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?