What is art? is a question about those activities that are considered artistic.   Here are example of such activites:

Liberal education.   Starting in Middle Ages, from the 12th century extending into the present, universities brought together the sciences, mathematics, design studies (mainly architecture), philosophy, creative studies (painting and sculpture) and literature in a curriculum that was considered liberating.   Such an education aimed at liberating the mind, freeing it from a reliance on opinion and encouraging logical reasoning as a means of describing findings. 

A recent study of liberal education has identified the following outcomes of such an education:

knowledge of human culture and the natural world that includes the sciences (traditionally, chemistry, biology, physics and astronomy), mathematics (mainly, geometry and algebra reaching toward the more recent study of mathematical analysis, topological spaces and algebraic structures), humanities and the arts.

liberal arts skills include written and oral communication, critical and creative thinking, quantitative and informative literacy, individual and social responsibilities. For more about this, see

http://libres.uncg.edu/ir/uncg/f/umi-uncg-1627.pdf

Dialogical art:  Works of art typically provoke dialogue among viewers, typically in  response to a finished object.    In dialogical art projects, the conversation becomes an integral part of the work itself.   This form of art produces new and unanticipated forms of collaborative knowledge.   For more about this, see

http://ualresearchonline.arts.ac.uk/6063/1/MBRADFIELD_Thesis_SPREAD_24_06_13.pdf

Polyform art:  The term polyform refers to collections of repeated edge-connected shapes that cover a part or all of a Euclidean space.   The space can be either flat surface in the plane or a volume.   Recently, polyforms have been introduced by D. Markovic in paintings (see, for example, the attached image).   Polyform art provides a powerful vehicle for mathematics education, since it offers imaginative explorations of space using various combinations of geometric shapes full of colour, pleasing to eye, and encouraging an intuition of the infinite reaches of space as polyforms stretch toward the edges of a visual field.    For more about this, see

https://www.researchgate.net/profile/Markovic_Doko

Creative mode of knowing and learning:  Recently, art has been described as a creative mode of knowing.    See Section 8, starting on page 112, in

http://www.neurohistoriasztuki.umk.pl/pliki/danielleboutet.pdf

Art as communication:  In his essay What is art? Tolstoy contended that art is communication of the feelings of a creator of a work to an audience.   A measure of the value of a work of art is proportional the number of people influenced by the work.    For more about this, see

http://drum.lib.umd.edu/bitstream/1903/1474/1/umi-umd-1595.pdf

The problem with Tolstoy's view of art is that it is limited to emotion.   Obviously, there are a considerable number of elements reflected in works of art besides an effort to convey emotion.   Examples of non-emotive elements of works of art are the Desarguesian (projective) geometry of polyform art, the use of perspective in Italian art, the use of the ciascuro effect (illuminated objects in the presence of a light source), the spatial 3D geometry of vaulted ceilings in ancient cathedrals and mosques and the symmetries, elegance, vision and depth of various forms of Mathematics such as theorem of Pappus of Alexandria (ca. 3rd Century AD) that yields what is known as the Pappian line (see the attached drawing).

Even with all these points of view, it is apparent that there is no clear answer to the question What is art?   The same term is used to describe painting, sculpture and architecture, which appear to have something in common to merit our calling such works artistic.    But then this term is also use to describe pure mathematics.   To say that pure mathematics is a form of art suggests something quite different from what Tolstoy had in mind.    It is also the case that chemistry, physics and astronomy are forms of art.   In what sense, are these sciences forms of art?   So it appears that there are very different forms of art.  What are they.   What is meant by the term art in each of its forms?

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