How would a philosopher of math describe what happened when ancient mathematicians discovered (?) the number zero?

Ancient mathematicians (Babylonian, Egyptian, Greek) did not in fact discover the number zero. The discovery is thought to have been in India, which was the first place to treat zero like any other number (rather than as a placeholder), sometime between the 6th and the 10th century CE. It is thought that this advance required abstract thinking that was perhaps facilitated by Indian Philosophy. The discovery spread to the West through Arabic mathematics.

I think the answer will depend on which philosopher of math you ask. As you seem to recognize, some philosophers of math deny that numbers exist independently of us in such a way that their existence is genuinely discovered by us. Even philosophers of math who think that numbers are discovered might say that your question -- "What happened?" -- is an empirical historical or psychological question rather than a philosophical one. In any case, you'll find relevant material in the SEP entry on "Philosophy of Mathematics" at this link.

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