Meaning and Formalism in Mathematics

This essay is an exploration of possible sources (psychological, not mathematical) of mathematical ideas. After a short discussion of plationism and constructivism, there is a brief review of some suggestions for these sources that have been put forward by various researcher (including this author). These include: mental representations, deductive reasoning, metaphors, natural language, and writing computer programs.The problem is then recast in terms of the relation between meaning and formalism. On one hand, formalism can be seen as a tool for expressing meaning that is already present in an individual's mind. On the other hand, and the discussion of this point is the main contribution of this paper, it is not only possible, but a standard activity of mathematicians, to use formalism to construct meaning and this can also be a source of mathematical ideas.Although using formalism to construct meaning is a very difficult method for students to learn, it may be that this is the only route to learning large portions of mathematics at the upper high school and tertiary levels. The essay ends with an outline of a pedagogical strategy for helping students travel this route.This revised version was published online in September 2005 with corrections to the Cover Date.