If something changes by an amount of zero, is it philosophically the same? Ofcourse!
It's time for a thought experiment. You've got a jar, and you've got marbles. You can put marbles in the jar.
Now say we add nothing to the jar, then two descriptions apply that normally would be mutually exclusive; a binary is broken down:
• The total number of marbles in the jar changes – by nothing.
• Change does not occur in the total number of marbles in the jar.
Normally, you can't have changing the amount of marbles in the jar be the same thing as not changing the amount of marbles in the jar. Those are two possible manifestations of a binary, they are naturally exclusive. But at the extremes, that definition is irrelevant. Verbs are the reason why you can have plays on words like this. How? Because reality is not dependent on verbage like 'change occurring'. Since 'changing nothing' and 'no changing' are the same idea, the verb is just a contrive – you don't need it to communicate the meaning. By reasoning at the extremes, we have realized that verbs are just fluff. And, ofcourse, the fact that nothingness can remain constant with the addition of nothing, but that it can also change by nothing, is equivalency. Anyhow, what defines equality? A difference of zero! A change/difference of the amount 'nothing' is just equality. And in the same way, an infinitesimal difference is none at all.