Difference between revisions of "Name"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | n0660201.png | ||
| + | $#A+1 = 9 n = 0 | ||
| + | $#C+1 = 9 : ~/encyclopedia/old_files/data/N066/N.0606020 Name | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| − | + | {{TEX|auto}} | |
| + | {{TEX|done}} | ||
| + | |||
| + | A linguistic expression to denote a specified object. The object denoted by a given name is called the denotation. In mathematics, names are widely used for specific mathematical objects, for example, $ e , \pi $ | ||
| + | for the well-known transcendental numbers, $ \sin $ | ||
| + | for the sine function, $ \emptyset $ | ||
| + | for the empty set. From such simple names one can form compound names, which name an object using the names of two objects. For example, $ \sin \pi $ | ||
| + | is another name for the number $ 0 $. | ||
| + | A name not only names the denotation, but it also expresses a definite meaning. Thus, the expressions | ||
| + | |||
| + | $$ | ||
| + | \lim\limits _ {n \rightarrow \infty } n ^ {1/n} \ \textrm{ and } \ \ | ||
| + | \sin | ||
| + | \frac \pi {2} | ||
| + | |||
| + | $$ | ||
are names for the number 1; however, their meanings are different. If in a compound name, some name occurring in it is replaced by a name having the same denotation, then the denotation of the compound name is unchanged. If in a compound name some name occurring in it is replaced by a synonym of it (that is, a name having the same meaning), then the meaning of the compound name is unchanged. | are names for the number 1; however, their meanings are different. If in a compound name, some name occurring in it is replaced by a name having the same denotation, then the denotation of the compound name is unchanged. If in a compound name some name occurring in it is replaced by a synonym of it (that is, a name having the same meaning), then the meaning of the compound name is unchanged. | ||
| − | Along with names, one uses in mathematics expressions containing variables. The expressions become names upon replacing the variables by names of objects in the range of values of the variables. Such expressions are called name forms. The expressions | + | Along with names, one uses in mathematics expressions containing variables. The expressions become names upon replacing the variables by names of objects in the range of values of the variables. Such expressions are called name forms. The expressions $ e ^ {x} $, |
| + | $ \int _ {0} ^ {x} ( \sin t ) d t / t $, | ||
| + | where $ x $ | ||
| + | is a variable for the real numbers, are examples of name forms. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Church, "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press (1956)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Church, "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press (1956)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Carnap, "Meaning and necessity" , Univ. Chicago Press (1947)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Carnap, "Meaning and necessity" , Univ. Chicago Press (1947)</TD></TR></table> | ||
Revision as of 08:02, 6 June 2020
A linguistic expression to denote a specified object. The object denoted by a given name is called the denotation. In mathematics, names are widely used for specific mathematical objects, for example, $ e , \pi $
for the well-known transcendental numbers, $ \sin $
for the sine function, $ \emptyset $
for the empty set. From such simple names one can form compound names, which name an object using the names of two objects. For example, $ \sin \pi $
is another name for the number $ 0 $.
A name not only names the denotation, but it also expresses a definite meaning. Thus, the expressions
$$ \lim\limits _ {n \rightarrow \infty } n ^ {1/n} \ \textrm{ and } \ \ \sin \frac \pi {2} $$
are names for the number 1; however, their meanings are different. If in a compound name, some name occurring in it is replaced by a name having the same denotation, then the denotation of the compound name is unchanged. If in a compound name some name occurring in it is replaced by a synonym of it (that is, a name having the same meaning), then the meaning of the compound name is unchanged.
Along with names, one uses in mathematics expressions containing variables. The expressions become names upon replacing the variables by names of objects in the range of values of the variables. Such expressions are called name forms. The expressions $ e ^ {x} $, $ \int _ {0} ^ {x} ( \sin t ) d t / t $, where $ x $ is a variable for the real numbers, are examples of name forms.
References
| [1] | A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956) |
Comments
References
| [a1] | R. Carnap, "Meaning and necessity" , Univ. Chicago Press (1947) |
Name. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Name&oldid=11327