can exploit constraints in the problem to reduce an exponential complexity problem into a polynomial complexity problem. In many cases, such as in Constraint Satisfaction Problems, one can dramatically reduce the search space by means of Constraint propagation, that is efficiently implemented in Constraint programming languages. The search space for problems can also be reduced by replacing the full problem with a simplified version. For example, in computer chess, rather than computing the full minimax tree of all possible moves for the remainder of the game, a more limited tree of minimax possibilities is computed, with the tree being pruned at a certain number of moves, and the remainder of the tree being approximated by a static evaluation function.

In cryptography
In cryptography, a brute-force attack involves systematically checking all possible keys until the correct key is found. This strategy can in theory be used against any encrypted data (except a one-time pad) by an attacker who is unable to take advantage of any weakness in an encryption system that would otherwise make his or her task easier.

The key length used in the encryption determines the practical feasibility of performing a brute force attack, with longer keys exponentially more difficult to crack than shorter ones. Brute force attacks can be made less effective by obfuscating the data to be encoded, something that makes it more difficult for an attacker to recognise when he has cracked the code. One of the measures of the strength of an encryption system is how long it would theoretically take an attacker to mount a successful brute force attack against it.

title: Brute-force search


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User:Quercusrobur/Ongoing projects & To Do list...

Some pages & projects I intend to spend some more time on when I get the opportunity...

Projects

 * Permaculture?????
 * Organic gardening????
 * WikiProject Horticulture and Gardening- this looks interesting...

Pages I intend to fiddle with or expand when I get time

 * Composting toilet (keep an eye on...)
 * Crass (fill out history- much more can go in)
 * Sex Pistols (always good for a tweak or two...)
 * Free improvisation including beginning some notes on some of the performers listed
 * Forest gardening (article needs expanding, also more pictures & links to add)
 * Quinoa (coud still expand... esp history stuff)
 * Judge Dredd (It would be fun to fill out the mythos of the JD universe a bit...)
 * Southend on Sea (its a crap town but its where I live...)
 * Westcliff on Sea (or more accurately i live here...)