The following is taken from the Introduction of Experimental Mathematics: A Discussion (J.Borwein, P.Borwein, R. Girgensohn, and S. Parnes).
Philosophers have frequently distinguished mathematics from the physical sciences. While the sciences were constrained to fit themselves via experimentation to the `real' world, mathematicians were allowed more or less free reign within the abstract world of the mind. This picture has served mathematicians well for the past few millennia but the computer has begun to change this. The computer has given us the ability to look at new and unimaginably vast worlds. It has created mathematical worlds that would have remained inaccessible to the unaided human mind, but this access has come at a price. Many of these worlds, at present, can only be known experimentally. The computer has allowed us to fly through the rarefied domains of hyperbolic spaces and examine more than a billion digits of but experiencing a world and understanding it are two very different phenomena. Like it or not, the world of the mathematician is becoming experimentalized.
The computers of tomorrow promise even stranger worlds to explore. Today, however, most of these explorations into the mathematical wilderness remain isolated illustrations. Heuristic conventions, pictures and diagrams developing in one sub-field often have little content for another. In each sub-field unproven results proliferate but remain conjectures, strongly held beliefs or perhaps mere curiosities passed like folk tales across the Internet. The computer has provided extremely powerful computational and conceptual resources but it is only recently that mathematicians have begun to systematically exploit these abilities. It is our hope that by focusing on experimental mathematics today, we can develop a unifying methodology tomorrow.