It is difficult to imagine mathematics without its symbolic language. It is especially difficult to imagine doing mathematics without using mathematical notation. Nevertheless, that is how mathematics was done for most of human history. It was only at the end of the sixteenth century that mathematicians began to develop systems of mathematical symbols . It is startling to consider how rapidly mathematical notation evolved. Viète is usually taken to have initiated this development with his Isagoge of 1591, and a recognisably (...) modern symbol system was available by the 1630s . In little more than a generation, mathematicians went from writing mathematics in natural language to manipulating symbols much as we do today. The manipulation of symbols became both a source of new mathematics and a mode of mathematical argument. This required profound conceptual changes in both mathematics and philosophy; antique conceptions of number, proof, language, and mathematics had to be replaced or at least suspended. Thus, the development of mathematical symbolism was sufficiently rapid and profound to justify the word ‘revolution’.Moreover, what began as a technical innovation in mathematics soon took on a wider philosophical significance. The last line of Viète's introduction reads ‘To solve every problem’. Viète presumably meant that his algebra would solve every mathematical problem . However, two generations after Descartes, the teenage Leibniz articulated the characteristically modern dream of a general algebra of thought . In this ‘universal characteristic’, conceptual errors would …. (shrink)