In summary, mathematics is the intellectual discovery of nature's infrastructure. It consists of theories about ideas that we sense with varying vagueness.
A theory begins with a number of ideas and assumptions, from which its story unfolds via the derivation of more and more logical consequences. The entire process happens within one's mind, relying fully on one's intuition.
The aim in mathematics is to reflect deep laws of nature; that's where its beauty comes from, and that's how it is connected with arts.
What is still missing is a clarification of the terms "logical consequence" and "fundamental idea", which were both used informally all along. Discussing these in detail will be the topic of another essay.