text:
ECHELON

ECHELON, originally a secret government code name, is a surveillance program (signals intelligence/SIGINT collection and analysis network) operated by the five signatory states to the UKUSA Security Agreement: Australia, Canada, New Zealand, the United Kingdom and the United States, also known as the Five Eyes.

Created in the late 1960s to monitor the military and diplomatic communications of the Soviet Union and its Eastern Bloc allies during the Cold War, the ECHELON project became formally established in 1971. By the end of the 20th century, it had greatly expanded.

Organization
The UKUSA intelligence community was assessed by the European Parliament (EP) in 2000 to include the signals intelligence agencies of each of the member states:
 * the Government Communications Headquarters of the United Kingdom,
 * the National Security Agency of the United States,
 * the Communications Security Establishment of Canada,
 * the Australian Signals Directorate of Australia, and
 * the Government Communications Security Bureau of New Zealand.

Public disclosures (1972–2000)
Former NSA analyst Perry Fellwock, under the pseudonym Winslow Peck, first blew the whistle on ECHELON to Ramparts in 1972, when he revealed the existence of a global network of listening posts and told of his experiences working there. He also revealed the existence of nuclear weapons in Israel in 1972, the widespread involvement of CIA and NSA personnel in drugs and human smuggling, and CIA operatives leading Nationalist Chinese (Taiwan) commandos in burning villages inside PRC borders.

In 1982, investigative journalist and author James Bamford wrote The Puzzle Palace, an in-depth history of the NSA and its practices, which notably leaked the existence of the eavesdropping operation Project SHAMROCK. Project SHAMROCK ran from 1945 to 1975, after which it evolved into ECHELON.

In 1988, Margaret Newsham, a Lockheed employee under NSA contract, disclosed the ECHELON surveillance system to members of Congress. Newsham told a member of the US Congress that the telephone calls of Strom Thurmond, a Republican US senator, were being collected by the NSA. Congressional investigators determined that "targeting of US political figures would not occur by accident, but was designed into the system from the start."

Also in 1988, an article titled "Somebody's Listening", written by investigative journalist Duncan Campbell in the New Statesman, described the signals intelligence gathering activities of a program code-named "ECHELON". Bamford described the system as the software controlling the collection and distribution of civilian telecommunications traffic conveyed using communication satellites, with the collection being undertaken by ground stations located in the footprint of the downlink leg.

A detailed description of ECHELON was provided by the New Zealand journalist Nicky Hager in his 1996 book Secret Power: New Zealand's Role in the International Spy Network. Two years later, Hager's book was cited by the European Parliament in a report titled "An Appraisal of the Technology of Political Control" (PE 168.184).

In March 1999, for the first time in history, the Australian government admitted that news reports about the top secret UKUSA Agreement were true. Martin Brady, the director of Australia's Defence Signals Directorate (DSD, now known as Australian Signals Directorate, or ASD) told the Australian broadcasting channel Nine Network that the DSD "does co-operate with counterpart signals intelligence organisations overseas under the UKUSA relationship."

In 2000, James Woolsey, the former Director of the US Central Intelligence Agency, confirmed that US intelligence uses interception systems and keyword searches to monitor European businesses.

Lawmakers in the United States feared that the ECHELON system could be used to monitor US citizens. According to The New York Times, the ECHELON system has been "shrouded in such secrecy that its very existence has been difficult to prove." Critics said the ECHELON system emerged from the Cold War as a "Big Brother without a cause".

European Parliament investigation (2000–2001)

The program's capabilities and political implications were investigated by a committee of the European Parliament during 2000 and 2001 with a report published in 2001. In July 2000, the Temporary Committee on the ECHELON Interception System was established by the European parliament to investigate the surveillance network. It was chaired by the Portuguese politician Carlos Coelho, who was in charge of supervising investigations throughout 2000 and 2001.

In May 2001, as the committee finalised its report on the ECHELON system, a delegation travelled to Washington, D.C. to attend meetings with US officials from the following agencies and departments:
 * US Central Intelligence Agency (CIA)
 * US Department of Commerce (DOC)
 * US National Security Agency (NSA)

All meetings were cancelled by the US government and the committee was forced to end its trip prematurely. According to a BBC correspondent in May 2001, "The US Government still refuses to admit that Echelon even exists."

In July 2001, the Committee released its final report. The EP report concluded that it seemed likely that ECHELON is a method of sorting captured signal traffic, rather than a comprehensive analysis tool. On 5 September 2001, the European parliament voted to accept the report.

The European Parliament stated in its report that the term ECHELON is used in a number of contexts, but that the evidence presented indicates that it was the name for a signals intelligence collection system. The report concludes that, on the basis of information presented, ECHELON was capable of interception and content inspection of telephone calls, fax, e-mail and other data traffic globally through the interception of communication bearers including satellite transmission, public switched telephone networks (which once carried most Internet traffic), and microwave links.

Confirmation of ECHELON (2015)
Two internal NSA newsletters from January 2011 and July 2012, published as part of Edward Snowden's leaks by the website The Intercept on 3 August 2015, for the first time confirmed that NSA used the code word ECHELON and provided some details about the scope of the program: ECHELON was part of an umbrella program with the code name FROSTING, which was established by the NSA in 1966 to collect and process data from communications satellites. FROSTING had two sub-programs: The European Parliament's Temporary Committee on the ECHELON Interception System stated, "It seems likely, in view of the evidence and the consistent pattern of statements from a very wide range of individuals and organisations, including American sources, that its name is in fact ECHELON, although this is a relatively minor detail". The US intelligence community uses many code names (see, for example, CIA cryptonym).
 * TRANSIENT: for intercepting Soviet satellite transmissions
 * ECHELON: for intercepting Intelsat satellite transmissions

Former NSA employee Margaret Newsham said that she worked on the configuration and installation of software that makes up the ECHELON system while employed at Lockheed Martin, from 1974 to 1984 in Sunnyvale, California, in the United States, and in Menwith Hill, England, in the UK. At that time, according to Newsham, the code name ECHELON was NSA's term for the computer network itself. Lockheed called it P415. The software programs were called SILKWORTH and SIRE. A satellite named VORTEX intercepted communications. An image available on the internet of a fragment apparently torn from a job description shows Echelon listed along with several other code names.

Britain's The Guardian newspaper summarized the capabilities of the ECHELON system as follows:

"A global network of electronic spy stations that can eavesdrop on telephones, faxes and computers. It can even track bank accounts. This information is stored in Echelon computers, which can keep millions of records on individuals.

Officially, however, Echelon doesn't exist."

Documents leaked by the former NSA contractor Edward Snowden revealed that the ECHELON system's collection of satellite data is also referred to as FORNSAT - an abbreviation for "Foreign Satellite Collection".

Intercept stations
First revealed by the European Parliament report (p. 54 ff) and confirmed later by the Edward Snowden disclosures the following ground stations presently have, or have had, a role in intercepting transmissions from Satellite and other means of communication:
 * RAF Little Sai Wan (Closed) (Hong Kong) Map
 * Australian Defence Satellite Communications Station (Geraldton, Western Australia) Map
 * RAF Menwith Hill (Yorkshire, England – Largest known ECHELON facility) Map
 * Misawa Security Operations Center (Oura, Misawa, Aomori, Tōhoku, Japan) Map
 * GCHQ Bude (formerly CSO Morwenstow) (Cornwall, UK) Map
 * Pine Gap (Outside Alice Springs, Northern Territory, Australia) Map
 * Sugar Grove (Closed) (West Virginia, US) Map
 * Yakima Training Center (Closed) (Washington State, US) Map
 * Buckley Space Force Base (Aurora, Colorado) Map
 * GCSB Waihopai (Marlborough, New Zealand) Map
 * GCSB Tangimoana (Manawatu-Wanganui, New Zealand) Map
 * CFS Leitrim (Ottawa, Ontario, Canada) Map
 * Teufelsberg (Closed 1992), *Berlin, Germany – Responsible for listening in to the Eastern Bloc.) Map

 * Ayios Nikolaos (British Sovereign Base area of Dhekelia, Cyprus – Cyprus)
 * Gibraltar (UK)
 * Diego Garcia (UK)
 * Bad Aibling Station (Bad Aibling, Germany – US)
 * relocated to Griesheim/Darmstadt in 2004.
 * Fort Eisenhower (Georgia, US)
 * CFB Gander (Newfoundland and Labrador, Canada)
 * Guam (Pacific Ocean, US)
 * Kunia Regional SIGINT Operations Center (Hawaii, US)
 * Lackland Air Force Base, Medina Annex (San Antonio, Texas, US)
 * RAF Edzell (Closed 1996) (Scotland)
 * RAF Boulmer (England)
 * SNICK International Processing Center (Seeb, Oman) Map

History and context
The ability to intercept communications depends on the medium used, be it radio, satellite, microwave, cellular or fiber-optic. During World War II and through the 1950s, high-frequency ("short-wave") radio was widely used for military and diplomatic communication and could be intercepted at great distances. The rise of geostationary communications satellites in the 1960s presented new possibilities for intercepting international communications. In 1964, plans for the establishment of the ECHELON network took off after dozens of countries agreed to establish the International Telecommunications Satellite Organization (Intelsat), which would own and operate a global constellation of communications satellites. In 1966, the first Intelsat satellite was launched into orbit. From 1970 to 1971, the Government Communications Headquarters (GCHQ) of Britain began to operate a secret signal station at Morwenstow, near Bude in Cornwall, England. The station intercepted satellite communications over the Atlantic and Indian Oceans. Soon afterwards, the US National Security Agency (NSA) built a second signal station at Yakima, near Seattle, for the interception of satellite communications over the Pacific Ocean. In 1981, GCHQ and the NSA started the construction of the first global wide area network (WAN). Soon after Australia, Canada, and New Zealand joined the ECHELON system. The report to the European Parliament of 2001 states: "If UKUSA states operate listening stations in the relevant regions of the earth, in principle they can intercept all telephone, fax, and data traffic transmitted via such satellites."

Most reports on ECHELON focus on satellite interception. Testimony before the European Parliament indicated that separate but similar UKUSA systems are in place to monitor communication through undersea cables, microwave transmissions, and other lines. The report to the European Parliament points out that interception of private communications by foreign intelligence services is not necessarily limited to the US or British foreign intelligence services. The role of satellites in point-to-point voice and data communications has largely been supplanted by fiber optics. In 2006, 99% of the world's long-distance voice and data traffic was carried over optical-fiber. The proportion of international communications accounted for by satellite links is said to have decreased substantially to an amount between 0.4% and 5% in Central Europe. Even in less-developed parts of the world, communications satellites are used largely for point-to-multipoint applications, such as video. Thus, the majority of communications can no longer be intercepted by earth stations; they can only be collected by tapping cables and intercepting line-of-sight microwave signals, which is possible only to a limited extent.

Concerns
British journalist Duncan Campbell and New Zealand journalist Nicky Hager said in the 1990s that the United States was exploiting ECHELON traffic for industrial espionage, rather than military and diplomatic purposes. Examples alleged by the journalists include the gear-less wind turbine technology designed by the German firm Enercon and the speech technology developed by the Belgian firm Lernout & Hauspie.

In 2001, the Temporary Committee on the ECHELON Interception System recommended to the European Parliament that citizens of member states routinely use cryptography in their communications to protect their privacy, because economic espionage with ECHELON has been conducted by the US intelligence agencies.

American author James Bamford provides an alternative view, highlighting that legislation prohibits the use of intercepted communications for commercial purposes, although he does not elaborate on how intercepted communications are used as part of an all-source intelligence process.

In its report, the committee of the European Parliament stated categorically that the Echelon network was being used to intercept not only military communications, but also private and business ones. In its epigraph to the report, the parliamentary committee quoted Juvenal, "Sed quis custodiet ipsos custodes. ("But who will watch the watchers"). James Bamford, in The Guardian in May 2001, warned that if Echelon were to continue unchecked, it could become a "cyber secret police, without courts, juries, or the right to a defence".

Alleged examples of espionage conducted by the members of the "Five Eyes" include:

 * On behalf of the British Prime Minister Margaret Thatcher, the Communications Security Establishment allegedly spied on two British cabinet ministers in 1983.
 * The US National Security Agency spied on and intercepted the phone calls of Diana, Princess of Wales right up until she died in a Paris car crash with Dodi Fayed in 1997. The NSA currently holds 1,056 pages of classified information about Princess Diana, which has been classified as top secret "because their disclosure could reasonably be expected to cause exceptionally grave damage to the national security ... the damage would be caused not by the information about Diana, but because the documents would disclose 'sources and methods' of US intelligence gathering". An official said that "the references to Diana in intercepted conversations were 'incidental'," and she was never a 'target' of the NSA eavesdropping.
 * UK agents monitored the conversations of the 7th Secretary-General of the United Nations, Kofi Annan.
 * US agents gathered "detailed biometric information" on the 8th Secretary-General of the United Nations, Ban Ki-moon.
 * In the early 1990s, the US National Security Agency intercepted the communications between the European aerospace company Airbus and the Saudi Arabian national airline. In 1994, Airbus lost a $6 billion contract with Saudi Arabia after the NSA, acting as a whistleblower, reported that Airbus officials had been bribing Saudi officials to secure the contract. As a result, the American aerospace company McDonnell Douglas (now part of Boeing) won the multibillion-dollar contract instead of Airbus.
 * The United States defense contractor Raytheon won a US$1.3 billion contract with the Government of Brazil to monitor the Amazon rainforest after the US Central Intelligence Agency (CIA), acting as a whistleblower, reported that Raytheon's French competitor Thomson-Alcatel had been paying bribes to get the contract.
 * In order to boost the United States position in trade negotiations with the then Japanese Trade Minister Ryutaro Hashimoto, in 1995 the CIA eavesdropped on the conversations between Japanese bureaucrats and executives of car manufacturers Toyota and Nissan.

Workings
[[File:Echelon-yakima-diagram.jpg|upright=1.35|thumb|System diagram of the ECHELON satellite intercept station of the NSA at the Yakima Research Station (YRS)

TOPCO = Terminal Operations Control

CCS = Computer Control Subsystem

STEAMS = System Test, Evaluation, Analysis, and Monitoring Subsystem

SPS = Signal Processing Subsystem

TTDM = Teletype Demodulator]] The first United States satellite ground station for the ECHELON collection program was built in 1971 at a military firing and training center near Yakima, Washington. The facility, which was codenamed JACKKNIFE, was an investment of ca. 21.3 million dollars and had around 90 people. Satellite traffic was intercepted by a 30-meter single-dish antenna. The station became fully operational on 4 October 1974. It was connected with NSA headquarters at Fort Meade by a 75-baud secure Teletype orderwire channel.

In 1999 the Australian Senate Joint Standing Committee on Treaties was told by Professor Desmond Ball that the Pine Gap facility was used as a ground station for a satellite-based interception network. The satellites were said to be large radio dishes between 20 and 100 meters in diameter in geostationary orbits. The original purpose of the network was to monitor the telemetry from 1970s Soviet weapons, air defence and other radars' capabilities, satellites' ground stations' transmissions and ground-based microwave communications.

Examples of industrial espionage
In 1999, Enercon, a German company and leading manufacturer of wind energy equipment, developed a breakthrough generator for wind turbines. After applying for a US patent, it had learned that Kenetech, an American rival, had submitted an almost identical patent application shortly before. By the statement of a former NSA employee, it was later claimed that the NSA had secretly intercepted and monitored Enercon's data communications and conference calls and passed information regarding the new generator to Kenetech. However, later German media reports contradicted this story, as it was revealed that the American patent in question was actually filed three years before the alleged wiretapping was said to have taken place. As German intelligence services are forbidden from engaging in industrial or economic espionage, German companies have complained that this leaves them defenceless against industrial espionage from the United States or Russia. According to Wolfgang Hoffmann, a former manager at Bayer, German intelligence services know which companies are being targeted by US intelligence agencies, but refuse to inform the companies involved.

title: ECHELON


text:
Talk:ECHELON

Request that this article be write protected?
Given the tendency of conspiracy theorists and over-enthusiastic amateur political analysts to resort to less than academic methods, shouldn't this article be at least a little protected from anonymous editing? — Preceding unsigned comment added by 123.243.215.92 (talk) 02:58, 9 August 2013 (UTC)

 * 144.22 longitude
 * 45.25 latitude
 * Washington state
 * North American cast
 * W3y3E$
 * Christopher Mark Rew 174.174.41.46 (talk) 20:47, 16 November 2023 (UTC)
 * using a friend's phone to post. 00000007000100010041989 174.174.41.46 (talk) 20:48, 16 November 2023 (UTC)

Echelon locations
Most of the locations listed as candidate Echelon stations are picked at random from the nearest conspiracy theory site, and they've almost all been unsourced for 9 months or so. Plenty of time for some evidence to gestate. Several of the sites listed were Iron Horse sites, which wasn't even intercept, but direction finding, and some were old WWII Y-sites. There are very few ground stations which can be reliably attributed to either Bamford or Campbell, probably the only credible sources.

With that in mind I'm going to trim it down to the first section only, most of the third section only contain hangers for the black helicopters anyway. Even the first section is OTT as a couple of the sites listed certainly aren't related, but that's not verifiable.

ALR 14:27, 9 October 2007 (UTC)

I really don't think it is a good idea to be discussion potential/possible ECHELON locations on here at all. (Yes, I know the above comment is from 2007 -- I still don't think it's a good idea to have this info here.) WarFighter (talk) 17:33, 10 June 2013 (UTC)

 * Don't we all know where they are? Pine Gap in Australia is well known where it is, it is well guarded, it is also known that we never get told what it costs us, never get a referendum if we still want it. I have been rather perplexed to read that echelon is tasked with industrial espionage although we heavily criticise China for doing just that. There are locations in neighbourhoods where you wonder about things which only make sense if they are an eavesdropping facility. Looking around is all you need to do. I wanted to have a look here whether the current war on Huawei could be due to them not having a backdoor for echelon. But there is no hint here. 2001:8003:AC60:1400:CD43:D6F9:C250:6512 (talk) 02:42, 30 May 2019 (UTC)

Probably not but that would be editor ALR's problem. And it's not exactly like Wikipedia is written in secret code. Most likely the NSA and ECHELON can deal with anything put in its article. --Wlmg (talk) 18:31, 10 June 2013 (UTC)

Regards to NSA (and ATT) this is a very long and very interesting article. It explains locations and operating details. The information may not fit anywhere (I don't know, it is above my pay grade) but it is awesomely interesting to read. Good thing that we are the good guys. We have awesome technology. Read this and report back on what you think:
 * https://theintercept.com/2018/06/25/att-internet-nsa-spy-hubs/
 * It has pictures of main buildings in eight USA largest cities. -- AstroU (talk) 03:45, 26 June 2018 (UTC)

Infobox
Not sure why there is an NSA infobox in place for an international programme. Having a box for a single nation is monumentally misleading here. - SchroCat (talk) 21:47, 3 October 2013 (UTC)

 * ✅. I agree, so I removed it. --Jeff Ogden (W163) (talk) 23:00, 3 October 2013 (UTC)
 * I was approached by a strange vehicle while at the Bonneville Power Administration building in Vancouver, Washington. Located by The Westfield Shopping Center,(aka. Vancouver Mall. The approximate location of the mall is 45°39′29″N 122°35′1″W) After I saw that the man driving wanted to make contact with myself I slowly approached the vehicle to find a very polite man and asked him if there was anything that i could help him with. We spoke for several minutes and as he handed me some hand sanitizer to help out with my hygeine problem( Due to Being a transient individual) I asked who he was employed with and what the symbol on his uniform was for.
 * The reply that I received was that he was employed through Echelon. I had remembered hearing this name while being a young child back in the early 1990's but could not remember what it was about.
 * To be 100% honest I am homeless in Vancouver washington with absolutely ZERO advocacy for the problems that I have been facing. I feel this is in part because of the conspiracy theories that I have going through my head. So my name is Christopher Mark Rew D.O.B Wed,October 4th 1989 at 12:04a.m. and I was born in Memorial hospital on main st. in Vancouver, Washington.
 * After using reasonable deduction I have come to a finalized conclusion that I have been a Target for many different things. From the time that I was born until my 34 years of age currently I believe that there are Human individuals, along with other more magnificent creatures of this world that have managed to time travel and have wound up "lost in space" if you will. I feel like our "WINCO" area ( Washington, Idaho, Nevada, Colorado,and Oregon) have been a controlled variable in this equation. I have been implemented into the Watergate Scandal through Bonneville dam being blown in 2007 to help with washing away a mistake that the "God Particle" helped to produce. Van-alco where my grandfather was employed while i was a child had been fined with a 76 billion dollar fine for poluting the columbia river during production. At this time i was in juvenile prison with Vincent Woods. aka Anthony Grecco. come find me if you would like to know more about the papa johns predicitons on the presidential callender, Pizza Hut, Walking trees, intelligent automated vessels, Area 51 ( THE REAL ONE***** I205SB at Van Mall drive exit.) or N51 through RS Medical. I am also the person from the news article "Pharming a New Recepie for Disaster " KATU, KOIN6, FOX12, channel 13 news, The Columbian Newspaper,and also the Oregonian. FINN(3.57) Solar Winds, The COLD WAR, or anything else regarding music and EVERY reference for a Check point within time to assist with this definitive reality equation. W3Y3ES, ISOL8,HAVOC, PramwerC, Pinky, or the magic TV. I have no phone due to everything being stolen by police and also the local population. PLEASE HELP ME!!! 0000000700010041989000100030005150000531174748000 66.96.79.217 (talk) 21:46, 17 November 2023 (UTC)
 * 2024AD Vancouver "westfield shoppingtowne" mall 66.96.79.221 (talk) 20:35, 28 March 2024 (UTC)

Jam Echelon Day
Seriously, no mention of this? 14.202.186.241 (talk) 01:00, 9 August 2014 (UTC)

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Le informazioni sono troppo speculative
La pagina andrebbe rivista: alcune fonti non sono oggettive, altre contraddicono l'esistenza stessa del programma negli anni 70 82.132.225.110 (talk) 06:44, 22 October 2021 (UTC)

Sorry, wanted to post in the Italian page. Please close 82.132.225.110 (talk) 06:47, 22 October 2021 (UTC)

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

title: Talk:ECHELON


text:
Equation

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.

Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables.

The "=" symbol, which appears in every equation, was invented in 1557 by Robert Recorde, who considered that nothing could be more equal than parallel straight lines with the same length.

Description
An equation is written as two expressions, connected by an equals sign ("="). The expressions on the two sides of the equals sign are called the "left-hand side" and "right-hand side" of the equation. Very often the right-hand side of an equation is assumed to be zero. This does not reduce the generality, as this can be realized by subtracting the right-hand side from both sides.

The most common type of equation is a polynomial equation (commonly called also an algebraic equation) in which the two sides are polynomials. The sides of a polynomial equation contain one or more terms. For example, the equation

 * $$ Ax^2 +Bx + C - y = 0 $$

has left-hand side $$ Ax^2 +Bx + C - y $$, which has four terms, and right-hand side $$ 0 $$, consisting of just one term. The names of the variables suggest that $x$ and $y$ are unknowns, and that $A$, $B$, and $C$ are parameters, but this is normally fixed by the context (in some contexts, $y$ may be a parameter, or $A$, $B$, and $C$ may be ordinary variables).

An equation is analogous to a scale into which weights are placed. When equal weights of something (e.g., grain) are placed into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, an equal amount of grain must be removed from the other pan to keep the scale in balance. More generally, an equation remains in balance if the same operation is performed on its both sides.

Properties
Two equations or two systems of equations are equivalent, if they have the same set of solutions. The following operations transform an equation or a system of equations into an equivalent one – provided that the operations are meaningful for the expressions they are applied to:
 * Adding or subtracting the same quantity to both sides of an equation. This shows that every equation is equivalent to an equation in which the right-hand side is zero.
 * Multiplying or dividing both sides of an equation by a non-zero quantity.
 * Applying an identity to transform one side of the equation. For example, expanding a product or factoring a sum.
 * For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.

If some function is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called extraneous solutions. For example, the equation $$x=1$$ has the solution $$x=1.$$ Raising both sides to the exponent of 2 (which means applying the function $$f(s)=s^2$$ to both sides of the equation) changes the equation to $$x^2=1$$, which not only has the previous solution but also introduces the extraneous solution, $$x=-1.$$ Moreover, if the function is not defined at some values (such as 1/x, which is not defined for x = 0), solutions existing at those values may be lost. Thus, caution must be exercised when applying such a transformation to an equation.

The above transformations are the basis of most elementary methods for equation solving, as well as some less elementary ones, like Gaussian elimination.

Analogous illustration

An equation is analogous to a weighing scale, balance, or seesaw.

Each side of the equation corresponds to one side of the balance. Different quantities can be placed on each side: if the weights on the two sides are equal, the scale balances, and in analogy, the equality that represents the balance is also balanced (if not, then the lack of balance corresponds to an inequality represented by an inequation).

In the illustration, x, y and z are all different quantities (in this case real numbers) represented as circular weights, and each of x, y, and z has a different weight. Addition corresponds to adding weight, while subtraction corresponds to removing weight from what is already there. When equality holds, the total weight on each side is the same.

Parameters and unknowns
Equations often contain terms other than the unknowns. These other terms, which are assumed to be known, are usually called constants, coefficients or parameters.

An example of an equation involving x and y as unknowns and the parameter R is

 * $$ x^2 +y^2 = R^2 .$$

When R is chosen to have the value of 2 (R = 2), this equation would be recognized in Cartesian coordinates as the equation for the circle of radius of 2 around the origin. Hence, the equation with R unspecified is the general equation for the circle.

Usually, the unknowns are denoted by letters at the end of the alphabet, x, y, z, w, ..., while coefficients (parameters) are denoted by letters at the beginning, a, b, c, d, ... . For example, the general quadratic equation is usually written ax2 + bx + c = 0.

The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called solving the equation. Such expressions of the solutions in terms of the parameters are also called solutions.

A system of equations is a set of simultaneous equations, usually in several unknowns for which the common solutions are sought. Thus, a solution to the system is a set of values for each of the unknowns, which together form a solution to each equation in the system. For example, the system
 * $$\begin{align}

3x+5y&=2\\ 5x+8y&=3 \end{align} $$ has the unique solution x = −1, y = 1.

Identities
An identity is an equation that is true for all possible values of the variable(s) it contains. Many identities are known in algebra and calculus. In the process of solving an equation, an identity is often used to simplify an equation, making it more easily solvable.

In algebra, an example of an identity is the difference of two squares:

 * $$x^2 - y^2 = (x+y)(x-y) $$

which is true for all x and y.

Trigonometry is an area where many identities exist; these are useful in manipulating or solving trigonometric equations. Two of many that involve the sine and cosine functions are:

 * $$\sin^2(\theta)+\cos^2(\theta) = 1 $$

and

 * $$\sin(2\theta)=2\sin(\theta) \cos(\theta) $$

which are both true for all values of θ.

For example, to solve for the value of θ that satisfies the equation:

 * $$3\sin(\theta) \cos(\theta)= 1\,, $$

where θ is limited to between 0 and 45 degrees, one may use the above identity for the product to give:

 * $$\frac{3}{2}\sin(2 \theta) = 1\,,$$

yielding the following solution for θ:

 * $$\theta = \frac{1}{2} \arcsin\left(\frac{2}{3}\right) \approx 20.9^\circ.$$

Since the sine function is a periodic function, there are infinitely many solutions if there are no restrictions on θ. In this example, restricting θ to be between 0 and 45 degrees would restrict the solution to only one number.

Algebra
Algebra studies two main families of equations: polynomial equations and, among them, the special case of linear equations. When there is only one variable, polynomial equations have the form P(x) = 0, where P is a polynomial, and linear equations have the form ax + b = 0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis. Algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory. These equations are difficult in general; one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions.

Polynomial equations

In general, an algebraic equation or polynomial equation is an equation of the form

 * $$P = 0$$, or

 * $$P = Q$$

where P and Q are polynomials with coefficients in some field (e.g., rational numbers, real numbers, complex numbers). An algebraic equation is univariate if it involves only one variable. On the other hand, a polynomial equation may involve several variables, in which case it is called multivariate (multiple variables, x, y, z, etc.).

For example,
 * $$x^5-3x+1=0$$

is a univariate algebraic (polynomial) equation with integer coefficients and
 * $$y^4+\frac{xy}{2}=\frac{x^3}{3}-xy^2+y^2-\frac{1}{7}$$

is a multivariate polynomial equation over the rational numbers.

Some polynomial equations with rational coefficients have a solution that is an algebraic expression, with a finite number of operations involving just those coefficients (i.e., can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but equations of degree five or more cannot always be solved in this way, as the Abel–Ruffini theorem demonstrates.

A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root finding of polynomials) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).

Systems of linear equations

A system of linear equations (or linear system) is a collection of linear equations involving one or more variables. For example,
 * $$\begin{alignat}{7}

3x &&\; + \;&& 2y            &&\; - \;&& z  &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y            &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}$$ is a system of three equations in the three variables $P – Q = 0$. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by
 * $$\begin{alignat}{2}

x &\,=\,& 1 \\ y &\,=\,& -2 \\ z &\,=\,& -2 \end{alignat}$$ since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually.

In mathematics, the theory of linear systems is a fundamental part of linear algebra, a subject which is used in many parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in physics, engineering, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer simulation of a relatively complex system.

Analytic geometry
In Euclidean geometry, it is possible to associate a set of coordinates to each point in space, for example by an orthogonal grid. This method allows one to characterize geometric figures by equations. A plane in three-dimensional space can be expressed as the solution set of an equation of the form $$ ax+by+cz+d=0$$, where $$a,b,c$$ and $$d$$ are real numbers and $$x,y,z$$ are the unknowns that correspond to the coordinates of a point in the system given by the orthogonal grid. The values $$a,b,c$$ are the coordinates of a vector perpendicular to the plane defined by the equation. A line is expressed as the intersection of two planes, that is as the solution set of a single linear equation with values in $$\mathbb{R}^2$$ or as the solution set of two linear equations with values in $$\mathbb{R}^3.$$

A conic section is the intersection of a cone with equation $$x^2+y^2=z^2$$ and a plane. In other words, in space, all conics are defined as the solution set of an equation of a plane and of the equation of a cone just given. This formalism allows one to determine the positions and the properties of the focuses of a conic.

The use of equations allows one to call on a large area of mathematics to solve geometric questions. The Cartesian coordinate system transforms a geometric problem into an analysis problem, once the figures are transformed into equations; thus the name analytic geometry. This point of view, outlined by Descartes, enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.

Currently, analytic geometry designates an active branch of mathematics. Although it still uses equations to characterize figures, it also uses other sophisticated techniques such as functional analysis and linear algebra.

Cartesian equations
In Cartesian geometry, equations are used to describe geometric figures. As the equations that are considered, such as implicit equations or parametric equations, have infinitely many solutions, the objective is now different: instead of giving the solutions explicitly or counting them, which is impossible, one uses equations for studying properties of figures. This is the starting idea of algebraic geometry, an important area of mathematics.

One can use the same principle to specify the position of any point in three-dimensional space by the use of three Cartesian coordinates, which are the signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).

The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2 in a plane, centered on a particular point called the origin, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

Parametric equations
A parametric equation for a curve expresses the coordinates of the points of the curve as functions of a variable, called a parameter. For example,
 * $$\begin{align}

x&=\cos t\\ y&=\sin t \end{align}$$ are parametric equations for the unit circle, where t is the parameter. Together, these equations are called a parametric representation of the curve.

The notion of parametric equation has been generalized to surfaces, manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).

Diophantine equations
A Diophantine equation is a polynomial equation in two or more unknowns for which only the integer solutions are sought (an integer solution is a solution such that all the unknowns take integer values). A linear Diophantine equation is an equation between two sums of monomials of degree zero or one. An example of linear Diophantine equation is $x, y, z$ where a, b, and c are constants. An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.

Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it.

The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis.

Algebraic and transcendental numbers
An algebraic number is a number that is a solution of a non-zero polynomial equation in one variable with rational coefficients (or equivalently — by clearing denominators — with integer coefficients). Numbers such as $\pi$ that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.

Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying solutions of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

Differential equations
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. They are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in areas such as physics, chemistry, biology, and economics.

In pure mathematics, differential equations are studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form.

If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

Ordinary differential equations
An ordinary differential equation or ODE is an equation containing a function of one independent variable and its derivatives. The term "ordinary" is used in contrast with the term partial differential equation, which may be with respect to more than one independent variable.

Linear differential equations, which have solutions that can be added and multiplied by coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate, as one can rarely represent them by elementary functions in closed form: Instead, exact and analytic solutions of ODEs are in series or integral form. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, analytic solutions.

Partial differential equations
A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations, which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.

PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.

Types of equations
Equations can be classified according to the types of operations and quantities involved. Important types include:
 * An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree:
 * linear equation for degree one
 * quadratic equation for degree two
 * cubic equation for degree three
 * quartic equation for degree four
 * quintic equation for degree five
 * sextic equation for degree six
 * septic equation for degree seven
 * octic equation for degree eight
 * A Diophantine equation is an equation where the unknowns are required to be integers
 * A transcendental equation is an equation involving a transcendental function of its unknowns
 * A parametric equation is an equation in which the solutions for the variables are expressed as functions of some other variables, called parameters appearing in the equations
 * A functional equation is an equation in which the unknowns are functions rather than simple quantities
 * Equations involving derivatives, integrals and finite differences:
 * A differential equation is a functional equation involving derivatives of the unknown functions, where the function and its derivatives are evaluated at the same point, such as $$f'(x) = x^2$$. Differential equations are subdivided into ordinary differential equations for functions of a single variable and partial differential equations for functions of multiple variables
 * An integral equation is a functional equation involving the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from a differential equation primarily through a change of variable substituting the function by its derivative, however this is not the case when the integral is taken over an open surface
 * An integro-differential equation is a functional equation involving both the derivatives and the antiderivatives of the unknown functions. For functions of one variable, such an equation differs from integral and differential equations through a similar change of variable.
 * A functional differential equation of delay differential equation is a function equation involving derivatives of the unknown functions, evaluated at multiple points, such as $$f'(x) = f(x-2)$$
 * A difference equation is an equation where the unknown is a function f that occurs in the equation through f(x), f(x−1), ..., f(x−k), for some whole integer k called the order of the equation. If x is restricted to be an integer, a difference equation is the same as a recurrence relation
 * A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process

title: Equation


text:
Ethical naturalism

Ethical naturalism (also called moral naturalism or naturalistic cognitivistic definism) is the meta-ethical view which claims that:

 * 1) Ethical sentences express propositions.
 * 2) Some such propositions are true.
 * 3) Those propositions are made true by objective features of the world.
 * 4) These moral features of the world are reducible to some set of non-moral features.

Overview
The versions of ethical naturalism which have received the most sustained philosophical interest, for example, Cornell realism, differ from the position that "the way things are is always the way they ought to be", which few ethical naturalists hold. Ethical naturalism does, however, reject the fact-value distinction: it suggests that inquiry into the natural world can increase our moral knowledge in just the same way it increases our scientific knowledge. Indeed, proponents of ethical naturalism have argued that humanity needs to invest in the science of morality, a broad and loosely defined field that uses evidence from biology, primatology, anthropology, psychology, neuroscience, and other areas to classify and describe moral behavior.

Ethical naturalism encompasses any reduction of ethical properties, such as 'goodness', to non-ethical properties; there are many different examples of such reductions, and thus many different varieties of ethical naturalism. Hedonism, for example, is the view that goodness is ultimately just pleasure.

Ethical theories that can be naturalistic

 * Altruism
 * Consequentialism
 * Consequentialist libertarianism
 * Cornell realism
 * Ethical egoism/ Objectivism
 * Evolutionary ethics
 * Hedonism
 * Humanistic ethics
 * Natural law
 * Natural-rights libertarianism
 * Utilitarianism
 * Virtue ethics

Criticisms
Ethical naturalism has been criticized most prominently by ethical non-naturalist G. E. Moore, who formulated the open-question argument. Garner and Rosen say that a common definition of "natural property" is one "which can be discovered by sense observation or experience, experiment, or through any of the available means of science." They also say that a good definition of "natural property" is problematic but that "it is only in criticism of naturalism, or in an attempt to distinguish between naturalistic and nonnaturalistic definist theories, that such a concept is needed." R. M. Hare also criticised ethical naturalism because of what he considered its fallacious definition of the terms 'good' or 'right', saying that value-terms being part of our prescriptive moral language are not reducible to descriptive terms: "Value-terms have a special function in language, that of commending; and so they plainly cannot be defined in terms of other words which themselves do not perform this function".

Moral nihilism
Moral nihilists maintain that there are no such entities as objective values or objective moral facts. Proponents of moral science like Ronald A. Lindsay have counter-argued that their way of understanding "morality" as a practical enterprise is the way we ought to have understood it in the first place. He holds the position that the alternative seems to be the elaborate philosophical reduction of the word "moral" into a vacuous, useless term. Lindsay adds that it is important to reclaim the specific word "morality" because of the connotations it holds with many individuals.

Morality as a science
Author Sam Harris has argued that we overestimate the relevance of many arguments against the science of morality, arguments he believes scientists happily and rightly disregard in other domains of science like physics. For example, scientists may find themselves attempting to argue against philosophical skeptics, when Harris says they should be practically asking – as they would in any other domain – "why would we listen to a solipsist in the first place?" This, Harris contends, is part of what it means to practice a science of morality.

In modern times, many thinkers discussing the fact–value distinction and the is–ought problem have settled on the idea that one cannot derive ought from is. Conversely, Harris maintains that the fact-value distinction is a confusion, proposing that values are really a certain kind of fact. Specifically, Harris suggests that values amount to empirical statements about "the flourishing of conscious creatures in a society". He argues that there are objective answers to moral questions, even if some are difficult or impossible to possess in practice. In this way, he says, science can tell us what to value. Harris adds that we do not demand absolute certainty from predictions in physics, so we should not demand that of a science studying morality (see The Moral Landscape).

Physicist Sean Carroll believes that conceiving of morality as a science could be a case of scientific imperialism and insists that what is "good for conscious creatures" is not an adequate working definition of "moral". In opposition, John Shook, vice president of the Center for Inquiry, claims that this working definition is more than adequate for science at present and that disagreement should not immobilize the scientific study of ethics.

title: Ethical naturalism


text:
Ethical non-naturalism

Ethical non-naturalism (or moral non-naturalism) is the meta-ethical view which claims that:
 * 1) Ethical sentences express propositions.
 * 2) Some such propositions are true.
 * 3) Those propositions are made true by objective features of the world, independent of human opinion.
 * 4) These moral features of the world are not reducible to any set of non-moral features.

This makes ethical non-naturalism a non-definist form of moral realism, which is in turn a form of cognitivism. Ethical non-naturalism stands in opposition to ethical naturalism, which claims that moral terms and properties are reducible to non-moral terms and properties, as well as to all forms of moral anti-realism, including ethical subjectivism (which denies that moral propositions refer to objective facts), error theory (which denies that any moral propositions are true), and non-cognitivism (which denies that moral sentences express propositions at all).

Definitions and examples
According to G. E. Moore, "Goodness is a simple, undefinable, non-natural property." To call goodness "non-natural" does not mean that it is supernatural or divine. It does mean, however, that goodness cannot be reduced to natural properties such as needs, wants or pleasures. Moore also stated that a reduction of ethical properties to a divine command would be the same as stating their naturalness. This would be an example of what he referred to as "the naturalistic fallacy."

Moore claimed that goodness is "indefinable", i.e., it cannot be defined in any other terms. This is the central claim of non-naturalism. Thus, the meaning of sentences containing the word "good" cannot be explained entirely in terms of sentences not containing the word "good." One cannot substitute words referring to pleasure, needs or anything else in place of "good."

Some properties, such as hardness, roundness and dampness, are clearly natural properties. We encounter them in the real world and can perceive them. On the other hand, other properties, such as being good and being right, are not so obvious. A great novel is considered to be a good thing; goodness may be said to be a property of that novel. Paying one's debts and telling the truth are generally held to be right things to do; rightness may be said to be a property of certain human actions.

However, these two types of property are quite different. Those natural properties, such as hardness and roundness, can be perceived and encountered in the real world. On the other hand, it is not immediately clear how to physically see, touch or measure the goodness of a novel or the rightness of an action.

A difficult question
Moore did not consider goodness and rightness to be natural properties, i.e., they cannot be defined in terms of any natural properties. How, then, can we know that anything is good and how can we distinguish good from bad?

Moral epistemology, the part of epistemology (and/or ethics) that studies how we know moral facts and how moral beliefs are justified, has proposed an answer. British epistemologists, following Moore, suggested that humans have a special faculty, a faculty of moral intuition, which tells us what is good and bad, right and wrong.

Ethical intuitionists assert that, if we see a good person or a right action, and our faculty of moral intuition is sufficiently developed and unimpaired, we simply intuit that the person is good or that the action is right. Moral intuition is supposed to be a mental process different from other, more familiar faculties like sense-perception, and that moral judgments are its outputs. When someone judges something to be good, or some action to be right, then the person is using the faculty of moral intuition. The faculty is attuned to those non-natural properties. Perhaps the best ordinary notion that approximates moral intuition would be the idea of a conscience.

Another argument for non-naturalism
Moore also introduced what is called the open-question argument, a position he later rejected.

Suppose a definition of "good" is "pleasure-causing." In other words, if something is good, it causes pleasure; if it causes pleasure, then it is, by definition, good. Moore asserted, however, that we could always ask, "But are pleasure-causing things good?" This would always be an open question. There is no foregone conclusion that, indeed, pleasure-causing things are good. In his initial argument, Moore concluded that any similar definition of goodness could be criticized in the same way.

title: Ethical non-naturalism