The individual and the culture

Mathematics

How Mathematics Grows: The Role of the Individual and the Culture

Influences the Course of Mathematical Discovery

There are many influences on the course of mathematical discovery. The dominant forces influencing mathematical discovery are individual and culture, as pointed out by the reading. There is as much subjectivity in mathematical computation as there is science, which is part of the reason so much of mathematics is ill understood and often misinterpreted. As the reading points out those not well versed in mathematics or those with little training often have an imperfect notion of how it operates and how solutions or mathematical proofs must be derived. Often what seems correct is later found incorrect, and vice versa. This is evidenced by the example in the reading, whereby professor Hans Rademacher of the University of Pennsylvania, a leading theoretician at the time, mistakenly believed he solved Riemann's Hypothesis (p. 60) only to be disproved later. The individual and culture directly influence mathematical discovery. There isn't really a dichotomy between the individual and culture; they simply influence mathematics differently. For example, an individual who is brilliant and capable of great mathematical feats may at the same time think outside of the scope of what the culture he or she lives within may consider "ordinary." Thus this person's brilliance or folly will either be embraced or rejected depending on the fit with the culture at the time the solution or hypothesis is presented. The reading for example, points out the case of Hermann Grassman, whose work is today considered genius, however during his time was consider obscure and mystical because the work was not in line with the culture Grassman grew up in.

The creating and practice of mathematics are not the same. The practice of mathematics is more aligned with cultural norms and what is considered acceptable practice during the time mathematics is accomplished. Creation however, may involve the extremes of an individual, or an individual's ability to tap into a genius that may seem unusual or obscure at the time it is conceived or at the time an idea is realized.

Mathematics as the reading suggests posses many characteristics, being derived from individual genius but thriving only when the culture or wider community as described by the text accepts the principles and applications surrounding it (p. 64). Individual discovery alone however, is not responsible for mathematics. The text also notes that economic and social forces can stimulate mathematical discovery. Mathematical discovery however, is something that persists regardless of the historical time or culture dominating. The text notes that many theorists favor the "doctrine of culture" so one must acknowledge the influence culture has on discovery. The text suggests the dichotomy between the individual and the doctrine of the culture is false, an argument similar to that of mind over matter; one can conclude simply that individuals may matter temporarily but culture dominates the field of mathematics over wider time periods. That culture is inextricably linked to mathematical inquiry is a fact that can't be denied (Saxe, 1991). Wang (1963) notes that it is important however, to distinguish between mathematical discipline and application, suggesting that axioms may not be true in the physical world, and that individuals may influence mathematical possibility as much as culture. He notes Euclid's fifth postulate which was found later to be false, which Pasch (1882) pointed out when confirming the shortcomings in this once accepted axiomatization (Wang, 1963, p.2). Euclid's axioms were at first however, regarded as completed logical and necessary. However as times changed, especially during the Renaissance for example, controversy existed over the axiom and axiomatic systems in particular, giving rise to questions regarding the legitimacy of the axioms formulated by Euclid (Wang, 1963).