Handbook of Cognitive Mathematics ed. by Marcel Danesi (review)

Reviewed by: Handbook of Cognitive Mathematics ed. by Marcel Danesi Nathan Haydon Marcel Danesi (Ed) Handbook of Cognitive Mathematics Cham, Switzerland: Springer International, 2022, vii + 1383, including index For one acquainted with C.S. Peirce, it is hard to see Springer's recent Handbook of Cognitive Mathematics (editor: Marcel Danesi) through none other than a Peircean lens. Short for the cognitive science of mathematics, such a modern, scientific pursuit into the nature and study of mathematical practice would no doubt be found agreeable to Peirce. The fact that references to Peirce appear often throughout the Handbook is a welcome find, with Peirce's ideas being a key subject of half a dozen chapters, and where a reader of any other chapter may well find further connections to Peirce's ideas. After spending time with the Handbook, it is clear that cognitive mathematics has not only embraced some of Pierce's ideas but may be at an important forefront of Peirce studies. In the end, the field may well be an area a Peircean should pay attention to. The book itself is pitched as a reference volume to the field (p. v) with the necessary background to familiarize oneself with the aims and results. The connection to cognitive science may call to mind detailed cognitive models [Ch. 10–14], theories on the origins of numeracy and other theories behind the biological and evolutionary requirements of mathematical thought [Ch.15–18], and the like. These are all present. But a key theme of the Handbook is to situate mathematics not just within more traditional 'cognitive' faculties and the more formal, i.e. algebraic, presentations of mathematics, but also to place mathematics within other human faculties and practices, from the arts to language [Ch. 19–22], within education and learning more broadly [Ch. 23–26], and in relation to the significant, though at first perhaps less quantitative parts of mathematics, like the use of metaphor, gesture, analogy, [End Page 243] abstraction, as well as further cultural and ethnographic considerations [Ch. 5–8]. The Handbook has explicitly taken a broader, more interdisciplinary approach (p. vi–vii) towards the scientific aspects of mathematical practice—choosing to regulate the study not by antecedently drawn opinions about what mathematics is (or has traditionally been taken to be) but by what future quantifiable and diverse study may come to bear on the practice and have to say about those engaging in it. This broad interdisciplinarity has a pragmatist ring, where theory cannot so easily be separated from the normative, social, and, more generally, the more thoroughly human aspects that we encounter and employ when we engage in it. Two further commitments of cognitive mathematics steer us even closer towards Peirce's views. The first—going back, for example, to Lakoff and Núñez's Where Mathematics Comes From (2000), which is taken to be a key early work in shaping the field—is that mathematics is taken to be a language like any other and as such must be learned and situated within our other cognitive faculties (p. vi, Ch. 4). The second—and largely, one imagines, following from a concern for seeing how mathematics is actually practiced—is that mathematics is essentially diagrammatic, involving as it does experimenting in diagrams and employing other signs (one may think here even of the mathematician sketching equations or geometric figures on a sheet of paper). What is perhaps initially most noteworthy about Peirce's philosophy of mathematics is his early concern for both these aspects of the practice. The book (more like a tome of over 1300 pages) is too large to be covered here in all its aspects, hence the review will focus on those aspects a Peircean or Pragmatist may find of interest. In particular, this review focuses on several of the themes above—situating mathematics more broadly within our everyday (cognitive) lives, the necessarily diagrammatic, fallible, and creative aspects of the practice, and, finally, what these aspects may have to say about Peirce's philosophy more generally, particularly with respect to (scientific) inquiry. A Peircean reader may well begin with Pietarinen's "Pragmaticism as Philosophy of Cognitive Mathematics" [Ch. 39] and "Peirce on Mathematical...