# Variable or constant, and Dependence among variables

I've been working with symbolic Mathematics for a very long time, but I still have many small questions relating to the idea of a 'constant' and the idea of change with respect to variables, and their co-dependence. I will warn that this may be slightly difficult to express to you.

Primarily it is the following:

When we are using a symbolic relationship with variables (I'm going to use the most simple definition of the idea as a variable representing an unspecified number) that change, such as the following:

$$y=2x+1$$

Or (ignoring the binding operation of function definition):

$$4f(x)=x^2+\sin(x)$$ (function $$f$$ at the point $$x$$)

Such equations are designed to specify varying quantities, and here we have a relationship that holds for pairs of values $$(y,x)$$ and $$(f(x),x)$$ and in the context of this problem, we have a relation that holds for various values.

It is possible to discuss the situations where two independent values can have the same value, but we cannot draw a distinction between something like $$a=b$$ for two independent varying quantities $$a$$ and $$b$$ and a relation that holds for multiple pairs of values (such as $$y=2x+1$$) where the variables are dependent, however how do we deal with the fact that $$a$$ and $$b$$ are able to change, but in this context we're only discussing a relation between values that they are equal, which they do not need to be, but it is still misleading due to this relation, and almost seems to imply they are not independent.

This raises a point another common thing we do is having a formula such as $$y=2x+1$$ we might want to solve for a value, but when doing this we still use the original variable, almost like we are saying, 'in the context where $$x$$ is...' and then we solve the equation using the original variable $$x$$, this seems somewhat misleading, as the relation e.g. $$2x+1=3$$ is not a relation that can hold for the varying quantity $$x$$ as it can only be existentially quantified, this brings questions about how to represent the idea of change and context once again.

We also use the idea of a 'constant' and we might represent it with $$a$$, which brings the first question, how can it be 'constant' if we literally express it as a variable, any number can go into this expression, again, we can use the idea of context, in a given context we can say similarly have a value '$$a$$' that is 'constant' and talk about the 'point' $$x=a$$ and the limit 'at the point $$x=a$$, once again this idea causes an issue, if we want to talk about many 'contexts' defining $$x=a$$ is misleading as $$x$$ and $$a$$ are 'changing' and again that could imply that $$x$$ and $$a$$ are not independent, which is required if $$a$$ is to be 'constant'. Can we talk about scenarios for different values of a constant, analyzing and replacing the value? Do we need to change the way we talk about them?

How should I approach clearing up this confusing situation regarding notation, and dependence between variables, as a result of the use of 'constants'?

• This may help: math.stackexchange.com/questions/2738360/… Commented Feb 28, 2022 at 21:05
• Here is a question I asked that is very similar to this question. The answer provided me some conceptual satisfaction I think you will benefit from (be sure to read the comments under the answer as well). math.stackexchange.com/questions/3687044/… Commented Mar 1, 2022 at 16:20
• The previous addendum of this Question has been migrated to the OP's follow-up post. @TaylorRendon's comment above pertains to this new post. Commented Mar 4, 2022 at 15:23

1. In the sentence $$2x+7=5,$$ $$x$$ is a called variable. If the sentence is a constraint (i.e., is actually true, and we might be trying to solve it), then $$x$$ is more accurately called a constant with an unknown value!
2. In the equation $$x^2+px=3,$$ $$p$$ is called a parameter or arbitrary constant, or simply a constant to emphasise that its value is fixed within each set of conditions. But since its value is arbitrary and adjustable to generate a family of equations, $$p$$ is really a special variable, even as it can be framed as a constant!
The terminology for these placeholders is informal, and usage is guided by context and framing or intended signalling. In this example, the parameter $$k:=gMm$$ generally varies but is fixed in the context of the inverse proportionality of $$F$$ and $$d^2.$$ In this example, $$m_1,$$ which equals $$P,$$ instantiates $$m,$$ but the three objects have somewhat different ontological statuses.