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Posted 9 November 2011
Conventions and defaults
default
An interface to a computer program will have many possible choices for the user to make. In most
cases, the interface will use certain choices automatically when the user doesn't specify
them. One says the program defaults to those choices.
[
2.8in] I have spent a lot of time in both Minnesota and Georgia and
the remarks about skiing are based on my own observation. But these
usages are not absolute. Some affluent Georgians (including native Georgians) may refer to snow skiing
as "skiing", for example, and this usage can be intended as a kind of snobbery.
One wonders where the boundary line is. Perhaps people in Kentucky are confused on the issue.
Example
A word processing program may default to justified paragraphs and insert mode, but
allow you to pick ragged right and typeover mode.
This concept is a remarkably useful one in linguistic contexts. There is a sense in
which the word "ski" defaults to snow skiing in Minnesota and to water skiing in Georgia.
Similarly "CSU" defaults to Cleveland State University in northern Ohio and to Colorado
State University in parts of the west.
Default usage may be observed in many situations in mathematical discourse. Some examples from
my own experience:
Example
To algebraists, the "free group" on a set S is non-Abelian. To some topologists, this
phrase means the free Abelian group.
Example
In informal conversation among many analysts, functions are automatically continuous.
Example
"The group Z" usually means the group with Z (the set of integers) as underlying set and addition
as operation. There are of course many other group operations on Z. Indeed, the privileged nature of the
addition operation may be part of a mathematician's schema for Z.
See also theory of functions and number theory.
Remark
This meaning of "default" has only made it into dictionaries in the last ten years.
This usage does not carry a derogatory connotation.
Remark
As far as I can tell, English speaking mathematicians follow this sort of default linguistic behavior in
much the same way as English speakers do in general.
names from other languages
Mathematicians from many countries are mentioned in mathematical discourse, commonly to
give them credit for theorems or to use their names for a type of mathematical object. Two problems
for the student arise: Pronunciation and variant spellings.
Pronunciation
During the twentieth century, it gradually became an almost universal attitude among educated
people in the USA to stigmatize pronunciations of words from common European languages that are not
approximately like the pronunciation in the language they came from, modulo the phonologies of the
other language and English. This did not affect the most commonly-used words. The older practice was to
pronounce a name as if it were English, following the rules of English pronunciation. this shift in attitude
For example, today many mathematicians pronounce "Lagrange" the French way, and others,
including (in my limited observation) most engineers, pronounce it as if it were an English word, so that
the second syllable rhymes with "range". I have heard people who used the second pronunciation
corrected by people who used the first (this happened to me when I was a graduate student), but never
the reverse when Americans are involved.
Forty years ago nearly all Ph.D. students had to show mastery of two foreign languages; this
included pronunciation, although that was not emphasized. Today the language requirements in the USA
are much weaker, and educated Americans are generally weak in foreign languages. As a result,
graduate students pronounce foreign names in a variety of ways, some of which attract ridicule from older
mathematicians. (Example: the possibly apocryphal graduate student at a blackboard who came to the
last step of a long proof and announced, "Viola!", much to the hilarity of his listeners.) There are
resources on the internet that allow one to look up the pronunciation of common foreign names; these
may be found on the website of this handbook.
Remark
The older practice of pronunciation is explained by history: In 1100 AD, the rules of pronunciation of
English, German and French, in particular, were remarkably similar. Over the centuries, the sound
systems changed, and Englishmen, for example, changed their pronunciation of "Lagrange" so that the
second syllable rhymes with "range", whereas the French changed it so that the second vowel is
nasalized (and the "n" is not otherwise pronounced) and rhymes with the "a" in "father".
Transliterations from Cyrillic
Another problem faced by the mathematics graduate student is the spelling of foreign names. The
name of the Russian mathematician most commonly spelled "Chebyshev" in English is also spelled
Chebyshov, Chebishev, Chebysheff, Tschebischeff, Tschebyshev, Tschebyscheff and Tschebyschef.
(Also Tschebyschew in papers written in German.) The correct spelling of his name is , since he was
Russian and Russian used the Cyrillic alphabet. The only spelling in the list above that could be said to
have some official sanction is Chebyshev, which is used by the Library of Congress. This is discussed by
Philip J. Davis in Cite: Dav83. Other citations: ErdTur37147, For57b79, Ged81844, Gor58693, Hig44440,
Sch89141.
Remark
In spite of the fact that most of the transliterations show the last vowel to be an "e", the name in
Russian is pronounced approximately "chebby-SHOFF", accent on the last syllable.
German spelling and pronunciation
The German letters "ä", "ö" and "ü" may also be spelled "ae", "oe" and "ue" respectively. The letters
"ä", "ö" and "ü" are alphabetized in German documents as if they were spelled "ae", "oe" and "ue". It is far
better to spell "Möbius" as "Moebius" than to spell it "Mobius".
The letter "ö" represents a vowel that does not exist in English; it is roughly the vowel sound in "fed"
spoken with pursed lips. It is sometimes incorrectly pronounced like the vowel in "code" or in "herd".
Similar remarks apply to "ü", which is "ee" with pursed lips. The letter "ä" may be pronounced like the
vowel in "fed".
The German letter "ß" may be spelled "ss" and often is by Swiss Germans. Thus Karl Weierstrass
spelled his last name "Weierstraß". Students sometimes confuse the letter "ß" with "f" or "r". In English
language documents it is probably better to use "ss" than "ß".
Another pronunciation problem that many students run into are the combinations
"ie" and "ei". The first is pronounced like the vowel in "reed" and the second like the
vowel in "ride". Thus "Riemann" is pronounced REE-mon.
narrative style
The narrative style of writing mathematics is a style involving infrequent labeling; most commonly,
the only things labeled are definitions, theorems, proofs, and major subsections a few paragraphs to
a few pages in length. The reader must deduce the logical status of each sentence from connecting
phrases and bridge sentences. This is the way most formal mathematical prose is written.
Terminology
Students have difficulties of several types with narrative proofs.
The proof may leave out steps.
The proof may leave out reasons for steps.
The proof may instruct the reader to perform a calculation which may not be
particularly easy. See proof by instruction.
The proof may not describe its own structure, which must be determined by pattern
recognition. See proof by contradiction.
The proof may end without stating the conclusion; the reader is expected to
understand that the last sentence of the proof implies the conclusion of the theorem via
known facts. Example under pattern recognition gives a proof that two sides of a triangle are
equal that ends with "Then triangle ABC is congruent to triangle ACB... "; the reader must then see that
the congruence of these two triangles implies that the required sides are the same.
Contrast labeled style. References: This style is named and discussed in []. See [].
redundant
Redundancy in discourse
A given discourse is redundant if it contains words and expressions that could be
omitted without changing the meaning. As another example, consider the sentence
"The counting function of primes
π(x):=#{p≤x:p prime }
satisfies the formula π(x)~x/(logx)."
The phrase "the counting function of primes" is redundant, since the definition just
following that phrase says it is the counting function for primes. This example, adapted
from [], is in no way bad writing: the redundancy adds much to the reader's
understanding (for this reader, anyway).
Type labeling is another commonly occurring systematic form of redundancy.
Redundancy in definitions
Redundancy occurs in definitions in a different sense from the type of verbal redundancy just
discussed. In this case redundancy refers to including properties or constituent structures that can be
deduced from the rest of the definition.
Structure determines underlying set
An apparent systematic redundancy in definitions of mathematical structures occurs throughout
mathematics, in that giving the structure typically determines the underlying set, but the definition
usually mentions the underlying set anyway. (Rudin Cite: Rud66 point out this phenomenon on page 18.)
Example
A semigroup is a set S together with an associative binary operation ☆ defined on S. If you
say what ☆ is explicitly, what S is is forced - it is the set of first (or second) coordinates of the domain
of ☆. Similarly, if you give a topology, the underlying set is simply the maximal element of the
topology.
In practice, however, the specification of the set is commonly part of the definition of
the operation.
Example
up to isomorphism as the group with underlying
The cyclic group of order three is defined
set {0,1,2} and multiplication given by addition mod 3.
Addition mod 3 defines a binary operation on the set
{0,1,2,3,4,5}
as well, so the mention of the underlying set is necessary.
The point of this example is that if you give the operation extensionally, the operation does indeed
determine the underlying set, but in fact operations are usually given by a rule.
I have heard mathematicians say (but not seen in print) that an \ln{assertion} purporting to be a
\lni{mathematical definition} is not a definition if it is redundant. This is a \textit{very} unwise stance, since
it can be an unsolvable problem to determine if a particular definition is redundant. Nevertheless, for
reasons of efficiency in proof, irredundant definitions are certainly
desirable.
Other examples
There are some other examples where the definition is redundant and the redundancy cannot be
described as a matter of convention. For example, in defining a group one usually requires an identity
and that every element have a two-sided inverse; in fact, a left identity and left inverses with respect to
the left identity are enough. In this case it is properties, rather than data, that are redundant. See radial
concept.
Acknowledgments.
plural
Many authors form the plural of certain learned words using endings from the
language from which the words originated. Students may get these wrong, and may
sometimes meet with ridicule for doing so.
Plurals ending in a vowel
Here are some of the common mathematical terms with vowel plurals.
singul
plural
ar
autom auto
aton mata
polyhe polyh
dron edra
focus foci
locus loci
radius radii
formul formu
a lae
Linguists have noted that such plurals seem to be processed differently from s-plurals ([]). In
particular, when used as adjectives, most nouns appear in the singular, but vowel-plural nouns appear in
the plural: Compare "automata theory" with "group theory".
The plurals that end in a (of Greek and Latin neuter nouns) are often not recognized as plurals and
are therefore used as singulars. (This does not seem to happen with my students with the -i plurals and
the -ae plurals.) In the written literature, the -ae plural appears to be dying, but the -a and -i plurals are
hanging on. The commonest -ae plural is "formulae"; other feminine Latin nouns such as "parabola" are
usually used with the English plural. In the 1990-1995 issues of six American mathematics journals
(American Journal of Mathematics, American Mathematical Monthly, Annals of Mathematics, Journal of
the American Mathematical Society, Proceedings of the American Mathematical Society, Transactions of
the American Mathematical Society), I found 829 occurrences of "formulas" and 260 occurrences of
"formulae", in contrast with 17 occurrences of "parabolas" and and no occurrences of "parabolae". (There
were only three occurrences of "parabolae" after 1918.) In contrast, there were 107 occurrences of
"polyhedra" and only 14 of "polyhedrons".
Plurals in s with modified roots
sing plura
ular l
ma matri
trix ces
sim simp
plex lices
ver verti
tex ces
Students recognize these as plurals but produce new singulars for the words as back formations.
For example, one hears "matricee" and "verticee" as the singular for "matrix" and "vertex". I have also
heard "vertec".
Remark
It is not unfair to say that some scholars insist on using foreign plurals as a form of one-upmanship.
But students and young professors need to be aware of these plurals in their own self interest.
It appears to me that ridicule and put-down for using standard English plurals instead of foreign
plurals, and for mispronouncing foreign names, is much less common than it was thirty years ago.
However, I am assured by students that it still happens.
The use of plurals in mathematical English is discussed under collective plural and distributive
plural.
Acknowledgments, .