Origin of mathematics and the periodic table -- in human cognition?

Periodic Pattern of Human Knowing (Part #7)

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The question of the respect in which "mathematics" is independent of the human mind is an old one. It is a question which many mathematicians consider to be irrelevant to their interests. For others the order discovered by mathematics is simply an exemplification of some understanding of the divine order within which human cognition has emerged.

For example, Reuben Hersh (What is Mathematics, Really? 1997) notes that most philosophers of mathematics treat it as isolated, timeless, ahistorical, inhuman. Hersh argues the contrary, that mathematics must be understood as a human activity, a social phenomenon, part of human culture, historically evolved, and intelligible only in a social context. He reveals mathematics as seen by professionals, debunking many mathematical myths, and demonstrating how the "humanist" idea of the nature of mathematics more closely resembles how mathematicians actually work. This follows from his earlier collaborative investigation (Philip J. Davis and Reuben Hersh, The Mathematical Experience, 1981).

More recently Hersh has collected disparate essays on understandings of mathematics (18 Unconventional Essays on the Nature of Mathematics, 2005) that are tackling, from various points of view, the problem of giving an accounting of the nature, purpose, and justification of real mathematical practice -- mathematics as actually done by real live mathematicians. His concern is with the the nature of the objects being studied and what determines the directions and styles in which mathematics progresses (or, perhaps, degenerates).

The focus of several more recent studies (some cited by Hersh) indicates the challenge for the philosophy of mathematics at the crossroads of two schools of thought. On one side are the old school mathematicians who see mathematics as a foundation of science. On the other side is a small but growing group of scholars made up of cognitive psychologists, linguists, and neural biologists (and some mathematicians as well) who see mathematics as a function of the brain (John D. Barrow, Pi in the Sky: Counting, Thinking, and Being, 1994; Brian Butterworth, What Counts: how every brain is hardwired for math, 1999; Keith Devlin, The Math Gene: how mathematical thinking evolved and why numbers are like gossip, 2001; George Lakoff and Rafael Núñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000).

These new neurobiological / linguistic / cognitive theories promise help in understanding how mathematics is learnt and comprehended. (Ironically the provocative list of "co-" terms above highlights a sense in which mathematics is "born" from the matrix of a "space" that they contribute to defining, as the problematic history of the treatment of mathematical innovators by their peers shows only too clearly)

In particular George Lakoff and Rafael Núñez advocate a cognitive idea analysis of mathematics in terms of the human experiences, metaphors, generalizations, and other cognitive mechanisms giving rise to those ideas. Idea analysis is distinct from mathematics and cannot be performed by mathematicians unless they are trained in cognitive science. They are mainly concerned with proposing and establishing an alternative view of mathematics, one grounding the field in the realities of human biology and experience.

This emphasis accords with the interest in the cognitive status of a Periodic Table as a metaphor of a Periodic Pattern of Human Life. However the emphasis here is on the manner in which such a periodic table incorporates or embodies the challenges of learning. This implies a degree of self-reflexivity. There is an implication of one in the other.

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