Abstract:Jeremy Gray (2012) writes that Poincaré’s essay “On the foundations of geometry” (1898) “can be read as an early example of cognitive science” (49). In this paper, Poincaré suggests that our preference for Euclidean geometry originates in our constitution as an evolved kind. He argues not only that our experiences as rigid bodies in the world incline us towards one kind of geometry, but also that if our experiences had been different, we would have different inclinations. Further, our minds are not all cast in the same mold, to use an industrial metaphor from before the computer era. In his last public lecture in 1912, Henri Poincaré spoke at the inaugural meeting of Le Ligue Française d’Éducation Morale. In his talk, “L’Union Morale,” he said, “Let us guard against imposing uniform methods on all; that is unrealizable and, moreover, it is not desirable. Uniformity is death because it is a door closed to all progress…” One might read this remark as a rhetorical olive branch, offered by a scientist to an audience of wary humanists. Yet, Poincaré’s commitment to intellectual diversity is central to his view of mathematics as something alive that grows. This paper considers Poincaré’s interrelated conjectures that mathematicians come in different kinds and mathematical discovery occurs in stages. According to Poincaré, cognitive diversity is necessary for the growth of knowledge. He distinguishes between logicians, who correct and prove, and intuitionists, who hypothesize and predict: “The two sorts of minds are equally necessary for the progress of science; both the logicians and the intuitionalists have achieved great things that others could not have done.” Contemporaries of Poincaré also argued that different people had different kind of minds. But while Pierre Duhem contrasts Blaise Pascal’s espirit de finesse and esprit de géométrie as cultural kinds, Poincaré contrasts intuitifs and analysts as individual kinds of mathematical minds that may be born into any culture. Significantly, we can trace a path from these rudimentary speculations to contemporary cognitive neuroscience investigations into the genetic bases for visual reasoning, e.g., Delis hierarchical processing tasks. For Poincaré, there are differences between mathematicians as well as different kinds of cognition within an individual. He contrasts the conscious activities of mathematicians seeking answers to particular problems with how their unconscious minds mechanically sift through possible solutions until a plausible answer, worthy of conscious attention, is generated. Finally, Poincaré also considers different the phases of development that mathematical domains undergo as communities collaborate to make implicit relations between different branches of mathematics explicit.
Three discussion questions will be proposed for discussion. First, what insights may be drawn from considering the similarities and differences between Poincaré’s and Duhem’s accounts of “kinds of minds”? Second, given what we now know about cognitive neuroscience, does it make sense to insist upon a sharp division of labor between the philosophy of mathematics, on the one hand, and the psychology of mathematics on the other? Finally, what do recent studies in the natural and social sciences have to contribute to our understanding of the growth of knowledge, in general, and mathematics, in particular?
References:
Last Essays, p. 116. “Gardons-nous d’imposer à tous des moyens uniformes, cela est irréalisable, et d’ailleurs, cela n’est pas à désirer: l’uniformité, c’est la mort, parce que c’est la porte close à tout progress” (Dernières pensées, p. 256).
The Value of Science, p. 17. “Les deux sortes d’esprits sont également nécessaires aux progrès de la science; les logiciens, comme les intuitifs, ont fait de grandes choses qeu les autres n’auraient pas pu faire” (La valeur de la science, p. 10).
