Concepts at the Becoming of Mathematics

• In Event: Roundtable Session 15
  In Roundtable Session: What Is a Mathematical Concept?

Abstract

Objective and Theoretical Framework:

The prevalent uses of the word “concept” in our culture can be traced back, among many other paths, to Aristotele’s taxonomic use of ‘genre’ and ‘species’. This etymology sheds light on how mathematical concepts relate to questions of class membership. This image of concept uses a tree-diagram model, with nodes from which branches bifurcate towards other nodes, and so on. Human beings, for instance, are located at a proper node derived from ‘animal’ and bifurcated towards ‘rational.’ Each node encompasses a class of entities defined by a chain of genre-species. A class is constituted by “blocking” further bifurcations underneath; for example, the class of human beings is constituted by brushing aside such bifurcations as ‘male/female,’ ‘adult/child’ and infinite others. Each class correlates to a ‘concept.’ The web of predications proper to the entities of a certain class forms its concept.
This paper is inspired by Deleuze’s efforts to generate alternative approaches to this pervasive image of a concept and to show how this alternative approach allows for a new way of theorizing mathematics teaching and learning. Granting that this arboreal image of concept does “work” to address common issues, Deleuze highlights its superficiality and frequent thought-impoverishing blockages. A major instance of such blockage is the exclusion of the becoming of entities. Entities show up in this or that node without traces of their birth, genesis, or struggle to be there. Another blockage is the split up between the design of the arboreal diagrams and the actual empirical entities that populate them. The arboreal diagram seems to emanate detachedly from the things it organizes, such as the categories of “rationality” and “animality” given a priori, in order to properly define you or I as a human.

Scholarly Significance:

One can see these same blockages in our theorizing of mathematical concepts. The aim of this paper is to contribute towards an image of (mathematical) concept in which these two blockages — the exclusion of becoming and the split between the general and the particular — are lifted.

Data and Method:

The alternative image of concept we propose is the genesis of a crystal. We trace this genesis by conjuring up a supersaturated solution precipitating around one or more “seeds” immersed in it. The becoming of the crystal arises from open-ended encounters between a supersaturated solution and a set of seeds. It is crucial that these encounters are open-ended rather than deterministic. They are open ended because countless aspects participate, no matter how slight, and have unpredictable potential to shape the future crystal. The idea of supersaturated solution is broadened as a situated medium with “folds,” “gaps,” or “cracks” amenable to nurture the open-ended growth of particular utterances, gestures, or signs. This fissured situated medium is, we propose, a concept. Our argument will be elaborated through an analysis of an interaction between a pair of undergraduate students who engage with the concept of “limit” emerging, in part, from their past experience with calculus.

Author

• Ricardo Nemirovsky, San Diego State University