Exploring the Meaning of "Free Variables" in Maths

[0rthodontist]

Note: this could go in the philosophy of science but I'm more interested in the mathematical answer

Edit: no, I've changed my mind. I think this would be better placed in the philosophy of science forum.

The concept of a "free variable" is easy to understand: it's something that can vary while the other quantities remain fixed. But what does that really mean?

On a basic level, free and bound variables are defined in logic, and I have no confusion there. But is that the same kind of meaning as, say, the variable you are integrating or differentiating with respect to? Naively, it seems like there should be no such thing as a free variable, since every variable represents a distinct number. The concept of one thing being "free" while others are not doesn't make a lot of sense to me, except for the fact that it makes sense. So what does it mean in mathematics?

I have to admit that I haven't thought about this a whole lot, just from time to time.


Maybe my language isn't very precise in the above. I mean free as opposed to fixed, as in you might say let 2 sides of a triangle be fixed and the other side vary. I don't know if I was using the term "free" as it is generally used.

On second thought it seems that "free" (as I meant it) in an integral or derivative really means logically bound by the definition of integral or derivative. Is that right? And is that generally the case wherever the concept of one variable "varying" and the other being fixed appears?

 

[honestrosewater]

Don't forget that you need to specify what a variable varies over, i.e. which values it can take.

What if you wanted to prove that every natural number has some property? You couldn't check each case since there are infinitely many of them. So being able to say "for all x in N, x has the property P" is helpful, yes? That's at least one thing that I would consider variables to be: things in your language that allow you to make quantified statements (all x, some x, exactly one x, no x, most x, etc.) about the members of a set or class.

Note that, in logic, free and bound are used specifically to refer to whether or not a variable falls within the scope of a quantifier. In "for all x, x > y", where x and y are variables, x falls within the scope of the universal quantifier (for all), or x is bound by that quantifier, while y does not fall within the scope of any quantifier, or it is free. I'm not sure if that's how you meant to use them.

 

[tacman]

In mathematics, the term "free variable" refers to a variable that is not explicitly defined or constrained within a given equation or system of equations. It is a variable that is allowed to take on any value, and is not dependent on any other variables in the equation. This concept is important in understanding the nature of mathematical equations and the solutions they represent.

To better understand the meaning of free variables, let's look at an example. Consider the equation x + 2 = 5. Here, x is a free variable, as it is not defined or constrained by any other variables in the equation. It can take on any value, and when we solve for x, we get x = 3. This means that 3 is just one possible solution to the equation, as there are infinitely many values that x could take on.

In contrast, a bound variable is one that is explicitly defined or constrained within an equation. For example, in the equation y = 2x, both x and y are bound variables, as they are defined in relation to each other. For any value of x, there is only one corresponding value of y that satisfies the equation.

The concept of free variables is also important in calculus, particularly in the context of integration and differentiation. When we integrate a function, we are essentially finding the area under the curve. In this process, the variable of integration (usually denoted as dx) is considered to be a free variable, as it is not explicitly defined in the equation. This allows us to integrate with respect to any variable, as long as it is defined in terms of the variable of integration.

Similarly, in differentiation, the variable we are differentiating with respect to (usually denoted as dy/dx) is considered a free variable, as it is not explicitly defined in the equation. This allows us to find the rate of change of a function with respect to any variable, as long as it is defined in terms of the variable of differentiation.

In summary, free variables in mathematics refer to variables that are not explicitly defined or constrained within an equation or system of equations. They are essential in understanding the nature of mathematical equations and the solutions they represent.