Problems of Reification: Representations and Mathematical Objects

Had Bishop Berkeley, as many fine minds before and after him, not criticized the ill-defined concept of infinitesimal, mathematical analysis — one of the most elegant theories in mathematics — could have not been born. On the other hand, had Berkeley launched his attack through Internet, the whole foundational effort might have taken a few decades rather than one and one-half centuries. This is what we were reminded of when starting our discussion. Like Berkeley, we were dealing with a theory that works but is still in a need of better foundations. Unlike Berkeley and those after him, we had only a few months to finish, and we had e-mail at our disposal. Needless to say, the theory we were concerned with, called reification, was nothing as grandiose and central as mathematical analysis. It was merely one of several recently-constructed frameworks for investigating mathematical learning and problem solving. The example of Bishop Berkeley taught us there is nothing more fruitful than a good disagreement. Thus, we decided to play roles, namely to agree to disagree. Since we are, in fact, quite close to each other in our thinking, we sometimes had to polarize our positions for the sake of a better argument. The subject proved richer and more intricate than we could dream. Inevitably, our discussion led us to places we did not plan to visit. When scrutinizing the theoretical constructs, we often felt forced to go meta-theoretical and tackle such basic quandaries as what counts as acceptable theory — or why we need theory at all. Above all, we enjoyed ourselves. We also believe it was more than fun, and we hope we made some progress. Whether we did, and whether our fun may be shared with others, is for you to judge. The following excerpt appears in Research in Collegiate Mathematics Education. In it I speak about the " fiction " of multiple-representations of function—I do not speak about reification 2 Thompson & Sfard as such. However, I think what I said about functions is applicable in the more general case of reification, too: That we experience the subjective sense of " mathematical object " because we build abstractions of representational activity in specific contexts and form connections among those activities by way of a sort of " semantic identity. " We represent to ourselves aspects of (what we take to be) the same situation in multiple ways, …