I am reading ordinary differential equations from a book which says
Hence, $dy = f'(x) \Delta x$, we call $dy$ the differential of $y$. As the differential $dy$ is a function of two independent variables $x$ and $\Delta x$, we indicate this dependency by $(dy)(x, \Delta x)$.
Excuse me but what the hell is this? It goes even further by saying
The differential of $y$, written as $dy$ (or $df$) is defined by $$(dy)(x, \Delta x) = f'(x) \Delta x$$
And now I am completely confused. Do we define $dy$ twice? Or what is $(dy)(x, \Delta x)$? Is it a sort of relation as it says "we write a dependency in this way", so it is a relation in terms of set theory? What is it exactly?
Why do $dy$ and $(dy)(x, \Delta x)$ have the same definitions? Can we say that $dy = (dy)(x, \Delta x)$ then? If so, what is the point of this mess? Thank you if you read up to this point and I will be more thankful if you help me to understand what I am missing.