Understanding constants (and parameters) I'm very confused by the idea of representing a symbolic constant as in a 'general way'. Primarily, it is the idea of representing a 'constant' with a symbol that has no fixed value. It's something that has always annoyed me since school. A classic example is:
$F(t)=ma(t)$ where $m$ is a constant
In Logic, a 'constant' in any language is a symbol that has a particular interpretation, when we use it we understand that it has to essentially be a another name for a particular number, e.g having two names for the same thing, if $a$ is constant and $a=10$  we know that '$a$' and '$10$' are both names for the same thing, and when we define our logical language this is defined.
In conventional Mathematics we don't seem to know whether we define constants in a way that they have a particular value or if we want to allow for various scenarios where they can differ, I often see things like 'for different values of the constant $a$', if we are considering different values surely it should be a variable?
If we limit the use of the term to a function, it makes more sense, we can define 'parameters' and in this case they are 'constant' for a specific function, i.e. to change their values defines $f$ to be a new function, so for a particular function $f$ then the parameter has to have a particular value, and we can treat it as a variable, but one that defines a new function, so in the context of a function if we have it as a 'constant'
$f(x)=ax$
But we tend to define 'constants outside of function definitions, and it becomes difficult to understand, another definition (given by one of my text books) is equally confusing:
'A constant represents an unknown, but a variable represents many numbers'
Here's another example from one of my engineering books
They start with a number that you 'think of', and say that it's unknown
'except to the person who thought of it', and describe this as a 'constant', he uses the letter $a$ for this number.
He then defines the basic rules of algebra such as:
$x(x+y)=x^2+xy$
And defines that $x$ and $y$ are variables, because they 'represent one, of a group of many numbers, even though this would be true for a given constant as well. Why isn't $a$ a variable then? It can vary if you think of another number?
If we have a constant $a$ and we are looking at the structure of something for a 'general' undefined value that can change from scenario to scenario, what is the difference between a 'generalized constant' and a variable? in the context of functions it changes how we can define them, but if we're allowing them to take different values depending on different static situations, surely they have the ability to change from scenario to scenario? In which case why are they 'constants'? Is it better to call them parameters?
Edit:
After a bit of reading I wonder if the following is a good way to view it:
Variable: Something I will use to map between elements of sets and look at different values, or assign different values, generally as arguments of functions, changes within a 'context'.
Parameter: Something that works very similar to a variable will be constant in a given 'context' so if we have a particular context we may change the value to 'jump' between them, or to categorize a family of functions or equations, 'defining a new context'.
Constant: A constant symbol, essentially take a parameter and give it a value, you're now in a particular context and it has a particular value, and that symbol refers to that value, so if $m=1$ at all times in our context $m$ is constant.
Parameters allow me to do things with the variables as if they were explicit numbers and get expressions that work in all 'contexts'
 A: You can think of it in terms of doing an experiment. In an experiment, you will change some quantities and keep some others constant.  The constant quantities are called constants of the experiment and the changed quantities are called variables of the experiment.
Of course, a variable like "pressure" might be a constant of one experiment and a variable of another.  And of course the value of the constant might be different in different experiments.
That might make "constant" seem like a bad name.  On the other hand, the reason we call it "constant" is because holding it constant during the experiment is what makes the experiment work.
A constant stays the same during the experiment to make the experiment work. A variable changes during the experiment to make the experiment work.  (A constant might have different values in different experiments, but that's changing between experiments, not during.)
To know which quantities are acting as variables and acting as constants, you have to know what the current experiment is. And of course you can have experiments within experiments.
