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 bootstrap below. and the discussion of 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. potent and state were fully domesticated in English but they had come into the language at an earlier date. The elements of these particular technical English words a re non-mathematical words in Latin/English, German/English and classical Greek respectively. When these particular words were formedandwere fully domesticated in English but they had come into the language at an earlier date.

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.

See 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

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

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.