I guess it would break against some basic ideas of math.
The thing is: in math a value - say 3 is unique. There is only
one number 3. All expressions referring to 3 refer to the same 3.
Otherwise sets will not be same if the elements are the same.
(definition of natural numbers by sets)
assignment is definition... it means defined as, used rather frequently...
Say you want to define the variable g as gravity:
g := 9.8 m/s^2
now we know anywhere in the following math, g means 9.8 m/s^2
Some people just write at the top of the work:
"g is defined as 9.8 m/s^2"
but most mathematicians hate English and prefer short hand symbols for everything.
This even occurs in functional math. We can define a function that can take a function, and then define which functions you may use with in said function. Then this can build to algorithms where those definitions can change.
step 1 : define formula f
step 2 : define dependent formula g
step 3 : perform f with input g
step 4 : if result is positive g becomes h, else done
step 5 : goto step 3
again, all of this would be in symbols... because mathematicians tend to prefer symbols.
Give you common examples that we all use constantly and take for granted.
pi := 3.14159...
e := 2.71828...
golden ratio := (1 + sqrt(5)) / 2
we don't usually need to write out that these are defined as such... but if we HAD to, then well... here it is.
Math seldom deals with a single constant like '3'. It tends to deal with number sets, represented in variables and restricted by the set said set is defined as. Those sets can be defined as a constant, a number set like R (all real numbers), functions, all sorts of things. These things are stored inside some variable representation so that we don't have to write out (1 + sqrt(5) / 2) every time we need to reference it in our formula.
This post has been edited by lordofduct: 16 October 2012 - 08:36 AM
So you don't mean identity?
That notation is not familiar to me, and I believe it's more like writer's notation than standard math.
Math seldom deals with a single constant like '3'. It tends to deal with number sets, represented in variables and restricted by the set said set is defined as.
I didn't get all of that, but that's where OO typicallu goes wrong: It tahes the class of elements as a set, and tries to define operations as properties of the elements, whereas in math and algebraic construction typically consists od a set of elements AND operations between them (separately from the set and elements).
Basically it's not even question of operations, but mapping from one or more sets to another set (sorry, I'm not too familiar with the english terms of math - mapping into a set/onto a set/...
What I ment about the definition of 3 and the sets was ZF.
Basically math doesn't describe changes but snapshots. Even changes are described as invariants in the change, or snapshot of snapshots where each of the snapshots have different value for some parameter (one of which may be called "time").
You just seem to have different idea of what the word "definition" means than it means in math.
Remember that math does not interpret the "meaning of the entities".
The consepts of math and the concepts of programming are different, even if they often look similar.
The definition in math is more close to textual replacement macro in programming.
The symbols below are usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics) but in some situations a different convention may be used.
x := y, y =: x or x ≡ y means x is defined to be another name for y, under certain assumptions taken in context.
That seems to me like usually ":=", "=:", or "≡" is used to express identity.
That is, alias names for the same thing.
To be more exact, math is almost all about rewriting rules, and the identity is exactly that:
A very strict and "unconditional" rewrite rule.