Mathematical
Words: Origins and Sources
Notes
for Earliest
Known Uses of Some of the Words of Mathematics
The Earliest Known Uses of Some of the Words of Mathematics pages provide information on the origins of many of the English mathematical words in use today. The story of a particular word may be very complicated. There has been an English word ALGEBRA for nearly 500 years and the word can be traced back another 700 years to an expression in Arabic. The word has been around for a long time, travelled through several languages and experienced great shifts in meaning. There are patterns in the history of particular words and there are also procedures for uncovering that history. These notes describe in general terms the origins of the modern vocabulary of mathematics and the sources of information on which the Earliest Uses entries are based.
_______________________________
Contents
In the Origins section of
these notes
·
There is a
general discussion of word formation followed by an
example—the creation of a modern English term.
·
There is an overview of the origins of the modern English lexicon and a sketch
of the development of mathematical
writing in English.
·
The
contributions of Old English and of Latin,
French, Greek and German
to the English mathematical lexicon are described in more detail.
|
|
Word-formation in
English today 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. 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. 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 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 other languages
and in forming technical terms:
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 and cipher (see ZERO) are
instances. 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 and HANDWAVING 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 journals, e.g. the British
Society for Research into Learning Mathematics and Journal
of Mathematical Behavior. |
|
|
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:
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. The origins story
of bootstrap is exceptionally tidy.
More often the
origins story is complex and/or only partially known
In its short
history bootstrap has not been subject to significant semantic drift.
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 (see here 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.
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 References The problems of
finding the right word for topological concepts in different languages is
discussed by
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.”
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.
|
English Mathematical words
English as a language of mathematics presents a paradox.
The explanation is that
The historical development of a specialised branch of language, such as mathematical language, belongs to the history of mathematics and to the history of language. There are many works on these subjects but very few on their intersection. For the history of mathematics I refer to Katz; for the history of English I refer to Algeo & Pyles, Crystal and Graddol. The Wikipedia: History of the English language is a useful summary.
Works on the history of mathematical
words in English are described below. The history of mathematics in the
English-speaking lands is relevant and so is the history of mathematics
education there. There appear to be no specialised studies of the former, for
the latter I refer to Howson. The
MacTutor series on the history
of mathematical education is also useful.
The following table gives some
information on the origins of a few of the words in use today.
A table like this would work for other
Western European languages but for Chinese, say, a complete reconstruction
would be required.
When and whence for some English mathematical words
|
|
|
|
|
|
|
|
|
|
1300 |
|
Latin 1100 |
|
|
|
|
|
|
1400 |
Latin 1100 Greek 300BC |
Latin 1100 |
|
Latin 1100 Greek 300BC |
|
|
|
|
1500 |
Greek 300BC |
|
Latin 1100 Arabic 800 |
|
|
|
|
|
1600 |
|
Greek 300BC Latin 1100 Arabic
800 Sanskrit 400 |
Latin 1100 Arabic 800 |
Greek 300 BC Greek 300BC Latin 1100 Arabic
800 Sanskrit 500 |
|
|
|
|
1650 |
|
|
Latin 1200 |
Latin 1600 |
|
|
|
|
1700 |
|
|
Latin French 1637 |
|
Latin |
Latin Dutch Latin Dutch |
|
|
1750 |
Latin ? Greek 300 BC |
|
|
|
|
|
|
|
1800 |
|
|
|
|
French |
|
|
|
1825 |
|
|
French |
|
|
French 1810 |
|
|
1850 |
|
Latin 1800 German |
German French 1815 |
|
|
French 1800 |
German |
|
1875 |
|
German |
|
French 1822 |
|
|
|
|
1900 |
German German |
|
|
|
French 1859 French German |
|
French |
|
1925 |
German German |
German 1900 |
German German 1900 German 1890 |
|
French/German |
German German |
French 1892 German 1907 German 1914 |
|
1950 |
|
|
|
|
|
|
French |
|
1975 |
|
|
French |
|
|
|
|
From this table of English words it is
striking
The colour coding indicates the
significant other language of the age.
Simplifying heroically
·
From 1300 to
the late C18 Latin was the important
language of Western European mathematics. In the beginning the main centre was
in
·
From the late
C18 to the late C19 French rivalled
Latin. In the beginning most of the French literature was produced in
·
From the late
C19 to the Second World War German rivalled French—both because so much mathematics was done in
·
Since the