The General, the Abstract, and the Generic in Advanced Mathematics

ion An abstraction process occurs when the subject focuses attention on specific properties of a given object and then considers these properties in isolation from the original. This might be done, for example, to understand the essence of a certain phenomenon, perhaps later to be able to apply the same theory in other cases to which it applies. Such application of an abstract theory would be a case of reconstructive generalization – because the abstracted properties are reconstructions of the original properties, now applied to a broader General, Abstract and Generic Guershon Harel & David Tall – 4 – domain. However, note that once the reconstructive generalization has occurred, it may then be possible to extend the range of examples to which the arguments apply through the simpler process of expansive generalization. For instance, when the group properties are extracted from various contexts to give the axioms for a group, this must be followed by the reconstruction of other properties (such as uniqueness of identity and of inverses) from the axioms. This leads to the construction of an abstract group concept which is a re-constructive generalization of various familiar examples of groups. When this abstract construction has been made, further applications of group theory to other contexts (usually performed by specialization from the abstract concept) are now expansive generalizations of the original ideas. The case of definition The process of formal definition in advanced mathematics actually consists of two distinct complementary processes. One is the abstraction of specific properties of one or more mathematical objects to form the basis of the definition of the new abstract mathematical object. The other is the process of construction of the abstract concept through logical deduction from the definition. The first of these processes we will call formal abstraction, in that it abstracts the form of the new concept through the selection of generative properties of one or more specific situations; for example, abstracting the vector-space axioms from the space of directed-line segments alone or from what it is noticed to be common to this space and the space of polynomials. This formal abstraction historically took many generations, but is now a preferred method of progress in building mathematical theories. The student rarely sees this part of the process. Instead (s)he is presented with the definition in terms of carefully selected properties as a fait accomplit. When presented with the definition, the student is faced with the naming of the concept and the statement of a small number of properties or axioms. But the definition is more than a naming. It is the selection of generative properties suitable for deductive construction of the abstract concept. The abstract concept which satisfies only those properties that may be deduced from the definition and no others requires a massive reconstruction. Its construction is guided by the properties which hold in the original mathematical concepts from which it was abstracted, but judgement of the truth of these properties must be suspended until they are deduced from the definition. For the novice this is liable to cause great confusion at the time. The newly constructed abstract object will then generalize the General, Abstract and Generic Guershon Harel & David Tall – 5 – properties embodied in the definition, because any properties that may be deduced from them will be part of it. Because of the difficulties involved in the construction process, this is a reconstructive generalization. Occasionally the process leads to a newly constructed abstract object whose properties apply only to the original domain, and not to a more general domain. For instance, the formal abstraction of the notion of a complete ordered field from the real numbers, or the abstraction of the group concept from groups of transformations. Up to isomorphism there is only one complete ordered field, and Cayley’s theorem shows that every abstract group is isomorphic to a group of transformations. In these cases the process leads to an abstract concept which does not extend the class of possible embodiments. We include these instances within the same theoretical framework for, though they fail to generalize the notion to a broader class of examples, they very much change the nature of the concept in question. The formal abstraction process coupled with the construction of the formal concept, when achieved, leads to a mental object that is easier for the expert to manipulate mentally because the precise properties of the concept have been abstracted and can lead to precise general proofs based on these properties. Formal abstraction leading to mathematical definitions usually serves two purposes which are particularly attractive to the expert mathematician: (a) Any arguments valid for the abstracted properties apply to all other instances where the abstracted properties hold, so (provided that there are other instances) the arguments are more general. (b) Once the abstraction is made, by concentrating on the abstracted properties and ignoring all others, the abstraction should involve less cognitive strain. These two factors make a formal abstraction a powerful tool for the expert yet – because of the cognitive reconstruction involved – they may cause great difficulty for the learner.