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The name of a mathematical object is a word or phrase in math English used to identify an object. A name is a special kind of description -- a one-word description.

Sources of names

Suggestive English words

A suggestive name is a a common English word or phrase, chosen to suggest its meaning. This means it is a type of metaphor.


In none of these examples is
the metaphorical meaning
exactly suitable to be
the mathe­matical definition.

Random English words

Most English words used in math are not suggestive. They are either chosen at random or were intended to suggest something but misfired in some way.


A group is a collection of math objects with a binary operation defined on it, subject to certain constraints. To many non-mathe­maticians, a "group" sounds like essentially what a mathe­matician calls a "set". So the word "group" does not suggest the existence of a binary operation.

The concept of group was one of the earliest mathe­matical concepts des­cribed as a set-with-structure. I believe that a group was origi­nally referred to as a "group of trans­forma­tions". May­be that phrase got shortened to "group" without anyone realizing what a disas­trous met­a­phor it caused.


A field in the algebraic sense is a structure which is not in any way suggested by the word "field". The German word for field in this sense is "Körper", which means "body". That is about as bad as "group", and I suspect it was motivated in much the same way. The name "Körper" may be due to Dedekind. I don't know who to blame for "field".

A field in the sense of an assignment of a scalar or a vector to every point in a space is a completely separate notion than that of field as an algebra. The concept was invented in the nineteenth century by physicists, but any math student is likely to see fields in this vector sense in several different courses.

Perhaps the second meaning of field was suggested by contour plowing.

The word "field" is also discussed in the Glossary.

Person’s name

A concept may be named after a person.


I have no idea why "Riemann" gets an ending when it is a manifold but not when it is a surface.

Made-up name

Some names are made up in a random way, not based on any other language. Googol is an example.

Named after notation


A mathematical object may be named by the typographical symbol(s) used to denote it. This is used both formally and in on-the-fly references.  

Some objects have standard names that are single letters (Greek or Roman), such as $e$, $i$ and $\pi$. There is much more about this in Alphabets.

Be warned that any letter can be given another definition. $\pi$ is also used to name a projection, $i$ is commonly used as an index, and $e$ means "energy" in physics.



A synecdoche is a name of part of something that is used as a name for the whole thing.


The Tochar­ians appear to have called a cart by their word for wheel several thousand years ago. See the blog post by Don Ringe.

Names from other languages

In English, many technical names are borrowed from other languages. It may be difficult to determine what the meaning in the old language has to do with the mathematical meaning.


Much of this information comes from The On-Line Etymological Dictionary. (Read its article about "sine".) See also my articles on secant and tangent.

I enjoy finding out about etymol­ogies, but I concede that knowing an ety­mol­ogy doesn't help you very much in under­standing the math.

Names made up from other languages' roots

A name may be a new word made out of (usually) Greek or Latin roots.


Mathematical names cause problems for students

The name may suggest the wrong meaning

This is discusses in detail in the article cognitive dissonance.

The name may not suggest any meaning

English is unusual among major languages in the number of technical words borrowed from other languages instead of being made up from native roots.  We have some, listed under suggestive names.  But how can you tell from looking at them what “parabola” or “homomorphism” mean?   This applies to concepts named after people, too: The fact that “Hausdorff” is German for a village near an estate doesn’t tell me what a Hausdorff space is.

The English word “carnivore” (from Latin roots) can be translated as “Fleischfresser” in German; to a German speaker, that word means literally “meat eater”.  So a question such as “What does a carnivore eat” translates into something like, “What does a meat-eater eat?” 

Chinese is another language that forms words in that way: see the discussion of “diagonal” in Julia Lan Dai’s blog.  (I stole the carnivore example from her blog, too.)

The result is that many technical words in English do not suggest their meaning at all to a reader not familiar with the subject.  Of course, in the case of “carnivore” if you know Latin, French or Spanish you are likely to guess the meaning, but it is nevertheless true that English has a kind of elitist stratum of technical words that provide little or no clue to their meaning and Chinese and German do not, at least not so much. This is a problem in all technical fields, not just in math.


There are two main reasons math students have difficulties in pronouncing technical words in math.

Most students have little knowledge of other languages

Forty years ago nearly all Ph.D. students had to show mastery in reading math in two foreign languages; this included pronunciation, although that was not emphasized. Today the language requirements in the USA are much weaker, and younger 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 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.

Pronunciation of words from other languages has become unpredictable

In English-speaking countries until the early twentieth century, the practice was to pronounce a name from another language as if it were English, following the rules of English pronunciation.

We still pronounce many common math words this way: “Euclid” is pronounced “you-clid” and “parabola” with the second syllable rhyming with “dab”.

But other words (mostly derived from people's names) are pronounced using the pronunciation of the language they came from, or what the speaker thinks is the foreign pronunciation. This particularly involves pronouncing "a" as "ah", "e" like "ay", and "i" like "ee".


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".

German spelling

The German letters "ä", "ö" and "ü" may also be spelled "ae", "oe" and "ue" respectively. It is far better to spell "Möbius" as "Moebius" than to spell it "Mobius".

The German letter "ß" may be spelled "ss" and usually is spelled that way by the Swiss. 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".

Transliterations from Cyrillic

 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 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.

The correct spelling of his name is "Чебышев" since he was Russian and the Russian language uses the Cyrillic alphabet.

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.  Now, that is a ridiculous situation, and it is the transliterators who are ridiculous, not Russian spelling, which in spite of that peculiarity about the Cyrillic letter “e” is much more nearly phonetic than English spelling.

Some other Russian names have variant spellings (Tychonov, Vinogradov) but Chebyshev probably wins the prize for the most.


Many authors form the plural of certain technical 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.


Plurals in s with modified roots









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".


It is not unfair to say that some scholars insist on using foreign plurals as a form of one-upmanship. 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.

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