In the course of debating SteveH over the last few days, I had my own Aha! moment. I'm beginning to realize that the investigations I've been writing for several decades for my students were rudimentary examples of a somewhat new approach to finding mathematical 'truth' known as Experimental Mathematics. The Wikipedia article gives an excellent account of this from which I will excerpt the following:
Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit."
In one sense, mathematicians have always practised experimental mathematics. Existing records of early mathematics, such as Babylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.
Experimental mathematics as a separate area of study re-emerged in the twentieth century, when the invention of the electronic computer vastly increased the range of feasible calculations, with a speed and precision far greater than anything available to previous generations of mathematicians. A significant milestone and achievement of experimental mathematics was the discovery in 1995 of the Bailey-Borwein-Plouffe formula for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous proof found.
 Objectives and uses
The objectives of experimental mathematics are "to generate understanding and insight; to generate and confirm or confront conjectures; and generally to make mathematics more tangible, lively and fun for both the professional researcher and the novice".
The uses of experimental mathematics have been defined as follows:
- Gaining insight and intuition.
- Discovering new patterns and relationships.
- Using graphical displays to suggest underlying mathematical principles.
- Testing and especially falsifying conjectures.
- Exploring a possible result to see if it is worth formal proof.
- Suggesting approaches for formal proof.
- Replacing lengthy hand derivations with computer-based derivations.
- Confirming analytically derived results.
High-speed computers allow for a different approach to 'proof', an idea that I used to find inconsistent with the formal structure of mathematical logic and truth. However, I'm beginning to open my mind to the possibilities afforded by technology and numerical analysis. In fact, I often wondered why, as a student, I was only shown the finished product, the formal abstract theorem, not the slightest hint of the trial and error and conjectures that were an integral part of the derivation. When I became a teacher, I continually reminded my students of this important facet of mathematical truth - the research aspect. Even though part of me continues to resist the idea of a computer-assisted proof by consideration of a large but finite number of cases, I cannot ignore the most remarkable result of this approach:
"The discovery in 1995 of the Bailey-Borwein-Plouffe formula for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous proof found."More to follow...