Way, way, way older.
Writing is an incredibly novel development. There are still today linguistic communities without writing, although they're becoming much rarer. Writing was unknown in parts of the world until quite recently - the Aborigines of Australia didn't have it, nor did many pre-Columbian civilisations.
It's a safe bet that writing first emerged out of a need for accounting and this thus closely tied to larger agricultural civilisations. That's why we find it in places like Mesopotamia and Egypt first. Of course, those weren't alphabets yet, but logosyllabic writing (characters could stand for either meanings or syllables). The alphabet is a specifically Phoenician innovation, although similar systems (such as abugidas), which are also phonetic, have emerged elsewhere.