Different Way of Using the Mind
January 11, 2011 Leave a comment
Mathematics has something to teach us, all of us, whether or not we like mathematics or use it very much. This lesson has to do with thinking, the way we use our minds to draw conclusions about the world around us. When most people think about mathematics they think about the logic of mathematics. They think that mathematics is characterized by a certain mode of using the mind, a mode I shall henceforth refer to as “algorithmic.” By this I mean a step-by-step, rule-based procedure for going from old truths to new ones through a process of logical reasoning. But is this really the only way that we think in mathematics? Is this the way that new mathematical truths are brought into being? Most people are not aware that there are, in fact, other ways of using the mind that are at play in mathematics.
After all, where do the new ideas come from? Do they come from logic or from algorithmic processes? In mathematical research, logic is used in a most complex way, as a constraint on what is possible, as a goad to creativity, or as a kind of verification device, a way of checking whether some conjecture is valid. Nevertheless, the creativity of mathematics—the turning on of the light switch—cannot be reduced to its logical structure.
Where does mathematical creativity come from? This book will point toward a certain kind of situation that produces creative insights. This situation, which I call “ambiguity,” also provides a mechanism for acts of creativity. The “ambiguous” could be contrasted to the “deductive,” yet the two are not mutually exclusive. Strictly speaking, the “logical” should be contrasted to the “intuitive.” The ambiguous situation may contain elements of the logical and the intuitive, but it is not restricted to such elements. An ambiguous situation may even involve the contradictory, but it would be wrong to say that the ambiguous is necessarily illogical.
Of course, it is not my intention to produce some sort of recipe for creativity. On the contrary, my argument is precisely that such a recipe cannot exist. This book directs our attention toward the problematic and the ambiguous because these situations so often form the contexts that produce creative insights. Normally, the development of mathematics is reconstructed as a rational flow from assumptions to conclusions. In this reconstruction, the problematic is avoided, deleted, or at best minimized.
What is radical about the approach is the assertion that creativity and understanding arise out of the problematic, out of situations I am calling “ambiguous.” Logic abhors the ambiguous, the paradoxical, and especially the contradictory, but the creative – mathematician welcomes such problematic situations because they raise the question, “What is going on here?” Thus the problematic signals a situation that is worth investigating.
The problematic is a potential source of new mathematics. How a person responds to the problematic tells you a great deal about them. Does the problematic pose a challenge or is it a threat to be avoided? It is the answer to this question, not raw intelligence, that determines who will become the successful researcher or, for that matter, the successful student.
Ref: How Mathematician Thinks