1.2. Variables, sets, functions and operations
Let us start mathematics by introducing some simple concepts from
the founding cycle, which may seem self-sufficient. It is natural
to start with a set theory not fully formalized as an
axiomatic theory.
Let us first explain what is a set, then we will complete the
picture with more concepts and explanations on the context of
foundations (model theory) and its main subtleties (paradoxes).
Constants
A constant symbol is a symbol denoting a unique object,
called its value. Examples: 3, ⌀, ℕ. Those of English
language are proper names and names with «the» (singular without
complement).
Free and bound variables
A variable symbol (or a variable), is a symbol
without a fixed value. Each possible interpretation gives it a
particular value and thus sees it as a constant.
It can be understood as limited by a box, whose inside has multiple
versions in parallel.
- From inside, the variable is seen as having a fixed value,
thus usable as a constant : it is called fixed.
- We call the variable free when we start going out and
find that this value is not uniquely determined as the only
possible one, and can thus vary.
- A variable is called bound when completely seen from
the «outside», where the diversity of its possible values is
considered as fully known (perceived), gathered and
ready to be processed as a whole.
The diverse «internal viewpoints», corresponding to the possible
fixed values, may be thought of as abstract «locations» in the
mathematical universe, while the succession of statuses of a symbol
(as a constant, a free variable or a bound variable), can be
seen as a first expression of the flow of time in mathematics: a
variable is bound when all the diverse "parallel locations inside
the box" (possible values) are past. All these places and
times are themselves purely abstract, mathematical entities.
Ranges and sets
The range of a variable, is the meaning it takes when seen
as bound: it is the «knowledge» of the totality of its possible or
authorized values (seen in bulk: unordered, ignoring their context),
that are called the elements of this range. This
«knowledge» is an abstract entity that can encompass infinities of
objects, unlike human thought. A variable has a range when it can be
bound, i.e. when an encompassing view over all its possible values
is given.
Any range of a variable is called a set.
Cantor defined a set as «a grouping into a whole of distinct
objects of our intuition or our thought». He explained to
Dedekind : «If the totality of elements of a multiplicity can be
thought of as «simultaneously existing», so that it can be
conceived as a «single object» (or «completed object»), I call it
a consistent multiplicity or a «set».» (We expressed this
«multiplicity» as that of values of a variable).
He described the opposite case as an «inconsistent multiplicity»
where «admitting a coexistence of all its elements leads to a
contradiction». But non-contradiction cannot suffice to
generally define sets: the non-contradiction of a statement does not
imply its truth (the opposite statement may be true but unprovable);
facts of non-contradiction can be themselves unprovable (incompleteness theorem);
and two separately consistent coexistences might contradict each
other (Irresistible
force paradox / Omnipotence
paradox).
A variable is said to range over a set, when it is bound
with this set as its range. Any number of variables can be
introduced ranging over a given set, independently of each other and of
other variables.
Systematically renaming a bound variable in all its box, into
another symbol not used in the same context (same box), with the
same range, does not change the meaning of the whole. In practice,
the same letter can represent several separate bound variables (with
separate boxes), that can take different values without conflict, as
no two of them are anywhere free together to compare their values.
The common language does this continuously, using very few variable
symbols («he», «she», «it»...)
Functions
A function is any object f behaving as a variable whose value
is determined by that of another variable called its argument; the
argument has a range called the domain of f and denoted
Dom f. Whenever its argument is fixed (gets a name, say
x, and a value in Dom f), f becomes a constant
(written f(x)).
In other words, f is made of the following data:
- A set called the domain of f, denoted Dom
f
- For each element x of Dom f,
an object written f(x), called the image of x
by f or value of f at x.
Operations
The notion of operation generalizes that of function, by
admitting a finite list of arguments (variables with given
respective ranges) instead of one. So, an operation gives a result (a value)
when all its arguments are fixed. The number n of arguments
of an operation is called its arity ; the operation is
called n-ary. It is called nullary if n=0
(it is a constant), unary if n=1 (it is a function),
binary if n=2, ternary if n=3...
Nullary operations are useless as their role is played by their
unique value; we shall see how to construct those with arity
> 1 by means of functions.
The value of a binary operation f on its fixed arguments
named x and y (i.e. its value when its arguments are
assigned the values of x and y), is denoted
f(x,y).
Generally, instead of symbols, the arguments are represented by the left and
right spaces in parenthesis, to be filled by any expression giving
them desired values.
Other languages:
FR : 1.2. Variables, ensembles, fonctions et opérations