What is the definition of $(dx)?$ I often see in calculus that we treat the symbols $dx$ and $dy$ as infinitesimal numbers but I don't really accept this definition because in the real number line there's no such thing as infinitesimals.
I read in Calculus by Michael Spivak that the $dx$ has no meaning [Page $264$].

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In MathJax
The symbol $dx$ has no meaning in isolation, any more that the symbol $x \rightarrow$ has any meaning, except in the context $\lim \limits_{x \to a} f(x).$

So Can someone explain it in explicit way and the true definition for it ?

the exact definition is this:

if $\mathbf{x}$ denotes the general point of $\mathbb{R^n}$, we denote
the $i^{th}$ projection function mapping $\mathbb{R^n}$ to
$\mathbb{R}$ by the symbol $x_i$. Then $dx_i$ equals the elementary
1-form $\tilde{\phi_i}$ in $\mathbb{R^n}$.

