## Posts Tagged ‘**How Mathematicians Think**’

## A dynamic organizing principle

The mathematical idea acts as a dynamic organizing principle on a certain body of mathematics. It is used as much to generate questions and further ideas as it is to answer questions, that is, to generate theorems. The activity that is generated by the idea is open-ended in the sense that future research is always possible. However, the potential of the idea may be optimized in certain mathematical results…A strong idea may have numerous optimal results associated with it. Moreover, this idea may be joined with other ideas to produce even stronger results. The whole matter is a dynamic flux of mathematical activity with occasional points of relative stability that we call theorems.

The usual sequence of definition, axiom, theorem, proof does not apply to the mathematical idea. The idea may not even precede the result. It is conceivable that the result comes first. In trying to understand the result…we may become aware of what we have called the “idea.” When we have isolated “what is really going on” we would then proceed to apply that principle as widely as our imaginations and knowledge will allow. I conclude that mathematics is not merely a body of facts arranged and justified by a stringent logical structure. The logical structure of mathematics gives theorems their stability, but when we remove logic as the focal point of mathematics and replace it by the “idea” we see that the seeming stability of mathematics is not absolute. The data can be structured in many different ways corresponding to what we are interested in and to the mathematical ideas that arise to do the structuring. These ideas arise in response to the question, “What is going on here?”