abstractmath.org
help with abstract math
Produced by Charles Wells. Home. Website Contents Website Index
Back to Languages Head
Posted 2 June
2008
The Symbolic Language of Math
The symbolic language of math is a distinct special-purpose language. Unlike mathematical English, it is not a variety of English. It has its own rules of grammar that are quite different from those of English. You can usually read expressions in the symbolic language in any math article written in any language.
The chapter More about the languages of math discusses topics that involve both the symbolic language and mathematical English.
Symbolic expressions
The symbolic language consists of symbolic expressions written in the way mathematicians traditionally write them. They occur in two types:
¨ Symbolic assertions. These are complete statements that stand alone as sentences. An assertion says something. Assertions play the same role in the symbolic language as assertions in math English.
¨ Symbolic terms. They are expressions that refer to some mathematical object. A term names something. Terms play the same role in the symbolic language that names do in math English. See remark for variations in usage of the word “term”.
Every expression is either a symbolic term or a symbolic assertion.
Examples
¨
" ".
This is a symbolic
assertion in
the symbolic language of mathematics. It is true.
“
” is a false symbolic assertion.
¨
The expression “ ” is a symbolic term. It is another name for the number
¨ The expression “<” is a symbolic term. It refers to the less-than relation, which is a mathematical object.
¨ “ ” (which means
) is a
symbolic term containing a variable x,
so it has variable
meaning depending on which value is substituted for x.
For example,
is another name for
. Talk about intensional
and extensional.
¨
is symbolic term with two variables. If
you substitute
¨
The expressions and
are symbolic assertions. See more about them in the section on intent.
Reading symbolic expressions
Distinguish between assertions and terms
An expression such as “ ” can be an assertion (it is saying
something) or a term (it is naming something).
¨
“ x < y” is an assertion a complete (true or false) statement.
¨
“ x + y” is a term an expression to be evaluated.
When you see a complicated assertion or term you have to be patient.
You must stop and unwind it. Example. Another example. Put some examples here.
|
|
Note on terminology
The names “symbolic assertion” and “symbolic term” are not standard usage in math. In mathematical logic, symbolic assertions may be called formulas or predicates and true assertions may be called propositions or sentences. All these words, as well as our use of “term”, can cause cognitive dissonance:
¨ Many people would refer to “ ” as “the formula for water”, but it is not a formula in sense of logic
because it does not make a statement.
¨ In everyday usage “proposition” may mean a statement to be debated, or a proposal for action, but in math logic the meaning is simply a true assertion.
¨
The name “term”
comes from mathematical logic. The
expression “ ” contains symbolic terms
and
,
which would in math English be called factors rather
than terms.
¨
Giving names to symbolic expressions to be written
talk about functions (for terms) and truth sets (for assertions)
Intent of symbolic assertions to be written
Grammar of the symbolic language
Symbols and arrangements
In symbolic expressions, the symbols and the arrangement of the symbols both communicate meaning. For example, as you know,
Examples
¨
“ ” , “
” and “
” all mean different things.
¨
“ ” is meaningless.
¨
“ ” and “
” mean
the same thing: that is one of the rules of understanding
symbolic expressions. If you took a
class in trig, you may have had this fact expressed explicitly or you may have
learned it by osmosis.
¨
Subexpressions
An expression may contain several subexpressions. The rules for forming expressions and the use of delimiters let you determine the subexpressions.
Examples
¨ The subexpressions in “ ” are “x”
and “
¨ Two of the subexpressions in “ ” are “
” and “
”. The
rules of algebra require “
” to be enclosed in parentheses, but 2x need not be, although it is not wrong
to write “
”.
Math English subexpressions
A phrase in math English can be a subexpression of a symbolic expression.
Example
The set could also be written as
.
Precedence
The expression xy + z means (xy) + z, not x(y + z). This is an illustration of the principle that in an algebraic expression, multiplication is performed first, then addition. We say multiplication has a higher precedence that addition.
When two operations have the same precedence, the operations should be done from left to right. The mnemonic “Please Excuse My Dear Aunt Sally” (PEMDAS) describes the order of the common operations:
¨ Parentheses
¨ Exponentiation
¨ Multiplication and Division
¨ Addition and Subtraction.
Unary operations (functions with one input) in math writing typically have low precedence.
Example
The expression sin(x+y) denotes the sine of the number x + y. In theory, the expression sin x + y
denotes the sum of y and the sine
of x, in other words, (sin x) + y. In practice, the expression sin x + y
bothers many people (including me) so they commonly include the
parentheses. Similarly, I always put the
parentheses in double exponents, for example or
instead of writing
. The expression
is supposed to denote
.
Wikipedia has an excellent detailed description of the precedence rules of algebra.
Irregular syntax in the symbolic language
The symbolic language of math has developed over the centuries the way natural languages do. In particular, the symbolic language, like English, has definite rules and it has irregularities.
Rules
¨ 
In English, the plural of a noun is normally formed by adding “s” or
“es” according to fairly precise rules.
(The plural of car is cars, the plural of loss is losses.)
¨ In the symbolic language, the symbol for a function is usually put to the left of the input (argument) and the input is put in parentheses. For example if f is the function defined by f(x) = x + 1, then the value of f at 3 is denoted by f(3) (which of course evaluates to 4.)
Irregularities
Just as English has irregular plurals and past tenses (mouse/mice, hold/held), the symbolic language has irregular syntax for certain expressions.
¨ The symbol “!” for the factorial (MW, Wi) function is put after the argument, for example 6! = 720.
¨ The parentheses around the argument of a function are omitted for the trig and log (MW, Wi) functions, so we typically write sin x instead of sin(x). More about that here.