Our paper, written together by myself and Marcin Miłkowski, entitled Cognitive artifacts for geometric reasoning was published in Foundations of Science a few days ago (online first, open access). In our joint paper, we claim that explanations of geometric cognition should go beyond methodological individualism and take into account the role of distributed cognitive factors in the shaping of Euclidean geometry. In other words, we argue that abstract geometry that raised in ancient Greece cannot be satisfactorily explained only as a product of individual minds, or skull-bound cognitive processes. Instead, 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. In addition to this, we emphasize that in contrast to mental mechanisms of symbolic logical inference, mechanisms of diagrammatic inference are still weakly understood in cognitive science.
In the first part of the paper, we note that contemporary cognitive science of mathematics is focused on numbers and calculations much more than on the mental processing of geometry, which is reflected at least in bibliometric evidence. This does not, however, mean that cognitive scientists completely ignore geometry. The theory of cognitive systems of core geometric knowledge by Elizabeth Spelke is a creditable example that we discuss in the paper. In a nutshell, relying on developmental, neuroscientific, comparative, and evolutionarily data, Spelke claims that cognitive base of geometry consists of two phylogenetically ancient and ontogenetically early systems hardwired in the brain. We call these core systems, respectively, the system of layout geometry and the system of object geometry. The former is implemented in the hippocampus and surrounding structures of the vertebrate brain, and its primary function is supporting navigation in large-scale spatial layouts. The latter is localized in the lateral occipital complex and supports recognition of 2D shapes and 3D manipulable objects. According to Spelke, none of these systems, however, provide representations of geometric objects characterized by properties such as angle, sense, and length. Therefore, Euclidean representations involving all these geometric properties require a flexible combination of the core systems during ontogeny. Although Spelke agrees that the developmental shift toward a full-blooded representational system of geometry is mediated by cognitive artifacts, i.e., acquiring the language involving spatial expressions and using map-like scale objects, her account remains silent about shaping of geometric cognition in the historical time-scale.
Therefore, in the second part of our article, we explore the role of cognitive artifacts in the emergence of the Greek geometry, summarized in Euclid’s masterpiece, Elements. Drawing from the Reviel Netz’s approach called cognitive history, we look at properties and mutual relationships of two specific artifacts developed by Greek geometers. The first one is a lettered diagram, and the second one is the technical language composed of fixed strings of words, called–in line of a tradition of Homeric philological studies–formulae. Despite the fact that Babylonian, Egyptian, and Chinese geometers used diagrams before the Greeks, marking them with letters is a uniquely Greek invention. Letters associated with the points make diagrams something more than just auxiliary drawings that facilitate the initial recognition of the problem. On the one hand, thanks to letters Euclidean diagrams are strictly embedded in the discursive (textual) components of mathematical discourse that disambiguate theirs understanding. On the other hand, diagrams allow us to understand the text, because they determine geometric points. One of the crucial discoveries about Greek geometry is that diagrams are full-blooded deductive components of proofs, and without them many statements would lose their truth-value. Furthermore, according to Netz, a fine-grained logical network established by diagrams and linguistic formulae is sufficient to make geometric reasoning compelling and it’s results universally valid. Last but not least, as the researcher claims, lettered diagrams serve as a substitute for mathematical ontology, which means that the geometer does not have to participate in ontological discussions about the semantics of diagrams.
Although we fully agree with Netz about the role of the cognitive artifacts in the shaping of the deductive practices in Greek mathematics, we simultaneously point out two problems. The first one is associated with a thesis that the diagram serves as a substitute for the ontology of geometry. In our opinion, the use of the geometric cognitive artifacts should be investigated from both an epistemological and an ontological point of view. We claim that while using diagrams (and formulae as well) epistemologically non-neutral (in Netz’s line), but ontologically neutral (contrary to Netz). Diagrams constrain the permitted steps in the proof (and thus they indeed contribute to establishing deductive practices), but at the same time do not constrain ontological commitments of geometry. In other words, a mathematician is capable of proving proofs characterized by the necessity of subsequent inferences and the generality of the outputs, however, he or she has still the free choice of mathematical ontology. We ground this thesis in the historical setting, showing that debates about the ontology of geometry existed in ancient times. For instance, the constructive school of Menaechemus accepted the literal interpretation of Euclidean constructions, while Speusippus claimed that constructions should be considered only as heuristic devices, that allow mathematicians to grasp Platonic geometric ideas. To sum up, we claim that deductive practices of Greeks were ontologically neutral.
The second problem with Netz’s cognitive history is that although it elucidates the emergence of two hallmark epistemic properties of Euclidean geometry, namely, necessity and generality of proof, it, however, lacks a substantial account of geometric operations that are performed using these cognitive artifacts. Looking at Proposition 1 of the Book 1 of Euclid’s Elements we show that linguistic formulae supplemented by the diagram indeed reflects inference steps leading to a universal result in a necessity-preserving way, but all the internal, or cognitive, work remains hidden beneath these external representations. By referring to Peirce’s and Magnani’s works we claim that investigation of cognitive foundations of geometric reasoning should go beyond the deductive product that is easy to observe in the final proof and elucidate abductive operations (or manipulative abductions in Magnani’s terms) performed on the diagram. In our opinion, further studies on geometric cognition should look not only at the context of justification but explain the context of discovery, in an empirically grounded way.
Even though our investigation is far from complete, especially regarding cognitive operations involved in geometric reasoning, we stress that cognitive artifacts play a non-negligible role in the emergence of full-blooded geometric cognition not only in the developmental time-scale but also in the historical one. Therefore, our message is that the studies on geometric cognition should take into account not only individual cognitive factors but also distributed cognitive practices facilitated, or even constituted, by external devices.
The above and related problems are further elucidated in my forthcoming book entitled Foundations of Geometric Cognition, which will be published in October 2019 by Routledge.