Latin terms and phrases in math
Mathematics is an ancient discipline, and consequently it has picked up a good deal of terminology over the years that is not commonly used in ordinary discourse. Phrases and terms from Latin make up a large part of this terminology, and reading mathematical texts—especially more advanced ones—is made easier if one is equipped with knowledge of these terms in advance.
We review below the Latin terms most commonly used in mathematics, and follow with a more extensive list of such terms and phrases as one may run into more rarely or in other contexts. The pronunciations given are not the “correct” Latin pronunciations, but instead reflect common usage in English speaking countries.
Note that when Latin or other non-English words are used in writing, they should be italicized except where they are abbreviated as single letters. E.g., “His next remark was a non sequitur.”
TERMS USED IN MATHEMATICS
ad infinitum (AHD-in-fin-ITE-um)
Literally, “to infinity,” indicates that a process or operation is to be carried out endlessly.
a fortiori (ah-FOR-tee-OHR-ee)
“With stronger reason.” If every multiple of two is even, then a fortiori every multiple of four is even.
a posteriori (AH-paws-TEER-ee-OHR-ee)
“From effect to cause.” A thing is known a posteriori if it is known from evidence or empirical reasoning.
a priori (AH-pree-OHR-ee)
A thing is known a priori if it is evident by logic alone from what is already known.
See exempli gratia.
exempli gratia (ex-EMP-lee GRAH-tee-uh)
“For example.” Usually abbreviated to ‘e.g.’ and often confused with ‘i.e.’ Example: “Many real numbers cannot be expressed as a ratio of integers, e.g., the square root of two.”
id est (id EST)
Literally, “that is.” Usually abbreviated ‘i.e.’ and often confused with ‘e.g.’ Example: “She won the race, i.e., she crossed the finish line first.” The decision whether to use ‘i.e.,’ or ‘e.g.’ should be based on whether “that is” or “for example” is what is wanted in the sentence.
See id est.
ipso facto (IP-soh FAK-toh)
Literally, “by that very fact.” Example: “Lie group representations are useful in characterizing quantum mechanical phenomena, and they are ipso facto an important part of a physicist’s mathematical training.”
See nota bene
nota bene (NOH-tuh BAY-nay)
Literally, “note well.” Usually abbreviated ‘n.b.’, this is a way of saying, “take note of this.”
per impossibile (pehr ihm-paws-SEE-bee-lay)
“As is impossible.” Qualifies a proposition that cannot be true.
See quod erat demonstrandum.
See quod erat faciendum.
quod erat demonstrandum (KWAWD eh-RAHT dem-on-STRAHND-um)
“That which was to have been proved.” Traditionally placed at the end of proofs, the QED is now usually indicated by a small square. A few students have clung to use of the traditional letters, in the hope they might be interpreted as “quite elegantly done.”
quod erat faciendum (KWAWD eh-RAHT FAH-kee-END-um)
“That which was to have been shown.” Abbreviated QEF, it was traditionally used to mark the end of a solution or calculation. It is rarely used now. (Impress your professor by putting it at the end of exam problems.)
OTHER COMMON LATIN TERMS
ab initio (AHB in-IT-ee-oh)
From the beginning.
ad hoc (add-HOK)
For the immediate purpose. An ad hoc committee is appointed for some specific purpose, after completing which it is dissolved.
ad hominem (add HOM-in-um)
“To the man.” An argument is ad hominem when it attacks the opponent personally rather than addressing his arguments.
ad nauseam (add NAWS-ee-um)
Something continues ad nauseam when it goes on so long you become sick of it.
alma mater (ALL-muh MAH-ter)
Your alma mater is the university or college which granted your degree.
An alum, as it is sometimes shortly said, is a former member/student of a university or college. (The ‘us’ ending is masculine, the ‘a’ ending feminine. The plurals are alumni and alumnae, respectively.)
See anno Domini.
anno Domini (AN-noh DOM-in-ee)
“In the year of Our Lord.” Indicates that a date is given in the western or Gregorian calendar, in which years are counted roughly from the birth of Christ. Most contemporary writers would use the non-culture-specific indicator CE (‘current era’) in place of A.D.
bona fide (BONE-uh FIDE)
“In good faith.” One’s bona fides are documents or testimonials establishing one’s credentials or honesty.
carpe diem (CAR-pay DEE-um)
“Seize the day.” A motto which says to live in the now, and/or to not waste time or opportunity.
Approximately. Used with dates, e.g., Euclid wrote the Elements circa 300 BCE.
“Compare.” Usually abbreviated cf. and often used in footnotes, this indicates that one should compare the present passage or statement with the one referred to.
cum laude (coom LOUD-ay)
“With praise.” Used on degree certificates to indicate exceptional academic standing.
de facto (day FAK-toh)
“In actual fact.” Used to indicate that, whatever may be believed or legislated, the reality is as indicated here. E.g., she’s the de facto leader of the union.
de jure (day JHOOR-ay)
“In law, or according to the rules.” Contrast to de facto.
That settles it. Literally, “I have spoken.”
(feminine: emerita) Indicates someone who has served out his or her time and retired honorably. E.g., she is now professor emerita.
Literally, “error/errors,” this term in fact refers to the corrections included in a paper or book after it is published to correct minor errors in the text.
et al. (ETT ALL)
Abbreviation of et alia, meaning “and others.” Used to indicate an unstated list of contributing authors following the main one, for instance.
et cetera (ETT SET-er-ah)
And so forth. Note the pronunciation—there is no “eks” sound.
ex post facto (eks post FAK-toh)
“After the fact.”
“In the same place.” Used in footnotes to indicate that the reference is the same as the preceding one(s).
in re (IN RAY)
“In regards to.” Often used to head formal correspondence. When only re is written, it should be translated as “regarding.”
inter alia (IN-ter ALL-ee-uh)
Among other things.
in toto (in TOH-TOH)
in vacuo (in VAK-yoo-oh)
Literally, “in a vacuum.” Should be taken to mean “in the absence of other conditions or influences.” E.g., nobody achieves maturity in vacuo.
magna cum laude (MAG-nuh coom LOUD-ay)
With great praise. See cum laude.
modus operandi (MODE-us op-ehr-AWN-dee)
Manner or method of work characterizing a particular person’s professional habits.
mutatis mutandis (myoo-TAH-tis myoo-TAHN-dis)
With necessary changes. “Mutatis mutandis, this proof applies in more general cases.”
non sequitur (nahn-SEK-wit-ter)
“Not following.” Used to indicate a statement or conclusion that does not follow from what has gone before.
per se (per SAY)
“In and of itself.” Example: “This argument does not force the conclusion per se, but with this added premise the result would follow.”
post hoc, ergo propter hoc (POST hawk air-go PROP-ter hawk)
“After, therefore because of.” A common fallacy in reasoning, in which causality is ascribed to preceding conditions which were in fact irrelevant to the supposed effect.
post scriptum (post SKRIP-tum)
“Written after.” Indicates an afterword or footnote to a main text, and is often used in written correspondence (where it is abbreviated p.s.).
prima facie (PRIME-uh FAYSH-uh)
“On its face.” Indicates that a conclusion is indicated (but not necessarily proved) from the appearance of things.
pro forma (proh FOR-muh)
“For form’s sake.” E.g., “It was a pro forma interview—the decision to hire her had already been made.”
“In the capacity of.” For example, “He is really very personable, but qua chairman he can be direct and even gruff.”
quod vide (kwawd VEE-day)
“Which see.” Usually abbreviated q.v., this is a scholarly way of directing the reader to a reference.
See quod vide.
sensu lato (SEN-soo LAH-toh)
“In the broad sense.” Indicates a general or loose meaning of a term or phrase.
sensu stricto (SEN-soo STRIK-toh)
“In the strict sense.” Indicates the strict or precise meaning of a term or phrase.
sine qua non (SIN-ay kwah NAHN)
“That without which nothing.” Indicates an essential element or condition.
summa cum laude (SOOM-uh coom LOUD-ay)
With greatest praise. See cum laude.
tabula rasa (TAB-yoo-lah RAH-sah)
“Blank Slate.” Often refers to a person who has not yet formed prejudices or preconceptions on a given matter.
Word-for-word. Indicates a precise transmission of a phrase, discussion, or text.
Usually abbreviated viz., this is translated as “namely.” For example, “The math club picked a new president, viz., Carl.”