Cognitive psychology and comprehension

Potential Psychosocial Significance of Monstrous Moonshine: an exceptional form of symmetry as a Rosetta stone for cognitive frameworks (Part #6)

---

[Parts: First | Prev | Next | Last | All] [Links: To-K | From-K | From-Kx | Refs ]

---

It is to be expected that neither specialists in group theory, nor those in number theory, would have any interest in the psychological dimensions of comprehension. Conway's recognition that groups like the Monster are "psychologically rather difficult to grasp" is an exceptional admission that is not considered relevant to the actual challenge of the quest for the Monster, nor for understanding it. Curiously mathematicians are happy, informally, to recognize and admire an attribute termed "brilliance" but the implications of its absence are simply deplored. Comprehension capacity is not a feature of mathematics -- despite the kinds of arguments formulated elsewhere (Einstein's Implicit Theory of Relativity -- of Cognitive Property? Unexamined influence of patenting procedures, 2007).

However, as carefully reviewed in detail by Eduard Prugovecki (Historical and Epistemological Perspectives on Developments in Relativity and Quantum Theory, 1992):

The founders of relativity theory and of quantum mechanics were as concerned with the epistemological aspects and mathematical consistency of these theories, as they were with their empirical accuracy as reflected by experimental tests. In fact, some of them gave to epistemological scope and soundness preference over immediately apparent agreement with experiment, since they were acutely aware that all raw empirical data are submitted to a considerable amount of theoretical analysis and interpretation, before they are eventually released for publication. Of necessity, all such interpretations reflect the experimentalists' conscious or subconscious biases. Hence, the outcome is prone to various kinds of errors, ranging from systematic ones, due to the faulty design of apparatus or erroneous analysis of the raw data, to the subtle ones, due to misinterpretation or unwarranted extrapolation....

Unfortunately, after the Second World War this attitude towards epistemology and foundational issues in quantum physics became reversed2, as leading physicists of the post-war generation obviously decided that, contrary to the opinions of their great predecessors, it was legitimate to secure 'the adaptation of the theory to the facts by means of additional artificial assumptions'. Thus, soon after the 'triumph' of renormalization theory, Dirac (1951) felt compelled to point out in print that: 'Recent work by Lamb, Schwinger and Feynman and others has been very successful... but the resulting theory is an ugly and incomplete one.'

In the psychosocial domain there is a real challenge to formulating, representing and comprehending simple sets. This has been discussed in some detail elsewhere (Representation, Comprehension and Communication of Sets: the role of number, International Classification, 1978-1979). The "sets" in question may be formulations of principles, values, strategic elements, or the like -- whether for governance, as essential to religious doctrine, or in the organization of any enterprise. Many "models" elaborated and used by academics, and/or for purposes of management, are defined in terms of sets of concepts, perhaps presented as a list or a matrix. Governance, for example, is conducted through a set of ministries and/or departments -- possibly in the light of a set of electoral commitments or principles. Such sets may be composed of sub-sets.

The branch of mathematics known as group representation theory does not consider the cognitive challenge of comprehending a group as it may be variously represented. This is to some degree addressed more generally through category theory which deals, at a higher order of abstraction with mathematical structures and relationships between them -- notably the equivalence of categories and isomorphism of categories. The issue of how understanding of such abstractions is to be enabled is not however their concern.

However the 15,000 page proof is clearly an embarrassment as Ronan implies:

The proof of the Classification has come a long way from the time when a handful of experts believed in it, to the point where is it being written for future generations of mathematicians to understand. This is the role of the great Revision project, which will form a basis on which we can continue to strive for a better understanding of it all. (pp 214-5)

The work on this Revision project was initiated in 1982 by Daniel Gorenstein and has been continued, since his death in 1992, by Richard Lyons and Ronald Solomon and is expected to be completed in 2010.

---

[Parts: First | Prev | Next | Last | All] [Links: To-K | From-K | From-Kx | Refs ]