Should Art be intentional and representational to be concidered art?

In an attempt to challenge myself to define art, I became very confused and eventually gave up on that ill-fated attempt. In every way I found a definition, I also found a way to contradict myself. So I came up with the idea, that art can be virtually anything. I don't believe the term beautiful is necessary in a definition in art. I had this same discussion in my design class.

During that discussion I was shown a picture of what I thought was just a urinal, marked R. Mutt. Then later it was explained the urinal in the picture was actually a sculpture submitted to an art exhibition by the great Marcel Duchamp. He used that already made urinal he grabbed off the assembly line, slapped R. Mutt on the front, and named it The Fountain and presented it with his argument of " what is art?". He won his argument and the urinal was indeed accepted into the exhibition.

It seems that it's a given, you see a painting, a drawing, any type of conventional artwork, it's easy to point a finger and say, "now that's art!". However, what about things in nature? It's not intended, but can nature really "intend" something? I say yes and no, nature can't because it doesn't have a mind, but flowers are intended to be colorful. In nature it seems it can't be considered art until somebody paints it, or takes a picture of it and mounts it and names it something. I disagree, I think the art can be the subject in the composition rather than the composition itself.

I strongly believe that art cannot be defined. I don't always believe it needs to be beautiful, because beauty is subjective to the viewer anyways, I mean try to define beauty. Throughout history this question has been asked over, and over again. Still to this day, I have yet to hear a definition that I would take to my grave as the correct definition, but who am I to decide that? Isn't that what makes art so interesting anyways?