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Word-formation in English today

Generalities

All forms of human activity tend to give rise to a specialised vocabulary which participants use in conjunction with the general vocabulary of whatever language they happen to speak. The special language of mathematics contains technical terms and symbols—for the latter see Earliest Uses of Various Mathematical Symbols—and there is also an unofficial language of slang expressions. 

Mathematics does not have millions of technical terms like chemistry or biology but even so the number of terms in use is very great. General purpose mathematical dictionaries—even large ones—usually cover only a small fraction of them. The ISI Multilingual Glossary of Statistical Terms has around 5000 entries and it does not include all the statistical terms in use. (There are around 500 statistical terms in Earliest Uses.)

In some sciences there are central bodies responsible for creating or for approving terms, e.g. the IUPAC, as in some countries there are bodies responsible for the condition of the language, e.g. the Académie française. In mathematics word-formation is an individual initiative and it is up to other individuals whether they adopt the new word. Getting a new term established can be quite a process. In E Pluribus Boojum the physicist David Mermin recalls the 5 years he spent on the term boojum.

Word-formation may be open to all but only a few are successful. Much of our language of geometry comes from a single known source, Euclid's Elements. In the C19 J. J. Sylvester wrote, “Perhaps I may without immodesty lay claim to the appellation of Mathematical Adam, as I believe that I have given more names (passed into general circulation) of the creatures of mathematical reason than all the other mathematicians of the age combined.” (quoted in Parshall). Karl Pearson, Ronald Fisher and John Tukey were responsible for many statistical terms, as a search for their names on Earliest Uses will show. John Conway is another word-smith with a taste for the flamboyant.  

Earliest Uses records the births of living terms but terms also die. Old literature and old dictionaries are full of words no longer used. Some dead words are remembered because they once mattered—a good example is FLUXION—or for their association with great names—see the senses of MODULUS associated with De Moivre and Legendre. Some are remembered for their curiosity value like RADIOGRAM, an example of a still-birth—a word announced but never subsequently used. Conway’s UNLESSS may be another example.

Do’s but mostly don’ts in word-formation

Word-formation in mathematics appears not to have been studied systematically either by mathematicians or linguists. Words do not make themselves and the process of word-formation is often quite unpredictable. There may be a gap for a word and the gap stays unfilled: the word RADIUS filled a gap but mathematicians managed for centuries without a word additional to DIAMETER. Once it is decided to fill a gap there is usually a choice of term: several acceptable words may suggest themselves and it may seem a matter of chance which is adopted; RANDOM VARIABLE was a case where chance did decide the issue. Unacceptable words will also suggest themselves. Once the choice is made there is a powerful tendency to stick with it—even if the term is unacceptable!

Guides on how to write mathematics sometimes have advice on how words should be made—more often how they should not be made. The advice—sometimes contradictory—usually reflects dissatisfaction with the existing language. 

·        Do not introduce new terms is the first principle of Halmos (p. 40)

(1) Avoid technical terms, and especially the creation of new ones, wherever possible. (2) Think hard about the new ones you must create; consult Roget; and make them as appropriate as possible.

One of Sylvester’s friends told him, “terminology … is your strength as well as your weakness. You have too much of propensity to create new words. It would be well for you to forget about Greek.” (quoted in Parshall). Over appropriateness opinions often differ. Some now familiar terms originally seemed totally wrong: see the objections to the terms MATRIX and STATISTIC.

·       Do not try to undo past mistakes warns Boas (p. 728)

If you think you can invent better words than those that are currently in use, you are undoubtedly right. However, you are rather unlikely to get many people except your own students to accept your terminology; and it is unkind to make it hard for your students to understand anyone else's writing.

There is a saying “possession is nine tenths of law.”

·       You must try to undo past mistakes says Steenrod (p. 6), if you can

An author of a research monograph that is first in its area has the opportunity and the obligation to replace poor by good terminology. … The name that a research worker attaches to a new concept is usually chosen before the scope and thrust of the concept is fully understood, so his choice may be an unhappy one.

Naturally opinions differ over whether the moment has passed.

·       Boas (p. 728) complains about ambiguity (more politely, polysemy)

if you must create new words, you can at least take the trouble to verify that they are not already in use with different meanings. It has not helped communication that DISTRIBUTION now means different things in probability and in functional analysis.

There are so many different mathematical communities that polysemy is probably the rule not the exception: see CHARACTERISTIC FUNCTION, MODULUS, NORM, etc.

·       The existence of different communities and different histories also explains the proliferation of synonyms, different names for the same thing: see e.g. MODULUS and ABSOLUTE VALUE, CHARACTERISTIC FUNCTION and INDICATOR FUNCTION, and the alternatives to the EIGEN terms.

Creating mathematical terms

Given that an existing technical term cannot be adapted or a collection of existing terms cannot be combined into a new phrase, how is a new term to be created?

Mathematical words are created in much the same way and undergo the same processes of change as words in ‘ordinary’ language. Algeo provides a taxonomy of word making. In Words, Words, Words Crystal traces the word “nice” back through two thousand years of evolution. Over the same period the word ANALYSIS has undergone similarly dramatic changes.

Algeo & Pyles ch. 10-12 describe the processes by which words are formed in ‘ordinary’ English. The same processes are involved in forming ordinary words in all languages—see Durkin—and in forming technical terms:

• Almost all words are formed from existing words, i.e. the new word has an etymon. An instance (unique?) of a mathematical word without etyma is GOOGOL. The OED gives a few other words without a root: some, like the science fiction word dalek, are intended to sound alien. That’s not the usual intention with mathematical terms—whatever students may think,

• Some mathematical words are existing non-technical words whose meaning has undergone some kind of metaphorical extension. See e.g. the entries CHAIN, WINDOW and FUZZY and the discussion of bootstrap below. Hersh discusses the relationship between mathematical language and ordinary languages. This is an issue that particularly worries teachers of mathematics.

• Other terms come from ordinary, non-technical language by specialising the meaning of an existing word: see e.g. ESTIMATION and SET. In AMENABLE there is an element of punning.

• The stock of ordinary non-technical words does not have to be limited to the ‘home’ language and these last two processes can be applied to words from foreign languages especially when these languages enjoy high prestige. However borrowing whole words is much less common than borrowing roots.

• Many terms are constructed by modifying or combining existing words or roots of existing words, e.g., IDEMPOTENT, EIGENSTATE and KURTOSIS. The elements of these particular technical English words are non-mathematical words in Latin/English, German/English and classical Greek respectively. When these particular words were formed potent and state were fully domesticated in English but they had come into the language at an earlier date.

• From new roots derivatives can be generated potentially without limit: E.g., TOPOLOGY combines with ALGEBRA, GROUP and SPACE to produce TOPOLOGICAL ALGEBRA, TOPOLOGICAL GROUP and TOPOLOGICAL SPACE.

• There are more artful ways of making new terms from old, such as the sly back-formation of statistic out of STATISTICS or the abbreviation of binary digit to BIT and directed graph to DIGRAPH.

• Many mathematical terms—from ABBE-HELMERT CRITERION to ZORN'S LEMMA—incorporate the names of people. The practice is called EPONYMY.

• A few terms are really nicknames e.g. PONS ASINORUM and THEOREMA EGREGIUM.  FERMAT'S LITTLE THEOREM combines eponymy and nickname.

• Many words are borrowed from other languages, entering with the idea the word expresses.  See below.

One peculiarity of this specialised language is the number of technical terms that are borrowed from the general language with some shading or alteration in meaning. In their Introduction to Topology T. W. Gamelin & R. E. Greene (1983, p. 80) state the theorem, “A compact Hausdorff space is normal.” All the words—apart from Hausdorff—are everyday words found in the smallest English dictionary, yet only two words have their everyday meaning, “a” and “is”. The rest are technical terms: the everyday words COMPACT  and NORMAL have been given mathematical meanings, while HAUSDORFF SPACE is an expression concocted by mathematicians.

Evidently there is little system in mathematical naming. The entry on PROBABILITY DISTRIBUTIONS, NAMES FOR shows how several principles can be at work in the same field. Schwartzman shows how 1500 or so mathematical words have been formed out of components from ordinary language. Occasionally there is movement in the other direction, when a mathematical term is used figuratively: ECCENTRIC, PARALLEL and cipher (see ZERO) are instances and so is the fashionable pre-fix cyber- from  CYBERNETICS. See also Charles Wells on Names.

Mathematicians use technical terms like those discussed above when they are doing mathematics and they also use ordinary language words in their ordinary senses. In addition there are terms used in discussing mathematics such as HARD AND SOFT MATHEMATICS, HANDWAVING and PATHOLOGICAL which are part of the occupational slang of mathematics.

References

• Karen Hunger Parshall James Joseph Sylvester, Johns Hopkins. 2006. Amazon. (See index entry “Sylvester, James Joseph, neologism and.”
• Reuben Hersh Math Lingo vs. Plain English: Double Entendre, American Mathematical Monthly, (1997), 48-51. Excerpt and Follow-up.
• Roget’s Thesaurus, the Wikipedia artcle on the famous reference book.
• Norman E. Steenrod, Paul R. Halmos, Menahem M. Schiffer, and Jean A. Dieudonné. How to Write Mathematics. American Mathematical Society, Providence, RI, USA, 1973. Amazon
• P. R. Halmos  How to write mathematics, L’Enseignement Mathématique, Vol.16 (1970).
• Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts Mathematical Writing.
• R. P. Boas Can We Make Mathematics Intelligible? American Mathematical Monthly, (1981), 727-731.
• Nicholas J. Higham Handbook of Writing for the Mathematical Sciences, SIAM, 1998 Second Edition. homepage. The large bibliography is available on-line.
• Bernard Ycart How to Write Mathematics has useful links.
• Charles Wells The Language of Math.
• Wikipedia Mathematical jargon discusses some of the occupational slang used by mathematicians.

The following do not treat mathematical words yet they offer insights into the way mathematical words are formed and how they change. 

• Richard Nordquist Introduction to Etymology: Word Histories.
• David Crystal Words, Words, Words, Oxford University Press. 2006. Amazon
• Ronald Carter Vocabulary, second edition Routledge 1998.   Amazon
• John Algeo The Taxonomy of Word Making, Word, 29, (1978), 122-31.
• John Algeo Fifty Years among New Words, Cambridge 1993. Amazon.

The use of words in mathematics is taken very seriously by those involved in teaching mathematics. There are societies and the journals include the British Society for Research into Learning Mathematics and Journal of Mathematical Behavior.

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An example: bootstrap

As an example of word-formation consider a modern term coined in English, BOOTSTRAP. For this term information is plentiful and the origins story is straightforward. Sources of information are described below.

BOOTSTRAP is a recent (1979) but already well-established term in statistics. The word had been in ordinary non-mathematical English since the C18 meaning “a strap sewn on to a boot to help in pulling it on” (OED). The technical statistical meaning (a method for making inferences based on simulation from a process derived from the sample itself) draws on the expression “to pull oneself up by one’s bootstraps.” If the idiom had been, “to pull oneself up by one’s (neck) tie,” we might have the term tie.    

Bootstrap is an unusually imaginative term but it is carefully chosen term and truer to Halmos’s advice, “think hard,” than most. It would also satisfy Boas for the term was not already in use with a different meaning. The expression to BOOT a computer probably has the same root but the meanings are too dissimilar to cause confusion.

Bootstrap illustrates some points of general interest, including:

• It is a new term for a new technique and as such is a typical addition to the lexicon. It is easier to live with an inadequate old term than overturn an established usage.
• The suggestion of flippancy, or, at least, informality makes the term very much of its time; cf. the contemporary coinages FUZZY and HAIRY BALL THEOREM. In earlier days dignity was expected: Karl Pearson’s (1857-1936) speciality was the Greek-based neologism like HISTOGRAM and HETEROSCEDASTICITY; Ronald Fisher (1890-1962) liked to take existing English words (usually with Latin roots) and stretch their meaning, such as VARIANCE and SUFFICIENT.  Bootstrap is more in the style of John Tukey (1915-2000) who liked to make technical terms from everyday words, e.g. JACKKNIFE and WINDOW.
• Bootstrap arrived as a singleton but there are also mass creations. A family of new words may be needed, or an old terminology may need reform. Pearson’s term KURTOSIS was part of a successful mass creation, while RADIOGRAM belonged to an abortive one.

We cannot usually hear the voice of the namers of the remote past but we can of more recent ones. Some have even described the process of finding the right word, especially when their choice is not an obvious one: see the entries FRACTAL, FUZZY, GYROID, PARAMETER and THRACKLE.

Naming is not just the business of the original namer. Others have to adopt the term and keep it going. Every time a word is used there is an opportunity to re-name it. There is a powerful tendency to respect the innovation—provided the concept is useful—and thus Sylvester’s term MATRIX survived the criticism of Lewis Carroll who thought block would be better. Yet terms do get replaced.  When a term is translated into another language there is an opportunity to deviate more or less from the original author’s intentions: EXHAUSTIVE ESTIMATION an English translation of a French translation of an English original presented two opportunities for creative translation.

Bootstrap story

The origins story of bootstrap is exceptionally tidy.

• The first use can be dated to a specific publication by a specific author which announced the idea and with it the term. 

More often the origins story is complex and/or only partially known

•  Bootstrap was not a new word but the new meaning was clearly announced. Sometimes terms slip into use or existing terms slip into new meanings.

• The word bootstrap (in this new sense) arrived with the idea which was ‘fully formed.’  Sometimes the creator author tries out a number of words before settling on the ‘right’ one; see e.g. SET and MAXIMUM LIKELIHOOD. It is often not clear whether the tryouts have exactly the same meaning as the final term.

• The same idea can arrive in different minds and be expressed in different words. See the pairs DIFFERENTIAL and FLUXION or SEMI-INVARIANT and CUMULANT. To what extent these terms represent the same idea is another difficult matter. There were no parallel creations of bootstrap.

• There can also simultaneous coinings or re-coinings, where the same term is applied to the same thing by more than one person independently: see e.g. the term NORMAL. For ‘headline’ words the phenomenon is unusual but with derived terms it is probably the norm.

• With old words our first sight of them is not necessarily a “first use.” What we see depends on the thoroughness of the search and ultimately on the documents that survive. We first see naught and cipher (see under ZERO) when a non-mathematical author makes a (non-mathematical) point by means of a mathematical expression familiar to his readers—a use that cannot have been a first use. Much of our knowledge of the vocabulary of Middle English (C14) comes from the writing of Geoffrey Chaucer: see e.g. ARGUMENT, DEGREE and EPICYCLE. Sometimes Chaucer was writing science but more often he was writing about people doing science or mathematics and it is not clear whether he was using their words or making up his own.

In its short history bootstrap has not been subject to significant semantic drift.

• Some words like TRIANGLE have preserved their original meaning across centuries.

• Most words have been subject to greater or less movement: in the course of centuries ANALYSIS, CONTINUITY and SPACE have been subject to huge changes but OPEN SET, HILBERT SPACE, MONTE CARLO and MULTICOLLINEARITY have experienced significant change in the course of a few years.

Of course further investigation of bootstrap might uncover earlier uses in pre-publication writing and in recorded conversations and possibly even show that other terms were tried and discarded.

As well as going back to before the date of publication it is possible to go forwards and notice that the term is not used in exactly the same way now as it was in 1979. A historian writing in 2020 (or even now) would be interested in both the backward and forward stories.  

Bootstrap into other languages

As translation into English has been so important building up the English mathematical lexicon (see below) it is interesting to see how bootstrap has gone from English into other languages.

It is a very tricky word and an obvious expedient—in the short run at least—is to take the word or the phrase bootstrap method into the language: ALGEBRA is a remnant of such a phrase. English has often operated in this way, borrowing wholesale from Latin, French and Greek.

The translations of bootstrap method given in the ISI Multilingual Glossary illustrate a variety of more permanent arrangements.

· Literally translate both words: the Afrikaans skoenlusmetode. OPEN SET came into English after this fashion.

· Adopt bootstrap as a loan word and translate method: the Italian metode bootstrap. EIGENVALUE is an English word formed in this way.

·       Convey something of the nature of the method using ‘local’ words: the Spanish método autosuficiente.

·       Convey something of the allusion to impossible self-locomotion in the expression “to pull oneself up by one’s bootstraps”: the French la méthode de Cyrano (Accromath Été-Automne 2008 p. 5) or the Chinese 自 助 法. In the translations the essence of the original joke—you can't pull yourself up by your own bootstraps—is retained but the accidentals are lost.

·       A language may support more than one translation. French also has méthode de bootstrap.

·       No language appears to have taken the EPONYMY option and called the method after its inventor, méthode d’Efron, say. In this particular case one of the main disadvantages of eponymy—the conflict between different local traditions over who should be honoured—is avoided: for such conflict see the entries PASCAL’S TRIANGLE GREGORY'S SERIES or CAUCHY-SCHWARZ INEQUALITY.

English writers of the Late Middle Ages and Renaissance had the same options when they translated Latin words. See Graddol (176).

Because of the idiom in which it is embedded, bootstrap is an especially tricky word but translation is always tricky.

• Words taken from ordinary language can rarely be translated automatically. The English word SET and its French and German ‘equivalents’ all have several other meanings and the bundles of meanings vary from language to language; the words are not generally interchangeable.   

• It is different with the specially constructed single-meaning technical term, such as ENTROPY. The translations show the algorithm in action: take the Greek root and apply the local affix.

Of course the translator is just one more person who makes choices: choices were made by the person who originally coined the term and the people who sanctioned its use by adopting it. Occasionally something is gained in translation: the most famous example is the mis-translation WITCH OF AGNESI. For another famous mistranslation see SINE.

Words have come into English as singletons and also when an entire work—a book or an article—has been translated. Billingsley’s C16 translation from the Greek of Euclid below may have brought more words into English mathematics than any other single work. It is mentioned in over 400 entries in the OED although not all of these involve mathematical words and not all are first uses. Some other translations appear repeatedly in Earliest Uses. They make a curious collection: De Proprietatibus Rerum by Bartholomæus Anglicus a C13 Latin encyclopaedia translated by John Trevisa at the end of the C14 brought many Latin terms into English—see ANGLE, DIAMETER, POINT and TRIANGLE. Lacroix’s Traité du calcul différentiel et integral written at the end of the C18 and translated in the early C19 brought many Continental calculus terms into English—see DIFFERENTIAL CALCULUS, VARIABLE and POLAR COORDINATES. Many entire works have been translated into English but most often they have made their impact on the language before the translation has appeared; a C20 example is Bourbaki’s Eléments de mathématique and the terms INJECTION, SURJECTION & BIJECTION.

References

The problems of finding the right word for topological concepts in different languages is discussed by 

• B. Barton, Lichtenberk, F. & Reilly, I. The Language of Topology: A Turkish Case Study, Applied General Topology, 6(2), (2005), 107-117. pdf

Translation is of such cultural and commercial significance that it has given rise to an academic subject, Translation Studies. The following books give an idea of the range of the subject, although neither discusses mathematics. The textbook by Baker discusses the choice of word as “equivalence at the word level.”

• Daniel Weissbort, Astradur Eysteinsson Translation - Theory and Practice. A Historical Reader. Oxford University Press. 2006. Amazon.
• Mona Baker In Other Words: A Coursebook on Translation, Routledge. 1992. Amazon

Modern translators of old texts face different and very difficult problems. Three hundred years ago when Bernoulli wrote his Ars Conjectandi the (i.e. our) language of PROBABILITY was in its early stages of formation. Sylla prefaces her translation by describing the problem of choosing the right words.

• Edith Dudley Sylla (Translator) Jacob Bernoulli The Art of Conjecturing, together with Letter to a Friend on Sets in Court Tennis, Johns Hopkins University Press. 2006. Amazon.

Mathematics in English: behind the table  

Before 1300—off the table

In 1300 three languages were in use in England—French, Latin and English (in the form of Early English). There was not much demand for English mathematical words.

• The known story of English begins around 400 AD when parts of England were occupied by settlers from the regions that are now northwest Germany and the Northern Netherlands.

·        Old English or Anglo-Saxon was a diverse group of dialects, reflecting the varied origins of these invaders.

·        The conversion to Christianity around 700 made Latin the language of religion.

·        The Christian missionaries introduced the Latin alphabet (plus a few additional letters, such as þ (thorn) that have now disappeared) and a literary tradition developed with Old English works written in the new alphabet.

·        Viking settlers whose language was Old Norse (Wikipedia Viking) English also left their mark on the language.

·        Norman French became the language of government after the conquest of 1066.

• The first named native English speakers in the annals of mathematics appear in the C12—writing in Latin, see below.

• Mathematics was taught in the universities which had appeared in the C13. Across Europe Latin was the language of the universities.

For this period see the MacTutor pages The teaching of mathematics in The Dark Ages and The teaching of mathematics in Britain in the Mediaeval Ages.

1300-1700

By the middle of the C14 French was no longer an official language in England but it remained important because ideas from the rest of the world generally came through France. By the C14 English was the language of lowland Scotland. With the Reformation of the C16 Latin ceased to be the language of religion. Scholars wrote in Latin but a market developed for mathematical works in English—especially after the introduction of printing. Many words were borrowed from Greek and Latin. Towards the end of the period a brilliant group of mathematicians appeared of whom the best known is Newton.

• By the early C14 manuals of arithmetic written in English were appearing. Surviving documents (manuscripts) are scarce: one of these, The crafte of nombrynge, is described in the entry ADDITION; see also DIGIT and DIVISION.

• The language of the C14 is called Middle English: see Crystal. See the entries on ANGLE and TRIANGLE for samples of English text from around 1400. There are some letters no longer used, e.g. þ (thorn), and the spelling is somewhat different but with some imagination the texts can be understood.

• Following the introduction of printing (in the late C15) there was a spectacular expansion in mathematical works in English. The C16 author Robert Recorde favoured the creation of an English mathematical vocabulary but the new English words of the C16 were mainly classical loans like HEXAGON or EQUILATERAL; see Howson and Greek below. Recorde’s writings were the first English writings to have an impact abroad; he is remembered for a symbol—the equals sign (see Signs of Relation and (Katz 355).

·        Works of high science written for publication were still written in Latin. The most famous, Newton’s Philosophiae Naturalis Principia Mathematica, appeared in 1687; an English translation followed in 1729. The scholars wrote in Latin and so the first English use of a term was often due, not to a scholar, but to a ‘tradesman’—see the entries SINE and COSINE. However in the late C17 English mathematicians were writing to each other in English and publishing translations of their Latin works: for instances, see the entries SERIES and COMBINATION.

See the MacTutor pages The teaching of mathematics in the Renaissance and The teaching of mathematics in Britain in the Seventeenth Century. 

1700 to the present

In this period—for the first time—important original work appears in English. Mathematics in English is not the same as mathematics done by the English for Scottish and Irish mathematicians also wrote in English and the British Empire took English all over the world. In the C20 the centre of gravity of mathematics in English shifted to the United States of America. In the early part of this period the most important mathematical work in the European tradition was being done on the Continent of Europe. Before the C20 the only mathematicians who wrote in English lived in English-speaking countries. Today English is used by anybody anywhere.

• Since the mid-C18 Anglophone mathematicians have published their main scientific work in English. However the period 1750-1850 was not noted for original work in English.

• In the C19 the reputation of mathematics in English was founded on applied mathematics/mathematical physics. The leading applied mathematicians were not English but Scottish or Irish: for examples of their activity see the entries NABLA STOKES’ THEOREM and VECTOR. Benjamin Peirce’s term IDEMPOTENT was an early American contribution to the language of pure mathematics.

• In the period 1750-1950 most of the new terms in pure mathematics were created in French and later in German. In the early C19 English writers took up calculus in the Continental manner; see the entry DIFFERENTIAL CALCULUS, DERIVATIVE. At the end of the century writers of the first English books on COMPLEX ANALYSIS or GROUP THEORY had to invent an English vocabulary. There is an appendix in Burnside’s Theory of Groups of Finite Order (1897) here showing how his terms correspond to the French and German originals and Harkness & Morley’s A Treatise on the Theory of Functions (1893) has a multi-lingual glossary (p. 501).

• As the table shows, as we approach the present day, the time elapsing between the original coinage and the English translation gets shorter and the proportion of terms originally coined in English grows.

• Modern history is reflected in three of the statistical terms coined in English “around 1950.” (EMPIRICAL) BAYES and ADMISSIBLE were coined in the United States, the first by a native speaker of English and the second by a refugee from the German-speaking world. AUTOREGRESSION was coined in Sweden by a Swede writing in English and it illustrates the rise of English as an international language.

I have been referring to English mathematical words without questioning the notion of Englishness.  Linguists are familiar with the varieties of English around the world and their differences in vocabulary, spelling, pronunciation, etc.; see ch. 9 of Algeo and Pyles. Differences in the mathematical languages of the varieties of English seem trivial.

American English and British English are the two most important national forms and Earliest Uses entries indicate some differences. Billion once had a different meaning in the two varieties of English (10¹² in British English and 10⁹ in American English) but now British English generally conforms to American; see MILLION. Different abbreviations for MATHEMATICS continue to be used, viz. Math and Maths. Modern university mathematics is very international but school mathematics is less so and differences in usage may persist: see the entry TRAPEZIUM and TRAPEZOID. When everyday words are given a mathematical sense some of the zest is lost if the word is not an everyday one in the other variety; in British English the word JACKKNIFE is not an everyday word and has currency only as a technical term in Statistics. A difference in pronunciation mathematicians will notice is that of the Greek letters Beta, Zeta, etc.

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The contribution of Old English       

The English, Latin, French, Greek and German languages belong to the family of Indo-European languages (Wikipedia) and share a common ancestry. Old English is a Germanic language; specifically it belongs to the Anglo-Frisian branch of the West Germanic line. See Crystal and Algeo & Pyles.

Only a minority of the words of Modern English are derived from Old English. Crystal p. divides the vocabulary of everyday modern English so.

• Old English provides the core of the most basic words, not only the grammatical words—articles, prepositions and pronouns—but such content words as mother and brother,

• 50-70% of the common words come from French or Latin.

The overlay of French and Latin words is so heavy that modern English sometimes has three ways of expressing the same basic idea: Crystal (48 & 124) illustrates the possibility with the triples rise/mount/ascend and kingly/royal/regal. “The Old English word is usually the more popular one, with the French more literary, and the Latin more learned.” 

There was a literary tradition in Old English—the Anglo-Saxon Chronicle of the C9 has been described as the first continuous history of any Western people in their own language. But there was no mathematical literature and therefore no specialised mathematical vocabulary for Latin and French words to augment or to displace.

A form of the popular/learned division applies to mathematical words. Very few of Schwartzman’s 1500 mathematical words are labelled “native English.” There is a rare run of such words: on, one, onto and open. These words illustrate some general points about the Old English presence in the modern language.

• These are among the most frequently used words in the language and are learnt well before a child goes to school and ‘learns mathematics’.

• The ‘school’ words for discussing numbers and space, including NUMBER and SPACE, are more recent additions to English and are usually adaptations of foreign words—from Latin or French.

• One and the other Old English number words have survived. But they were not used as components in the words created as equivalents to the classical Greek terms when the latter were put into English in the C16. Instead the Greek or Latin forms of the numbers were adopted: thus TRIANGLE, QUADRILATERAL and PENTAGON. Well into the C19 the classical languages also provided the ingredients when words were sought for new ‘numerical’ concepts–see e.g. QUATERNION and OCTONION.

• More recently the full resources of the language have been used in forming mathematical terms and the Old English words have figured in such constructions as OPEN SET and ONTO FUNCTION from the C20.  The first is a translation from the similarly colloquial expression in French.

Behind Schwartzman’s label “native English” is often a complicated story in which an ancestor word has been greatly modified. The OED entry for “one” relates how the word “an” used by writers of some version of Old English in the C9 was transformed into the word “one” which became the standard but not until the C16: Harper gives the basic story. Harper also treats the other number words.

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The contribution of Latin       

Latin (Wikipedia), as the language of the Roman Empire, had a profound influence on all the languages of Western Europe. It had most influence on the languages that are its direct descendants, the Romance languages (Wikipedia). One of these, French, had a very strong influence on English.

In all parts of Western Europe Latin was used in conjunction with the local language for religious, political and scholarly purposes. For aspects of medieval learning see QUADRIVIUM and LOGIC.  

If we count by years, Latin has been the most important language of mathematics in the European tradition. Yet as a language of mathematics, Latin presents a paradox:

• The native speakers of Latin, the classical Romans and the people of the Western Empire, are notorious as non-contributors to mathematics. Katz gives them one page in his account of the “Final Chapters of Greek Mathematics.”

• Yet for more than 1000 years non-native speakers published their mathematics in Latin and in that time Latin was the language of Western mathematics.

Latin only became a dead language in mathematics at the end of the C19. Before then new works were published in Latin and new words were created or old ones given new meanings.

·        In the time of the Roman Empire some Greek mathematical terms were taken into Latin. 

·        Later Latin translations of ancient Greek and contemporary Arabic works made those works became known in Western Europe. The first Englishmen recorded in histories of mathematics were translators of Arabic into Latin: Adelard of Bath (1075-1164) and Robert of Chester (fl. 1144–1150). For traces of their activity see the entries ALGEBRA , ALGORITHM FRACTION and SINE. (Katz 244 & 290-1)

·        Original work came later and it too was written in Latin. The Italian mathematicians of the medieval and early Renaissance periods, Fibonacci, Tartlaglia and Cardano, all wrote in Latin. One English contributor was Thomas Bradwardine (ca. 1295-1349); see the entry RATIO (Katz, 315).

• Latin continued to be the essential language in Western Europe and most of the new mathematics of the C17 and early C18 centuries was published in Latin. Leibniz, Huygens, the Bernoullis and Euler all wrote in Latin. See the entries CALCULUS, DIFFERENTIAL CALCULUS, MAXIMUM and PROBABILITY.

• The Italians and the French were the first to make the break with Latin: Galileo’s Dialogo sopra i due massimi sistemi del mondo appeared in 1632 and Descartes’ La Géométrie in 1637.

• In England Latin long continued to be the language of learning. The Artis logicae compendium a treatise on logic by Henry Aldrich was published in 1691 and was the main text in England for 150 years. It was translated into English only at the beginning of the C19.

• The last piece in Latin to appear in the English science journal Philosophical Transactions of the Royal Society (p. 477) was a communication from a Swedish astronomer in 1784.

• Until well into the C19 German authors published in Latin.   A glance at the dates of works with titles beginning with De (= “On”) on DML shows how long Latin lasted as an important medium.

The death of Latin

By the end of the C19 Latin had ceased to be a living language for mathematicians and ceased to be the international language for mathematics.

• At the beginning of the C20 English French and German were the accepted international languages. The Proceedings of the Imperial Academy of Japan founded in 1912 took articles in these languages (and Japanese). The international journal Acta Mathematica (published in Sweden) took articles in French and German. The journal Mathematische Annalen published articles in English and French as well as in German. Italian work on TENSORS reached a German audience through the medium of French.

• Some mathematicians looked to artificial languages (Esperanto was the best known) to succeed Latin. One of these was Peano who also tried to revive Latin in the form of a simplified Latino sine flexione.

• A century after the demise of the old Latin, English rather than an invented language has become the new Latin.

Gray describes the linguistic situation at the turn of the C20.

Latin mathematics and English words

The table above illustrates how from ADDITION and ARGUMENT in the C13 to MODULUS in the C19 large numbers of Latin words passed into English.

The author expressing new ideas in Latin or translating from Greek or Arabic had the usual options

• Use an existing word and stretch its meaning as Gauss did with residuum; see RESIDUE in number theory.

• Create a new Latin word; as ALGEBRA came from Arabic.

In the early centuries the words were created because the author was writing Latin text; later when the mathematician was writing in his own language he might construct a word from Latin roots.

Mathematicians writing in their own language often chose Latin roots for the words they invented. Examples from English-speaking writers include QUATERNION and IDEMPOTENT. The the C20 graph theory terms ARBORESCENCE and ARBORICITY both have Latin roots but the components were long established in English and the first word was already in English. 

Latin references

The first two references are introductory surveys of Latin and its role in Western history. The next one is an on-line resource for beginning students of Latin and Greek.

• Tore Janson A Natural History of Latin, Oxford University Press 2004.  Amazon
• Nicholas Ostler Ad Infinitum: A Biography of Latin, Harper 2007. Amazon
• N. S. Gill Etymology of Geometry Terms
• Barnabas Hughes “Mathematics and Geometry” pp. 348-54 of F. A. C. Mantello, A. G. Rigg (eds.) Medieval Latin: an introduction and bibliographical guide, Catholic University of America Press, 1996. Google Books.

• Perseus classics has two on-line Latin-English dictionaries, Lewis & Short and Lewis.
• Wikipedia List of Latin words with English derivatives
• Wikipedia Latin translations of the 12th century.
• Latin Therapy: Bibliography of Latin Language Resources

• Ostler Empires of the Word chapters 7 and 8.

• Jeremy J. Gray (2002) Languages for mathematics and the language of mathematics in a world of nations, pp. 201-228 of K. Parshall & A. Rice (eds.) Mathematics unbound: the evolution of an international mathematical community, 1800-1945 American and London Mathematical Societies.