I shall attempt to show the relationships that hold among these four approaches by referring the reader to Figure 1. Using this chart, a research director can assign different aspects of a research project to different researchers on the basis of the methodological skill in which they excel. The researcher can also assign values to the predictability factor of each methodology--whether 'high', 'medium', or 'low'.

Inventions and innovations which rely predominantly on the application of a known technology would have the highest predictability of success in relation to outcome. For example, the aforementioned R&D staff of an American automobile factory could safely predict that their electronic fuel injection unit would be successful because the technology on which it was based was already widely in use. Innovations which rely predominantly on exploration, however, would have the least predictability of success in relation to outcome. A case in point is the discovery of the ceramic high temperature superconductor.

Inventions and innovations which rely predominantly on logical reasoning would have a moderate predictability of success in relation to outcome. For example, both Edison and the Wright brothers could anticipate, with only a moderate degree of probability, that their inventions and innovations--respectively, the phonograph and the airplane--would actually work. On the other hand, inventions and innovations which rely predominantly on mathematical calculation tend to have a medium to high predictability factor. Einstein's theory of relativity, for example, although mathematically proven, still requires additional experimental proof. In summary, as Figure 1 suggests, every invention and innovation can be analyzed both according to the kind of methodological skill required to implement it and the degree of predictability that the innovation will yield a desired outcome.

A scientist’s job is to uncover the unknowns of nature. Basically there are three types of scientists, namely: (1) logicians such as Pythagoras, Thomas Edison, the Wright brothers, and Sir Isaac Newton and (2) mathematicians such as Albert Einstein, and (3) scholars such as those who, by exploring, uncover a certain gene which causes a specific form of cancer, or those who, by applying known knowledge, have invented the electronic fuel injection unit. Each scientist type has their own way of training and each capable to handle their own type of problems.

In the early history of human inventions and innovations, exploration and logical reasoning were the main methodologies used. Most Greek scholars were also good logicians, and mathematics was still in its infancy. This situation continued until the middle ages, which witnessed a rapid development in mathematics. Since then, mathematics has continued to gain in importance. From Newton’s time till the end of 19th century, the main methodologies were exploration, logical reasoning, and mathematical calculation. At the beginning of 20th century, education became compulsory, which forced students to memorize known knowledge rather than engage in "free thinking". Research centers were established which hired only those having academic credentials. By this time, a lot of knowledge had accumulated and people began to apply it. On the other hand, logical reasoning became increasingly neglected both by the school system and by research establishments. Yet logical reasoning has an important place in research. Many problems which we have failed to solve by other methodologies can be solved by logical reasoning. This can be demonstrated in the practical problem-solving exercise that follows.