Mathematics is derived from first principles - abstractions of perceived objects. There is an a priori set of conditions and rules of derivation to generate all of mathematics. The key lies in providing first principles. Where do they come from? Are there rules for generating them?
It is clear that the use of the laws of logic to generate postulates is useful, however, logic leads inevitably to incompleteness and inconsistency. But all of this begins with initial concepts. The point, the natural numbers, the set and its elements, the line, the plane, each one generates a new area of mathematics.
It can be postulated that the drive to create mathematics is the drive to be "right" or "clear". This, in my view, leads to the comparison with poetry.
Poetry is based upon language that arises from immediate experience (either physical or mental). This is not the same as "abstraction" but rather, perhaps, "extraction". The experience of life in the world leads to language that conveys that experience, in toto, to the reader. The reader directly experiences the poem and the poem is generated from the experience of the poet.
However, this must be a careful process, since there is such an importance in clarity. One wishes to be "clear". Hence there is a connection to the drive to develop mathematics.
Analysis and Synthesis Edit
Mathematics seeks to analyze and develop relationships between abstractions. Poetry synthesizes language-in-experience to take a given experience and "put it into" the reader. This is a synthetic process - the joining of language and experience.
In all fairness, there is no way to truly compare the experience of the poet to the experience of the reader reading the poem. Even if the poet and the reader would use different language to discuss a given experience, it is not clear that they would "mean" different things. Additionally, they could have different experiences and convey them using the same language. So we see, there is a kind of faith involved. This is as much the case with logic in mathematics. The rules of logic indicate a kind of faith, as well.
In the end, we see that mathematics and poetry at least have in common the desire to be clear about whatever the subject is - sometimes, regarding abstraction, and sometimes about experiences. In either case, the emphasis is on associating language with experience. Either the drive is to abstract, intellectually, the qualities of experience, or it is to abstract, experientially, the qualities of experience. The difference would be that to abstract mathematically from the world, is to create something derived from many common experiences, whereas to abstract experientially is to derive from specific occurrences.
Mathematics comes from the desire to abstract, while poetry comes from the desire to convey particularities, even when those particularities lie primarily within the imagination of the poet.
Poetry, it would seem, is the result of inward experience being expressed outwardly. Mathematics, on the other hand, is the result of outward experience being expressed outwardly. In other words, Poetry results from experience within the body - body speech. Mathematics abstracts from objects in the world to create its subject.
This, of course, involves naming. Any naming which involves body experience is food for poetry, while naming which exists outside the experience of the body is the contribution for mathematics.
There is a mind-as-body and a body-as-mind. The self must exist somewhere between the two. There must be overlap then. Perhaps mathematics exists without the experience while poetry exists with only the naming of its subjects as abstraction connected with the experience it seeks to express.