Mathematics may well be the most widely misunderstood branch of the sciences. Young people contemplating careers in science find it difficult to imagine what a research mathematician really does. One common image features a mathematician programming a computer to do difficult calculations. Another pictures a lone mathematician working in isolation on ideas so abstruse that no normal person could comprehend them. Neither of these images comes close to capturing the spirit of mathematical inquiry. It is certainly true that many modern mathematicians use computers to perform numerical and geometrical experiments. But this experimental phase is only one component of the mathematical experience, and the use of computers in this phase is the exception, not the rule. Nor is it true that mathematicians work in isolation. Indeed, a distinctive feature of mathematics is the open sharing of ideas within a community nurtured by a common language, shared values, and shared goals. All too often, mathematics is presented to students as a highly polished and well organized collection of definitions, algorithms, and theorems. The long struggle of many individuals that culminated in this “finished product” remains a hidden secret. Students rarely learn of the dynamic nature of mathematics, nor do they see the creative side. They do not come to understand that mathematics is a thriving field of research activity which is progressing faster today than at any other time in its distinguished history.
These misunderstandings may stem from the fact that mathematics deals so heavily in ideas. In mathematics, perhaps more than in any other science, research is an activity of the mind. The primary goal of the mathematician is to understand – to discover the essential ingredients of complex systems in order to render them simple, to find order within apparent chaos, to draw analogies between different structures, and to find connections between seemingly disparate branches of mathematics and science. To make interesting new contributions in the field of mathematics requires a healthy mix of creativity, experience, and hard work. We aim to engage young people in the struggle to understand an intricate collection of significant mathematical ideas. PROMYS participants come with unbounded energy and are anxious to grapple with challenging ideas. At the beginning of their investigations, they may sometimes feel lost and perplexed. But through carefully designed problem sets, we hope to subtly direct PROMYS students along productive paths towards understanding— to suggest that they experiment with examples and formulate conjectures, to encourage them to ask good questions, and to help them realize that through careful thought they can penetrate formidable obstacles and invent their own answers to difficult questions. The attitudes acquired through this experience will be even more valuable than the particular topics mastered.
Professor Glenn Stevens
Director of PROMYS