One theory of art that Ovid would certainly have known is found in Aristotle's view of art as mimesis, or imitation. In relation to the competition of Athena and Arachne, it seems necessary to distinguish between art as imitation and another kind of art.
The first kind -- Arachne's -- offers us imitation as fidelity to appearance -- making a careful copy of something. If you are a very good imitator, your copy can be said to rival Nature. Trompe l'oeil art is, in a sense, the ultimate in imitative success, since it actually fools us into thinking something is real, when it's an artistic illusion. Arachne's art is of this kind -- not only do her images rival those of nature, but there is a double rivalry, because what she's imitating is the power of the gods to imitate natural things - bulls, golden showers, horses, etc. Just as Zeus successfully impersonated a bull and seduced Europa, so Arachne's tapestry seduces the viewer into believing one is actually seeing Zeus as bull seducing Europa.
Of course this imitation of divine imitation is also a distinct echo of Ovid's tale of Zeus and Europa which ended Book 2 of the Metamorphoses, so there is a mirroring of imitative reflection that verges on a mise en abyme. The endless mirroring suspends the viewer in an undecidable predicament, which nonetheless requires a decision. Think of the final scene of Orson Welle's The Lady from Shanghai, where the characters shooting the guns have to tell, but can't tell, if they're aiming at the actual person, or at a reflection:
The labyrinth of Arachne's tapestry leads one into a world where all is imitation, cheat, illusion, and virgins are forever being seduced by clever divine rapists.
Despite the undeniable similarity of Arachne's subject matter to that of the very book in which she appears, we should at least look at Athena's image before deciding that the theory of art as mimesis in Arachne's sense of it is Ovid's own.
Clearly Ovid is setting up an opposition between Athena and Arachne to at least offer an alternative theory of art; so what can Athena's tapestry tell us?
At first glance, her image seems very much in the same vein of imitation. Athena has presented the story of how she won a contest with Poseidon at Athens. We see Poseidon striking water from the Acropolis, and then Athena striking the rock and giving the Athenians the precious olive tree, a living source of health and wealth, of culture and strength.
One thing about the goddess's image should be clear: the excellence of the work does not lie solely in its verisimilitude. Doubtless the Athena in the tapestry resembled the goddess who wove her, but that's not what really matters here. What matters is the act that that this is an image of -- the act of making, creation, poesis. Athena didn't merely put a copy of an olive tree in Athens, as if the city could have found another one elsewhere. She is putting something brand new into the world. The gift of the goddess is not something anyone else could have given the city, it is a novum, a thing so extraordinary that even the gods marvel at it.
The story Athena tells in her tapestry culminates in the people choosing her as their patron, an act that is marked by naming the city, and themselves, after Athena. Not only is there a new kind of tree, but a new word. Athena's image is about this non-mimetic creative power, the poetic power of naming.
There's another difference between Arachne's mode of art and that of Athena, and it has to do with how, or to what, each directs our attention. Arachne's art is essentially about itself. It says, "look at how well I have feigned this story of a god feigning to be a bull." Athena's image is not very interested in making a faithful copy of something, because it's concerned with something that is fundamentally other than copying. It's interested in the powers of imagination. As an image, it points beyond itself, it tells us not to look at how well it's copied some event, but instead to think about an act of origination, the origin of "Athens."
How are we to understand what Ovid is telling us about art? It seems that there are certainly two kinds, or theories, of what art can be, but which is to be preferred? Would it not seem that the similarity of Arachne's images to events described in his poem would tilt the balance toward the imitative art of the girl? Or is this yet another twist of Ovidian irony, in which he's suggesting that if we read his poem as an imitative work of fancy, we are getting it all wrong? Is Ovid perhaps giving us a hint about how his poem is to be read? Or is he just offering a kind of sampler of aesthetics, saying, "here are two kinds of art"?
Given that these two modes of art seem in some sense to be opposed -- in one, the image is about its own intrinsic "imageness," in the other the image points to something beyond mere imitation -- perhaps a decision is important, and not simply for aesthetic reasons. Note the relation of each artist here to the theological, for example. And then there's the relation of all of this to hubris.
The violent climax of Ovid's story might make us suspect that the two modes of "art" -- in the larger sense to which we have been led -- are not simply opposite, but fundamentally incompatible. At this point we have to ask whether the brutal conclusion to the contest resolves the enigma posed by the conflicting webs of Athena and Arachne, or destroys any hope of doing so.