The crux of Conrad Keely's argument against abstract art lies in this vague point:

"This led me to wonder, were we taught how to appreciate abstract art? If so, it would appear that its appeal is intellectual rather than intuitive."

By the word intuitive, I am not sure whether Keely refers to subconscious brain function or to the innate genetic mannerisms of the human brain. Regardless, I will show how abstract art appeals to both realms of quality-detection, and can do so with equal or greater success as classical art.  Keely is correct in noting that poor abstract art, or that is not generally well received by human observers, exists. I'm sure Keely has also seen plenty of classical pieces that are shit.  Abstract art, done well, appeals to the so-called intuitive functions of the brain as well as the learned intellectual sense. That Keely doubts this fact says more of his own personal interest in art, not of the way of the world.

Consider the art of Jackson Pollock. Ignore, for Keely's sake and for the sake of the scientific method, his experiment with the four year old girl. Although I would never use the results of such an experiment as research data, I have a proposition for Keely. Show the same four year old girl two pieces of abstract splatter paint art: one of Pollock's and one of an unknown artist. I do not wish to subjectively predict that the girl will choose Pollock's piece every time, but I do not have to... because she will.  The little girl is intuitively inclined to choose Pollock's piece as the more appealing piece each time she is confronted with a choice. The reason is purely neuropsychological. Human brains have an innate set of shape detection programs. Special clumps of shape detection neurons specializing in picking out circles, dots, lines, various simple curves and angles, fractal tree patterns, generic human faces, and a secret little ratio called "the golden ratio." Thousands of years swinging in the trees probably explains the fractal tree pattern recognition ability of the human brain, just as trans-generation evolutionary experience in a lush organic world influenced the mind's draw towards anything with a size ratio of about 1 : 1.6.  Since non-man-made objects on the earth, such as leaves, seashells, and minerals, tend to grow in a size ratio averaging at about the ratio of one to one point six, the human mind has taken on the ability to detect such a distinction without being taught to do so. 

Pollock followed the golden ratio in his work. Even though he may not have known about shape detecting sets of neurons, or even the golden ratio itself, while creating his art, he had a built-in golden ratio detector similar to anyone in his audience. If the art didn't appeal to him, he would destroy the art. This trial and error positive feedback loop left the world with artistic gems honed in on an ancient subconscious pleasure center. 

A similar neurological effect has tantalized onlookers for centuries in Japan's Ryoanji Garden. Instead of appealing to the golden ratio detecting feature of the human brain, the four hundred year old rock garden taps into the tree detecting neurons of the mind. The Ryoanji Garden is in fact highly abstract art, yet it appeals to the mind's intuitive senses, making it a success

Researchers have traced the intuitive appeal of classical music performed by a human piano player versus a computer simulated piece to simple mathematical reasons. While romanticists and other uninformed listeners believe the superiority of human performed music is due to the unique beauty and creativity of the human mind, one can alter robotic pieces of music with simple equations to mimic the "human touch." It turns out a human performer will simply filter the piece of music through a handful of simple mathematical functions. Working with multiple pattern filters simultaneously, the musician, be it a computer or a human, can create a piece of art more appealing to the human listener.

Just as classical music utilizes the intuitive easter eggs, if you will, of the human mind, so do abstract artists. The abstract artist’s vehicle for achieving the result may seem a bit unconventional to classical enthusiasts such as Keely, but few can deny the similar mathematical power of all “good art,” be it a canvas splattered with seemingly random bursts of paint or a curiously fascinating portrait.

Thanks to ac_hyper for the first reference