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Variable mathematical objects


VARIABLE MATHEMATICAL OBJECTS

In many mathematical texts, the variable $x$ may denote a real number, although which real number may not be specified. This is an example of a variable mathematical object. This point of view and terminology is not widespread, but I think it is worth understanding because it provides a deeper understanding of some aspects about how math is done.

Specific and variable mathematical objects


It is useful to distinguish between specific math objects and variable math objects.

Examples of specific math objects

  • The number $42$ (the math object represented as “42” in base $10$, “2A” in hexadecimal and “XLII” as a Roman numeral) is a specific math object. It is an abstract math object. It is not any of the representations just listed — they are just strings of letters and numbers.
  • The ordered pair $(4,3)$ is a specific math object. It is not the same as the ordered pair $(7,-2)$, which is another specific math object.
  • The sine function $\sin:\mathbb{R}\to\mathbb{R}$ is a specific math object. You may know that the sine function is also defined for all complex numbers, which gives another specific math object $\sin:\mathbb{C}\to\mathbb{C}$.
  • The group of symmetries of a square is a specific math object. (If you don’t know much about groups, the link gives a detailed description of this particular group.)

Variable math objects

Math books are full of references to math objects, typically named by a letter or a name, that are not completely specified. Some mathematicians call these variable objects (not standard terminology). The idea of a variable mathe­mati­cal object is not often taught as such in under­graduate classes but it is worth pondering. It has certainly clari­fied my thinking about expres­sions with variables.

Examples

  • If an author or lecturer says “Let $x$ be a real variable”, you can then think of $x$ as a variable real number. In a proof you can’t assume that $x$ is any particular real number such as $42$ or $\pi$.
  • If a lecturer says, “Let $(a,b)$ be an ordered pair of integers”, then all you know is that $a$ and $b$ are integers. This makes $(a,b)$ a variable ordered pair, specifically a pair of integers. The lecturer will not say it is a variable ordered pair since that terminology is not widely used. You have to understand that the phrase “Let $(a,b)$ be an ordered pair of integers” implies that it is a variable ordered pair just because “a” and “b” are letters instead of numbers.
  • If you are going to prove a theorem about functions, you might begin, "Let $f$ be a continuous function", and in the proof refer to $f$ and various objects connected to $f$. This makes $f$ a variable mathematical object. When you are proving things about $f$ you may use the fact that it is continuous. But you cannot assume that it is (for example) the sine function or any other particular function.
  • If someone says, “Let $G$ be a group” you can think of $G$ as a variable group. If you want to prove something about $G$ you are free to use the definition of “group” and any theorems you know of that apply to all groups, but you can’t assume that $G$ is any specific group.

Terminology

A logician would refer to the symbol $f$, thought of as denoting a function, as a vari­able, and likewise the symbol $G$, thought of as denoting a group. But mathe­maticians in general would not use the word “vari­able” in those situa­tions.

How to think about variable objects

The idea that $x$ is a variable object means thinking of $x$ as a genuine mathematical object, but with limitations about what you can say or think about it. Specifically,

Some assertions about a variable math object
may be neither true nor false.

Example

The statement, “Let $x$ be a real number” means that $x$ is to be regarded as a variable real number (usually called a “real variable”). Then you know the following facts:

  • The statement “${{x}^{2}}$ is not negative” is true.
  • The assertion “$x=x+1$” is false.
  • The assertion “$x\gt 0$” is neither true nor false.
Example

Suppose you are told that $x$ is a real number and that ${{x}^{2}}-5x=-6$.

  • You know (because it is given) that the statement “${{x}^{2}}-5x=-6$” is true.
  • By doing some algebra, you can discover that the statement “$x=2$ or $x=3$” is true.
  • The statement “$x=2$ and $x=3$” is false, because $2\neq3$.
  • The statement “$x=2$” is neither true nor false, and similarly for “$x=3$”.
  • This situation could be described this way: $x$ is a variable real number varying over the set $\{2,3\}$.
Example

This example may not be easy to understand. It is intended to raise your consciousness.

A prime pair is an ordered pair of integers $(n,n+2)$ with the property that both $n$ and $n+2$ are prime numbers.

Definition: $S$ is a PP set if $S$ is a set of pairs of integers with the property that every pair is a prime pair.

  • “$\{(3,5),(11,13)\}$ is a PP set” is true.
  • “$\{(5,7),(111,113),(149,151)\}$ is a PP set” is false.

Now suppose $SS$ is a variable PP set.

  • “$SS$ is a set” is true by definition.
  • “$SS$ contains $(7,9)$” is false.
  • “$SS$ contains $(3,5)$” is neither true nor false, as the examples just above show.
  • “$SS$ is an infinite set”:
    • This is certainly not true (see finite examples above).
    • This claim may be neither true nor false, or it may be plain false, because no one knows whether there is an infinite number of prime pairs.
    • The point of this example is to show that “we don’t know” doesn’t mean the same thing as “neither true nor false”.

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Names of mathematical objects

This is a revision of the abstractmath.org article on names.

The name of a mathematical object is a word or phrase in math English used to identify an object. A name plays the same role that symbolic terms play in the symbolic language.

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.

Examples

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

  • “Curve”, “point”, “line”, “slope“, “circle” and many other English words are used in elementary math with precise meanings that more or less fit their everyday meanings.
  • Connected subspace (of a topological space). When you draw a picture of a connected set it looks “connected”.
  • “Set” suggests a collection of things and provides a reasonable metaphor for its mathe­matical meaning. Both the abstractmath article on sets and the Wikipedia article on sets give you insight on why this metaphor cannot be entirely accurate.
  • 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.

    Groups

    A group is a collection of math objects with a binary operation defined on it subject to certain constraints. The binary operation is much more impor­tant than the underlying set! To many non-mathe­maticians, a “group” sounds like essentially what a mathe­matician calls a “set”.

    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.

    Fields

    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 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.

    Examples

    • L’Hôpital’s Rule
    • Hausdorff space
    • Turing machine
    • Riemann surface
    • Riemannian manifold
    • Pythagorean Theorem
    • 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 oter language. Googol is an example.

      Named after notation

      Symbols

      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.

      Expressions

      • The multiplication in a Lie Algebra is called the “Lie bracket”. It is written “$[v,w]$”.
      • In quantum mechanics, a vector $\vec{w}$ may be notated “$|w\rangle$” and called a “ket”. Another vector $\vec{v}$ induces a linear operator on vectors that is denoted by “$\langle v|$”, which is called a “bra”. The action of $\langle v|$ on $|w\rangle$ is the inner product $\langle v|w\rangle$, which suggested the “bra” and “ket” terminology (from “bracket”). You can blame Paul Dirac for this stuff.
      • In 1985, Michael Barr and I published a book in category theory called Triples, Toposes and Theories. Immediately after that everyone in category theory started saying “monad” for what had been called “triple”. (The notation for a triple, er, monad, is of the form “$(T,\eta,\mu)$”.)
      • Synecdoche

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

        Examples

      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.

      Examples

    • Matrix. This is the Latin word for “uterus”. I suppose the analogy is with “container”.
    • Parabola. “Parabola” is a word borrowed from Greek in late Latin, meaning something like “comparison”. The parabola $y=x^2$ “compares” a number with its square: it curves upward because the area of a square grows faster than the length of its side. “Parable” is from the same word.
    • Algebra. This comes from an Arabic word meaning the art of setting joints, or more generally “restore”. It came through Spanish where it once meant “surgical procedure” but that meaning is now obsolete.

    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.

    Examples

    • Homomorphism. “Homo” in Greek is a root meaning “same” and “morphism” comes from a root referring to shape.
    • Quasiconformal. “Quasi” is a Latin word meaning something like “as if”. It is a prefix mathematicians use a bunch. It usually implies a weakening of the constraints that define the word it is attached to. A map is conformal if it preserves angles in a certain sense, and it is quasiconformal then it does not preserve angles but it does take circles into ellipses in a certain restricted sense (which conformal maps also do). So it replaces a constraint by a weaker constraint.

    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.

    Pronunciation

    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”.

    Examples
    • Euler (oiler)
    • Fourier (foo-ree-ay)
    • Lagrange (second a pronounced “ah”)
    • Lie (lee)
    • Riemann (ree-monn)

    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 often is 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 mot 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.

Plurals

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.

singular plural
automaton automata
polyhedron polyhedra
focus foci
locus loci
radius radii
formula formulae
parabola parabolae
  • 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”.  No one says groups theory.  I used to say “automaton theory” but people looked at me funny.
  • The plurals that end in a (of Greek and Latin neuter nouns) are often not recognized as plurals and are therefore used as singulars.  That is how “data” became singular.  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, 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

    singular

    plural

    matrix

    matrices

    simplex

    simplices

    vertex

    vertices

    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”.

    Remarks

    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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