Statement.

The need to aggregate preferences occurs in many disciplines: in welfare economics, where one attempts to find an economic outcome which would be acceptable and stable; in decision theory, where a person has to make a rational choice based on several criteria; and most naturally in electoral systems, which are mechanisms for extracting a governance-related decision from a multitude of voters' preferences.  The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options (outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of preferences on the set of outcomes. We are searching for a ranked voting electoral system, called a "social welfare function" ("preference aggregation rule"), which transforms the set of preferences ("profile" of preferences) into a single global societal preference order. Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide among, then it is impossible to design a social welfare function that satisfies all these conditions (assumed to be a reasonable requirement of a fair electoral system) at once:  A later (1963) version of Arrow's theorem replaced the monotonicity and non-imposition criteria with:  This later version is more general, having weaker conditions. The axioms of monotonicity, non-imposition, and IIA together imply Pareto efficiency, whereas Pareto efficiency (itself implying non-imposition) and IIA together do not imply monotonicity.

Independence of irrelevant alternatives (IIA).

The IIA condition has three purposes (or effects):  Arrow's death-of-a-candidate example (1963, page 26) suggests that the agenda (the set of feasible alternatives) shrinks from, say, X = {a, b, c} to S = {a, b} because of the death of candidate c. This example is misleading since it can give the reader an impression that IIA is a condition involving "two" agenda and "one" profile. The fact is that IIA involves just "one" agendum ({x, y} in case of pairwise independence) but "two" profiles. If the condition is applied to this confusing example, it requires this: Suppose an aggregation rule satisfying IIA chooses b from the agenda {a, b} when the profile is given by (cab, cba), that is, individual 1 prefers c to a to b, 2 prefers c to b to a. Then, it must still choose b from {a, b} if the profile were, say: (abc, bac); (acb, bca); (acb, cba); or (abc, cba).  In different words, Arrow defines IIA as saying that the social preferences between alternatives x and y depend only on the individual preferences between x and y (not on those involving other candidates).

Formal statement of the theorem.

Let be a set of outcomes, a number of voters or decision criteria. We shall denote the set of all full linear orderings of by.  A (strict) social welfare function (preference aggregation rule) is a function which aggregates voters' preferences into a single preference order on.  An -tuple of voters' preferences is called a "preference profile". In its strongest and simplest form, Arrow's impossibility theorem states that whenever the set of possible alternatives has more than 2 elements, then the following three conditions become incompatible:

Informal proof.

Based on two proofs appearing in "Economic Theory". For simplicity we have presented all rankings as if ties are impossible. A complete proof taking possible ties into account is not essentially different from the one given here, except that one ought to say "not above" instead of "below" or "not below" instead of "above" in some cases. Full details are given in the original articles.  We will prove that any social choice system respecting unrestricted domain, unanimity, and independence of irrelevant alternatives (IIA) is a dictatorship. The key idea is to identify a "pivotal voter" whose ballot swings the societal outcome. We then prove that this voter is a partial dictator (in a specific technical sense, described below). Finally we conclude by showing that all of the partial dictators are the same person, hence this voter is a dictator.

Part one: There is a "pivotal" voter for B over A.

Say there are three choices for society, call them A, B, and C. Suppose first that everyone prefers option B the least: everyone prefers A to B, and everyone prefers C to B. By unanimity, society must also prefer both A and C to B. Call this situation "profile 0".  On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each "i" let "profile i" be the same as "profile 0", but move B to the top of the ballots for voters 1 through "i". So "profile 1" has B at the top of the ballot for voter 1, but not for any of the others. "Profile 2" has B at the top for voters 1 and 2, but no others, and so on.  Since B eventually moves to the top of the societal preference, there must be some profile, number "k", for which B moves above A in the societal rank. We call the voter whose ballot change causes this to happen the pivotal voter for B over A. Note that the pivotal voter for B over A is not, a priori, the same as the pivotal voter for A over B. In part three of the proof we will show that these do turn out to be the same.  Also note that by IIA the same argument applies if "profile 0" is any profile in which A is ranked above B by every voter, and the pivotal voter for B over A will still be voter "k". We will use this observation below.

Part two: The pivotal voter for B over A is a dictator for B over C.

In this part of the argument we refer to voter "k", the pivotal voter for B over A, as "pivotal voter" for simplicity. We will show that pivotal voter dictates society's decision for B over C. That is, we show that no matter how the rest of society votes, if Pivotal Voter ranks B over C, then that is the societal outcome. Note again that the dictator for B over C is not a priori the same as that for C over B. In part three of the proof we will see that these turn out to be the same too. In the following, we call voters 1 through "k − 1", "segment one", and voters "k + 1" through "N", "segment two". To begin, suppose that the ballots are as follows:  Then by the argument in part one (and the last observation in that part), the societal outcome must rank A above B. This is because, except for a repositioning of C, this profile is the same as "profile k − 1" from part one. Furthermore, by unanimity the societal outcome must rank B above C. Therefore, we know the outcome in this case completely.  Now suppose that pivotal voter moves B above A, but keeps C in the same position and imagine that any number (or all!) of the other voters change their ballots to move B below C, without changing the position of A. Then aside from a repositioning of C this is the same as "profile k" from part one and hence the societal outcome ranks B above A. Furthermore, by IIA the societal outcome must rank A above C, as in the previous case. In particular, the societal outcome ranks B above C, even though Pivotal Voter may have been the "only" voter to rank B above C. By IIA, this conclusion holds independently of how A is positioned on the ballots, so pivotal voter is a dictator for B over C.

Part three: There exists a dictator.

In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying parts one and two to the other pairs of candidates). First, the pivotal voter for B over C must appear earlier (or at the same position) in the line than the dictator for B over C: As we consider the argument of part one applied to B and C, successively moving B to the top of voters' ballots, the pivot point where society ranks B above C must come at or before we reach the dictator for B over C. Likewise, reversing the roles of B and C, the pivotal voter for C over B must be at or later in line than the dictator for B over C. In short, if "k" denotes the position of the pivotal voter for X over Y (for any two candidates X and Y), then we have shown  Now repeating the entire argument above with B and C switched, we also have  Therefore, we have  and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election.

Interpretations.

Although Arrow's theorem is a mathematical result, it is often expressed in a non-mathematical way with a statement such as "no voting method is fair", "every ranked voting method is flawed", or "the only voting method that isn't flawed is a dictatorship". These statements are simplifications of Arrow's result which are not universally considered to be true. What Arrow's theorem does state is that a deterministic preferential voting mechanism—that is, one where a preference order is the only information in a vote, and any possible set of votes gives a unique result—cannot comply with all of the conditions given above simultaneously.  Various theorists have suggested weakening the IIA criterion as a way out of the paradox. Proponents of ranked voting methods contend that the IIA is an unreasonably strong criterion. It is the one breached in most useful electoral systems. Advocates of this position point out that failure of the standard IIA criterion is trivially implied by the possibility of cyclic preferences. If voters cast ballots as follows: then the pairwise majority preference of the group is that A wins over B, B wins over C, and C wins over A: these yield rock-paper-scissors preferences for any pairwise comparison. In this circumstance, "any" aggregation rule that satisfies the very basic majoritarian requirement that a candidate who receives a majority of votes must win the election, will fail the IIA criterion, if social preference is required to be transitive (or acyclic). To see this, suppose that such a rule satisfies IIA. Since majority preferences are respected, the society prefers A to B (two votes for A > B and one for B > A), B to C, and C to A. Thus a cycle is generated, which contradicts the assumption that social preference is transitive.  So, what Arrow's theorem really shows is that any majority-wins electoral system is a non-trivial game, and that game theory should be used to predict the outcome of most voting mechanisms. This could be seen as a discouraging result, because a game need not have efficient equilibria; e.g., a ballot could result in an alternative nobody really wanted in the first place, yet everybody voted for.

Remark: Scalar rankings from a vector of attributes and the IIA property.

The IIA property might not be satisfied in human decision-making of realistic complexity because the "scalar" preference ranking is effectively derived from the weighting—not usually explicit—of a "vector" of attributes (one book dealing with the Arrow theorem invites the reader to consider the related problem of creating a scalar measure for the track and field decathlon event—e.g. how does one make scoring 600 points in the discus event "commensurable" with scoring 600 points in the 1500 m race) and this scalar ranking can depend sensitively on the weighting of different attributes, with the tacit weighting itself affected by the context and contrast created by apparently "irrelevant" choices. Edward MacNeal discusses this sensitivity problem with respect to the ranking of "most livable city" in the chapter "Surveys" of his book "MathSemantics: making numbers talk sense" (1994).

Other possibilities.

In an attempt to escape from the negative conclusion of Arrow's theorem, social choice theorists have investigated various possibilities ("ways out"). These investigations can be divided into the following two:

Approaches investigating functions of preference profiles.

This section includes approaches that deal with Since these two approaches often overlap, we discuss them at the same time. What is characteristic of these approaches is that they investigate various possibilities by eliminating or weakening or replacing one or more conditions (criteria) that Arrow imposed.

Infinitely many individuals.

Several theorists (e.g., Kirman and Sondermann) point out that when one drops the assumption that there are only finitely many individuals, one can find aggregation rules that satisfy all of Arrow's other conditions.  However, such aggregation rules are practically of limited interest, since they are based on ultrafilters, highly non-constructive mathematical objects. In particular, Kirman and Sondermann argue that there is an "invisible dictator" behind such a rule. Mihara shows that such a rule violates algorithmic computability. These results can be seen to establish the robustness of Arrow's theorem.

Limiting the number of alternatives.

When there are only two alternatives to choose from, May's theorem shows that only simple majority rule satisfies a certain set of criteria (e.g., equal treatment of individuals and of alternatives; increased support for a winning alternative should not make it into a losing one). On the other hand, when there are at least three alternatives, Arrow's theorem points out the difficulty of collective decision making. Why is there such a sharp difference between the case of less than three alternatives and that of at least three alternatives?  "Nakamura's theorem" (about the core of simple games) gives an answer more generally. It establishes that if the number of alternatives is less than a certain integer called the Nakamura number, then the rule in question will identify "best" alternatives without any problem; if the number of alternatives is greater or equal to the Nakamura number, then the rule will not always work, since for some profile a voting paradox (a cycle such as alternative A socially preferred to alternative B, B to C, and C to A) will arise. Since the Nakamura number of majority rule is 3 (except the case of four individuals), one can conclude from Nakamura's theorem that majority rule can deal with up to two alternatives rationally. Some super-majority rules (such as those requiring 2/3 of the votes) can have a Nakamura number greater than 3, but such rules violate other conditions given by Arrow.

Pairwise voting.

A common way "around" Arrow's paradox is limiting the alternative set to two alternatives. Thus, whenever more than two alternatives should be put to the test, it seems very tempting to use a mechanism that pairs them and votes by pairs. As tempting as this mechanism seems at first glance, it is generally far from satisfying even Pareto efficiency, not to mention IIA. The specific order by which the pairs are decided strongly influences the outcome. This is not necessarily a bad feature of the mechanism. Many sports use the tournament mechanism—essentially a pairing mechanism—to choose a winner. This gives considerable opportunity for weaker teams to win, thus adding interest and tension throughout the tournament. This means that the person controlling the order by which the choices are paired (the agenda maker) has great control over the outcome. In any case, when viewing the entire voting process as one game, Arrow's theorem still applies.

Domain restrictions.

Another approach is relaxing the universality condition, which means restricting the domain of aggregation rules. The best-known result along this line assumes "single peaked" preferences.  Duncan Black has shown that if there is only one dimension on which every individual has a "single-peaked" preference, then all of Arrow's conditions are met by majority rule. Suppose that there is some predetermined linear ordering of the alternative set. An individual's preference is "single-peaked" with respect to this ordering if he has some special place that he likes best along that line, and his dislike for an alternative grows larger as the alternative goes further away from that spot (i.e., the graph of his utility function has a single peak if alternatives are placed according to the linear ordering on the horizontal axis). For example, if voters were voting on where to set the volume for music, it would be reasonable to assume that each voter had their own ideal volume preference and that as the volume got progressively too loud or too quiet they would be increasingly dissatisfied. If the domain is restricted to profiles in which every individual has a single peaked preference with respect to the linear ordering, then "simple" aggregation rules, which include majority rule, have an "acyclic" (defined below) social preference, hence "best" alternative. In particular, when there are odd number of individuals, then the social preference becomes transitive, and the socially "best" alternative is equal to the median of all the peaks of the individuals (Black's median voter theorem). Under single-peaked preferences, the majority rule is in some respects the most natural voting mechanism.  One can define the notion of "single-peaked" preferences on higher-dimensional sets of alternatives. However, one can identify the "median" of the peaks only in exceptional cases. Instead, we typically have the destructive situation suggested by McKelvey's Chaos Theorem: for any "x" and "y", one can find a sequence of alternatives such that "x" is beaten by "x" by a majority, "x" by "x", up to "x" by "y".

Relaxing transitivity.

By relaxing the transitivity of social preferences, we can find aggregation rules that satisfy Arrow's other conditions. If we impose "neutrality" (equal treatment of alternatives) on such rules, however, there exists an individual who has a "veto". So the possibility provided by this approach is also very limited.  First, suppose that a social preference is "quasi-transitive" (instead of transitive); this means that the strict preference formula_2 ("better than") is transitive: if formula_3 and formula_4, then formula_5. Then, there do exist non-dictatorial aggregation rules satisfying Arrow's conditions, but such rules are "oligarchic". This means that there exists a coalition L such that L is "decisive" (if every member in L prefers x to y, then the society prefers x to y), and each member in L "has a veto" (if she prefers x to y, then the society cannot prefer y to x).  Second, suppose that a social preference is "acyclic" (instead of transitive): there do not exist alternatives formula_6 that form a "cycle" (formula_7). Then, provided that there are at least as many alternatives as individuals, an aggregation rule satisfying Arrow's other conditions is "collegial". This means that there are individuals who belong to the intersection ("collegium") of all decisive coalitions. If there is someone who has a veto, then he belongs to the collegium. If the rule is assumed to be neutral, then it does have someone who has a veto.  Finally, Brown's theorem left open the case of acyclic social preferences where the number of alternatives is less than the number of individuals. One can give a definite answer for that case using the "Nakamura number". See limiting the number of alternatives.

Relaxing IIA.

There are numerous examples of aggregation rules satisfying Arrow's conditions except IIA. The Borda rule is one of them. These rules, however, are susceptible to "strategic manipulation" by individuals.  See also Interpretations of the theorem above.

Relaxing the Pareto criterion.

Wilson (1972) shows that if an aggregation rule is non-imposed and non-null, then there is either a dictator or an inverse dictator, provided that Arrow's conditions other than Pareto are also satisfied. Here, an "inverse dictator" is an individual "i" such that whenever "i" prefers "x" to "y", then the society prefers "y" to "x".

Remark.

Amartya Sen offered both relaxation of transitivity and removal of the Pareto principle. He demonstrated another interesting impossibility result, known as the "impossibility of the Paretian Liberal" (see liberal paradox for details). Sen went on to argue that this demonstrates the futility of demanding Pareto optimality in relation to voting mechanisms.

Social choice instead of social preference.

In social decision making, to rank all alternatives is not usually a goal. It often suffices to find some alternative. The approach focusing on choosing an alternative investigates either "social choice functions" (functions that map each preference profile into an alternative) or "social choice rules" (functions that map each preference profile into a subset of alternatives).  As for social choice functions, the Gibbard–Satterthwaite theorem is well-known, which states that if a social choice function whose range contains at least three alternatives is strategy-proof, then it is dictatorial.  As for social choice rules, we should assume there is a social preference behind them. That is, we should regard a rule as choosing the maximal elements ("best" alternatives) of some social preference. The set of maximal elements of a social preference is called the "core". Conditions for existence of an alternative in the core have been investigated in two approaches. The first approach assumes that preferences are at least "acyclic" (which is necessary and sufficient for the preferences to have a maximal element on any "finite" subset). For this reason, it is closely related to relaxing transitivity. The second approach drops the assumption of acyclic preferences. Kumabe and Mihara adopt this approach. They make a more direct assumption that individual preferences have maximal elements, and examine conditions for the social preference to have a maximal element. See Nakamura number for details of these two approaches.

Rated electoral system and other approaches.

Arrow originally rejected cardinal utility as a meaningful tool for expressing social welfare, and so focused his theorem on preference rankings, but later stated that a cardinal score system with three or four classes "is probably the best".  Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and transitivity) on the set of alternatives. This means that if the preferences are represented by a utility function, its value is an "ordinal" utility in the sense that it is meaningful so far as the greater value indicates the better alternative. For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as having 1000, 100.01, 100, 0, which in turn is the same as having 99, 98, 1,.997. They all represent the ordering in which a is preferred to b to c to d. The assumption of "ordinal" preferences, which precludes "interpersonal comparisons" of utility, is an integral part of Arrow's theorem.  For various reasons, an approach based on cardinal utility, where the utility has a meaning beyond just giving a ranking of alternatives, is not common in contemporary economics. However, once one adopts that approach, one can take intensities of preferences into consideration, or one can compare (i) gains and losses of utility or (ii) levels of utility, across different individuals. In particular, Harsanyi (1955) gives a justification of utilitarianism (which evaluates alternatives in terms of the sum of individual utilities), originating from Jeremy Bentham. Hammond (1976) gives a justification of the maximin principle (which evaluates alternatives in terms of the utility of the worst-off individual), originating from John Rawls.  Not all voting methods use, as input, only an ordering of all candidates. Methods which don't, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") electoral system, can be viewed as using information that only cardinal utility can convey. In that case, it is not surprising if some of them satisfy all of Arrow's conditions that are reformulated. Range voting is such a method. Whether such a claim is correct depends on how each condition is reformulated. Other rated electoral system which pass certain generalizations of Arrow's criteria include approval voting and majority judgment. Note that Arrow's theorem does not apply to single-winner methods such as these, but Gibbard's theorem still does: no non-defective electoral system is fully strategy-free, so the informal dictum that "no electoral system is perfect" still has a mathematical basis.  Finally, though not an approach investigating some kind of rules, there is a criticism by James M. Buchanan, Charles Plott, and others. It argues that it is silly to think that there might be "social" preferences that are analogous to "individual" preferences. Arrow (1963, Chapter 8) answers this sort of criticism seen in the early period, which come at least partly from misunderstanding.  A multi-pronged refutation of Arrow's theorem was published by philosophy Professor Howard DeLong in 1991. He challenges the theorem on the basis that Arrow wrongly assumes Preference is transitive property and that Collective Preference is the same as summing up individual preferences. He also claims that Arrow's model fails to model democracy as it exists in the real world as the model ignores the possibility of consensual temporary dictatorships (i.e.: Greek tyrants in times of war) and the effect of allowing lotteries to decide tie breakers and to avoid the problem of the tyranny of the majority, the example used being a group of campers at a summer camp, 8 of whom prefer cake, 7 who prefer ice cream, but funds are limited to one choice or the other on a weekly basis. Under the collective preference of majority rule each week the group would select cake. By drawing lots the choices would more accurately reflect the preferences of the collective than the majority.

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: "unrestricted domain", "non-dictatorship", "Pareto efficiency", and "independence of irrelevant alternatives". The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book "Social Choice and Individual Values". The original paper was titled "A Difficulty in the Concept of Social Welfare".

Le théorème d'impossibilité d'Arrow, également appelé « paradoxe d'Arrow » (du nom de l'économiste américain Kenneth Arrow), est une confirmation mathématique, dans certaines conditions précises, du paradoxe soulevé et décrit dès 1785 par Nicolas de Condorcet. Supposons que chaque électeur ne puisse exprimer son opinion que de manière qualitative, en indiquant comment il classe les unes par rapport aux autres les options envisagées. Entre deux options, l'électeur indique celle qu'il préfère ou s'il est indifférent entre les deux, par contre il ne peut pas exprimer l'intensité de sa préférence. Dans ce cadre, il n'existe pas de processus de choix social indiscutable, qui permette d'exprimer une hiérarchie des préférences cohérente pour une collectivité à partir de l'agrégation des préférences individuelles exprimées par chacun des membres de cette même collectivité. Pour Condorcet, il n'existe pas de système "simple" assurant cette cohérence. Arrow tente de démontrer, sous réserve d'acceptation de ses hypothèses, qu'il n'existe "pas du tout" de système assurant la cohérence, hormis celui où le processus de choix social coïncide avec celui d'un seul individu, parfois surnommé dictateur, indépendamment du reste de la population.


Biography.

Early years.

Henri de Saint-Simon was born in Paris as a French aristocrat. His grandfather's cousin had been the Duke of Saint-Simon. "When he was a young man, being of a restless disposition... he went to America where he entered into American service and took part in the siege of Yorktown under General Washington."  From his youth, Saint-Simon was highly ambitious. He ordered his valet to wake him every morning with, "Remember, monsieur le comte, that you have great things to do." Among his early schemes was one to connect the Atlantic and the Pacific oceans by a canal, and another to construct a canal from Madrid to the sea.  During the American Revolution, Saint-Simon joined the Americans, and believed that their revolution signaled the beginning of a new era. He fought alongside the Marquis de Lafayette between 1779 and 1783, and was imprisoned by British forces. After his release, he returned to France to study engineering and hydraulics at the Ecole de Mézières.  At the beginning of the French Revolution in 1789, Saint-Simon quickly endorsed the revolutionary ideals of liberty, equality and fraternity. In the early years of the revolution, Saint-Simon devoted himself to organizing a large industrial structure in order to found a scientific school of improvement. He needed to raise some funds to achieve his objectives, which he did by land speculation. This was only possible in the first few years of the revolution because of the growing instability of the political situation in France, which prevented him from continuing his financial activities and indeed put his life at risk. Saint-Simon and Talleyrand planned to profiteer during the Terror by buying the Cathedral of Notre-Dame, stripping its roof of metal, and selling the metal for scrap. Saint-Simon was imprisoned on suspicion of engaging in counter-revolutionary activities. He was released in 1794 at the end of the Terror. After he recovered his freedom, Saint-Simon found himself immensely rich due to currency depreciation, but his fortune was subsequently stolen by his business partner. Thenceforth he decided to devote himself to political studies and research. After the establishment of the Ecole Polytechnique in 1794, a school established to train young men in the arts of sciences and industry and funded by the state, Saint-Simon became involved with the new school.

Life as a working adult.

When he was nearly 40 he went through a varied course of study and experiment to enlarge and clarify his view of things. One of these experiments was an unhappy marriage in 1801 to Alexandrine-Sophie Goury de Champgrand, undertaken so that he might have a literary salon. After a year, the marriage was dissolved by mutual consent. The result of his experiments was that he found himself completely impoverished, and lived in penury for the remainder of his life. The first of his numerous writings, mostly scientific and political, was "Lettres d'un habitant de Genève", which appeared in 1802. In this first work, he called for the creation of a religion of science with Isaac Newton as a saint. Around 1814 he wrote the essay "On Reconstruction of the European Community" and sent it to the Congress of Vienna. He proposed a European kingdom, building on France and the United Kingdom.  In 1817, in a treatise entitled "L'Industrie", he began to propound his socialist views, which he developed further in "L'Organisateur" (1819), a periodical on which Augustin Thierry and Auguste Comte collaborated. One of Saint-Simon's major beliefs was that the world should be linked with canals.  "L'Industrie" caused a sensation, but brought few converts. A couple of years later in his writing career, Saint-Simon found himself ruined, and was forced to work for a living. After a few attempts to recover his money from his former partner, he received financial support from Diard, a former employee, and was able to publish in 1807 his second book, "Introduction aux travaux scientifiques du XIX siècle". Diard died in 1810 and Saint-Simon found himself poor again, and this time also in poor health. He was sent to a sanatorium in 1813, but with financial help from relatives he had time to recover his health and gain some intellectual recognition in Europe. In February 1821 "Du système industriel" appeared, and in 1823–1824 "Catéchisme des industriels".

Death and legacy.

On March 9, 1823, disappointed by the lack of results of his writing (he had hoped they would guide society towards social improvement), he attempted suicide in despair. Remarkably, he shot himself in the head six times without succeeding, losing his sight in one eye.  Finally, very late in his career, he did link up with a few ardent disciples. The last and most important expression of his views is, which he left unfinished.  He was buried in Le Père Lachaise Cemetery in Paris, France.

Ideas.

Industrialism.

In 1817 Saint-Simon published a manifesto called the "Declaration of Principles" in his work titled "L'Industrie" ("Industry"). The Declaration was about the principles of an ideology called industrialism that called for the creation of an industrial society led by people within what he defined as the industrial class. The industrial class, also referred to as the working class, was defined as including all people engaged in productive work that contributed to society, emphasizing scientists and industrialists, but including engineers, businesspeople, managers, bankers, manual workers, and others.  Saint-Simon said the primary threat to the needs of the industrial class was another class he referred to as the idling class, that included able people who preferred to be parasitic and benefit from the work of others while seeking to avoid doing work. He saw the origins of this parasitic activity by idlers in what he regarded as the natural laziness of humanity. He believed the principal economic roles of government were to insure that productive activity in the economy is unhindered and to reduce idleness in society.  In the Declaration Saint-Simon strongly criticized any expansion of government intervention into the economy beyond these two principal economic roles, saying that when the government goes beyond these roles, it becomes a "tyrannical enemy of industry" and that the industrial economy will decline as a consequence of such excessive government intervention. Saint-Simon stressed the need for recognition of the merit of the individual and the need for hierarchy of merit in society and in the economy, such as society having hierarchical merit-based organizations of managers and scientists to be the decision-makers in government. His views were radical for his time. He built on Enlightenment ideas which challenged church doctrine and the older regime with the idea of progress from industry and science  Heavily influenced by the absence of social privilege he saw in the early United States, Saint-Simon renounced his aristocratic title and came to favor a form of meritocracy, becoming convinced that science was the key to progress and that it would be possible to develop a society based on objective scientific principles. He claimed that feudal society in France and elsewhere needed to be dissolved and transformed into an industrial society. As such, he invented the conception of the industrial society.  Saint-Simon's economic views and ideas were influenced by Adam Smith whom Saint-Simon deeply admired, and referred to him in praise as "the immortal Adam Smith". He shared with Smith the belief that taxes needed to be much reduced from what they were then in order to have a more just industrial system. Saint-Simon desired the minimization of government intervention into the economy to prevent disruption of productive work. He emphasized more emphatically than Smith that state administration of the economy was generally parasitic and hostile to the needs of production. Like Adam Smith, Saint-Simon's model of society emulated the scientific methods of astronomy, and said "The astronomers only accepted those facts which were verified by observation; they chose the system which linked them best, and since that time, they have never led science astray.".  Saint-Simon reviewed the French Revolution and regarded it as an upheaval driven by economic change and class conflict. In his analysis he believed that the solution to the problems that led to the French Revolution would be the creation of an industrial society where hierarchy of merit and respect for productive work would be the basis of society, while ranks of hereditary and military hierarchy would lessen in importance in society because they were not capable to lead a productive society.  Karl Marx propaganda efforts identified Saint-Simon as being among whom he called the "utopian socialists", though historian Alan Ryan regards certain followers of Saint-Simon, rather than Saint-Simon himself, as being responsible for the rise of utopian socialism that based itself upon Saint-Simon's ideas. Ryan also distinguishes between Saint-Simon's conceptions and Marxism's, as Saint-Simon did not promote independent working-class organization and leadership as a solution to capitalist societal problems, nor did he adhere to the Marxist definition of the working class as excluded by fundamental private property law from control over the means of production. Unlike Marx, Saint-Simon did not regard class relations, vis the means of production, to be an engine of socio-economic dynamics but rather the form of management. Furthermore, Saint-Simon was not critical of capitalists as exclusive owners, collaborators, controllers, and decision-makers. Rather, he regarded capitalists as an important component of the "industrial class." Ryan further suggests that by the 1950s it was clear that Saint-Simon had presaged the "modern" understanding of industrial society.

Feudalism and aristocracy.

In opposition to the feudal and military system—the former aspect of which had been strengthened by the restoration—he advocated a form of technocratic socialism, an arrangement whereby industrial chiefs should control society - similar to Plato's philosopher kings. In place of the medieval church, spiritual direction of society should fall to the men of science. Men who are fitted to organize society for productive labour are entitled to rule it. The conflict between labour and capital emphasized by later socialism is not present in Saint-Simon's work, but it is assumed that the industrial chiefs, to whom the control of production is to fall, shall rule in the interest of society. [SOURCE] Later on, the cause of the poor receives greater attention until, in his greatest work, "Nouveau Christianisme (The New Christianity)", it takes on the form of a religion. This development of his ideas occasioned his final quarrel with Comte.

Religious views.

Prior to the publication of the "Nouveau Christianisme", Saint-Simon had not concerned himself with theology. In this work he starts from a belief in God, and his object in the treatise is to reduce Christianity to its simple and essential elements. He does this by clearing it of the dogmas and other excrescences and defects that he says gathered round the Catholic and Protestant forms of it. He propounds as the comprehensive formula of the new Christianity this precept: "The whole of society ought to strive towards the amelioration of the moral and physical existence of the poorest class; society ought to organize itself in the way best adapted for attaining this end." This principle became the watchword of the entire Saint-Simon school of thought.

Influence.

During his lifetime the views of Saint-Simon had very little influence; he left only a few devoted disciples who continued to advocate the doctrines of their master, whom they revered as a prophet. The most acclaimed disciple of Saint-Simon was Auguste Comte. Others included Olinde Rodrigues, the favoured disciple of Saint-Simon, and Barthélemy Prosper Enfantin who together had received Saint-Simon's last instructions. Their first step was to establish a journal, "Le Producteur", but it was discontinued in 1826. The sect had begun to grow, and before the end of 1828 had meetings not only in Paris but in many provincial towns.  An important departure was made in 1828 by Amand Bazard, who gave a "complete exposition of the Saint-Simonian faith" in a long course of lectures in Paris, which was well attended. His "Exposition de la doctrine de St Simon" (2 vols., 1828–1830), which is by far the best account of it, won more adherents. The second volume was chiefly by Enfantin, who along with Bazard stood at the head of the society, but who was superior in philosophical acumen and was prone to push his deductions to extremities. The revolution of July (1830) brought a new freedom to the socialist reformers. A proclamation was issued demanding community of goods, abolition of the right of inheritance and enfranchisement of women.  Early next year the school obtained possession of "Le Globe" through Pierre Leroux, who had joined the school. The school now counted among its number some of the ablest and most promising young men in France, many of the pupils of the École Polytechnique having caught its enthusiasm. The members formed themselves into an association arranged in three grades, and constituting a society or family, which lived out of a common purse in the Rue Monsigny. Before long dissensions began to arise in the sect. Bazard, a man of stolid temperament, could no longer work in harmony with Enfantin, who desired to establish an arrogant and fantastic sacerdotalism with lax notions as to marriage and the relations between the sexes. In the name of progress, Enfantin announced that the gulf between the sexes was too wide and this social inequality would impede rapid growth of society. Enfantin called for the abolition of prostitution and for the ability for women to divorce and obtain legal rights. This was considered radical for the time.  After a time Bazard seceded and many of the strongest supporters of the school followed his example. A series of extravagant entertainments given by the society during the winter of 1832 reduced its financial resources and greatly discredited it in character. They moved to Ménilmontant, to a property of Enfantin, where they lived in a communalistic society, distinguished by a peculiar dress. Although the monks of Enfantin's school were required to be celibate, rumors were spread that they engaged in orgies. Shortly after, the chiefs were tried and condemned for proceedings prejudicial to the social order and the sect was entirely broken up in 1832. Many of its members became famous as engineers, economists and men of business. Enfantin would go on to organize an expedition of the disciples to Constantinople, and then to Egypt, where he influenced the creation of the Suez Canal.  French feminist and socialist writer Flora Tristan (1803–1844) claimed that Mary Wollstonecraft, author of "A Vindication of the Rights of Woman", anticipated Saint-Simon's ideas by a generation.  In Fyodor Dostoyevsky's novel "The Possessed", 'Saint-Simonist' and 'Fourierist' are used as derogatory insults of others by many of the politically active characters.

Works.

Saint-Simon wrote various accounts of his views :

Claude Henri de Rouvroy, comte de Saint-Simon, often referred to as Henri de Saint-Simon (; 17 October 1760 – 19 May 1825), was a French political and economic theorist and businessman whose thought had a substantial influence on politics, economics, sociology and the philosophy of science.

Claude-Henri de Rouvroy, comte de Saint-Simon, né et mort à Paris (-), est un philosophe, économiste et militaire français, fondateur du saint-simonisme. Ses idées ont eu une postérité et une influence sur la plupart des philosophes du. Philanthrope et philosophe de l'industrialisme, il est le penseur de la société industrielle française, qui était en train de supplanter la société d’Ancien Régime, à la fin du siècle des Lumières. L'économiste André Piettre le décrit par la formule : « le dernier des gentilshommes et le premier des socialistes ».


Setup.

Before the game, a starting position is randomly determined and set up, subject to certain requirements. White's pieces (not pawns) are placed randomly on the first rank, following two rules:  Black's pieces are placed equal-and-opposite to White's pieces. (For example, if the white king is randomly determined to start on f1, then the black king is placed on f8.) Pawns are placed on the players' second as in standard chess.  After setup, the game is played the same as standard chess in all respects, with the exception of castling from the different possible starting positions for king and rooks.  There are 4 × 4 × 6 × 10 × 1 = 960 legal starting positions:  This provides an easy way to set up a game (e.g. using a die / re-roll if needed to get values in the range 1-4 or 1-5).

Castling rules.

As in standard chess, each player may castle once per game, moving both the king and a rook in a single move; however, the castling rules were reinterpreted in Fischer random chess to support the different possible initial positions of king and rook. After castling, the final positions of king and rook are exactly the same as in standard chess, namely:  Castling prerequisites are the same as in standard chess, namely:  A recommended way to castle that is always unambiguous is to first move the king outside the playing area next to its final square, then move the rook to its final square, then move the king to its final square. It may also be useful for the player to state "I am about to castle" before castling.

Observations.

In some starting positions, squares can remain occupied during castling that would be required to be vacant under standard rules. Castling a-side (0-0-0) could still be possible despite the home rank a-, b-, or e-file squares being occupied, and similarly for the e- and h-files for h-side castling (0-0). In other positions, it can happen that the king or rook does not move during the castling maneuver since it already occupies its destination square – e.g., an h-side rook that starts on the f-file; in this case, only the king moves. No initial position would allow a castling where neither piece moves, as the king must start between the rooks.  Another unusual possibility is for castling to be available as the first move of the game, as happened in the 11th game of the tournament match between Hikaru Nakamura and Magnus Carlsen, Fischer Random Blitz 2018. The starting position had kings at f1/f8 and h-side rooks at g1/g8. Both players took the opportunity to castle on the first move (1.0-0 0-0).

Theory.

The study of openings in Fischer random chess is in its infancy, but fundamental opening principles still apply, including: protect the king, control the central squares (directly or indirectly), and develop rapidly, starting with the less valuable pieces. Unprotected pawns may also need to be dealt with quickly. The majority of starting positions have unprotected pawns, and some starting positions have up to two that can be attacked on the first move (see diagram). The Stockfish program rates the Fischer random chess opening positions between 0.1 and 0.5 pawns advantage for White, while the mean value for the same in standard chess is 0.2.  It has been argued that two games should be played from each starting position, with players alternating colors, since the advantage offered to White by some initial positions may be greater than in standard chess. For example, in some Fischer random chess starting positions White can attack an unprotected black pawn on the first move, whereas in standard chess it takes two moves for White to attack, and there are no unprotected pawns.

History.

Fischer random chess is a variant of shuffle chess, which had been suggested as early as 1792 with games played as early as 1842. Fischer's modification "imposes certain restrictions, arguably an improvement on the anarchy of the fully randomized game in which one player is almost certain to start at an advantage". Fischer started to develop his new version of chess after the 1992 return match with Boris Spassky. The result was the formulation of the rules of Fischer random chess in September 1993, introduced formally to the public on June 19, 1996 in Buenos Aires, Argentina.  Fischer's goal was to eliminate what he considered the complete dominance of openings preparation in classical chess, replacing it with creativity and talent. His belief about Russians fixing international games also provided motivation. In a situation where the starting position was random it would be impossible to fix every move of the game. Since the "opening book" for 960 possible opening systems would be too difficult to devote to memory, the players must create every move originally. From the first move, both players must devise original strategies and cannot use well-established patterns. Fischer believed that eliminating memorized book moves would level the playing field.  During the summer of 1993 Bobby Fischer visited László Polgár and his family in Hungary. All of the Polgar sisters, Judit Polgár, Susan Polgar, and Sofia Polgar played many games of Fischer Random chess with Fischer. At one point Sofia beat Fischer three games in a row. Fischer was not pleased when the father, Lazlo, showed Fischer an old chess book that described what appeared to be a forerunner of Fischer Random chess. The book was written by Izidor Gross and published in 1910. Bobby then changed the rules of his variation in order to make it different.

Tournaments.

Mainz Championships.

"Note: None of the Mainz championships were recognized by FIDE. Furthermore, they were all played with rapid time controls."

Computers.

In 2005, chess program The Baron played two Fischer random chess games against Chess960 World Champion Peter Svidler; Svidler won 11⁄2–1⁄2. The chess program Shredder, developed by Stefan Meyer-Kahlen of Germany, played two games against Zoltán Almási from Hungary; Shredder won 2–0.

Matches.

In 2018, a Fischer random chess match between reigning classical World Chess Champion Magnus Carlsen and the unofficial Fischer random chess world champion Hikaru Nakamura was held in Oslo. The match consisted of 8 rapid and 8 blitz games, with the rapid games counting double. Each position was used in two games, with colors reversed. Carlsen prevailed with a score of 14–10.  From 11 to 14 September 2018, the Saint Louis Chess Club held a Fischer random chess event. The playing format included five individual matches, each pair of players facing the same five different starting positions, with 6 rapid games (counting 2 points each) and 14 blitz games (counting 1 point each). "The players and scores:" Veselin Topalov (141⁄2–111⁄2) defeated Garry Kasparov; Hikaru Nakamura (14–12) defeated Peter Svidler; Wesley So (141⁄2–111⁄2) defeated Anish Giri; Maxime Vachier-Lagrave (171⁄2–81⁄2) defeated Sam Shankland; Levon Aronian (171⁄2–81⁄2) defeated Leinier Dominguez.

World Championship.

The first world championship in Fischer random chess officially recognized by FIDE were announced April 20, 2019, and ended on November 2, 2019. Wesley So defeated Magnus Carlsen 13.5–2.5 in the last round, and became the first world champion in Fischer Random Chess.  During the announcement FIDE president Arkady Dvorkovich commented: "It is an unprecedented move that the International Chess Federation recognizes a new variety of chess, so this was a decision that required to be carefully thought out. But we believe that Fischer Random is a positive innovation: It injects new energies and enthusiasm into our game, but at the same time it doesn't mean a rupture with our classical chess and its tradition. It is probably for this reason that Fischer Random chess has won the favor of the chess community, including the top players and the world champion himself. FIDE couldn't be oblivious to that: It was time to embrace and incorporate this modality of chess."

Naming.

The variant has held a number of different names. It was originally known as "Fischerandom", the name given by Fischer. Fischer random chess is the official term, used by FIDE.  Hans-Walter Schmitt, chairman of the Frankfurt Chess Tigers e.V. and an advocate of the variant, started a brainstorming process for creating a new name, which had to meet the requirements of leading grandmasters; specifically, the new name and its parts: The effort culminated in the name choice "Chess960" – derived from the number of different possible starting positions. Fischer never publicly expressed an opinion on the name "Chess960".  Reinhard Scharnagl, another proponent of the variant, advocated the term "FullChess". Today he uses FullChess, however, to refer to variants which consistently embed standard chess (e.g. Chess960, and some new variants based on the extended 10×8 piece set in Capablanca Chess). He recommends the name Chess960 for the variant in preference to Fischer random chess.

Coding games and positions.

Recorded games must convey the Fischer random chess starting position. Games recorded using the Portable Game Notation (PGN) can record the initial position using Forsyth–Edwards Notation (FEN), as the value of the "FEN" tag. Castling is notated the same as in standard chess (except PGN requires letter "O" not number "0"). Note that not all chess programs can handle castling correctly in Fischer random chess games. To correctly record a Fischer random chess game in PGN, an additional "Variant" tag (not "Variation" tag, which has a different meaning) must be used to identify the rules; the rule named "Fischerandom" is accepted by many chess programs as identifying Fischer random chess, though "Chess960" should be accepted as well. This means that in a PGN-recorded game, one of the PGN tags (after the initial seven tags) would look like this: [Variant "Fischerandom"].  FEN is capable of expressing all possible "starting" positions of Fischer random chess; however, unmodified FEN cannot express all possible positions of a Chess960 game. In a game, a rook may move into the back row on the same side of the king as the other rook, or pawn(s) may be underpromoted into rook(s) and moved into the back row. If a rook is unmoved and can still castle, yet there is more than one rook on that side, FEN notation as traditionally interpreted is ambiguous. This is because FEN records that castling is possible on that side, but not "which" rook is still allowed to castle.  A modification of FEN, X-FEN, has been devised by Reinhard Scharnagl to remove this ambiguity. In X-FEN, the castling markings "KQkq" have their expected meanings: "Q" and "q" mean a-side castling is still legal (for White and Black respectively), and "K" and "k" mean h-side castling is still legal (for White and Black respectively). However, if there is more than one rook on the baseline on the same side of the king, and the rook that can castle is not the outermost rook on that side, then the file letter (uppercase for White) of the rook that can castle is used instead of "K", "k", "Q", or "q"; in X-FEN notation, castling potentials belong to the outermost rooks by default. The maximum length of the castling value is still four characters. X-FEN is upwardly compatible with FEN, that is, a program supporting X-FEN will automatically use the normal FEN codes for a traditional chess starting position without requiring any special programming. As a benefit all 18 pseudo FRC positions (positions with traditional placements of rooks and king) still remain uniquely encoded.  The solution implemented by chess engines like Shredder and Fritz is to use the letters of the columns on which the rooks began the game. This scheme is sometimes called Shredder-FEN. For the traditional setup, Shredder-FEN would use HAha instead of KQkq.

Views of grandmasters.

Fischer's proposed "new chess" has elicited various comments from grandmasters.

Related variants.

Non-random setups.

The initial setup need not necessarily be random. The players or a tournament setting may decide on a specific position in advance, for example. Tournament Directors prefer that all boards in a single round play the same random position, as to maintain order and abbreviate the setup time for each round.  Edward Northam suggests the following approach for allowing players to jointly create a position without randomizing tools: First, the back ranks are cleared of pieces, and the white bishops, knights, and queen are gathered together. Starting with Black, the players, in turn, place one of these pieces on White's back rank, where it must stay. The only restriction is that the bishops must go on opposite-color squares. There will be a vacant square of the required color for the second bishop, no matter where the previous pieces have been placed. Some variety could be introduced into this process by allowing each player to exercise a one-time option of moving a piece already on the board instead of putting a new piece on the board. After all five pieces have been put on the board, the king must be placed on the middle of the three vacant back rank squares that remain. Rooks go on the other two.  This approach to the opening setup has much in common with Pre-Chess, the variant in which White and Black, alternately and independently, fill in their respective back ranks. Pre-Chess could be played with the additional requirement of ending up with a legal Fischer random chess opening position. A chess clock could even be used during this phase as well as during normal play.  Without some limitation on which pieces go on the board first, it is possible to reach impasse positions, which cannot be completed to legal Fischer random chess starting positions. Example: Q.RB.N.N If the players want to work with all eight pieces, they must have a prior agreement about how to correct illegal opening positions that may arise. If the bishops end up on same color squares, a simple action, such as moving the a-side bishop one square toward the h-file, might be agreeable, since there is no question of preserving randomness. Once the bishops are on opposite-color squares, if the king is not between the rooks, it should trade places with the nearest rook.

Chess480.

In "Castling in Chess960: An appeal for simplicity", John Kipling Lewis proposes alternative castling rules which Lewis has named "Orthodoxed Castling". The preconditions for castling are the same as in Chess960, but when castling, [...] the king is transferred from its original square two squares towards (or over) the rook, then that rook is transferred to the square the king has just crossed (if it is not already there). If the king and rook are adjacent in a corner and the king cannot move two spaces over the rook, then the king and rook exchange squares.  Unlike Fischer random chess, the final position after castling in Chess480 will usually "not" be the same as the final position of a castling move in traditional chess. Lewis argues that this alternative better conforms to how the castling move was historically developed.  Lewis has named this chess variation "Chess480"; it follows the rules of Chess960 with the exception of the castling rules. Although a Chess480 game can start with any of 960 starting positions, the castling rules are symmetrical (whereas the Chess960 castling rules are not), so that mirror-image positions have identical strategies; thus there are only 480 "effectively" different positions. The number of starting positions could be reduced to 480 without losing any possibilities, for example by requiring the white king to start on a light (or dark) square.  There are other claims to the nomenclature "Chess480"; Reinhard Scharnagl defines it as the white queen is always to the left of the white king.  David O'Shaughnessy argues in "Castling in Chess480: An appeal for sanity" that the Chess480 rules are often not useful from a gameplay perspective. In about 66% of starting positions, players have the options of castling deeper into the wing the king started on, or castling into the center of the board (when the king starts on the b-, c-, f-, or g-files). From Wikipedia article Castling: "Castling is an important goal in the early part of a game, because it serves two valuable purposes: it moves the king into a safer position away from the center of the board, and it moves the rook to a more active position in the center of the board." An example of poor castling options is a position where the kings start on g1 and g8 respectively. There will be no possibility of "opposite-side castling" where each player's pawns are free to be used in, as the kings' scope for movement is very restricted (it can only move to the h- or e-file). These "problem positions" play well with Chess960 castling rules.

Others.

There are several other variants based on randomization of the initial setup. "Randomized Chess, in one or other of its many reincarnations, continues to attract support even, or perhaps especially, that of top players."

References.

Bibliography

Fischer random chess, also known as Chess960, is a variation of the game of chess invented by former world chess champion Bobby Fischer. Fischer announced this new game variation on June 19, 1996, in La Plata, Argentina. Fischer random chess employs the same board and pieces as standard chess, but the starting position of the pieces on the players' is randomized, following certain rules. The random setup makes gaining an advantage through the memorization of openings impracticable; players instead must rely more on their spontaneous talent and creativity over the board.

Les échecs aléatoires Fischer (en anglais, "Fischer Random Chess" ou "Chess 960") sont une variante du jeu d'échecs dans laquelle l'emplacement initial des pièces de la première et de la dernière rangée est tiré au sort, et identique pour les deux camps. Ils sont nommés d'après le joueur d'échecs américain Bobby Fischer.


Composition.

A wide variety of materials have been described as soil conditioners due to their ability to improve soil quality. Some examples include biochar, bone meal, blood meal, coffee grounds, compost, compost tea, coir, manure, straw, peat, sphagnum moss, vermiculite, sulfur, lime, hydroabsorbant polymers, and biosolids. Many soil conditioners come in the form of certified organic products, for people concerned with maintaining organic crops or organic gardens. Soil conditioners of almost every description are readily available from online stores or local nurseries as well as garden supply stores.

Purpose.

Soil structure.

The most common use of soil conditioners is to improve soil structure. Soils tend to become compacted over time. Soil compaction impedes root growth, decreasing the ability of plants to take up nutrients and water. Soil conditioners can add more loft and texture to keep the soil loose.

Soil nutrients.

For centuries people have been adding things to poor soils to improve their ability to support healthy plant growth. Some of these materials, such as compost, clay and peat, are still used extensively today. Many soil amendments also add nutrients such as carbon and nitrogen, as well as beneficial bacteria.  Additional nutrients, such as calcium, magnesium and phosphorus, may be augmented by amendments as well. This enriches the soil, allowing plants to grow bigger and stronger.

Cation exchange.

Soil amendments can also greatly increase the cation exchange capacity (CEC) of soils. Soils act as the storehouses of plant nutrients. The relative ability of soils to store one particular group of nutrients, the cations. The most common soil cations are calcium, magnesium, potassium, ammonium, hydrogen, and sodium.  The total number of cations a soil can hold, its total negative charge, is the soil's cation exchange capacity. The higher the CEC, the higher the negative charge and the more cations that can be held and exchanged with plant roots, providing them with the nutrition they require.

Water retention.

Soil conditioners may be used to improve water retention in dry, coarse soils which are not holding water well. The addition of organic material for instance can greatly improve the water retention abilities of sandy soils and they can be added to adjust the pH of the soil to meet the needs of specific plants or to make highly acidic or alkaline soils more usable. The possibility of using other materials to assume the role of composts and clays in improving the soil was investigated on a scientific basis earlier in the 20th century, and the term soil conditioning was coined. The criteria by which such materials are judged most often remains their cost-effectiveness, their ability to increase soil moisture for longer periods, stimulate microbiological activity, increase nutrient levels and improve plant survival rates.  The first synthetic soil conditioners were introduced in the 1950s, when the chemical hydrolysed polyacrylonitrile was the most used. Because of their ability to absorb several hundred times their own weight in water, polyacrylamides and polymethacrylates (also known as hydroabsorbent polymers, superabsorbent polymers or hydrogels) were tested in agriculture, horticulture and landscaping beginning in the 1960s.  Interest disappeared when experiments proved them to be phytotoxic due to their high acrylamide monomer residue. Although manufacturing advances later brought the monomer concentration down below the toxic level, scientific literature shows few successes in utilizing these polymers for increasing plant quality or survival. The appearance of a new generation of potentially effective tools in the early 1980s, including hydroabsorbent polymers and copolymers from the propenamide and propenamide-propenoate families, opened new perspectives.

Application.

Soil conditioners may be applied in a number of ways. Some are worked into the soil with a tiller before planting. Others are applied after planting, or periodically during the growing season. Soil testing should be performed prior to applying a soil conditioner to learn more about the composition and structure of the soil. This testing will determine which conditioners will be more appropriate for the available conditions.

Ecological concerns.

While adding a soil conditioner to crops or a garden can seem like a great way to get healthier plants, over-application of some amendments can cause ecological problems. For example, salts, nitrogen, metals and other nutrients that are present in many soil amendments are not productive when added in excess, and can actually be detrimental to plant health. (See fertilizer burn.) Runoff of excess nutrients into waterways also occurs, which is harmful to the water quality and through it, the environment.

A soil conditioner is a product which is added to soil to improve the soil’s physical qualities, usually its fertility (ability to provide nutrition for plants) and sometimes its mechanics. In general usage, the term "soil conditioner" is often thought of as a subset of the category soil amendments (or soil improvement, soil condition), which more often is understood to include a wide range of fertilizers and non-organic materials.

Un amendement est un matériau apporté à un sol pour améliorer sa qualité agricole. Les amendements sont donc utilisés en agriculture et pour le jardinage pour améliorer les terres et les rendre plus productives. L’un des amendements les plus connus est la chaux, utilisée pour réduire l’acidité des sols.


Definitions.

Given two metric spaces ("X", "d") and ("Y", "d"), where "d" denotes the metric on the set "X" and "d" is the metric on set "Y", a function "f" : "X" → "Y" is called Lipschitz continuous if there exists a real constant "K" ≥ 0 such that, for all "x" and "x" in "X", Any such "K" is referred to as a Lipschitz constant for the function "f". The smallest constant is sometimes called the (best) Lipschitz constant; however, in most cases, the latter notion is less relevant. If "K" = 1 the function is called a short map, and if 0 ≤ "K" < 1 and "f" maps a metric space to itself, the function is called a contraction.  In particular, a real-valued function "f" : "R" → "R" is called Lipschitz continuous if there exists a positive real constant K such that, for all real "x" and "x", In this case, "Y" is the set of real numbers R with the standard metric "d"("y", "y") = |"y" − "y"|, and "X" is a subset of R.  In general, the inequality is (trivially) satisfied if "x" = "x". Otherwise, one can equivalently define a function to be Lipschitz continuous if and only if there exists a constant "K" ≥ 0 such that, for all "x" ≠ "x", For real-valued functions of several real variables, this holds if and only if the absolute value of the slopes of all secant lines are bounded by "K". The set of lines of slope "K" passing through a point on the graph of the function forms a circular cone, and a function is Lipschitz if and only if the graph of the function everywhere lies completely outside of this cone (see figure).  A function is called locally Lipschitz continuous if for every "x" in "X" there exists a neighborhood "U" of "x" such that "f" restricted to "U" is Lipschitz continuous. Equivalently, if "X" is a locally compact metric space, then "f" is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of "X". In spaces that are not locally compact, this is a necessary but not a sufficient condition.  More generally, a function "f" defined on "X" is said to be Hölder continuous or to satisfy a Hölder condition of order α > 0 on "X" if there exists a constant "M" ≥ 0 such that for all "x" and "y" in "X". Sometimes a Hölder condition of order α is also called a uniform Lipschitz condition of order α > 0.  If there exists a "K" ≥ 1 with then "f" is called bilipschitz (also written bi-Lipschitz). A bilipschitz mapping is injective, and is in fact a homeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz.

Lipschitz manifolds.

Let "U" and "V" be two open sets in R. A function "T" : "U" → "V" is called bi-Lipschitz if it is a Lipschitz homeomorphism onto its image, and its inverse is also Lipschitz.  Using bi-Lipschitz mappings, it is possible to define a Lipschitz structure on a topological manifold, since there is a pseudogroup structure on bi-Lipschitz homeomorphisms. This structure is intermediate between that of a piecewise-linear manifold and a smooth manifold. In fact a PL structure gives rise to a unique Lipschitz structure; it can in that sense 'nearly' be smoothed.

One-sided Lipschitz.

Let "F"("x") be an upper semi-continuous function of "x", and that "F"("x") is a closed, convex set for all "x". Then "F" is one-sided Lipschitz if for some "C" and for all "x" and "x".  It is possible that the function "F" could have a very large Lipschitz constant but a moderately sized, or even negative, one-sided Lipschitz constant. For example, the function  has Lipschitz constant "K" = 50 and a one-sided Lipschitz constant "C" = 0. An example which is one-sided Lipschitz but not Lipschitz continuous is "F"("x") = "e", with "C" = 0.

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the "Lipschitz constant" of the function (or "modulus of uniform continuity"). For instance, every function that has bounded first derivatives is Lipschitz continuous.

En analyse mathématique, une application lipschitzienne (du nom de Rudolf Lipschitz) est une application possédant une certaine propriété de "régularité" qui est plus forte que la continuité. Intuitivement, c'est une fonction qui est limitée dans sa manière d'évoluer. Tout segment reliant deux points du graphe d'une telle fonction aura une pente inférieure, en valeur absolue, à une constante appelée "constante de Lipschitz".


History.

Diesel's idea.

In 1878, Rudolf Diesel, who was a student at the "Polytechnikum" in Munich, attended the lectures of Carl von Linde. Linde explained that steam engines are capable of converting just 6–10% of the heat energy into work, but that the Carnot cycle allows conversion of much more of the heat energy into work by means of isothermal change in condition. According to Diesel, this ignited the idea of creating a highly efficient engine that could work on the Carnot cycle. Diesel was also exposed

The first diesel engine.

Diesel sought out firms and factories that would build his engine. With the help of Moritz Schröter and, he succeeded in convincing both Krupp in Essen and the Maschinenfabrik Augsburg. Contracts were signed in April 1893, and in early summer 1893, Diesel's first prototype engine was built in Augsburg. On 10 August 1893, the first ignition took place, the fuel used was petrol. In winter 1893/1894, Diesel redesigned the existing engine, and by 18 January 1894, his mechanics had converted it into the second prototype. On February 17, 1894, the redesigned engine ran for 88

Operating principle.

Characteristics.

The characteristics of

Cycle of the diesel engine.

The diesel internal combustion engine differs from the gasoline powered Otto cycle by using highly compressed hot air to ignite the fuel rather than using a spark plug ("compression ignition" rather than "spark ignition").  In the diesel engine, only air is initially introduced into the combustion chamber. The air is then compressed with a compression ratio typically between 15:1 and 23:1. This high compression causes the temperature of the air to rise. At about the top of the compression stroke, fuel is injected directly into the compressed air in the combustion chamber. This may be into a (typically toroidal) void in the top of the piston or a "pre-chamber" depending upon the design of the engine. The fuel injector ensures that the fuel is broken down into small droplets, and that the fuel is distributed evenly. The

Efficiency.

Due to its high compression ratio, the diesel engine has a high efficiency, and the lack of a throttle valve means that the charge-exchange losses are fairly low, resulting in a low specific fuel consumption, especially in medium and low load situations. This makes the diesel engine very economical. Even though diesel engines have a theoretical efficiency of 75%, in practice it is much lower. In his 1893 essay

Major advantages.

Diesel engines have several

Fuel injection.

Diesel engines rely on internal mixture formation, which means that they require a fuel injection system. The fuel is injected directly into the combustion chamber, which can be either a segmented combustion chamber or an unsegmented combustion chamber. Fuel injection with the latter is referred to as "direct injection" (DI), whilst injection into the former is called "indirect injection" (IDI). In diesel engine terminology, indirect injection does not mean fuel injection into the inlet manifold or anywhere else outside the cylinder or combustion chamber: in fact, the definition of the diesel engine excludes such injection methods. For creating the fuel pressure, diesel engines usually have an injection pump. There are several different types of injection pumps and methods for creating a fine air-fuel mixture. Over the years many different injection methods have been used. These can be described as the following:

Torque controlling.

Due to the way diesel engines work, a vital component of all diesel engines is a mechanical or electronic governor which regulates the torque of the engine and thus idling speed and maximum speed by controlling the rate of fuel delivery. This means a change of formula_2. Unlike Otto-cycle engines, incoming air is not throttled. Mechanically governed fuel injection systems are driven by the engine's gear train. These systems use a combination of springs and weights to control fuel delivery relative to both load and speed. Modern electronically controlled diesel engines control fuel delivery by

Types of fuel injection.

Air-blast injection.

Diesel's original engine injected fuel with the assistance of compressed air, which atomised the fuel and forced it into the engine through a nozzle (a similar principle to an aerosol spray). The nozzle opening was closed by a pin valve lifted by

Indirect injection.

An indirect diesel injection system (IDI) engine delivers fuel into a small chamber called a swirl chamber, precombustion chamber, pre chamber or ante-chamber, which is connected to the cylinder by a narrow air passage. Generally the goal of the pre chamber is to create increased turbulence for better air / fuel mixing. This system also allows for a smoother, quieter running engine, and because fuel mixing is assisted by turbulence, injector pressures can be lower. Most IDI systems use a single orifice injector. The

Helix-controlled direct injection.

Direct injection Diesel engines inject fuel directly into the cylinder. Usually there is a combustion cup in the top of the piston where the fuel is sprayed. Many different methods of injection can be used. Usually, an engine with helix-controlled mechanic direct injection has either an inline or a distributor injection pump. For each engine cylinder, the corresponding plunger in the fuel pump measures out the correct amount of fuel and determines the timing of each injection. These engines use injectors that are very precise spring-loaded valves that open and

Unit direct injection.

Unit direct injection, also known as Pumpe-Düse ("pump-nozzle"), is a high pressure fuel injection system that injects fuel directly into the cylinder of the engine. In this system the injector and the pump are combined into one unit positioned over each cylinder controlled

Common rail direct injection.

Common rail (CR) direct injection systems, unlike other injection systems, do not have a combined pressure creation and injection apparatus. A high-pressure injection pump creates a constant pressure, not depending on the engine speed or fuel mass injected. A buffer, the so-called rail, saves this pressure. This allows fuel injection at any given moment, even multiple injections in a very short amount of time. The Electronic Diesel Control unit (EDC) controls both rail pressure and injections depending on several different parameters of the engine. The injectors of older CR systems have solenoid-driven plungers for lifting the injection needle, whilst newer CR injectors use plungers driven by piezoelectric actuators, that have fewer moving mass and therefore allow even more injections in a very short period of time. The injection pressure of modern CR systems ranges from 140 MPa to 270 MPa.

Types.

There are several different ways of categorising diesel engines, based on different design characteristics:  "Source"

By cylinder bore.

"Source":

By cylinder arrangement.

Regular cylinder configurations such as straight (inline), V, and boxer (flat) configurations can be used for diesel engines. The inline-six-cylinder design is the most prolific in light- to medium-duty engines, though inline-four engines are also

By engine speeds.

Günter Mau categorises diesel engines by their rotational speeds into three groups:  "Source"

High-speed engines.

High-speed engines are used to power trucks (lorries), buses, tractors, cars, yachts, compressors, pumps and small electrical generators. As of 2018, most high-speed engines have direct

Medium-speed engines.

Medium-speed engines are used in large electrical generators, ship propulsion and mechanical drive applications such as large compressors or pumps. Medium speed diesel engines operate on either diesel fuel or heavy fuel oil by direct injection in the same manner as low-speed engines. Usually, they are four-stroke engines with trunk pistons.  The power output of medium-speed diesel engines can be as high as 21,870 kW,

Low-speed engines.

Low-speed diesel engines are usually very large in size and mostly used to power ships. There are two different types of low-speed engines that are commonly used: Two-stroke engines with a crosshead, and four-stroke engines with a regular trunk-piston. Two-stroke engines have a limited rotational frequency and their charge exchange is more difficult, which means that they are usually bigger than four-stroke engines and used to directly power a ship's propeller. Four-stroke engines on ships are usually used to power an electric generator. An electric motor powers the propeller. Both types are usually very undersquare. Low-speed diesel engines (as used in ships and other applications where overall engine weight is relatively unimportant) often have an effective efficiency of up to 55%. Like medium-speed engines, low-speed engines are started with compressed air, and they use heavy oil as their primary fuel.

Two-stroke engines.

Two-stroke diesel engines use only two strokes instead of four strokes for a complete engine cycle. Filling the cylinder with air and compressing it takes place in one stroke, and the power and exhaust strokes are combined. The compression in a two-stroke diesel engine is similar to the compression that takes place in a four-stroke diesel engine: As the piston passes through bottom centre and starts upward, compression commences, culminating in fuel injection and ignition. Instead of a full set of valves, two-stroke diesel engines have simple intake ports, and exhaust ports (or exhaust valves).

Dual-fuel diesel engines.

So-called dual-fuel diesel engines or gas diesel engines burn two different types of fuel "simultaneously", for instance, a gaseous fuel and diesel

Diesel engine particularities.

Torque and power.

Torque is a force applied to a lever at a right angle multiplied by the lever length. This means that the torque an engine produces depends on the displacement of the engine and the force that the gas pressure inside the cylinder applies to the piston, commonly referred to as "effective piston pressure":  Power is the

Mass.

The average diesel engine has a poorer power-to-mass ratio than the Otto engine. This is because the diesel must operate at lower engine speeds. Due to

Emissions.

As diesel engines burn a mixture of fuel and air, the exhaust therefore contains substances that consist of the same chemical elements, as fuel and air. The main elements of air are nitrogen (N) and oxygen (O), fuel consists of hydrogen (H) and carbon (C). Burning the fuel will result in the final stage of oxidation. An "ideal diesel engine", (a hypothetical model that we use as an example), running on an ideal air-fuel mixture, produces an exhaust that consists of carbon dioxide (CO), water (HO), nitrogen (N), and the remaining oxygen (O). The combustion process in a real engine differs from an ideal engine's combustion process, and due to incomplete combustion, the exhaust contains additional substances, most notably, carbon monoxide (CO), diesel particulate matter (PM), and due

Noise.

The distinctive noise of a diesel engine is variably called diesel clatter, diesel nailing, or diesel knock. Diesel clatter is caused largely by the way the fuel ignites; the sudden ignition of the diesel fuel when injected into the combustion chamber causes a pressure wave, resulting

Cold weather starting.

In general, diesel engines do not require any starting aid. In cold weather however, some diesel engines can be difficult to start and may need preheating depending on the combustion chamber design. The minimum starting temperature that allows starting without pre-heating is 40 °C for precombustion chamber engines, 20 °C for swirl chamber engines, and 0 °C for direct injected engines. Smaller engines with a displacement of less than 1 litre per cylinder usually have glowplugs, whilst larger heavy-duty engines have flame-start systems.  In the past, a wider variety of cold-start methods were used. Some engines, such as Detroit Diesel engines used a system to introduce small amounts of

Supercharging and turbocharging.

As the diesel engine relies on manipulation of formula_2 for torque controlling and speed regulation, the intake air mass does not have to precisely match the injected fuel mass (which would be formula_20). diesel engines are thus ideally suited for supercharging and turbocharging. An additional advantage of the diesel engine is the lack of fuel during the compression stroke. In diesel engines, the fuel is injected near top dead centre (TDC), when the piston is near its highest position. The fuel then ignites due to compression heat. Preignition, caused by the artificial turbocharger compression increase during the compression stroke, cannot occur.  Many diesels are therefore turbocharged and some are both turbocharged and supercharged. A turbocharged engine can

Fuel and fluid characteristics.

In diesel engines, a mechanical injector system vaporises the fuel directly into the combustion chamber (as opposed to a Venturi jet in a carburetor, or a fuel injector in a manifold injection system vaporising fuel into the intake manifold or intake runners as in a petrol engine). This "forced vaporisation" means that less-volatile fuels can be used. More crucially, because only air is inducted into the cylinder in a diesel engine, the compression ratio can be much higher as there is no risk of pre-ignition provided the injection process is accurately timed. This means that cylinder temperatures are much higher in a diesel engine than a petrol engine, allowing less volatile fuels to be used.  Therefore, diesel engines can operate on a huge variety of different fuels. In general, fuel for diesel engines should have a proper viscosity, so that the injection pump can pump the fuel to the injection nozzles without causing damage to itself or corrosion of the fuel line. At injection, the fuel should form a good fuel spray, and it should not have a coking effect upon the injection nozzles. To ensure proper engine starting and smooth operation, the fuel should be willing to ignite and hence not cause a high ignition delay, (this means that the fuel should have a high cetane number). Diesel fuel should also have a high lower heating value.  Inline mechanical injector pumps generally tolerate poor-quality or bio-fuels better than distributor-type pumps. Also, indirect injection engines generally run more satisfactorily on fuels with a high ignition delay (for instance, petrol) than direct injection engines. This is partly because an indirect injection engine has a much greater'swirl' effect, improving vaporisation and combustion of fuel, and because (in the case of vegetable oil-type fuels) lipid depositions can condense on the cylinder walls of a direct-injection engine if combustion temperatures are too low (such as starting the engine from cold). Direct-injected engines with an MAN centre sphere combustion chamber rely on fuel condensing on the combustion chamber walls. The fuel starts vaporising only after ignition sets in, and it burns relatively smoothly. Therefore, such engines also tolerate fuels with poor ignition delay characteristics, and, in general, they can operate on petrol rated 86 RON.

Fuel types.

In his 1893 work "Theory and Construction of a Rational Heat Motor", Rudolf Diesel considers using coal dust as fuel for the diesel engine. However, Diesel just "considered" using coal dust (as well as liquid fuels and gas); his actual engine was designed to operate on petroleum, which was soon replaced with regular petrol and kerosene for further testing purposes, as petroleum proved to be too viscous. In addition to kerosene and petrol, Diesel's engine could also operate on ligroin.  Before diesel engine fuel was standardised, fuels such as petrol, kerosene, gas oil, vegetable oil

Gelling.

DIN 51601 diesel fuel was prone to "waxing" or "gelling" in cold weather; both are terms for the solidification of diesel oil into a partially crystalline state. The crystals build up in the fuel system (especially in fuel filters), eventually starving the engine of fuel and causing it to stop running. Low-output electric heaters in fuel tanks and

Safety.

Fuel flammability.

Diesel fuel is less flammable than petrol, because its flash point is 55 °C, leading to a lower risk of fire caused by fuel in a vehicle equipped with a diesel engine.  Diesel fuel can

Cancer.

Diesel exhaust has been classified as an IARC Group 1

Engine runaway (uncontrollable overspeeding).

See diesel

Applications.

The characteristics of diesel have different advantages for different applications.

Passenger cars.

Diesel engines have long been popular in bigger cars and have been used in smaller cars such as superminis in Europe since the 1980s. They were popular in larger cars earlier, as the weight and cost penalties were less noticeable. Smooth operation as well as high low end torque are deemed important for passenger cars and small commercial vehicles. The introduction of electronically controlled fuel injection significantly improved the smooth torque generation, and starting in the early 1990s, car manufacturers began offering

Commercial vehicles and lorries.

In 1893, Rudolf Diesel suggested that the diesel engine could possibly power ‘wagons’ (lorries). The first lorries with diesel engines were brought to market in 1924.  Modern diesel engines for lorries have to be both extremely reliable and very fuel efficient. Common-rail direct injection, turbocharging and four valves per cylinder are standard. Displacements range from 4.5 to 15.5 litres, with power-to-mass

Railroad rolling stock.

Diesel engines for locomotives are built for continuous operation and may require the ability to use poor quality fuel in some circumstances. Some locomotives use two-stroke diesel engines. Diesel engines have eclipsed steam engines as the prime mover on all non-electrified railroads in the industrialised world. The first diesel locomotives appeared in 1913, and diesel multiple units soon after. Many modern diesel locomotives are actually diesel-electric locomotives: the diesel engine is used to power an electric generator that in turn powers electric traction motors with no mechanical connection between diesel engine and traction. While electric locomotives have

Watercraft.

The requirements for marine diesel engines vary, depending on the application. For military use and medium-size boats, medium-speed four-stroke diesel engines are most suitable. These engines usually have up to 24 cylinders and come with power outputs in the one-digit Megawatt region. Small boats may use lorry diesel engines. Large ships use extremely efficient, low-speed two-stroke diesel engines. They can reach efficiencies of

Aviation.

Diesel engines had been used in aircraft before World War II, for instance, in the rigid airship "LZ 129 Hindenburg", which was powered by four Daimler-Benz DB 602 diesel engines, or in several Junkers aircraft, which had Jumo 205 engines installed. Until the late 1970s, there has

Non-road diesel engines.

Non-road diesel engines are commonly used for construction equipment. Fuel efficiency, reliability and ease of maintenance are very important for such engines, whilst high power output and quiet

Stationary diesel engines.

Stationary diesel engines are commonly used for electricity generation, but also for powering refrigerator compressors, or other types of compressors or pumps. Usually, these engines run permanently, either with mostly partial load, or intermittently, with full load. Stationary diesel engines powering electric generators that put out

Low heat rejection engines.

A special class of prototype internal combustion piston engines has been developed over several decades with the goal of improving efficiency by reducing heat loss. These engines are variously called adiabatic engines; due to better approximation of adiabatic expansion; low heat rejection engines, or high temperature

Future developments.

In mid-2010s literature, main development goals for future diesel engines are described as improvements of exhaust emissions, reduction of fuel consumption, and increase of lifespan (2014). It is said that the diesel engine, especially the diesel engine for commercial vehicles, will remain the most important vehicle powerplant until the mid-2030s. Editors assume that the complexity of the diesel engine will increase further (2014). Some editors expect a future convergency of diesel and Otto engines' operating principles due to Otto engine development steps made towards homogeneous charge compression ignition (2017).

The diesel engine (also known as a compression-ignition or CI engine), named after Rudolf Diesel, is an internal combustion engine in which ignition of the fuel is caused by the elevated temperature of the air in the cylinder due to the mechanical compression (adiabatic compression). This contrasts with spark-ignition engines such as a petrol engine (gasoline engine) or gas engine (using a gaseous fuel as opposed to petrol), which use a spark plug to ignite an air-fuel mixture.

Le moteur Diesel, appelé également moteur à allumage par compression, est un moteur à combustion et explosion dont la mise en rotation est provoqué par l'injection de carburant dans la chambre de combustion et par un phénomène d'auto-inflammation lié aux températures élevées dans la chambre de combustion. Celles-ci sont atteintes grâce à un fort taux de compression (rapport volumétrique de ), permettant d'obtenir une température de. Pour le démarrage des petits moteurs, des bougies de préchauffage sont souvent utilisées pour permettre un meilleur démarrage à froid, en créant un point chaud dans la chambre de combustion. Pour les moteurs de bateaux et les gros moteurs fixes à fioul lourd, on chauffe celui-ci à haute température pour permettre le démarrage.


Classification.

Scientific classification of Elephantidae taxa embraces an extensive record of fossil specimens, over millions of years, some of which existed until the end of the last ice age. Some species were extirpated more recently. The discovery of new specimens and proposed cladistics have resulted in systematic revisions of the family and related proboscideans.  Elephantids are classified informally as the elephant family, or in a paleobiological context as elephants and mammoths. The common name elephant primarily refers to the living taxa, the modern elephants, but may also refer to a variety of extinct species, both within this family and in others. Other members of the Elephantidae, especially members of "Mammuthus", are referred to by the common name mammoth.  The family diverged from a common ancestor of the mastodons of Mammutidae. The classification of proboscideans is unstable and has been frequently revised.  The following cladogram shows the placement of the genus "Mammuthus" among other proboscideans, based on hyoid characteristics:  The systematics of the living subspecies and species of the modern elephants has undergone several revisions. A list of the extant Elephantidae includes:

Evolutionary history.

Although the fossil evidence is uncertain, by comparing genes, scientists have discovered evidence that elephantids and other proboscideans share a distant ancestry with Sirenia (sea cows) and Hyracoidea (hyraxes). These have been assigned, along with the extinct demostylians and embrithopods, to the clade Paenungulata. In the distant past, members of the various hyrax families grew to large sizes, and the common ancestor of all three modern families is thought to have been some kind of amphibious hyracoid. One hypothesis is that these animals spent most of their time under water, using their trunks like snorkels for breathing. Modern elephants have this ability and are known to swim in that manner for up to six hours and.  In the past, a much wider variety of genera and species was found, including the mammoths and stegodons.

The Elephantidae are a family of large, herbivorous mammals collectively called elephants and mammoths. These are terrestrial large mammals with a snout modified into a trunk and teeth modified into tusks. Most genera and species in the family are extinct. Only two genera, "Loxodonta" (African elephants) and "Elephas" (Asiatic elephants), are living.

Les Éléphantidés (Elephantidae), ou plus simplement les éléphants, sont une famille de mammifères de l'ordre des Proboscidiens. C'est la seule famille de cet ordre qui existe encore actuellement. Elle comptait de très nombreuses espèces aujourd'hui disparues, et notamment les mammouths du genre †"Mammuthus". Ce sont tous de très grands animaux dotés d'une trompe caractéristique. La plupart des espèces d'éléphantidés sont éteintes et cette famille ne regroupe plus que trois espèces aujourd'hui : l'éléphant de savane et l'éléphant de forêt, du genre "Loxodonta", autrefois regroupés sous l’expression « éléphant d'Afrique », et l'éléphant d'Asie, du genre "Elephas", anciennement appelé parfois éléphant "indien". Ils se distinguent les uns des autres par leurs caractéristiques anatomiques, les éléphants d'Asie étant en général plus petits avec des oreilles plus petites, ou encore par une différence du bout de la trompe. L'éléphant barrit, son cri est appelé un barrissement (ces deux termes valent aussi pour le rhinocéros).