Towards a Periodic Table of Ways of Knowing

Explores the possibility of a periodic table in relation to ways of knowing and learning.

Annex 1 of Pattern of Human Knowing: implication of the Periodic Table as metaphor of elementary order (2009)

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Introduction

This document tentatively explores a possible periodic organization of the 64 main categories of the Mathematics Subject Classification (MSC) in the light of arguments presented in the main paper. It then offers insights from various authors concerning the role of metaphor in relation to "doing" mathematics and to comprehending it. These arguments are seen as a justification for considering the possibility of associating distinct sets of metaphors with each of the main subject classes of the MSC -- or even of detecting what metaphors have already been used in this way.

Various insights from the recent compilation (Denis H. Rouvray et al., The Mathematics of the Periodic Table, 2005) are then briefly presented to highloight the extent to which mathematics is in process of reframing understandings of "element" and their periodicity in any ordering. This evolving understanding of an "element" -- due to new possibilities of distinguishing it within mathematical abstractions -- is relevant to any effort to the distinction of any fundamental "category" (as in the MSC). In contrast with past assumptions regarding the concreteness of "chemical elements", there is a shift from assertion of the nature of the reality constituted by "elements" to hypothesizing their nature in terms of new abstractions of every more generic insights. As noted, this occurs in a period in which the cognitive role of metaphor in relation to mathematical understanding is of increasing significance.

Contrasting examples of comprehensive periodic classifications from other cultures are then presented, namely the I Ching of Chinese culture and the All-Embracing Net of Buddhist culture. The former has been the subject of extensive mathematical commentary and the latter is relevant in therms of formal logical distinctions.

Although the implications are not explored here, it is appropriate to note that the Mathematics Subject Classification clusters at its heighest level 64 mathematical disciplines (although only 63 are indicated below). The I Ching is ordered in terms of 64 hexagrams. The All-Embracing Net distinguishes 62 explicit views, although 2 further views may be considered implicit. The argument might be made that such a degree of correspondence between otherwise disparate approach to order derives from constraints on human cognitive capacity which merits exploration towards a possible Periodic Table of Ways of Knowing.

Such seeming coincidences may indeed be interpreted in terms of some underlying order in objective reality. More intriguing is the extent to which such patterns may primarily signal a certain limit in the capacity of the mind to order any disparate set of entities. This has been the theme of previous explorations (Representation, Comprehension and Communication of Sets: the Role of Number, 1978; Patterns of N-foldness; comparison of integrated multi-set concept schemes as forms of presentation, 1984; Examples of Integrated, Multi-set Concept Schemes, 1984). The first noted the potential implications of the most cited study in psychology by George A. Miller (The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychological Review, 63, 1856, pp. 81-97). This would tend to affect the manner in which clusters and periods were well defined and appropriately bounded. Of course the following 8x8 patterns could then be said to result from (7+1)x(7+1).

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