What does "for art to come up with some vocation" mean?
1. Representational theories of art treat the work of the artist as similar to that of the scientist. Both, so to speak, are involved in describing the external world. But by the nineteenth century, any comparison between the scientist and the artist was bound to make the artist look like a poor relation in terms of making discoveries about the world or holding a mirror up to nature. ( ② ) Here, science clearly had the edge. ( So, there was a social pressure for art to come up with some vocation that both distinguished it from science and, at the same time, made it equal in stature to science. ) The notion that art specialized in the expression of the emotions was particularly attractive in this light.

Question

What does "for art to come up with some vocation" mean?
1. Representational theories of art treat the work of the artist as similar to that of the scientist. Both, so to speak, are involved in describing the external world.  But by the nineteenth century, any comparison between the scientist and the artist was bound to make the artist look like a poor relation in terms of making discoveries about the world or holding a mirror up to nature. ( ② ) Here, science clearly had the edge. ( So, there was a social pressure for art to come up with some vocation that both distinguished it from science and, at the same time, made it equal in stature to science. ) The notion that art specialized in the expression of the emotions was particularly attractive in this light.

Answer

Here, “vocation” means “purpose” or “raison d’etre.”   FYI, although “raison d’etre” is French, it is used in English as a loan phrase and means “reason for being” or “reason for existing.”   The idea of the sentence is that art needed to find some way to justify its existence by claiming to serve some important purpose.

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Meaning You Aught To Take a Vacation

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It seems to be using definition #3 here https://www.merriam-webster.com/dictionary/vocation  but that is very rare. It is usually used to mean the 2nd definition, or sometimes the first.