In a 2019 paper, Mateusz Hohol and Marcin Miłkowski argue that current explanations of geometric cognition should move beyond methodological individualism and take into account the influence of broader cognitive factors, such as cultural and societal influences, in the development of this area of mathematics. They suggest that the abstract geometry developed in ancient Greece cannot be fully understood by solely looking at the actions of individual minds.
The authors have conducted extensive research on cognitive geometry and have not only provided an introduction to the subject but have also highlighted the need for further exploration. They demonstrate that the development of Greek mathematics, as represented in Euclid’s Elements, was propelled by the utilization of two closely related cognitive artifacts: the use of lettered diagrams and the creation of linguistic formulae. Together, these artifacts formed the professional language of geometry.
We propose that cognitive artifacts, i.e., diagrams and well-structured language, scaffolded visuospatial capacities of our brains, and contributed to building a unique cognitive niche within Euclidean geometry, originated as a result of collective thinking and problem-solving.Mateusz Hohol
According to them, this case of Greek geometry illustrates that explanations of geometric reasoning must move beyond the limitations of methodological individualism to account for how the shared use of artifacts has become stabilized over time. This practice, as they suggest, has also played a significant role in shaping our understanding of what constitutes a mathematical proof. Traditionally, it has been assumed that proofs are not only deductively correct but also remain consistent across different individuals who share the same cognitive practice.
The cognitive artifacts present in Greek geometry restricted the possible inferential operations, making these proofs inter-subjectively verifiable and convincing. By focusing on the cognitive operations involving artifacts, the authors also emphasize that the mental mechanisms contributing to these operations are not yet well understood, in contrast to those mechanisms driving symbolic logical inference.
In their opinion, the use of geometric cognitive artifacts should be investigated from both an epistemological and an ontological point of view: a mathematician has the ability to demonstrate proofs that are characterized by the requirement of subsequent inferences and the broad applicability of the outcomes, however, they still retain the freedom of choice in their mathematical ontology.
The authors emphasize that cognitive artifacts play a crucial role in the emergence of robust geometric cognition, not only in terms of developmental time-scale but also in historical context. They assert that studies on geometric cognition should consider not only individual cognitive factors, but also the cognitive practices facilitated or even formed by external devices.
Cognitive Artifacts for Geometric Reasoning, Mateusz Hohol, Marcin Miłkowski
Published: May 2019