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{{Use dmy dates|date=July 2013}}
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In [[mathematics]] '''equality''' is a relationship between two quantities or, more generally two [[mathematical expression]]s, asserting that the quantities have the same value or that the expressions represent the same [[mathematical object]]. The equality between ''A'' and ''B'' is written ''A'' = ''B'', and pronounced ''A'' equals ''B''. The symbol "=" is called an "[[equals sign]]".
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'''Glottochronology''' (from [[Attic Greek]] γλῶττα “tongue, language” and χρóνος “time”) is that part of [[lexicostatistics]] dealing with the chronological relationship between languages.<ref>Sheila Embleton (1992). HISTORICAL LINGUISTICS: Mathematical concepts. In W. Bright (Ed.), International Encyclopedia of Linguistics, page 131</ref>
==Etymology==
The [[etymology]] of the word is from the Latin ''[[wikt:aequalis#Latin|aequālis]]'' (“equal”, “like”, “comparable”, “similar”) from ''[[wikt:aequus#Latin|aequus]]'' (“equal”, “level”, “fair”, “just”).


The idea was developed by [[Morris Swadesh]] under two assumptions: First that there exists a relatively stable "basic vocabulary" (therefore called "[[Swadesh list]]s") in all languages of the world, and secondly that any replacements happen in a way analogical to that in [[radioactive decay]] in constant percentages per time elapsed. Meanwhile there exist many different methods, partly extensions of the Swadesh method, now more and more under the biological assumptions of replacements in genes. However, Swadesh's technique is so well known that, for many people, 'glottochronology' refers to it alone.<ref>Sheila Embleton: HISTORICAL LINGUSITICS: Mathematical concepts. In: W. Bright (ed., International Encyclopedia of Linguistics, 1992: 133)</ref><ref>Holm, Hans J. (2007). The new Arboretum of Indo-European 'Trees'; Can new algorithms reveal the Phylogeny and even Prehistory of IE?. Journal of Quantitative Linguistics 14-2:167–214</ref>
==Types of equalities==
===Identities===
{{main|Identity (mathematics)}}
When ''A'' and ''B'' may be viewed as [[function (mathematics)|functions]] of some variables, then ''A''&nbsp;=&nbsp;''B'' means that ''A'' and ''B'' define the same function. Such an equality of functions is sometimes called an [[identity (mathematics)|identity]]. An example is (''x''&nbsp;+&nbsp;1)<sup>2</sup>&nbsp;=&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;2''x''&nbsp;+&nbsp;1.


==Methodology==
===Equalities as predicates===
When ''A'' and ''B'' are not fully specified or depend on some [[Variable (mathematics)|variables]], equality is a [[proposition (mathematics)|proposition]], which may be true for some values and false for some other values. Equality is a [[binary relation]], or, in other words, a two-arguments [[predicate (mathematical logic)|predicate]], which may produce a [[truth value]] (''false'' or ''true'') from its arguments. In [[computer programming]], its computation from two expressions is known as [[relational operator|comparison]].


===Word list===
===Congruences===
The original method presumed that the core vocabulary of a language is replaced at a constant (or constant average) rate across all languages and cultures, and can therefore be used to measure the passage of time. The process makes use of a list of lexical terms. Lists were compiled by Morris Swadesh and assumed to be resistant against borrowing (originally designed in 1952 as a list of 200 items; however, the refined 100 word list in Swadesh (1955)<ref name=swadesh1955>Swadesh, Morris. (1955). Towards greater accuracy in lexicostatistic dating. ''International Journal of American Linguistics'', ''21'', 121&ndash;137</ref> is much more common among modern day linguists). This core vocabulary was designed to encompass concepts common to every human language (such as personal pronouns, body parts, heavenly bodies, verbs of basic actions, numerals 'one' and 'two', etc.), eliminating concepts that are specific to a particular culture or time. It has been found that this ideal is not in fact possible and that the meaning set may need to be tailored to the languages being compared. Many alternative word lists have been compiled by other linguists, often using fewer meaning slots.
In some cases, one may consider as '''equal''' two mathematical objects that are only equivalent for the properties that are considered. This is, in particular the case in [[geometry]], where two [[geometric shape]]s are said equal when one may be moved to coincide with the other. The word '''congruence''' is also used for this kind of equality.


The percentage of [[cognate]]s (words that have a common origin) in these word lists is then measured. The larger the percentage of cognates, the more recently the two languages being compared are presumed to have separated.
===Equations===
An [[equation]] is the problem of finding values of some variables, called ''unknowns'', for which the specified equality is true. ''Equation'' may also refer to an equality relation that is satisfied only for the values of the variables that one is interested on. For example ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;=&nbsp;1 is the ''equation'' of the [[unit circle]]. There is no standard notation that distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriate interpretation from the semantic of expressions and the context.


===Glottochronologic constant===
===Equivalence relations===
[[Robert Lees (linguist)|Robert Lees]] obtained a value for the "glottochronological constant" ('''r''') of words by considering the known changes in 13 pairs of languages using the 200 word list.  He obtained a value of 0.805&nbsp;±&nbsp;0.0176 with 90% confidence. For the 100 word list Swadesh obtained a value of 0.86, the higher value reflecting the elimination of semantically unstable words. This constant may be related to the retention rate of words by:-
{{main|Equivalence relation}}
Viewed as a relation, equality is the archetype of the more general concept of an [[equivalence relation]] on a set: those binary relations that are [[reflexive relation|reflexive]], [[symmetric relation|symmetric]], and [[transitive relation|transitive]].
The identity relation is an equivalence relation. Conversely, let ''R'' be an equivalence relation, and let us denote by ''x<sup>R</sup>'' the equivalence class of ''x'', consisting of all elements ''z'' such that ''x R z''. Then the relation ''x R y'' is equivalent with the equality ''x<sup>R</sup>''&nbsp;=&nbsp;''y<sup>R</sup>''. It follows that equality is the smallest equivalence relation on any set ''S'', in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element).


:<math>L = 2\ln(r) </math>
==Logical formalizations of equality==
There are several formalizations of the notion of equality in [[mathematical logic]], usually by means of axioms, such as  the first few [[Peano axioms]], or the [[axiom of extensionality]] in [[Zermelo–Fraenkel set theory|ZF set theory]]). There are also some [[mathematical logic|logic systems]] that do not have any notion of equality. This reflects the [[undecidable problem|undecidability]] of the equality of two [[real number]]s defined by formulas involving the [[integer]]s, the basic [[arithmetic operation]]s, the [[logarithm]] and the [[exponential function]]. In other words,
there cannot exist any [[algorithm]] for deciding such an equality.


where ''L'' is the rate of replacement, ln is the logarithm to base e, and ''r'' is the glottochronological constant
==Logical formulations==
Equality is always defined such that things that are equal have all and only the same properties. Some people{{who|date=January 2014}} define equality as congruence. Often equality is just defined as [[Identity (philosophy)|identity]].


===Divergence time===
A stronger sense of equality is obtained if some form of [[Identity of indiscernibles|Leibniz's law]] is added as an [[axiom]]; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms.  The axiom states that two things are equal if they have all and only the same [[Property (philosophy)|properties]]. Formally:
The basic formula of glottochronology in its shortest form is:-
: [[Given any]] ''x'' and ''y'', ''x'' = ''y'' [[material conditional|if]], given any [[Predicate (mathematics)|predicate]] ''P'', ''P''(''x'') [[if and only if]] ''P''(''y'').


: <math>t = \frac{\ln(c)}{-L} </math>
In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original.


where ''t'' = a given period of time from one stage of the language to another, ''c'' = proportion of wordlist items retained at the end of that period, and ''L'' = rate of replacement for that word list.
Instead of considering Leibniz's law as an axiom, it can also be taken as the ''definition'' of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become [[theorem]]s.
If a=b, then a can replace b and b can replace a.


By testing historically verifiable cases where we have knowledge of ''t'' through non-linguistic data (e. g. the approximate distance from Classical Latin to modern Romance languages), Swadesh arrived at the empirical value of approximately 0.14 for ''L'' (meaning that the rate of replacement constitutes around 14 words from the 100-wordlist per millennium).
==Some basic logical properties of equality==
The substitution property states:
* [[For any]] quantities ''a'' and ''b'' and any expression ''F''(''x''), [[material conditional|if]] ''a'' = ''b'', then ''F''(''a'') = ''F''(''b'') (if both sides make sense, i.e. are [[well-formed formula|well-formed]]).
In [[first-order logic]], this is a [[schema (logic)|schema]], since we can't quantify over expressions like ''F'' (which would be a [[functional predicate]]).


===Results===
Some specific examples of this are:
Glottochronology was found to work in the case of Indo-European, accounting for 87% of the variance. It is also postulated to work for Hamito-Semitic (Fleming 1973), Chinese (Munro 1978) and Amerind (Stark 1973; Baumhoff and Olmsted 1963).  For the latter, correlations have been obtained with radiocarbon dating and blood groups as well as archaeology.
* For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'', then ''a'' + ''c'' = ''b'' + ''c'' (here ''F''(''x'') is ''x'' + ''c'');
Note that the approach of Gray and Atkinson,<ref>''Language-tree divergence times support the Anatolian theory of Indo-European origin, Russell D. Gray & Quentin D. Atkinson, Nature 426, 435&ndash;439'' 2003</ref> after their own words, have nothing to do with "glottochronology".
* For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'', then ''a'' − ''c'' = ''b'' − ''c'' (here ''F''(''x'') is ''x'' − ''c'');
* For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'', then ''ac'' = ''bc'' (here ''F''(''x'') is ''xc'');
* For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'' and ''c'' [[Division by zero|is not]] [[0 (number)|zero]], then ''a''/''c'' = ''b''/''c'' (here ''F''(''x'') is ''x''/''c'').


==Discussion==
The reflexive property states:
The concept of language change is old and its history is reviewed in Hymes (1973) and Wells (1973).  Glottochronology itself dates back to the mid-20th century.<ref name=swadesh1955/><ref>Swadesh, Morris (1972). What is glottochronology? In M. Swadesh, ''The origin and diversification of languages'' (pp.&nbsp;271–284). London: Routledge & Kegan Paul.</ref><ref>Lees, Robert. (1953). The basis of glottochronology. ''Language'', ''29'' (2), 113&ndash;127.</ref>  An introduction to the subject is given in Embleton (1986)<ref>Embleton, Sheila M. (1986). ''Statistics in Historical Linguistics'' [Quantitative linguistics, vol. 30]. Bochum: Brockmeyer. ISBN 3-88339-537-4. &ndash; State of the art up to then.
:[[For any]] quantity ''a'', ''a'' = ''a''.
</ref> and in McMahon and McMahon (2005).<ref>McMahon, April and McMahon, Robert (2005) ''Language Classification by Numbers.'' Oxford: Oxford University Press (in particular p. 95)</ref>


Glottochronology has been controversial ever since, partly owing to issues of accuracy, as well as the question of whether its basis is sound (see e.g. Bergsland 1958; Bergsland and Vogt 1962; Fodor 1961; Chretien 1962; Guy 1980).  These concerns have been addressed by Dobson et al. (1972), Dyen (1973)<ref name=dyen1973>Dyen, Isidore, ed. (1973). ''Lexicostatistics in genetic linguistics: Proceedings of the Yale conference, April 3&ndash;4, 1971''. La Haye: Mouton.</ref> and Kruskal, Dyen and Black (1973).<ref name=kruskal1973>Some Results From the Vocabulary Method of Reconstructing Language Trees, Joseph B. Kruskal, Isidore Dyen and Paul Black, Lexicostatistics in Genetic Linguistics, Isidore Dyen (editor), Mouton, The Hague, 1973, pp. 30-55</ref> The assumption of a single-word replacement rate can distort the divergence-time estimate when borrowed words are included (Thomason and Kaufman 1988). Chrétien purported to disprove the mathematics of the Swadesh-model. At a conference at Yale in 1971 his criticisms were shown to be invalid. See the published proceedings under Dyen (1973)<ref name=dyen1973/> The same conference saw the application of the theory to [[Creole language]] (Wittmann 1973).
This property is generally used in [[mathematical proof]]s as an intermediate step.
An overview of recent arguments can be obtained from the papers of a conference held at the McDonald Institute in 2000.<ref name=renfrew2002>Renfrew, C., McMahon, A., & L. Trask, Eds. (2000). Time Depth in Historical LInguistics. Cambridge, England: The McDonald Institute for Archaeological Research.
</ref> These presentations vary from "Why linguists don't do dates" to the one by [[Sergei Starostin|Starostin]] discussed above.{{clarify|date=November 2011}}
Since its original inception, glottochronology has been rejected by many linguists, mostly Indo-Europeanists of the school of the traditional [[comparative method]]. Criticisms have been answered in particular around three points of discussion.


* Criticism levelled against the higher stability of lexemes in Swadesh lists alone (Haarmann 1990) misses the point, because a certain amount of losses only enables the computations (Sankoff 1970).
The symmetric property states:
* Traditional glottochronology did presume that language changes at a stable rate.
* [[For any]] quantities ''a'' and ''b'', [[material conditional|if]] ''a'' = ''b'', then ''b'' = ''a''.
:Thus, in Bergsland & Vogt (1962), the authors make an impressive demonstration, on the basis of actual language data verifiable by extra-linguistic sources, that the "rate of change" for [[Icelandic language|Icelandic]] constituted around 4% per millennium, whereas for closely connected [[Riksmal]] (Literary Norwegian) it would amount to as much as 20%. (Swadesh's proposed "constant rate" was supposed to be around 14% per millennium).
:This and several other similar examples effectively proved that Swadesh's formula would not work on all available material&mdash;a serious accusation considering that evidence that can be used to "calibrate" the meaning of ''L'' (i. e. language history recorded during prolonged periods of time) is not overwhelmingly large in the first place.
:It is highly likely that the chance of replacement is in fact different for every word or feature ("each word has its own history", among hundreds of other sources:<ref>Kirk JM, St Anderson, & JDA Widdowson, 1985 Studies in Linguistic Geography: The Dialects of English in Britain and Ireland. London: Croom Helm</ref>).
:This global assumption has been modified and downgraded to single words even in single languages in many newer attempts (see below).
*A serious argument is that language change arises from socio-historical events which are of course unforeseeable and, therefore, uncomputable.
:New methods developed by Gray & Atkinson are claimed to avoid these issues, but are still seen as controversial, primarily since they often produce results that are incompatible with known data and because of additional methodological issues.


==Modified glottochronology==
The transitive property states:
* [[For any]] quantities ''a'', ''b'', and ''c'', [[material conditional|if]] ''a'' = ''b'' [[and (logic)|and]] ''b'' = ''c'', then ''a'' = ''c''.


Somewhere in between the original concept of Swadesh and the rejection of glottochronology in its entirety lies the idea that glottochronology as a formal method of linguistic analysis becomes valid with the help of several important modifications.  Thus, inhomogeneities in the replacement rate were dealt with by Van der Merwe (1966)<ref>van der Merwe, N. J. 1966 "New mathematics for glottochronology", Current Anthropology 7: 485--500</ref> by splitting the word list into classes each with their own rate, while Dyen, James and Cole (1967)<ref>Dyen, I., James, A. T., & J. W. L. Cole 1967 "Language divergence and estimated word retention rate", <Language 43: 150--171</ref> allowed each meaning to have its own rate. Simultaneous estimation of divergence time and replacement rate was studied by Kruskal, Dyen and Black.<ref name=kruskal1973/>
The [[binary relation]] "[[approximation|is approximately equal]]" between [[real number]]s or other things, even if more precisely defined, is not transitive (it may seem so at first sight, but many small [[Difference (mathematics)|differences]] can add up to something big).
However, equality [[almost everywhere]] ''is'' transitive.


Brainard (1970) allowed for chance cognation and drift effects was introduced by Gleason (1959). Sankoff (1973) suggested introducing a borrowing parameter and allowed synonyms.
Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitution and reflexive properties are assumed instead.


A combination of these various improvements is given in Sankoff's "Fully Parameterised Lexicostatistics".  In 1972 Sankoff in a biological context developed a model of genetic divergence of populations. Embleton (1981) derives a simplified version of this in a linguistic context. She carries out a number of simulations using this which are shown to give good results.
==Relation with equivalence and isomorphism==
{{See also|Equivalence relation|Isomorphism}}


Improvements in statistical methodology related to a completely different branch of science &ndash; [[Phylogenetics|changes in DNA over time]] &ndash; have sparked a recent renewed interest. These methods are more robust than the earlier ones because they calibrate points on the tree with known historical events and smooth the rates of change across these. As such, they no longer require the assumption of a constant rate of change ([http://language.psy.auckland.ac.nz/publications/index.php?pub=Gray_and_Atkinson2003Nature Gray & Atkinson 2003]).
In some contexts, equality is sharply distinguished from ''[[equivalence relation|equivalence]]'' or ''[[isomorphism]].''<ref>{{Harv|Mazur|2007}}</ref> For example, one may distinguish ''[[fraction (mathematics)|fractions]]'' from ''[[rational number]]s,'' the latter being equivalence classes of fractions: the fractions <math>1/2</math> and <math>2/4</math> are distinct as fractions, as different strings of symbols, but they "represent" the same rational number, the same point on a number line. This distinction gives rise to the notion of a [[quotient set]].


===Starostin's method===
Similarly, the sets
:<math>\{\text{A}, \text{B}, \text{C}\} \,</math> and <math>\{ 1, 2, 3 \} \,</math>


Another attempt to introduce such modifications was performed by the Russian linguist [[Sergei Starostin]], who had proposed that
are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements, and thus isomorphic, meaning that there is a [[bijection]] between them, for example
:<math>\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3.</math>


* systematic [[loanword]]s, borrowed from one language into another, are a disruptive factor and have to be eliminated from the calculations; the one thing that really matters is the "native" replacement of items by items from the same language. The failure to notice this factor was a major reason in Swadesh's original estimation of the replacement rate at under 14 words from the 100-wordlist per millennium, when the real rate is, in fact, much slower (around 5 or 6). Introducing this correction effectively cancels out the "Bergsland & Vogt" argument, since a thorough analysis of the Riksmal data shows that its basic wordlist includes about 15&ndash;16 borrowings from other Germanic languages (mostly [[Danish language|Danish]]) &ndash; exclusion of these elements from the calculations brings the rate down to the expected rate of 5&ndash;6 "native" replacements per millennium;
However, there are other choices of isomorphism, such as
* the rate of change is not really constant, but actually depends on the time period during which the word has existed in the language (i. e. chances of lexeme X being replaced by lexeme Y increase in direct proportion to the time elapsed – the so-called "aging of words", empirically understood as gradual "erosion" of the word's primary meaning under the weight of acquired secondary ones);
:<math>\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,</math>
* individual items on the 100 wordlist have different stability rates (for instance, the word "I" generally has a much lower chance of being replaced than the word "yellow", etc.).


The resulting formula, taking into account both the time dependence and the individual stability quotients, looks as follows:
and these sets cannot be identified without making such a choice – any statement that identifies them "depends on choice of identification". This distinction, [[Isomorphism#Relation_with_equality|between equality and isomorphism]], is of fundamental importance in [[category theory]], and is one motivation for the development of category theory.
 
: <math>t = \sqrt \frac{\ln(c)}{-Lc}</math>
 
In this formula, &minus;''Lc'' reflects the gradual slowing down of the replacement process due to different individual rates (the less stable elements are the first and the quickest to be replaced), whereas the square root represents the reverse trend &ndash; acceleration of replacement as items in the original wordlist "age" and become more prone to shifting their meaning. The formula is obviously more complicated than Swadesh's original one, but, as shown in Starostin's work, yields more credible results than the former (and more or less agrees with all the cases of language separation that can be confirmed by historical knowledge). On the other hand, it shows that glottochronology can really only be used as a serious scientific tool on language families the historical phonology of which has been meticulously elaborated (at least to the point of being able to clearly distinguish between cognates and loanwords).
 
===Time-depth estimation===
The problem of time-depth estimation was the subject of a conference held by the McDonald Institute in 2000.  The published papers<ref name=renfrew2002/> give an idea of the views on glottochronology at the time.  These vary from "Why linguists don't do dates" to the one by Starostin discussed above. Note that in the referenced Gray and Atkinson paper, they hold that their methods can not be called "glottochronology", by incorrectly confining this term to its original method.


==See also==
==See also==
*[[Lexicostatistics]]
*[[Equals sign]]
*[[Dolgopolsky list]]
*[[Inequality (mathematics)|Inequality]]
*[[Leipzig–Jakarta list]]
*[[Logical equality]]
*[[Swadesh list]]
*[[Extensionality]]
*[[Mass lexical comparison]]
*[[Basic English]]
*[[Historical linguistics]]
*[[Proto-language]]
*[[Cognate]]
*[[Indo-European studies]]


==References==
==References==
<references/>
{{Reflist}}
 
{{Refbegin}}
==Bibliography==
* {{Citation | first = Barry | last = Mazur | authorlink = Barry Mazur | title = When is one thing equal to some other thing? | date = 12 June 2007 | url = http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf }}
* Arndt, Walter W. (1959). The performance of glottochronology in Germanic. ''Language'', ''35'', 180&ndash;192.
* {{Cite book
* [[Knut Bergsland|Bergsland, Knut]]; & Vogt, Hans. (1962). On the validity of glottochronology. ''Current Anthropology'', ''3'', 115&ndash;153.
| authorlink = Saunders Mac Lane
* Brainerd, Barron (1970).  A Stochastic Process related to Language Change.  ''Journal of Applied Probability'' 7, 69&ndash;78.
| first = Saunders
* Callaghan, Catherine A. (1991). Utian and the Swadesh list. In J. E. Redden (Ed.), ''Papers for the American Indian language conference, held at the University of California, Santa Cruz, July and August, 1991'' (pp.&nbsp;218–237). Occasional papers on linguistics (No. 16). Carbondale: Department of Linguistics, Southern Illinois University.
| last = Mac Lane
* Campbell, Lyle. (1998). ''Historical Linguistics; An Introduction'' [Chapter 6.5]. Edinburgh: Edinburgh University Press. ISBN 0-7486-0775-7.
|author2=[[Garrett Birkhoff]]  
* Chretien, Douglas (1962). The Mathematical Models of Glottochronology.  ''Language'' 38, 11&ndash;37.
| title = Algebra
* Crowley, Terry (1997). An introduction to historical linguistics. 3rd ed. Auckland: Oxford Univ. Press. pp.&nbsp;171–193.
| publisher = American Mathematical Society
* Dyen, Isidore (1965). "A Lexicostatistical classification of the Austronesian languages." ''International Journal of American Linguistics'', Memoir 19.
| year = 1967}}
* [http://language.psy.auckland.ac.nz/publications/index.php?pub=Gray_and_Atkinson2003Nature Gray, R.D. & Atkinson, Q.D. (2003): "Language-tree divergence times support the Anatolian theory of Indo-European origin." ''Nature'' 426-435-439.]
{{Refend}}
* Gudschinsky, Sarah. (1956). The ABC's of lexicostatistics (glottochronology). ''Word'', ''12'', 175&ndash;210.
* Haarmann, Harald. (1990). "Basic vocabulary and language contacts; the disillusion of glottochronology. In ''Indogermanische Forschungen '' 95:7ff.
* Hockett, Charles F. (1958). ''A course in modern linguistics'' (Chap. 6). New York: Macmillan.
* Hoijer, Harry. (1956). Lexicostatistics: A critique. ''Language'', ''32'', 49&ndash;60.
* Holm, Hans J. (2003). The Proportionality Trap. Or: What is wrong with lexicostatistical Subgrouping.''Indogermanische Forschungen'', ''108'', ''38&ndash;46''.
* Holm, Hans J. (2005). Genealogische Verwandtschaft. Kap. 45 in ''Quantitative Linguistik; ein internationales Handbuch. Herausgegeben von R.Köhler, G. Altmann, R. Piotrowski'', Berlin: Walter de Gruyter.
* Holm, Hans J. (2007). The new Arboretum of Indo-European 'Trees'; Can new algorithms reveal the Phylogeny and even Prehistory of IE?. ''Journal of Quantitative Linguistics'' 14-2:167&ndash;214
* Hymes, Dell H. (1960). Lexicostatistics so far. ''Current Anthropology'', ''1'' (1), 3&ndash;44.
* [[John McWhorter|McWhorter, John]]. (2001). ''The power of Babel''. New York: Freeman. ISBN 978-0-7167-4473-3.
* Nettle, Daniel. (1999). Linguistic diversity of the Americas can be reconciled with a recent colonization. in ''PNAS'' 96(6):3325&ndash;9.
*Sankoff, David (1970). "On the Rate of Replacement of Word-Meaning Relationships." ''Language'' 46.564&ndash;569.
* Sjoberg, Andree; & Sjoberg, Gideon. (1956). Problems in glottochronology. ''American Anthropologist'', ''58'' (2), 296&ndash;308.
* Starostin, Sergei. Methodology Of Long-Range Comparison. 2002. [http://starling.rinet.ru/Texts/method.pdf pdf]
* Thomason, Sarah Grey, and Kaufman, Terrence. (1988). ''Language Contact, Creolization, and Genetic Linguistics''. Berkeley: University of California Press.
* Tischler, Johann, 1973. Glottochronologie und Lexikostatistik [Innsbrucker Beiträge zur Sprachwissenschaft 11]; Innsbruck.
*Wittmann, Henri (1969). "A lexico-statistic inquiry into the diachrony of Hittite." ''Indogermanische Forschungen'' 74.1&ndash;10.[http://www.nou-la.org/ling/1969a-lexstatHitt.pdf]
*Wittmann, Henri (1973). "The lexicostatistical classification of the French-based Creole languages." ''Lexicostatistics in genetic linguistics: Proceedings of the Yale conference, April 3&ndash;4, 1971'', dir. Isidore Dyen, 89&ndash;99. La Haye: Mouton.[http://www.nou-la.org/ling/1973f-lexstatFC.pdf]
* [[George Kingsley Zipf|Zipf, George K.]] (1965). ''The Psychobiology of Language: an Introduction to Dynamic Philology.'' Cambridge, MA: M.I.T.Press.
 
==External links==
*[[wikt:Swadesh list|Swadesh list]] in Wiktionary.
* [http://linguistlist.org/issues/5/5-1168.html Discussion with some statistics]
* [http://www.specgram.com/CLIV.1/08.phlogiston.cartoon.jiu.html A simplified explanation of the difference between glottochronology and lexicostatistics.]
* [http://www.elinguistics.net/ Queryable experiment: quantification of the genetic proximity between 110 languages - with trees and discussion]
{{Chronology}}


[[Category:Historical linguistics]]
{{DEFAULTSORT:Equality (Mathematics)}}
[[Category:American inventions]]
[[Category:Mathematical logic]]
[[Category:Language comparison]]
[[Category:Mathematical relations]]
[[Category:Quantitative linguistics]]
[[Category:Elementary arithmetic]]
[[Category:Statistical natural language processing]]

Revision as of 01:14, 10 August 2014

30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí. In mathematics equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign".

Etymology

The etymology of the word is from the Latin aequālis (“equal”, “like”, “comparable”, “similar”) from aequus (“equal”, “level”, “fair”, “just”).

Types of equalities

Identities

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. When A and B may be viewed as functions of some variables, then A = B means that A and B define the same function. Such an equality of functions is sometimes called an identity. An example is (x + 1)2 = x2 + 2x + 1.

Equalities as predicates

When A and B are not fully specified or depend on some variables, equality is a proposition, which may be true for some values and false for some other values. Equality is a binary relation, or, in other words, a two-arguments predicate, which may produce a truth value (false or true) from its arguments. In computer programming, its computation from two expressions is known as comparison.

Congruences

In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties that are considered. This is, in particular the case in geometry, where two geometric shapes are said equal when one may be moved to coincide with the other. The word congruence is also used for this kind of equality.

Equations

An equation is the problem of finding values of some variables, called unknowns, for which the specified equality is true. Equation may also refer to an equality relation that is satisfied only for the values of the variables that one is interested on. For example x2 + y2 = 1 is the equation of the unit circle. There is no standard notation that distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriate interpretation from the semantic of expressions and the context.

Equivalence relations

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set: those binary relations that are reflexive, symmetric, and transitive. The identity relation is an equivalence relation. Conversely, let R be an equivalence relation, and let us denote by xR the equivalence class of x, consisting of all elements z such that x R z. Then the relation x R y is equivalent with the equality xR = yR. It follows that equality is the smallest equivalence relation on any set S, in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element).

Logical formalizations of equality

There are several formalizations of the notion of equality in mathematical logic, usually by means of axioms, such as the first few Peano axioms, or the axiom of extensionality in ZF set theory). There are also some logic systems that do not have any notion of equality. This reflects the undecidability of the equality of two real numbers defined by formulas involving the integers, the basic arithmetic operations, the logarithm and the exponential function. In other words, there cannot exist any algorithm for deciding such an equality.

Logical formulations

Equality is always defined such that things that are equal have all and only the same properties. Some peopleTemplate:Who define equality as congruence. Often equality is just defined as identity.

A stronger sense of equality is obtained if some form of Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same properties. Formally:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original.

Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems. If a=b, then a can replace b and b can replace a.

Some basic logical properties of equality

The substitution property states:

  • For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if both sides make sense, i.e. are well-formed).

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functional predicate).

Some specific examples of this are:

  • For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);
  • For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);
  • For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);
  • For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states:

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step.

The symmetric property states:

  • For any quantities a and b, if a = b, then b = a.

The transitive property states:

  • For any quantities a, b, and c, if a = b and b = c, then a = c.

The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined, is not transitive (it may seem so at first sight, but many small differences can add up to something big). However, equality almost everywhere is transitive.

Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitution and reflexive properties are assumed instead.

Relation with equivalence and isomorphism

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In some contexts, equality is sharply distinguished from equivalence or isomorphism.[1] For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions and are distinct as fractions, as different strings of symbols, but they "represent" the same rational number, the same point on a number line. This distinction gives rise to the notion of a quotient set.

Similarly, the sets

and

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements, and thus isomorphic, meaning that there is a bijection between them, for example

However, there are other choices of isomorphism, such as

and these sets cannot be identified without making such a choice – any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism, is of fundamental importance in category theory, and is one motivation for the development of category theory.

See also

References

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