When I teach mathematics, I am constantly reminding the students that one of the characteristics that makes math difficult to understand is that the language used is so compact and context sensitive. When we use terms such as group, ideal and field or symbols such an integral sign it is a summary of many words, concepts or actions. Some phrases and symbols also have different meanings depending on the context. Therefore, one cannot understand mathematics without having a complete understanding of the language.
As the coverage of mathematics has expanded, there has been a commensurate increase in the terms and phraseology of mathematical language. Fortunately, there has also been a corresponding compaction of many concepts. A simple example is the comparison between Roman numerals and the Arabic place notation now universally used. Another example that Kvasz describes is the language of algebra, using examples from recent antiquity where the problem and solution are expressed verbally; the superiority of modern notation is obvious.
Progress in mathematics can be measured many ways and is subject to differing forms of interpretation. However, the changes and improvements in the use of language and symbols is a neutral and therefore effective way of measuring mathematical progress. Kvasz puts forward solid and irrefutable arguments that mathematics is fundamentally a rich, dynamic and complex language that all its practitioners must learn and progress in mathematics generally requires expansion of the language. From this book, you will also gain an appreciation for the enormous power that it gives you in the expression of ideas.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
Preface.- Introduction.- Re-codings as the first pattern of change in mathematics.- Historical description of re-codings.- Philosophical reflections on re-codings.- Relativizations as the second pattern of change in mathematics.- A Historical description of relativizations in synthetic geometry.- Historical description of relativizations in algebra.- Philosophical reflections on relativizations.- Re-formulations as a third pattern of change in mathematics.- Re-formulations and concept-formation.- Re-formulations and problem-solving.- Re-formulations and theory-building.- Mathematics and change.- The question of revolutions in mathematics (Kuhn).- The question of mathematical research programs (Lakatos).- The question of stages of cognitive development (Piaget).- Notes.- Bibliography