Poetic Logic and the Origins of the Mathematical Imagination

In 1940, two American mathematicians, Kasner and Newman, published a book titled Mathematics and the Imagination which was met with resounding success. The main claim of the book was that mathematics was a product of the imagination, arguably linking the two for the first time in the history of mathematics and even psychology. The ideas in Kasner and Newman's work coincided largely with those of the eighteenth-century Italian philosopher Giambattista Vico (1688-1744), sometimes called the "philosopher of the imagination." Vico put forth his views on how the imagination (which he called the fantasia) was at the root of all discoveries and inventions, and that its manifestation was through a form of thinking and expression that he called "poetic logic." One of the main products of poetic logic can be discerned in metaphor, which until the work of Lakoff and Núñez, in Where Mathematics Comes From, was hardly ever considered to be intrinsic to the creation of mathematical ideas. Missing from Kasner and Newman and Lakoff and Núñez is the notion of poetic logic, which would go a long way towards interconnecting metaphor, mathematics, and language in their origins.

This book treats Vico's theory of poetic logic for the first time as the originating force in mathematics, transforming instinctive counting and spatial perception into poetic (metaphorical) symbolism that dovetails with the origin of language. It looks at current work on mathematical cognition (from Lakoff and Núñez to Butterworth, Dehaene, and beyond), matching it against the poetic logic paradigm. In a sense, it continues from where Kasner and Newman left off, connecting contemporary research on the mathematical mind to the idea that the products of early mathematics were virtually identical to the first forms of poetic language, as documented in the founding myths and symbols of early cultures. As such, this book informs the current research on mathematical cognition from a different angle, by looking back at a still relatively unknown philosopher within mathematics.

The aim of this volume is to look broadly at what constitutes the mathematical mind through the Vichian lens of poetic logic. Vico was among the first to suggest that the essential nature of mind could be unraveled indirectly by reconstructing the sources of its "modifications" (his term for "creations"); that is, by examining the creation and function of symbols, words, and all the other uniquely human artifacts--including mathematics--the mind has allowed humans to establish "the world of civil society," Vico's term for culture and civilization.

The book is of interest to cognitive scientists working on math cognition. It presents the theory of poetic logic as Vico articulated it in his book The New Science, examining its main premises and then applying it to an interpretation of the ongoing work in math cognition. It will also be of interest to the general public, since it presents a history of early mathematics through the lens of an idea that has borne fruit in understanding the origin of language and symbols more broadly.

1940年，兩位美國數學家卡斯納（Kasner）和紐曼（Newman）出版了一本名為《數學與想像力》的書籍，取得了巨大的成功。該書的主要觀點是數學是想像力的產物，可以說是數學和心理學史上首次將兩者聯繫起來。卡斯納和紐曼的觀點在很大程度上與18世紀意大利哲學家維科（Giambattista Vico，1688-1744）的觀點相吻合，他有時被稱為“想像力的哲學家”。維科提出了他對想像力（他稱之為“幻想”）如何成為所有發現和發明的根源的看法，並且他稱之為“詩意邏輯”的一種思維和表達形式。詩意邏輯的主要產物可以在隱喻中看出，直到拉科夫（Lakoff）和努涅斯（Núñez）在《數學的本質》中的工作之前，隱喻很少被認為是創造數學思想的內在因素。卡斯納和紐曼以及拉科夫和努涅斯的著作中缺少詩意邏輯的概念，這將有助於在它們的起源中相互聯繫隱喻、數學和語言。

這本書首次將維科的詩意邏輯理論視為數學中的起源力量，將本能的計數和空間感知轉化為詩意（隱喻）象徵，與語言的起源相契合。它對當前關於數學認知的研究（從拉科夫和努涅斯到巴特沃思、德哈恩等）進行了研究，並將其與詩意邏輯範式進行了對比。從某種意義上說，它延續了卡斯納和紐曼的研究，將當代對數學思維的研究與早期數學的產物相連接，這些產物與早期文化的創世神話和符號中記錄的詩意語言的最初形式幾乎相同。因此，這本書以一個不同的角度回顧了對數學認知的當前研究，通過回顧數學中一位仍然相對未知的哲學家。

本書的目的是通過詩意邏輯的維科觀點廣泛探討數學思維的本質。維科是最早提出通過重建其“修改”（他對“創造”的術語）的來源間接解開心靈本質的本質的人之一；也就是通過研究符號、詞語和其他獨特的人類工具（包括數學）的創造和功能，心靈使人類能夠建立“文明社會的世界”，維科對文化和文明的術語。

這本書對從事數學認知研究的認知科學家具有興趣。它以維科在他的著作《新科學》中表達的詩意邏輯理論為基礎，檢視其主要前提，然後將其應用於對數學認知的持續研究的解釋。對於一般公眾來說，這本書也很有趣，因為它通過一個觀念的歷史，從更廣泛的角度來理解語言和符號的起源。

Marcel Danesi is Professor Emeritus of Anthropology at the University of Toronto. He has written extensively on the relation between mathematics and symbol systems, including how puzzles and problems in mathematics are part of a more general dialectic frame of mind for grasping the nature of reality. Among his works in the field are Language and Mathematics (2018) and Ahmes' Legacy (2020). He also founded the CogSci Network at the Fields Institute for Research in Mathematical Sciences, which consists of internationally renowned researchers in the field of cognitive mathematics.

Marcel Danesi是多倫多大學人類學名譽教授。他在數學和符號系統之間的關係上有豐富的著作，包括數學中的謎題和問題如何成為理解現實本質的更一般辯證思維框架的一部分。他在這個領域的作品包括《語言與數學》（2018年）和《阿姆斯的遺產》（2020年）。他還在數學科學研究領域的Fields研究所創辦了CogSci Network，該網絡由國際知名的認知數學研究者組成。