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GLOSSARY
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Posted 16 April 2009
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calculate
To calculate may mean to perform symbol manipulation on an expression, usually with the intent to arrive at another, perhaps more satisfactory, expression. Compute is used in much the same way. Neither word necessarily indicates that the procedure was done on a computer or that a known algorithm was used.
Examples
¨ “An easy calculation
shows that the equation factors into linear factors over the reals.” The calculation would probably go like
this: “
,
and we know that
”. “We
know” is not something that would occur in a program or algorithm!
¨ “Let us compute
the roots of the equation .”
If you do this using the quadratic
formula then that really is a computation using an algorithm. But you could also try factoring
and trying
and
and the first one works. This is not an algorithm but many
mathematicians would still call it a computation or calculation. (This remark about word usage is not based
on lexicographical research, but only my own observations.)
call
Call may be used to form a definition.
Examples
¨ “A monoid is called a group if every element has an inverse.”
¨ “Let .
We call g the conjugate of f by h.”
¨ “We call an
integer even if it is divisible
by
Remark
Some object to using “call” with an adjective, as in the third example above. However, this usage has been in the language for centuries.
Example
“…from henceforth all generations shall call me blessed.” King James Bible (published in 1611), Luke 1:48. (This usage also occurs in Genesis chapter 30.)
case
The Roman alphabet, the Greek alphabet, and the Cyrillic alphabet have two forms of letters, “capital” or uppercase, A, B, C, etc, and lowercase, a, b, c, etc.
Case distinction always matters in mathematics. It is common for example to use a capital letter to name a mathematical structure and the same letter in lowercase to name an element of the structure. This means that when you take notes at a lecture in abstract math, you must write “S” when the lecturer writes “S” and “s” when the lecturer writes “s”!
Other variations in font and style may also be significant. See fraktur, boldface and blackboard bold.
cases
A concept is defined by cases if it is dependent on a parameter and the defining expression is different for different values of the parameter. This is also called a disjunctive definition or split definition. Example
Let the function be defined by
Then for example, f(1.9) = 2.9 and f(2.01) = 1.01. This defines one function f.
Note that the absolute
value function is defined by cases.
Disjunctive definitions may seem unnatural at first. This may be because real life definitions are rarely disjunctive. One exception is the concept of strike in baseball (swing and miss or didn’t swing when the ball was in the zone or one of several other possibilities).
category
The word category is used with two unrelated meanings in math (Baire and Eilenberg-Mac Lane). It is used with still other meanings by linguists and cognitive scientists.
character
¨ A character in computing science is a letter or
symbol in an alphabet, or its representation.
More here.
¨ A character in group theory is the trace of a linear
representation. More here.