The Nature of Advanced Mathematical Thinking

ion is the isolation of specific attributes of a concept so that they can be considered separately from the other attributes. Abstraction is often coupled with generalization. But the two are by no means synonymous. For instance, the solution of linear equations in two variables may be seen as a generalization to the process of solving linear equations in three variables. Although one may argue that there is an implicit abstraction of the solution process, the more general process is, in this case, certainly no more abstract. On the other hand, it sometimes happens that when an abstraction occurs, the properties abstracted are such that they uniquely determing the original concept. One example is the abstraction of the notion of a complete ordered field from the real numbers. Another is the abstraction of the group concept from groups of transformations – Cayley’s theorem shows that every abstract group is isomorphic to a group of transformations – so abstract groups are no more general than transformation groups. However these latter examples are singularities in the process of abstraction. In general (!) abstraction serves two purposes: (a) Any arguments which apply to the abstracted properties apply to other instances where the abstracted properties hold, so (provided that there are other instances) the arguments are more general. (b) By concentrating on the abstracted properties and ignoring all others, the abstraction should involve less cognitive strain. In the latter case, however, although mathematically there is concentration only on the salient properties, cognitively there are obstacles to overcome. There is a clear cognitive difference between generalizations and abstractions. A generalization involves the expansion of a cognitive schema: the generalization sets the particular cases in a broader context which enhances their properties without violating them in any way. If there are difficulties, the difficulties lie in the comprehension of the generalization. The mental process of abstraction involves a reconstruction of the cognitive schema: any properties of the abstraction (which may also be properties of the original concept) must be deduced from the abstracted properties alone and seen not to depend on any implicit assumptions concerning other properties of the original concept. This is almost always likely to be accompanied by a period of confusion as the cognitive structure is reorganized. An example is seen in linear algebra. The generalization is to Rn, so that all the processes are involved with n-tuples of “real numbers” where the latter are just familiar decimals, without any abstraction of the field properties of R. The