Poetic Logic and the Origins of the Mathematical Imagination (Mathematics in Mind) 3031315812, 9783031315817

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Mathematics in Mind

Marcel Danesi

Poetic Logic and the Origins of the Mathematical Imagination

Mathematics in Mind Series Editor Marcel Danesi, University of Toronto, Canada Editorial Board Members Norbert Hounkonnou, University of Abomey-Calavi, Benin Louis H. Kauffman, University of Illinois at Chicago, USA Dragana Martinovic, University of Windsor, Canada Melanija Mitrović

, University of Niš, Serbia

Yair Neuman, Ben-Gurion University of the Negev, Israel Rafael Núñez, University of California, USA Anna Sfard, University of Haifa, Israel David Tall, University of Warwick, UK Kumiko Tanaka-Ishii, Kyushu University, Japan Shlomo Vinner, Hebrew University, Israel

The monographs and occasional textbooks published in this series tap directly into the kinds of themes, research findings, and general professional activities of the Fields Cognitive Science Network, which brings together mathematicians, philosophers, and cognitive scientists to explore the question of the nature of mathematics and how it is learned from various interdisciplinary angles. Themes and concepts to be explored include connections between mathematical modeling and artificial intelligence research, the historical context of any topic involving the emergence of mathematical thinking, interrelationships between mathematical discovery and cultural processes, and the connection between math cognition and symbolism, annotation, and other semiotic processes. All works are peer-reviewed to meet the highest standards of scientific literature.

Marcel Danesi

Poetic Logic and the Origins of the Mathematical Imagination

Marcel Danesi Toronto, ON, Canada

ISSN 2522-5405 ISSN 2522-5413 (electronic) Mathematics in Mind ISBN 978-3-031-31581-7 ISBN 978-3-031-31582-4 (eBook) https://doi.org/10.1007/978-3-031-31582-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Because the language of mathematics is so precise, it is ideally suited to defining concepts for which such a consensus exists. (Armand Borel 1923–2003)

“The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.” These words were written by Bertrand Russell (1907), arguably modernity’s greatest mathematical logician. His pairing of mathematics and poetry would thus seem to be an oxymoron, but the two do indeed spring from the same source—the human imagination. More specifically, mathematics and poetry are products of a singular faculty of mind that the Italian philosopher Giambattista Vico called poetic logic in his revolutionary philosophical treatise, la scienza nuova (first edition, 1725), which Donald Verene (1981) has renamed, appropriately, a “science of the imagination.” It is this faculty that will be examined in this book in order to address the question of what the mathematical imagination is—a goal that Vico himself did not pursue directly and which, as far as can be told, has never been considered in any cohesive analytical way since the publication of his treatise. Among the related questions that will guide such an examination are the following: How did the mathematical imagination originate? Is it different than, say, the imagination that is used by artists when they paint or by writers when they devise their stories? Is thinking about number concepts different than thinking about geometric forms? The key insight from the scienza nuova is that poetic logic is the source of the fantasia, which is more than imagination (the formation of images of things)—it is a blend of imagination and fantasy, undergirding the ability to create scenarios within the mind and then use them ingeniously to grasp the meaning of things. The fantasia is thus a meaning-making capacity that allows us to make sense of the world. In this book, the terms imagination and fantasia are used somewhat interchangeably, for the reason that there is no fantasia without imagination and vice versa. Moreover, it should be mentioned from the outset that there are three processes involved in the operation of poetic logic—in addition to the fantasia, there is the ingegno (ingenuity) and the memoria (creative recollection). The end

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result of these operations is rational logic, the ability to work with the discoveries made possible by the fantasia “formally,” that is to give them a form or locus within the ever-evolving system of mathematics. This book is organized sequentially around these aspects—Chap. 1 deals with the imagination, Chap. 2 with ingenuity, Chap. 3 with memory, Chap. 4 with metaphor (a product of the imagination), and finally logic in Chap. 5. As Vico scholar Isaiah Berlin emphasized in his book, Against the Current (2013), to grasp the meaning of mathematics in a Vichian framework, one must first understand what he meant by his verum-factum principle (“the truth is made”). This asserts that mathematical truths are understandable “precisely because humans themselves have made mathematics” (Berlin 2013: 23). So, the only practicable way of answering what the mathematical imagination is would be to examine the actual mathematics that humans have made throughout history. These will allow us to observe the manifestations of the fantasia in this particular area of human meaningmaking and specifically of how the mathematical imagination might have originated. This was the approach, actually, taken by Edward Kasner and James Newman in their remarkable popular book, Mathematics and the Imagination, in which they unwittingly put forth the Vichian notion of the fantasia as the source of fables in the mind, which they called a fairyland (Kasner and Newman 1940: 8): [Mathematics] is a veritable fairyland, a fairyland which is strange, but makes sense, if not common sense. From the ordinary point of view mathematics deals with strange things. . .but mostly it deals with familiar things in a strange way.

To the best of my knowledge, an approach to the mathematical imagination using a Vichian framework has never been undertaken as such. Perhaps the reason is that Vico himself maintained that it is not possible to study poetic logic directly, since the mind cannot study itself. Nevertheless, he suggested that we could certainly gain a good understanding of what the human mind is by studying one of the most imaginative products of poetic logic—metaphor. As remarkable as that insight was in the eighteenth century, it is only today that metaphor has finally started to catch the attention of cognitive scientists and mathematicians. In this framework, mathematical concepts are, literally, as Reed (1994) calls them, “figures of thought.” In their ground-breaking book, Where Mathematics Comes From (2000), George Lakoff and Rafael Nuñez have provided an in-depth analysis of the relation between metaphor and mathematics, demonstrating how metaphorical thinking is an inbuilt feature of the conceptual system underlying both mathematics and language. Indeed, there could be no mathematics without language which provides the fables (discourses) for grasping mathematical ideas. And the reason why the two are interrelated is the fact that they are based on the same metaphorical structures, and thus, ideas in mathematics can be framed in language via metaphor. The massive amount of data collected on the learning of mathematics, ever since, suggests very strongly that many abstract mathematical concepts, if not most, are encoded and knowable primarily as “metaphorized ideas.” The source of these ideas is ascribed to what Lakoff and Núñez call image schemas. In a Vichian framework, these would correspond to what the Italian philosopher called imaginative universals—that is,

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mental images that are formed via the experience of things through the fantasia. In personal debates and discussions with the two cognitive scientists at the Fields Institute for Research in Mathematical Sciences (Bockarova et al. 2012, 2015), a consensus was reached that Vico clearly needed to be introduced into the debate on the origins of the mathematical imagination. This book is the practical response to this need. In Vico’s times, Cartesianism dominated the academic scene of the day, causing the older tradition, centered on the study of the humanities, to lose ground to an emphasis on logically rigid views of the mind. Vico’s concept of fantasia came forward to show why such views were untenable as such, since they ignored how ideas were formed in the first place. So, the origination of mathematical ideas in a Vichian model of mind is akin to the origination of poetic ideas, as Russell seems to have intimated as well (above). Both are guided by conscious experiences that have been transformed into ideas. The sense of number (called numerosity) may be instinctive, but its conversion into numeration is not—it requires a mind capable of literally creating a correlation between the two. The sense of space (spatiality) may be instinctive as well, but its conversion into geometry is not—it too involves the mind’s ability to construct a connection between the two. The connecting linkage manifests itself in metaphor, such as the envisioning of numbers as objects in containers (to be discussed throughout this book). Evelyn Lamb (2018) makes the following relevant observation, which can be considered a point-of-departure on the metaphorical journey into the mathematical imagination (no pun intended) that I intend to take in this book: In mathematics, a single object or idea might take different forms. A quadratic equation, for example, can be understood in terms of its algebraic expression, perhaps y = x2 + 3x - 7, or in terms of its graph, a parabola. Henri Poincaré, a French polymath who laid the foundations of two different fields of mathematics in the early 1900s, described mathematics as “the art of giving the same name to different things.” Likewise, poets create layers of meaning by utilizing words and images that have multiple interpretations and associations. Both mathematicians and poets strive for economy and precision, selecting exactly the words they need to convey their meaning.

Toronto, ON, Canada 2023

Marcel Danesi

Contents

1

Imagination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Ingenuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3

Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

4

Metaphor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5

Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

ix

List of Figures

Fig. 1.1 Fig. 1.2 Fig. 1.3 Fig. 1.4 Fig. 1.5 Fig. 1.6 Fig. 1.7 Fig. 1.8 Fig. 1.9 Fig. 1.10 Fig. 1.11 Fig. 1.12 Fig. 1.13

Prehistoric bone etchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unfolding the cube .. .. . .. . .. . .. .. . .. . .. . .. . .. .. . .. . .. . .. .. . .. . .. . .. . .. A hypercube in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A model of an engineering problem . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . Poetic logic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Problem in Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A tricky problem in geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resolving the problem .. . . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . The Ishango Bone .. . .. . . .. . .. . . .. . . .. . .. . . .. . . .. . .. . . .. . . .. . .. . . .. . .. . A Chinese abacus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Königsberg bridges map . . .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . Graph version of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The topology of a donut and coffee mug . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 5 6 8 14 16 16 17 20 21 26 27 31

Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14

Thales’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thales’s proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Archimedes’ polygon method for π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Estimation of π in the Rhind Papyrus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear image schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Half sequence arrangement . .. . .. . .. .. . .. .. . .. . .. .. . .. .. . .. . .. .. . .. .. . Distinct solutions to the eight queens problem . . . . . . . . . . . . . . . . . . . . . Simple example of the Four-Color Theorem . . . . . . . . . . . . . . . . . . . . . . . Complex example of the Four-Color Theorem . . . . . . . . . . . . . . . . . . . . . Directed graph of the Collatz conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fourteen/Fifteen Puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intersecting lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eratosthenes’ estimation of the Earth’s circumference . . . . . . . . . . . . The discovery of √2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 36 37 38 39 39 41 45 45 49 50 52 60 61

Fig. 3.1 Fig. 3.2 Fig. 3.3

Version of Abu Al-Wafa’s puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zero on the number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The real number line . . . . .. . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . .. . .

71 73 74 xi

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List of Figures

Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 3.8 Fig. 3.9 Fig. 3.10 Fig. 3.11 Fig. 3.12 Fig. 3.13 Fig. 3.14 Fig. 3.15 Fig. 3.16 Fig. 3.17 Fig. 3.18

Babylonian numerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagrammatic proof of the Pythagorean theorem . . . . . . . . . . . . . . . . . . Argand diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler versus Venn diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peircean existential graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Legendre’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Looping structure of poetic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagram of Dudeney’s puzzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flattened diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal’s triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal’s triangle and the Fibonacci numbers . . . . . . . . . . . . . . . . . . . . . . . The golden ratio . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . .. . . . . .. . . Sierpinski triangle . .. . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . Plimpton 322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 78 79 80 81 81 82 84 84 85 86 87 88 89 90

Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9 Fig. 4.10 Fig. 4.11 Fig. 4.12 Fig. 4.13 Fig. 4.14 Fig. 4.15 Fig. 4.16 Fig. 4.17 Fig. 4.18

Vico’s stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Placement of “Now” on a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . “Before” and “After” on the line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cartesian plane . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A circle in the Cartesian plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some members of the source-path-goal family . . . . . . . . . . . . . . . . . . . . . Representing the derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A representation of Euclid’s fifth postulate . . . . . . . . . . . . . . . . . . . . . . . . . The Beltrami model . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. . . . . .. . . . .. . Clustering ICM for “Number” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A radiation ICM based on the “Path” image schema . . . . . . . . . . . . . . Galileo’s one-to-one correspondence . .. . . . . . . . .. . . . . . . .. . . . . . . . .. . . . Cantor’s elaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cantor’s sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cantor’s cardinality demonstration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cantor’s array with binary numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cantor’s diagonal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wallis’s illustration of the number line . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 100 103 103 104 105 108 108 109 110 116 117 117 118 119 120 122

Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8

Carroll’s Diagrammatic Logic (1896) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A quaternion table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Primes on Ibn Khallikan’s chessboard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kissing Number Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circle packing problem . .. . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . .. . . . Mathematical thinking scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schröder’s Reversible Staircase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impossible Staircase (Penrose and Penrose 1958) . . . . . . . . . . . . . . . . .

134 137 142 144 144 145 152 152

Chapter 1

Imagination

It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances. (Hannah Arendt 1906–1975)

Prologue Based on anthropological data pertaining to over 30 hunter-gatherer societies, archeologist Karenleigh Overmann (2013) was able to establish that prehistoric people counted with their fingers around 40,000 years ago, even before they (likely) had developed words for them. In effect, the fingers were used as proto-numerals, as were other kinds of counting signs and artifacts found in prehistoric cultures— pebbles, knots, marks, tallies, etc. (Ifrah 1981; Graslund 1987). Finger-counting is still an instinctive behavior—no doubt a residue from our evolutionary history (Neugebauer 1952) and reflecting an innate ability to connect something tangible or visible (fingers) with something that is purely conceptual (number). The connective capacity was called semiosis by logician Charles Peirce (1931–1958). It is guided by the conversion of sensory-experiential information, such as quantities of objects, into inner images of the objects by the imagination, which are then converted by the brain into sign behavior, such as finger-counting or etching tallies onto some stick or bone. In turn, this behavior allows us to reflect on the experiential information autonomously, that is, without the objects being physically present for the senses to perceive. Semiosis manifests itself early on in childhood. When children come into contact with a new object, their instinctive reaction is to explore it with their senses—that is, to handle it, taste it, smell it, listen to any sounds it makes, and visually observe its features. This exploratory phase of identifying the object constitutes an innate sensory stage. This produces an image of the object in the brain that allows infants to recognize the same object subsequently without having, each time, to examine it over again “from scratch” with their senses (although they might explore its physical © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Danesi, Poetic Logic and the Origins of the Mathematical Imagination, Mathematics in Mind, https://doi.org/10.1007/978-3-031-31582-4_1

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1 Imagination

qualities further for various other reasons). Now, as children grow, they start to engage more and more in behavior that displaces this sensory phase—they start pointing to the object and imitating any sounds it makes, rather than just handling it, tasting it, etc. These imitations and indications are the child’s first attempts at representing the object abstractly (Morris 1938, 1946). Thereafter, the repertoire of such sign-based activities increases dramatically, as children learn how to refer to the world through the language to which they are exposed in cultural context, rather than through the senses alone. Now, the same kind of sign-based displacement process characterizing childhood (ontogenetic) development can be imagined as having occurred phylogenetically in the prehistoric origination of number signs and thus of numeration. Over time, as people started using the signs and manipulating them for various reasons even beyond numeration, the brain started producing images of novel or uncommon ideas, which transcended the primordial functions of the imagination (so to speak). As Kasner and Newman (1940: xiv) insightfully observe, mathematics as a system of signs evolved from such “creative faculties beyond even imagination and intuition.” As will be argued throughout this book, those faculties have a name—the fantasia, defined as the ability to reconstitute the images gathered from sensory experience to produce novel imaginary constructs of the world that can then be used to understand that very world on human terms. The fantasia is thus the ultimate source of art, narrative, poetry, mathematics, and other creations of the human brain. It gives poetic form to things, manifesting itself typically in metaphorical signs and structures (as will be discussed subsequently). It is to be noted that the ancient Greek word for poetry—from poiein (“to create”)—referred to something made by humans so that they can grasp the essential meaning of things in their own peculiar ways. As discussed in the preface, it was the Neapolitan philosopher Giambattista Vico who used this notion to encapsulate how humans carry out inferences about the world. The fantasia is thus a blend of imagination and fantasy, having its roots in the Greek notion of phantasia, a term referring to the ability to convert pieces of information from sensory experience into thoughts (Long 2001). The difference between the Greek and Vichian versions is that phantasia referred to the images themselves as the basis of thought forms (Aristotle 350 BCE); the fantasia, on the other hand, refers to our ability to play with the images creatively, building on them to craft novel ideas and systems of thought. Vico assigned that crafting ability to the ingegno (ingenuity), which he saw as a faculty in itself that varies from person to person, akin to the ability of artists or poets to create visions of the world from their imaginations; it can be conceived as a form of “applied imagination.” In this framework, prehistoric objects, cave drawings, and bones, such as the following prehistoric animal horns, are products of the ingegno, the creative-inventive faculty of poetic logic (Youmans 1894: 646) (Fig. 1.1): The top one shows a fish etched on a reindeer horn; the one below shows a squatting stag engraved on a stag horn; the third from the top shows a reindeer running after an animal depicted on a reindeer horn; the fourth one shows two auroch heads sculpted on an auroch bone; and the last one shows a human figure, an eel, a horse head, and three rows of marks etched on some unknown bone. Such artifacts

Prologue

3

Fig. 1.1 Prehistoric bone etchings. (Wikimedia Commons)

are evidence of how humans have always had the ability of converting impressions into ingenious creations (literally) that allow them to reflect upon the impressions themselves. The same faculty of invention was behind the creation of the first numeration artifacts, consisting of tallies etched into animal bones (Flegg 2002). These include the Lebombo Bone discovered in the Lebombo Mountains of Eswatini, which is around 44,000 years old, on which 29 distinct notches can be seen; the Wolf Bone, discovered at Vestonice, Moravia, which is around 30,000 years old, on which 55 tally marks can be seen; and the Ishango Bone dated to around 18,000 to 20,000 BCE, which will be discussed below. These bones are concrete evidence of how early humans devised their systems of representing numbers ingeniously, through iconic constructions (similarity between the form of a sign and its meaning): one tally = one object, two tallies = two objects, three tallies = three objects, and so on. As time passed, and early civilizations developed numeral systems for carrying out everyday tasks, the iconic signs became gradually part of a system of thinking that came to be known as mathematics, which was praised early on as a high art form, much like poetry, allowing humans to encode and detect patterns of number and space in the world. It is at that point that what we now call the mathematical mind emerged to guide the development of the discipline of mathematics. As one of the first mathematicians (in this sense), Pythagoras believed that the mathematical

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imagination would ultimately reveal the obscure secrets of the universe, founding a cult to study the connectivity between numbers, geometrical properties, music, and the cosmos—a connectivity that came to be known as the “music of the spheres.” Pythagoras claimed that the movements of celestial bodies constituted a form of music that is not audible, but which displays the same pattern of numerical ratios that musical harmonies possess. Some of that ancient Pythagorean sense of discovery of hidden pattern continues to reverberate within us today every time we solve some difficult problem or get the point of some arcane mathematical concept. As Peirce (1923) emphasized, there is a strong sense in us that mathematical ideas enfold larger realities concealed within them. The same can be said of poetry—one can read a poem by, say, Emily Dickinson over and over and still find more meanings in it. This might be the reason why Russell paired mathematics with poetry (as mentioned in the preface), perhaps because they display a similar kind of creative exploration of the world, and as such should be integrated into everyday life: “What is best in mathematics deserves not merely to be learnt as a task, but to be assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement” (Russell 1907). As the nineteenth-century mathematician Karl Weierstrass also keenly understood, prefiguring Russell: “A mathematician who is not also something of a poet will never be a complete mathematician” (in Bell 1937: xix).

The Mathematical Fantasia Vico introduced his notion of the fantasia in his book Scienza nuova (1725), translated in English as the New Science (Bergin and Fisch 1984), which is fundamentally a treatise on the “science of the imagination,” as Donald Verene (1981) has phrased it, with the term scienza referring to “knowledge of any kind” (its original etymological meaning). Vico did not discuss the mathematical imagination in the book as such; yet his notion of fantasia is a central one for unlocking how it works, as will be discussed throughout this book. Since it cannot be accessed directly, or studied in any empirical-objective way, the best approach to the mathematical fantasia is, as Kasner and Newman showed in their popular book, to put it on display through the many ingenious intellectual artifacts it has produced throughout mathematical history. Long before the Kasner and Newman book, another popular one, Flatland: A Romance of Many Dimensions, written by the preacher and literary critic Edwin A. Abbott in 1884, showed how the mathematical imagination is ingrained in all of us, allowing us to even envision the fourth dimension. It does so by engaging us directly in imagining it via analogy (Hofstadter 1979; Hofstadter and Sander 2013). The characters of the novel are geometrical figures living in a two-dimensional universe, called Flatland. Flatlanders see each other edge-on, that is, as points or lines, even though, from the vantage point of an observer in three-dimensional space (called Sphereland) looking down upon them, they are actually lines, circles, squares, triangles, etc. To grasp how Flatlanders “see” the world, Flatland can be

The Mathematical Fantasia Fig. 1.2 Unfolding the cube

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Cube (three-dimensional folded version)

Cube (two-dimensional unfolded version = six squares)

imagined as the surface of a table. If one crouches to look at a square piece of paper lying on the table, with the eyes level with the table’s surface, it will appear as a line, or if one vertex is proximate to the line of sight, as a point. The only way to see it as a square is to view it from above the table (in Sphereland). Now, the question becomes: Can the Flatland and Sphereland perspectives be represented geometrically so as to show any relation? The answer involves showing how a threedimensional figure can be unfolded into a two-dimensional one, and vice versa. Consider a cube made up of six sides, which are, geometrically speaking, six Flatland squares. The cube can be transformed into a Flatland figure, by unfolding its sides onto a flat surface (Fig. 1.2): The resulting Flatland figure can, of course, be just as easily transformed back into a three-dimensional cube by folding the six squares together again. Now comes the key insight that Abbott’s book allows us to contemplate: Can an analogous method be envisioned whereby a hypothetical four-dimensional figure (a hypercube) can be unfolded into a three-dimensional one? The “unfolded” hypercube would consist of eight cubes, and it can be represented as shown below (Fig. 1.3): We cannot actually see the hypercube in four-dimensional space, of course. But the three-dimensional representation invites us to envision it nonetheless as a real entity, given the unfolding analogy involved between Flatland and Sphereland figures. In all this scenario, it is the fantasia that leads us to use our “inner eye” to “paint” the resulting visions, so to speak: “For when we wish to give utterance to our understanding of spiritual [abstract] things, we must seek aid from our imagination

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Fig. 1.3 A hypercube in three dimensions. (Creative Commons)

to explain them and, like painters, form human images of them” (Vico in Bergin and Fisch 1984: 139). Once the thought experiment has occurred, it becomes the source for subsequent novel ideas and thoughts. We cannot, of course, fold the hypercube into its four-dimensional form. The imagination is not clairvoyance—like the Flatlanders, we Spherelanders can only imagine (literally) what a four-dimensional world might look like. So, whether the hypercube really is an unfolded fourdimensional figure or not in any physical sense is actually beside the point. The fact that it can be envisioned shows what the fantasia permits us to do—to see in the mind what the eyes cannot see outside of it. Now, from this type of simple mind experiment, we can envisage a geometry in n-dimensions and develop the mathematics behind it without any direct sensation of what the mathematics represents, since it unfolds within the imagination itself. As Freiberger (2006) has aptly observed, because it puts on display how the imagination is at the root of ideation in mathematics, Flatland remains is one of the best introductions to the notion of higher dimensions: This is an extremely well-thought-out story; every aspect of life in Flatland is accounted for, from housing and climate to the way in which Flatlanders recognise each other’s shape (being unable to see their world from above). Abbott’s descriptions of how the square manages to imagine a three-dimensional world are a great guide to how we might go about imagining four (or more) dimensions. In fact, the square even conjectures that a fourdimensional world might exist, much to the annoyance of the sphere which considers itself supreme. Abbott’s analogy is clear and strong, and will make sense to the most unmathematically-minded reader. It’s a beautiful defense of mathematical thought and its power to open doors to fascinating new worlds.

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Abbott’s book showed concretely how we can ascend from plane geometry to solid geometry, and thus to grasp the difficulties of understanding higher geometries. Actually, decades before Flatland, the psychologist and physiologist Gustave Fechner wrote his own small story, Space Has Four Dimensions (1846), under the pseudonym of Dr. Mises. In it, a two-dimensional creature was a shadow person projected onto a screen by an opaque projector. He could interact with other shadows, but he could not conceive of a direction perpendicular to his screen. Fechner suggested that for such a creature, time would be a third dimension, which cannot be comprehended spatially (Fellner and Lindren 2011). After the publication of Flatland, physicist Hermann von Helmholtz also wrote his own story about a two-dimensional creature constrained to move along a surface, trying to come to an understanding of the intrinsic geometry of his own world without the benefit of a three-dimensional perspective. He examined the geometrical implications of his own story in 1878, in his article “On the Origin and Significance of the Axioms of Geometry,” published in the periodical Mind. Similar geometric ideas were also being contemplated by mathematicians, starting with Gauss’s 1827 General Investigations of Curved Surfaces. But, as mathematician James Sylvester asserted in his 1870 article, “A Plea for the Mathematician,” without the stories, perhaps, the pursuit of n-dimensional geometry may not have gained momentum. Over a century after Flatland, mathematician Ian Stewart wrote Flatterland (2001) as a sequel bringing in ideas that had fomented in n-dimensional geometry since 1884, featuring a geometrical character who is able to travel to any space in the “Mathiverse,” the set of all imaginable worlds. The novel also brings in themes from topology and hyperbolic geometry, as well as notions from quantum physics. The point of Stewart’s novel is that anyone can imagine geometric constructs without knowing the complex mathematics behind them. The reason is that the fantasia is an innate ability we all possess, allowing us to see from the inside what may be outside. Vico saw the fantasia as part of a faculty that he called poetic logic, defined as the capacity to not only imagine unfamiliar things ( fantasia), but also to express or construct ideas and artifacts from the inner visions, which he called the ingegno, and, finally, to be able to reconstruct the visions for expanded imaginative explorations (memoria). This tripartite faculty can be seen to operate in Flatland as follows: the fantasia is involved in stimulating the contemplation of a world beyond the third dimension; this is brought about by the insight of an ingenious unfolding technique (the ingegno) and finally, after reading the book our memory attempts to reconstruct the experience in different ways so that we can grasp the dimensionality principles involved ever more deeply (memoria). Vico did not discard the kind of rational logic that can be seen to operate in such argumentation as the syllogism or even in geometric proofs. But for Vico it is a point-of-arrival, not a point-of-departure. Natural discoveries, such as the one brought about by the thought experiment involved in Flatland, followed a “poetic” route that started from imaginative thinking, progressing gradually, and with significant effort, to rational modes of understanding. Arriving at the very idea of an n-dimensional geometry, could not occur in a vacuum; it took the work of the imagination to literally envisage a relation between dimensions. This type of unconscious inference starts, actually, through what Vico called poetic wisdom, an innate sense that things are a certain way; poetic logic then

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A

A´ C

900 [AB - (AA´ + BB´)] = length of tunnel

B´

B Fig. 1.4 A model of an engineering problem

explores this sense through the fantasia. So, in the Flatland scenario, the starting point is a sense that there are points of contact between two-dimensional and threedimensional spaces (poetic wisdom); after which, poetic logic stimulates the fantasia to produce an inner vision that is translated by the ingegno into a short story. As these visions accumulate, eventually rational logic steps in to stabilize them into a system of knowledge—namely, n-dimensional geometry. The fantasia is what liberates human beings from the constraints imposed on them by biology. It allows humans “to know from the inside” by extending “what is made to appear from sensation beyond the unit of its appearance and to have it enter into connection with all else that is made by the mind from sensation,” as Verene (1981: 101) puts it. The debate on the relation between the imagination and thought goes back to Plato, who separated the image (eikon) from the idea (eidos). This set in motion the tendency to view rational thinking as separate from mental imagery. Descartes reinforced the Platonic model by claiming that mental images proceeded without logic, and so they could not be studied scientifically. Paradoxically, as Verene (1981) has pointed out, Descartes’ own ideas unfolded in the form of highly suggestive and creative imagery. What Plato, Descartes, and all philosophers fixated on eidos forgot was that thinking starts via eikon, that is, by imagining something before concrete ideas of what to do can be contemplated. Consider the following engineering task. Suppose that a tunnel must be dug through the middle of a mountain. Since the length of the tunnel cannot be measured physically, the Pythagorean theorem suggests a plan for doing so without direct measurement. Point A on one side of the boulder and point B on the other are chosen such that both points remain visible from C to the right. C is chosen so that angle ACB is a right angle (90°). Then, by aligning A with A´ (the entrance to the mountain on one side) and B with B´ (the entrance to the mountain on the other side) the required length can be seen to be A´B´ (Fig. 1.4):

The Mathematical Fantasia

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The problem is now solved easily with simple geometry. We start by measuring AC and BC, and then, plug the values into the Pythagorean formula AB2 = AC2 + BC2, which will yield a measure for AB. Then we measure the distances AA´ and BB´, subtracting them from AB to get the length of A´B´, which is the length required to dig a tunnel through the mountain. As this shows, before applying known geometry (eidos), it was necessary to imagine a geometrical scenario to which it could be applied in a task-specific way (eikon). Overall, the solution to this problem entails the operation of poetic logic at the start (imagining the scenario in an ingenious representational way), which then leads to a use of the established logic of geometrical method. This simple example thus shows how poetic logic allows us to eliminate trial-and-error physical intervention and solve problems imaginatively before we tackle them in real terms. There is no science or engineering without the mathematical imagination. The mathematical fantasia manifests itself in many ways, sometimes even producing a shock effect among mathematicians as it comes up with something new or unorthodox. A classic case in point is Kurt Gödel’s (1931) famous proofs, such as his incompleteness one, which was inspired by Cantor’s (1891) diagonal proof. Cantor had shown ingeniously that there are infinite sets of numbers which cannot be put into a one-to-one correspondence with the set of natural numbers—to be discussed in Chap. 4. Suffice it to say here that Cantor’s proof shocked the mathematical world at the time: Was it a proof or some kind of bizarre demonstration leading nowhere logically? It would have to be either accepted as a novel way of proving something, or else discarded completely. As it turned out, the diagonal method became intrinsic to mathematics opening up new vistas for understanding mathematics itself. It also introduced an image schema (diagonalization) into proof systems, as Lakoff and Núñez (2000) called it (Chap. 4), becoming the basis for an even more shocking proof. Before Gödel, it was taken for granted that every proposition within a logical system could be either proved or disproved within it. But Gödel argued that such a system will invariably contain a proposition within it, akin to a Cantorian diagonal number, that is true but unprovable (Danesi 2002: 146): Consider a mathematical system that is both correct—in the sense that no false statement is provable in it—and contains a statement “S” that asserts its own unprovability in the system. S can be formulated simply as: “I am not provable in system T.” What is the truth status of S? If it is false, then its opposite is true, which means that S is provable in system T. But this goes contrary to our assumption that no false statement is provable in the system. Therefore, we conclude that S must be true, from which it follows that S is unprovable in T, as S asserts. Thus, either way, S is true, but not provable in the system.

Gödel showed this more technically by representing statements in the system with natural numbers (Gödel numbers). In this way, the properties of the statements— such as whether they are true or false—are equivalent to determining whether the properties of the Gödel numbers match the properties of the statements. Gödel then used the diagonalization technique to demonstrate that one statement (a Gödel number) in the system was unprovable. Without going into details here, suffice it to say that the unprovable statement appeared in a diagonal of Gödel’s tabulated

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statements. As a result, Gödel’s demonstration showed that mathematical systems are incomplete and undecidable, shocking the mathematical universe at the time. The point here is that, after his flash of insight based on Cantor’s diagonal proof, Gödel then used analogical thinking (numbers for propositions) to show incompleteness and undecidability within the system. Numbers exist independently of numeral systems; but once the systems are used, they allow us to explore numbers as abstractions and thus to envision what ideas or problems there may be within them. This was intuited by the Pythagoreans, who did not distinguish between numeration (the symbols themselves) and numerology (the broader meanings that numbers seem to bear). The Gödelian diagonal argument shows how the ingegno of mathematicians guides the utilization of their inner visions ( fantasia) to devise or explore something ingeniously (literally). Whereas the fantasia transforms sensory inferences into inner visions, the ingegno carries out the creative handiwork of turning the vision into ideas and systems of thought. The brains of all animals have the capacity to form memorable images. This is a survival function. But nonhuman animals lack the ability to transform their bodily based images into invented ideas and structures in the same way as humans. As Vico phrased it: “The human mind is naturally inclined by the senses to see itself externally in the body” (in Bergin and Fisch 1984: 236). In this Vichian scenario, the ingegno is the source of diagrams such as the Cantorian one and, at a more practical level, the engineering one above. As Charles Pierce (1931–1958, volume 4: 353) cogently remarked in discussing how we make diagrams of things, “the act of inference consists in observing a relation between parts of that diagram that had not entered into the design of its construction.” The final touch to the process of “mathematizing something” is to give it a “stable logical form,” such as a proof or an equation, which evokes mathematical memoria (memory). To reiterate, this is not just a retrieval or recall function of the mind, but a recreative one, and thus itself a part of poetic logic. It allows mathematicians to envision different aspects of mathematical information as an integrated whole, rather than as disparate elements. As will be discussed further in Chap. 4, the fantasia transforms bodily based hunches and intuitions into image schemas, a term introduced by Mark Johnson (1987) and George Lakoff (1987). An example is the container image schema, which is the source of the concept of set in mathematics (Lakoff and Núñez 2000), defined as a collection (container) of mathematical objects of any kind (numbers, symbols, points in space, lines, variables, etc.). The container schema is the product of imagining something abstract (a mathematical set) in terms of something concrete (a container). Johnson (1987: 79) defines image schemas as “those recurring structures of, or in, our perceptual interactions, bodily experiences and cognitive operations.” Lakoff (1987: 444) defines them as largely unconscious processes that portray locations, movements, shapes, etc., in the mind. Langacker (1987) has designated them “cognitive routines”—mentally prepackaged assemblages that can be employed in essentially automatic fashion without attending to the details of their composition. If we examine the mathematical lexicon of set theory, we can easily see

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the use of metaphors based on the container schema: numbers are in a set; numbers can be taken out of sets; they are part of a set; and so on. It is not possible to pinpoint any specific neural region for the fantasia, since it arises from the interactions of various parts of the brain, although image schemas have been found to reside in certain neural spaces (for example, Rohrer 2005). But such neural collocations do not explain how the imagination produces them in the first place. Whatever the case, without this inner creative force, there would be no discoveries, no mathematics. It is the originating force of all human ideas, inventions, and discoveries. Vico (in Bergin and Fisch 1984: 129) describes it as a corporeal imagination, prefiguring so-called embodied cognition theories in current cognitive science: In such fashion the first children of nascent mankind. . .created things according to their own ideas. . .by virtue of a wholly corporeal imagination. And because it was quite corporeal, they did it with marvelous sublimity; a sublimity such and so great that it excessively perturbed the very persons who by feigning did the creating, for which they were called “poets,” which is Greek for “makers.”

The main premise of embodied cognition theory is fundamentally Vichian, being based on the notion that human thought is shaped by aspects of the body beyond perception and cognition (the fantasia). Without going into the many points of contact between Vichian ideas and embodied cognition theory, suffice it to say here that they have converged on a basic principle of human life—thoughts and their representations are extensions of the human body, allowing it to do more than it was programmed to do by biology.

Poetic Wisdom As mentioned briefly above, the starting point for the construction of ideas is poetic wisdom, which Vico defines as follows (in Bergin and Fisch 1984: 128): Poetic wisdom, the first wisdom of the world, must have begun with a metaphysic not rational and abstract like that of learned men now, but felt and imagined as that of these first men must have been, who, without power of ratiocination, were all robust sense and vigorous imagination. . .This metaphysic was their poetry, a faculty born with them (for they were furnished by nature with these senses and imaginations); born of their ignorance of causes, for ignorance, the mother of wonder, made everything wonderful to men who were ignorant of everything.

Poetic wisdom is Vico’s term for our innate need to make sense of things. It is behind the vague intuitions about things that come up every so often, suggesting that there may be something behind them, for which we have no available concept. As a classicist, Vico undoubtedly grafted his notion of poetic wisdom from an ancient Greek idea, best articulated by Aristotle in his Metaphysics (350 BCE), that intuitive wisdom was the basis for exploring why things are a certain way, which is deeper than just knowing that things are a certain way. It is an intuitive sense of what needs to be done and to do it without being told what to do. Poetic wisdom awakens the

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faculty of poetic logic, which then attempts to convert intuitive understanding into creative images and proceed from there onto a path of discovery. The discussion of poetic wisdom is found in the second and largest section of the New Science. Vico’s objective is to argue that hunches and guesses are guided by an instinctive search for patterns and possibilities, so as to make sense out of the chaotic flux of impressions that assail the brain. The search for Order in Chaos is the driving force behind poetic wisdom and, arguably, the founding spirit of mathematics itself. This spirit is evident in Pythagoras’s establishment of a community in Crotone Italy to explore mathematics intuitively and systematically at the same time, in the belief that mathematics would allow for the uncovering of the many secrets of the universe, because it mirrored the structures inherent in the universe. Pythagoras himself showed how the structure of music mirrored the structure of the cosmos and even the mind in the form of mathematical ratios. As Stuart Isacoff (2003) has argued, western musical traditions came about from the Pythagorean legacy that the “natural structure” of harmony is mathematical and reflects the harmony of the emotions in humans. Thus, the reason as to why we react to, say, the symphonies of Beethoven so emotionally, is due to its mathematical symmetry. Ian Stewart (2008: 9) has eloquently summarized the Pythagorean theory of a “mathematical cosmos” as follows: The main empirical support for the Pythagorean concept of a numerical universe comes from music, where they had noticed some remarkable connections between harmonious sounds and simple numerical facts. Using simple experiments they discovered that if a plucked string produces a note with a particular pitch, then a string half as long produces an extremely harmonious note, now called the octave. A string two-thirds as long produces the next most harmonious note, and one three-quarters as long also produces a harmonious note. These two numerical aspects of music are traced to the physics of vibrating strings, which move in patterns of waves. The number of waves that can fit into a given length of string is a whole number, and these whole numbers determine the simple numerical ratios. If the numbers do not form a simple ratio then the corresponding notes interfere with each other, forming discordant ‘beats’ which are unpleasant to the ear. The full story is more complex, involving what the brain is accustomed to, but there is a definite physical rationale behind the Pythagorean discovery.

Indirect support for the Pythagorean wisdom came forward dramatically in 1865, when English chemist John Newlands discovered that arranging the elements according to atomic weight produced an incredible finding—those with similar properties occurred at every eighth element like the octaves of music. This finding came to be called the Law of Octaves, which led, in turn, to the development of the Periodic Law of chemical elements. Interestingly, Newlands’s law was ridiculed by some of his contemporaries, and the Society of Chemists did not accept his work for publication at first (Bryson 2004: 141–142). The Pythagorean mathematical model of the universe actually revealed unexpected faults. When the ratios between certain string vibrations are set, other ratios are thrown off, thus producing dissonances. The Pythagoreans knew about these defects in their paradigm, but kept them secret. To banish the dissonances, in the eighteenth century, the keyboard of the piano (clavier) was tempered by breaking the octave into equal parts, so that all harmonies sounded in tune—an event attributed to Johann Sebastian Bach’s Well-Tempered Clavier (1742). It was thus a human invention (the ingegno) that rectified the Pythagorean defect, thus establishing a musical tradition that continues to this day. This episode

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illustrates that discovery and invention may be two sides of the same mental process, so to speak (discussed subsequently). In essence, Pythagoras’s approach to mathematics was, at its core, guided by poetic wisdom. Like Vico, he must have realized that “because the origins of all things must by nature have been crude: for all these reasons we must trace the beginnings of poetic wisdom to a crude metaphysics [abstract understanding]” (Vico in Bergin and Fisch 1984: 100). As an inner force, poetic wisdom was what led Pythagoras on his search to find relations among numbers, geometrical figures, and their manifestations in the universe, using poetic logic as his subsequent guide. As philosopher George Henry Lewes noted in 1875, (Lewes 1875: 369), the mind is constantly in search of “a co-operation of things” in the world, which manifests itself by the “inner dialectic” that spurs us on to look for answers to our intuitive questions. As the medieval philosopher, Peter Abelard, put it in his Sic et Non, “Constant and frequent questioning is the first key to wisdom. . .and by inquiry we perceive the truth” (cited in Graves 1918: 53).

Poetic Logic As a tripartite faculty, consisting of a dynamic interaction among the fantasia, the ingegno, and the memoria, the notion of poetic logic provides a concrete framework for grasping how the mathematical imagination might work in all kinds of tasks, such as those involving visualizing a pattern in some practical problem. It is the fantasia that generates an image of the situation that allows us to view it internally from various mental angles, so to speak. To illustrate this, consider the well-known river-crossing problem devised by the eighth-century scholar, teacher, and theologian, Alcuin of York, in his collection of mathematical problems, Propositiones ad acuendos juvenem (c. 782) (in Burkholder 1993): A certain traveler needed to take a wolf, a she-goat and a load of cabbage across a river. However, he could only find a boat which would carry two of these [at a time—that is, himself and one other]. Thus, what rule did he employ so as to get all of them across unharmed [given that if the goat were left alone with the cabbage, it would eat the cabbage, and if the wolf were left alone with the goat, it would eat the goat]?

The solution hinges on visualizing how the first trip over can be realized successfully, after which the ingegno provides the logical (sequential) steps to do so. The traveler cannot start with the cabbage, since the wolf would eat the goat, nor the wolf, since the goat would then eat the cabbage. So, his only choice is to start with the goat. Once this critical decision is made, the rest of the problem is solved easily. The traveler goes across with the goat, drops it off, and goes back alone. When he gets to the original side, he could pick up either the wolf or the cabbage. Let us select the cabbage. He goes across with the cabbage to the other side, drops it off, but goes back to the original side with the goat (to avoid disaster). Back on the original side, he drops off the goat and goes over to the other side with the wolf. When there, he drops off the wolf safely with the cabbage. He travels back alone to

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Fig. 1.5 Poetic logic model

pick up the goat. He then travels to the other side with the goat and, together with the wolf and cabbage, continues on his journey safely. The problem itself was likely inspired by an actual experience or occurrence and thus detected as meaningful by an innate poetic wisdom—that is, a situation similar to that in the problem of getting items and animals across a river or some other obstacle in a practical way that avoids possible dangers, which is something that would make sense intuitively. This is what likely prompted Alcuin’s fantasia to produce an inner vision of the situation, in the form of an image schema (Lakoff 1987) involving a back-and-forth path constrained by a specific combinatory structure of elements (cabbage, goat, wolf). The image schema and the constraints guide the ingegno into envisioning a sequence of back-and-forth movements that satisfies the conditions posed by the situation. Once solved, we feel a sense of having gained an insight that may hide principles that can be used over and over again to solve similar problems—a function of the memoria. Among the latter one can see in Alcuin’s problem the roots of subsequent graph theory and combinatorics (combinations of objects in accordance with certain constraints). The whole process can be represented as follows (Fig. 1.5): As it turns out, the same problem shows up in diverse languages and cultures prior to Alcuin, which differ in detail but not in the overall imaginary scenario encoded by Alcuin’s version (Singmaster 1998), suggesting that we are dealing with a cross-cultural image schema derived from the same kind of experience, which Vico called an imaginative universal—a view of something that is both common and derived from similar experiences of the world. As Martha Ascher (1990: 26) has remarked, these “are expressions of their cultures and so variations will be seen in the characters, the settings, and the way in which the logical problem is framed.” Given the time frames in which they were devised, it is unlikely that they were mere adaptations of some previously known prototype, but parallel offshoots of the fantasia devised with the same type of poetic logic. As examples such as this one bring out, the fantasia is the primary capacity of poetic logic that allows us to generate an imaginary modeling of something in the mind, or more precisely, of our perception of reality. The English term figment of the imagination is a suitable one here, since it implies that the imagination literally contrives images of things—figment derives from Latin figmentum meaning “form, contrivance.” The cross-cultural appearance of river-crossing problems implies that

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people across the world tend to have the same kind of figment in the mind from a common experience. This is likely to be the form of consciousness that helped primordial humans cope with the first stages of sentient life: “the first founders of humanity applied themselves to sensory topics, by which they brought together those properties or qualities or relations of individuals and species which were, so to speak, concrete, and created from poetic genera” (Vico in Bergin and Fisch 1984: 495). The question of intelligence invariably comes up in any discussion of the mathematical fantasia. First, the term itself is too broad, as the psychologists Sternberg and Davidson (1982) found in their study of the relation between problem-solving and intelligence. As such, it literally means the ability to solve problems—no more, no less. A year later, Howard Gardner (1983) challenged the notion of a single intelligence, since it manifests itself in different ways according to medium or skill (music, drawing, logic, etc.). Again, even by broadening the definition of intelligence in this way, it still turns out to be an ability, not a faculty. The solution of Alcuin’s problem does indeed show intelligence; but this would be a characterization, not an explanation. The metaphors used in English (and other languages) in reference to intelligence suggest, etymologically, that it is nothing more than a convenient term for poetic logic, the faculty that transforms bodily based experiences into inner visual forms via the fantasia. As Eve Sweetser (1990) has cogently demonstrated, metaphorically transformed visual and physical sensations are, in fact, at the root of how we conceptualize intelligence, as evidenced by expressions such as grasp (something) and catch sight of, among myriad others. As Ong (1977: 134) has also remarked, the universality of such metaphorical strategies suggests that “we would be incapacitated for dealing with knowledge and intellection without massive visualist conceptualization, that is, without conceiving of intelligence through models applying initially to [physical experience].” This is why we conceptualize ideas as objects that we can approach, look at, touch, take apart, view from different perspectives, and so on. The solution to Alcuin’s problem also raises the question of the meaning of logic in mathematics, which is, essentially, any type of reasoning that allows us to envision a sequence of steps that hold together as a way to resolve a situation. Now, the initial form of logic used in the problem is poetic in the sense that it is a way of resolving a situation presented to us by the imagination, not proving it via deduction as such—hence it is part of the creative functions of the ingegno. Of course, logic can also be rational and used to prove something deductively (or in some other rational way), rather than discover a way to do something. But even so, when proving or solving something in mathematics, poetic logic is still involved unconsciously in different aspects of the proof or solution. Consider the following typical school problem in Euclidean geometry (Fig. 1.6): Prove that the angles of a triangle add up to 180°.

We start by considering established knowledge and previous theorems, including (a) angles on the opposite sides of a transversal between two parallel lines are equal and (b) a straight line is 180°. So, in the diagram above, angle A = angle 1 and angle

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Fig. 1.6 Problem in Euclidean geometry

Fig. 1.7 A tricky problem in geometry. (Based on Gardner 1994)

A

O

C

6

B 4 D

C = angle 2. Now, using (b), whereby angle 1 + angle B + angle 2 = 180°, and making the appropriate substitutions employing (b), we get angle A + angle B + angle C = 180°. Now, while this proof appears to reveal straightforward logical-deductive reasoning, there is still an imaginative component to it, since one must envision that (a) and (b) are essential aspects of the problem, not to mention to visualize the actual construction of the diagram involved in elaborating the proof. If one were to put imagination and logic (as so defined) in a binary relation, in this case, the functions of the two faculties would be complementary, not autonomous. Now, in contrast, consider the following well-known tricky problem, which is a version of a puzzle devised by Martin Gardner (1994: 44). At first glance, it would seem to be a problem that could be solved in the same way as the previous problem (Fig. 1.7): Given the dimensions of the radius OD (6 + 4 = 10), can the length of the diagonal AB in rectangle AOBC be determined?

As it turns out, it is impossible to solve this with the information that the diagram presents. So, it is here that the ingegno comes into play to literally help us “figure out” what to do. First, we recall that the diagonals of a rectangle are equal to each

Poetic Logic

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Fig. 1.8 Resolving the problem

A

O

C

6

B4 D

other in length. This suggests drawing the other diagonal (OC) of rectangle AOBC (Fig. 1.8): By doing this, we can now literally see that diagonal OC is also a radius of the circle. We know that the radii of a circle are equal. Line OBD is a radius as well and is equal to 10, as shown. Since line OC is a radius, it is thus also equal to 10. From this, we conclude that the other diagonal, AB, is equal to 10. This problem shows that in this case, poetic logic was more prominent than the systematic type of logic used in solving the problem above. So, in comparison, it is used to a greater degree, not equally as above. This shows that logic is a form of reasoning that can take various forms, of which the poetic form is of primary importance both in solving original problems and in resolving tricky ones such as Gardner’s puzzle. This suggests, in turn, that logic has at least two main forms, poetic and rational, “blended” together in the resolution of mathematical situations. The notion of blending, derived from cognitive science, is key to understanding how the mathematical imagination works, and will be discussed in more detail in Chap. 4 (Fauconnier and Turner 2002). Because of the ambiguity of the meaning of the term logic in western philosophical and mathematical traditions, the mathematician-writer Lewis Carroll took it head on in two marvelous books—Symbolic Logic (1896) and The Game of Logic (1886). The former consists of a formal diagrammatic treatment of the syllogism and the latter plays on it capriciously. Consider the following syllogism, for the sake of argument and illustration: Major premise: Minor premise: Conclusion:

All mammals are warm-blooded. Whales are mammals. Therefore, whales are warm-blooded.

The major premise states that a category has (or does not have) a certain characteristic, and the minor premise identifies a certain thing as a member (or not) of that category. The conclusion then affirms (or denies) that the thing has that characteristic. In Symbolic Logic Carroll gave the syllogism its first diagrammatic basis (discussed in Chap. 5). Carroll deeply understood that the tradition of

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Aristotle and Euclid in mathematics was a worthy one to maintain in an age of change and of uncertainty. So, he showed the importance of syllogistic methods of reasoning. But always the trickster, Carroll had previously poked fun at these very methods in his Game of Logic, understanding full well that there is no monolithic form of logic in the human mind—but many “logics,” including anti-logic. So, at the same time that he extolled the syllogism as a marvelous achievement, he also made fun of it as a human construct. In fact, he showed that one could make up valid syllogisms that were patently silly, which he called sillygisms, and thus devoid of any meaning. Examples of Carroll’s sillygisms are the following (in Fauvel, Flood, and Wilson 2013: 276): Major premise: Minor premise: Conclusion:

A prudent man shuns hyenas. No banker is imprudent. No banker fails to shun hyenas.

Major premise: Minor premise: Conclusion:

No bald creature needs a hairbrush No lizards have hair. No lizard needs a hairbrush.

Despite his trickster side, Carroll’s love of Euclidean geometry as a template of how logic unfolds in the mind is not only evident in his famous book on Euclid, Euclid and His Modern Rivals (1879), but in many of his musings, as found in his letters and diaries, where he turns Euclidean ideas into perplexing puzzles. In a diary entry dated December 19, 1898 (Wakeling 1994), he writes the following: Sat up last night till 4 a.m., over a tempting problem, sent me from New York, ‘to find three equal rational-sided right-angled triangles’ [with the same area]. I found two, whose sides are 20, 21, 29; 12, 35, 37; but could not find three.

As Gardner (1996: 22) remarks, there is an infinity of such triplets with a common area of 840: 40, 42, 58; 24, 70, 74; 15, 112, 113; and so on. Gardner goes on to observe that “Had Carroll doubled the sides of the two triangles he found, he would have obtained the first two triangles in the triplets just cited, from which it is easy to determine the third.” It seems that Carroll, a master inventor of logic problems, was stumped by the simple logic just described—a logic that required a dose of the ingegno to resolve. Aware of the ingenuity involved in all logical tasks, Carroll painted fantasy worlds in his children’s books where the imagination reigns supreme, as if it were a dream-like form of thought. The ancient Greek mathematicians were among the first to discuss the various forms of logic used in proofs, illustrated concretely by Euclid in his Elements (300 BCE), from induction and deduction to reductio ad absurdum (proof by contradiction). Induction involves reaching a general conclusion from observing a recurring pattern; deduction is reasoning about the consistency or structure of a recurring pattern; and contradiction inheres in showing that if something is not true then it would lead to a contradiction. Logic replaces the need to carry out practical activities over and over. The number of degrees in a plane triangle is 180° (above), as is suggested by measuring the angles of hundreds, perhaps thousands, of triangles.

The Origins Question

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Assuming that the measuring devices are precise and that errors are not made, we are bound to infer that the sum of the three angles adds consistently up to 180°. The expert mathematician would, however, claim that such thinking is not always reliable, because one can never be sure that some triangle may not crop up whose angles add up to more or less than 180°. To be sure that 180° is the sum for all triangles one must use a deductive method of demonstration. This inheres in applying already-proved concepts to the case-at-hand (as was done above), which show that this sum is necessarily so, no matter the type or size of triangle involved. However, as some of the problems discussed above show, this type of logic does not always apply. Aware of this, mathematicians themselves maintain that hunches and guesses play a central role in the origination of mathematical ideas. Charles Peirce (1931–1958), himself a mathematician and logician, emphasized that many, if not most, of our originating concepts are formed by a type of inferential-logical process that he called abduction. He described it as follows (Peirce 1931–1958, volume V: 180): The abductive suggestion comes to us like a flash. It is an act of insight, although of extremely fallible insight. It is true that the different elements of the hypothesis were in our minds before; but it is the idea of putting together what we had never before dreamed of putting together which flashes the new suggestion before our contemplation.

Abduction can be seen as the flash of insight that comes from the fantasia (Danesi 2004). So, mathematical discoveries come about through abduction, which is produced by the inner vision that the fantasia permits. After the initial phases, the more rational forms of logic come into play to organize the discoveries into systems of mathematical knowledge. Once an insight is attained, it becomes useful to “routinize” it, so that a host of related problems can be solved as a matter of course, with little time-consuming mental effort. Such routinization is a memory-preserving and time-saving strategy—it is part of memoria. It is the rationale behind all organized knowledge systems. Once abductive-imaginative thinking has done its job, so to speak, the rational part of the mind steps in to give its products stability through other forms of logical thinking. So, for instance, the Alcuin problem can now be made to be as complex as we desire (increasing the number of objects to be taken across and new conditions for doing so), but the “principle” learned from solving the original puzzle need not be discovered over and over again each time. Rather, we can now use a form of inductive (or perhaps even deductive) thinking to generalize a strategy for solving any puzzle of this type—no matter how many objects or constraints are involved (e.g., Csorba, Hurkens, and Woeginger 2012).

The Origins Question If poetic logic is the faculty that guides the mathematical imagination, then there must be some archeological evidence that it appeared early on in human history and even prehistory. The evidence is provided, arguably, by the etchings on prehistoric bones, mentioned above, which suggest that the ability to represent instinctive

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Fig. 1.9 The Ishango Bone. (Wikimedia, Science Museum of Brussels)

counting in some visual way is ancient. The bones show, in effect, a capacity to envision ideas (numbers) via symbols (tally marks). One bone in particular has been the target of much discussion among mathematicians; it is called the Ishango Bone, and actually involves two bones (probably the fibulae of a large feline mammal), left behind by the prehistoric ancestors of the Ishango people, who live on a Congolese lakeshore site in the north-eastern region of the Democratic Republic of Congo in Africa (Olivastro 1993: 5–11; Huylebrouck 2019: 153–166). The artifact was discovered in 1950 by Belgian geologist Jean de Heinzelin (see Heinzelin 1962) near Lake Edward in Zaire. It is now stored at the Royal Belgian Institute of Natural Sciences in Brussels. Carbon-dating estimates indicate that it is about 20,000 years old (Fig. 1.9): The bone has 168 etchings in total varying in orientation and length. No one really knows for certain what these represent. They could indicate the tallies of a game that was played by the ancestors of the Ishango. The bone could also be a calendar, a measuring stick, etc. Whatever its real purpose, the etchings show patterned symbolic thinking, which is clearly a creative force in the formation of early abstract thinking (Huizinga 1938). The most likely meaning of the bone is that it was some kind of mathematical tool (Everett 2017), with its tallies representing numerals via iconicity, that is, via a conceived similarity or analogy between the form of the numerals (the tallies) and their meaning: one tally = the quantity one, two tallies = the quantity two, and so on. So, the Ishango Bone reveals one of the earliest efforts to literally imagine a connection between tally marks (signs) and numerical concepts, indicating a shift from instinctive counting to abstraction. Without this shift, there would be no mathematics. The key notion here is iconicity as the originating mode in numeration. As Caterina and Gangle (2016) have argued, the appearance of iconic forms of representation in any system of knowledge, including mathematics and science, implies abduction, which, as discussed, is the spark that is prompted by the Vichian fantasia. It is thus indirect evidence of the workings of poetic logic in the prehistory of mathematics, as the Ishango Bone would suggest. Now, as subsequent archeology suggests, when tally signs for larger quantities were required, the original tally system became impracticable (Ifrah 1981). At that point, the abstraction process was extended in all likelihood to involve the concept of value, whereby some combination of signs (such as five tallies) would be equivalent in value to a new

The Origins Question

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Fig. 1.10 A Chinese abacus. (Public Domain)

singular sign that stood for them—that is, indexed them. Indexicality, as Peirce (1931–1958) defined it, involves relating signs to each other in some patterned way. This type of numeration became increasingly essential as signs for larger quantities were required, building upon the original iconic foundation. So, for instance, representing five tallies with a new sign (say a circle for the sake of argument) with the value of five, and then ten with, maybe, two circles, and so on, then a valuebased system emerges. Extrapolating the overall value of a numeral made up of tallies and circles requires a truly advanced form of abstraction, since it would be accomplished by adding up the different sign values. In a Vichian framework, the invention of increasingly complex numeration systems is tied to the ingegno, which, as discussed above, takes the conceptual forms created by the fantasia and develops them further through ingenuity and creativity. So, the flow from iconicity to indexicality mirrors the flow from fantasia to ingegno, the creative use of the imagination to come up with increasingly complex and ingenious abstract systems and devices to make counting efficient, routine, and less effortful. One of these was the abacus, traced to various ancient cultures in the second century BCE, from the Middle East and China to Mesoamerica and Africa. The Chinese version, known as suanpan, consists of columns of movable beads strung on wires, with each bead representing a digit and each wire a place value (Fig. 1.10): The column on the far right represents the “ones” values (1, 2, 3, ...), which in decimal terms is the 100 position; the next column (to the left) stands for the “tens” values (10, 20, 30,...), which in decimal terms is the 101 position; and so on. On each string, there are two parts: (1) two beads on top, each with the value of “five,” and (2) five beads below, each with the value of “one.” The number represented in the figure above is 6,302,715,408. Place-value systems, in the modern sense, are traced to the medieval era. It was the Hindu-Arabic decimal one that eventually became the principal one for purposes of numeration. It originated in India in the sixth or seventh century CE and introduced to Europe through the writings of Middle Eastern mathematicians, including al-Khwarizmi and al-Kindi, and then by Italian mathematician Leonardo Fibonacci in his Liber Abaci (1202). Fibonacci, who was born in Pisa, Italy,

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designed his book as a practical introduction to his fellow Italians of the HinduArabic number system, which he had learned to use during his travels along the Mediterranean coast, and which he realized could be used to solve what would otherwise constitute intractable problems with the Roman numeral system. Although the title is translated as the “Book of the Abacus,” Sigler (2002) maintains that the actual intent of the book was to show how mathematical computations could be carried out easily without the aid of an abacus (see also Ore 1948). The first two sections of the book deal with methods for converting Roman numerals to decimal ones and showing how these facilitate computation in areas such as currency and measurement. The construction of place-value systems could not have occurred, however, without the fantasia and the ingegno, which are always at work at different levels, from the earliest enactive stage of sensing pattern, passing through more complex stages, whereby the pattern is imagined in more abstract terms, leading to symbolism of truly the highest abstract order (the place-value system), which then allows for numeration to become independent as a system of symbolic knowledge—hence the emergence of a mathematical memoria. In the case of organizing tallies into groups for further signage, it is the image schema of the container that appears to be at play. When the container is filled with, say, five objects, then it is assigned the value five, and new containers are envisioned. The image schema eventually morphed to a linear layout in the case of place-value systems—that is, it involved envisioning numbers as increasing or decreasing in value along successive points on a line. This phylogenetic process appears to be reenacted in how children grasp number concepts (Mandler 2007; Small and Lin 2012). Even the work of Jean Piaget can be seen to reflect image-schematic number conceptualization, although it is not named as such (since Piaget’s work preceded that of image schema theory). In one experiment (Piaget 1952), he showed 5-year-old children two matching sets of six eggs placed in six egg-cups (the container schema in this case), asking the children whether there were as many eggs as egg-cups (or not). Invariably, the children would reply affirmatively, showing an understanding of numbers as objects in containers. Piaget would then take the eggs out of the cups, bunching them together, with the egg-cups left where they were previously, asking the children whether or not all the eggs could be put into the cups, one in each cup and none left over. The answer he typically received was “no.” Asked to count both eggs and cups, the child would correctly say that there was the same amount. But when asked if there were as many eggs as cups, the child would again answer “no.” The child, Piaget concluded, had not grasped the relational properties of numeration, which are not affected by changes in the positions of objects. Piaget showed, in effect, that 5-year-old children have not yet established in their minds the binary connection between numerals and numbers, although they certainly were able to conceptualize the notion of number equivalencies as based on the image schema of a container (the egg-cups). Recurrence and periodicity (the seasons, the length of days, etc.) is another image schema that lies at the core of the primordial mathematical fantasia. Calendars, for instance, are based on numeration signs that show recurrence in specific ways, according to number of days, weeks, months, years. It is this image schema that is the cognitive basis of sexagesimal systems (Neugebauer 1957), which involve cyclic

The Origins Question

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recurrence after the number 60. The Babylonians used the sexagesimal system for the calculation of time and angles, which is still practical not only because of the multiple divisibility of the number 60, but also because time involves 60 in various ways, such as 60 minutes equal to 1 hour. All this shows that the fantasia is adaptive to experiential and practical needs, subserving them in specific ways, producing image schemas of situations that are converted into ideas and thought systems. Over time, the different systems have led to the highly abstract notion of modular arithmetic, developed initially by Carl Friedrich Gauss in his Disquisitiones Arithmeticae (1801)—an offshoot of the cyclic image schema, whereby numbers are envisioned as “wrapping around” some circular object, reaching a certain value (the modulus). Yet another image schema that has played a role in the development of number conceptualization is that of binary numbers, which involves a full-versus-empty image schema, where the “1” is “full” and the “0” is empty. It was German mathematician Gottfried Wilhelm Leibniz who elaborated the first system based on binary numbers in 1703. But before Leibniz, the Chinese had come up with a binary system of their own, called the I Ching, devised during the Shang dynasty of ancient China (c. 1766–1027 BCE), which has traditionally been used for divination. It is constructed with two lines—a yin (broken line) and a yang (unbroken line). The lines are converted into numbers and then into symbolic answers to spiritual questions. Leibniz was apparently inspired by the I Ching to put forth his system of binary arithmetic, publishing it in a paper titled Explication de l’arithmétique binaire (“Explanation of Binary Arithmetic”). Once numeration systems had evolved, they could then be applied to contemplating increasingly abstract ideas of various kinds. Pythagoras had discovered, for instance, that plucking strings or striking objects produced harmonious sounds that could be described with specific numerical ratios—hence showing how fractions could be applied to a specific phenomenon for grasping what it meant, emotionally, aesthetically, and intellectually. He also reasoned that the planets moving in orbits also produced the same pattern of harmonious sounds, being part of the same physical world—a view that came to be known as musica universalis. The difference is that the sounds emitted by planetary orbital revolutions are imperceptible to the human ear, unlike human musical sounds which are audible. Legend has it that Pythagoras could actually hear the celestial sounds—a gift conferred on him by the Egyptian god Thoth. This is, of course, part of the lore that surrounded Pythagoras. What is true is that Pythagoras likely derived his musica universalis worldview from the Egyptians and the Chaldeans who described the celestial bodies as emitting a cosmic chant as they moved through the sky (Burkert 1972). Even in the Bible (Job 38.7), the beginning of time is described as the moment “when the stars of the morning sang together and all the sons of God raised a joyous sound.” Overall, the history of mathematics can be explained via poetic logic as it develops gradually into a system of rational logic, so as to make mathematical tasks routine and efficient. However, if innovation is required, or if it comes up via happenstance, then the mind resorts back to the fantasia, as discussed throughout this chapter. In this framework, the Ishango Bone provides evidence of the operation

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of the primordial (raw) imagination—a form of understanding that is based upon, and guided by, conscious bodily experiences that are transformed into abstractsymbolic ideas. Mathematics thus evolves from a poetical age, through a representational-symbolic one, and finally to a rationalistic one. What Is Mathematics? was the title of a significant book written for the general public by Courant and Robbins in 1941. The two authors the answer their question by illustration, showing what mathematics looks like and what it does, allowing us to come to our own conclusions as to what mathematics is. And perhaps this is the only plausible way to answer the conundrum of how mathematics originated—by examining the artifacts of the fantasia and the ingegno, such as the iconic, imageschematic nature of early numerations systems. It is the same kind of approach used in cybernetics with the notion of the mind as a “black box” (Wiener 1961: xi), that is, as an unknown entity that can only be reconstructed (hypothetically) by observing what it allows for and then working backward. Lakoff and Núñez also addressed the question of what is mathematics in 2000, making the claim that in the black box were image schemas that can be located in the same areas of the brain as language—a topic to be discussed in detail in Chap. 4. As numerical concepts became increasingly complex in antiquity, language became a means for “taming” them, since words for numbers can be organized lexically. In his work, The Sand Reckoner, Archimedes (see 1897) dealt perhaps for the first time with the dilemma of large numbers, asking the question: What is the upper limit for the number of grains that fit into the universe? Archimedes wanted to find, specifically, a way to name the upper limit for the number of grains of sand that will fit into the universe, which he estimated as 8 × 1063 grains. The numbernaming lexicon in use during his times could express numbers up to a myriad (10,000). Archimedes extended the word to name numbers up to a myriad myriads, or 108, separating numbers into orders. He labeled the numbers up to 108 the first order and called 108 itself the “unit of the second order.” Multiples of this unit then became the second order: 108 × 108 = 1016, which became the “unit of the third order,” whose multiples were the “third order,” and so on. The same kind of problem of naming large numbers was raised by Kasner and Newman in the book mentioned at the start of this chapter, Mathematics and the Imagination (1940), in which they used the word googol to represent the number 1 × 10100. As the authors indicate, the term was actually coined in 1920 by Kasner’s 9-year-old nephew. Kasner and Newman then introduced their own term, googolplex, defined as 10 to the googol power, or 10 multiplied by itself a googol number of times. It is impossible even to envision what number this represents, let alone write it out completely. It has, in fact, been estimated that there are more zeroes in a googolplex than there are particles in the known universe.

Psychological Validity

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Psychological Validity An obvious problem that emerges in using terms such as the mathematical fantasia, the ingegno, and memoria concerns their psychological validity. For example, what does the fantasia imply neurologically? If it is a creative image-making ability, is it located in areas that involve vision and perception? It has been found, significantly, that images, both novel and recalled, are processed in the visual cortex in the occipital lobe, spanning both cerebral hemispheres. The evidence suggests that they emerge in regions where experiential and perceptual processes are involved, indicating that the fantasia crystallizes from an integrated system in the brain for relating images to concepts and their meanings—a kind of blending of the regions to produce the abductive flash of insight (Fauconnier and Turner 2002; Brown 2018; Kutter et al. 2022). Research has shown that, while instinctive counting behaviors are found in other species, the abstraction of these abilities into systems of symbolic knowledge is undoubtedly a specific human ability, requiring the use of language, art (for drawing diagrams), and other creative faculties—thus involving neural blending of a more general kind. The work of Stanislas Dehaene (1997) is relevant in this area. He has brought forth experimental evidence to suggest that the human brain and that of some chimps both come with an innate aptitude for counting. The difference in the case of chimps is an inability to formalize this knowledge into systems of representation. So, humans and chimps possess a shared evolutionary number instinct, but not an abstract number sense. In neurological terms, it would appear that abstraction involves some form of inner spatial imagery blended with creative impulses, which may well be the basis of the fantasia—neuroimaging studies have revealed, in fact, that the association between number abstraction and spatial cognition shows up in regions of the parietal cortex as an integrated mode of thinking. In Vichian terms, it can thus be hypothesized that the fantasia is psychologically valid, constituting a kind of neural faculty for activating the blending process, producing inner visions of perceptual experiences; the ingegno then constitutes the neural process that activates the utilization of these images for purposes of inventing something or carrying out creative tasks. This process can be called bimodality (Danesi 2003), whereby the brain takes in unfamiliar information via the experiential (probing) right-hemisphere functions, transforming them into spatial forms of thought, which are then stabilized by processes of conceptualization, which likely have their primary locus in the left hemisphere. One of the more relevant findings from neuroscience that is of direct relevance to the discussion here is that the right hemisphere is, in fact, a crucial “point-of-departure” for processing novel stimuli: that is, for handling input for which there are no preexistent cognitive codes or programs available. In their often-quoted review of a large body of neuroscientific literature, Goldberg and Costa (1981) suggested that the main reason why this is so is because of the anatomical structure of the right hemisphere. Its greater connectivity with other centers in the complex neuronal pathways of the brain makes it a better “distributor” of new information. The left hemisphere, on the other hand, has a more sequentially organized neuronal-synaptic structure and, thus, finds it more difficult to

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assimilate information for which no previous categories exist. No contradictory evidence has come forth since Goldberg and Costa’s study. Psychologically, information is novel when it has never been interpreted beforehand or cannot be matched against existing categories of information. If poetic wisdom intuits a meaning in the novel information, it then activates the whole system of poetic logic, inducing the brain to explore the significance of the information first through the fantasia, and then, via the ingegno, to devise a way to grasp its meaning. So, critical path problems, such as the river-crossing one (above), present information at first that appears to have significance beyond the practical situation, enticing us to imagine the optimal sequence of paths needed for resolving it: What is the best way to cross a river on a boat with limited capacity carrying over certain things on it in a constrained way? What is the shortest and best route for doing so? These constitute an inner dialectic spurred on by the fantasia. A classic example of how this very same process was activated is Euler’s famous Königsberg Bridges Problem, one of most important problems of mathematical history, which he formulated in a famous 1736 paper that he presented to the Academy in St. Petersburg (published in 1741). It is worth revisiting here from the perspective of poetic logic, since its solution, when reconstructed, shows how the tripartite process might unfold. The problem is derived from the actual layout of bridges in the town of Königsberg through which the Pregel River runs. In the river, there are two islands connected with the mainland and with each other by seven bridges. It is said that the residents of the town would often debate whether or not they could cross each bridge once and only once, returning to the starting point, without having to double back at some point. No one had found a way to do it but, on the other hand, no one could explain why it seemed to be impossible. The inner dialectic that characterizes the problem is paraphrased below: In the town of Königsberg, is it possible to cross each of its seven bridges over the Pregel River, which connect two islands and the mainland, without crossing over any bridge twice?

Fig. 1.11 Königsberg bridges map. (Wikimedia Commons)

Psychological Validity

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Fig. 1.12 Graph version of the problem. (Wikimedia Commons)

Below is a map of Königsberg in Euler’s time, which shows the actual layout of the bridges and the Pregel river (Fig. 1.11): Euler became intrigued by the conundrum, seeing it as bearing hidden mathematical significance with his mathematical fantasia. After an abduction of how to represent the problem, Euler went on to prove that it was impossible to trace a path over the bridges without crossing at least one of them twice. He proved this with his mathematical ingegno by reducing the map of the area to a graphic outline form, such as the one below (Fig. 1.12): Is it possible to trace the following graph without lifting pencil from paper, and without tracing any edge twice?

This constitutes an abstract iconic representation of the situation, showing in outline form how it can be envisioned as a network of vertices and paths—namely, seven paths, representing the seven bridges, and four vertices, representing the land areas where the bridges converge. The graph thus allowed Euler to imagine the problem in terms of its essential structure, eliminating all the irrelevant features of the original map. It showed that the connectivity structure of the network was the relevant insight, since it is retained no matter how the paths are drawn as long as the network structure is retained. With the graph now at his disposal, Euler shifted modes to logical reasoning, proving that traversing the Königsberg network was impossible without doubling back on a path at some point. The reason for this had to do with the number of paths that converged at a vertex—it is not possible to traverse a network that has more than two odd vertices in it without having to double back over some of its paths. An even vertex is one in which an even number of paths converge (2n), whereas an odd vertex is one in which an odd number of paths converge (2n + 1). Euler proved this in a remarkably simple way: At an Even Vertex A network can have any number of even paths in it, because all the paths that converge at an even vertex are “used up” without having to double back on any one of them. For example, at a vertex with just two paths, one path is used to get to the vertex and another one to leave it. Both paths are thus used up without having to double back over either one of

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them. In a four path network, when we get to a vertex we can exit via a second path. Then, a third path brings us to the next vertex, and a fourth one gets us out. All paths are once again used up. At an Odd Vertex In an odd vertex network, on the other hand, there will always be one path that is not used up. For example, at a vertex with three paths, one path is used to get to the vertex and another one to leave it. But the third path can only be used to go back to the vertex. To get out, we must double back over one of the three paths. The same reasoning applies to any odd vertex network. Therefore, a network can have, at most, two odd vertices in it. And these must be the starting and ending vertices.

Euler’s graph constitutes a modeling of an image-schematic visualization of the Königsberg situation—that is, a visualization of the situation via an iconic extrapolation of its structure. While we praise Euler because of his ingenuity, the fact that we are able to understand it indicates that the same kind of imaginative process can be easily evoked in all of us—hence the psychological universality and validity of the fantasia. Now, having sensed a mathematical principle in the problem, Euler developed graph theory to study the features of graphs of all kinds. And because the actual way the paths were drawn was irrelevant, so long as the network structure was retained, the problem also provided the relevant key insight for the development of topology. Interestingly, Euler himself suggested that his insight was prompted by an idea that was discussed by Leibniz, which he called geometria situs, or geometry of position. After Euler’s solution, graph theory spread broadly, attracting major contributions from mathematicians. The point here is that by reconstructing Euler’s solution—hypothetically yet based on Euler’s own texts—we can envision how poetic logic works in mathematical discovery. As the American mathematician Keith Devlin (2005) has observed, in a way that reverberates with Vichian ideas, in their origin mathematical concepts seem to arise from imaginative thought, which is given a specific form (such as the graph above), and then siphoned for further exploration and theorization. As such, Reed (1994) appropriately characterizes mathematics as a mind-world made up of “cognitive figures,” which are (in more contemporary cognitive scientific terms) image schemas, which undergird all modes of proof and systematization of ideas. In synthesis, the process of understanding something starts with experiencing it as bearing significance, which is initially unknown, then envisioning it imaginatively, which leads to an abduction that prompts the mathematician’s ingenuity. The end result is a discovery that forms the basis of a new theoretical apparatus—graph theory. As Peirce (1878: 286) aptly put it, it is the practicality of the discovery that matters: “Consider the practical effects of the objects of your conception. Then, your conception of those effects is the whole of your conception of the object.” As Susac and Braeutigam (2014) found in their comprehensive review of the relevant neuroscience, mathematics will require “the coordinated action of many brain regions,” which involves the mapping of “distinguishable functional modules [schemas] onto anatomically separate brain regions.” So, for example, a model of something based on an unconscious image schema (such as the diagram Euler devised) extracts meaning from practical information, activating various regions of the brain in tandem. The underlying psychological mechanisms used in mathematics, as saliently illustrated by the reconstruction of Euler’s solution, emphasize the spatial structure of ideas in the brain, which are interconnected through a process of

Epilogue

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blending into holistic forms of understanding, such as the model used by Euler of a practical situation. Modeling leads to knowledge not because it is designed to be “knowledge-productive” in itself, but because it allows us to extract reality from its contexts of occurrence and give it an imaginative form, so that it can be studied on our own terms. The exploratory power of such modeling behavior, from geometry to quantum physics, suggests that we are probably programmed to discover things through our sign structures, given that they are products of the fantasia. In observing and modeling the facts of existence, we constantly stumble across hidden patterns in those facts. In a way, Euler’s discovery of graph theory brings us back to Flatland. To reiterate, the story depicts a two-dimensional world inhabited by geometric figures— some are line segments, while others are polygons with various numbers of sides. The narrator is a Square, who guides readers through some of the implications of life in two dimensions. Now, on New Year’s Eve, the Square dreams of visiting a one-dimensional world, Lineland, inhabited by lines and points—reminiscent of the paths and vertices of an Eulerian graph. The Linelanders see the Square as a set of points on a line. The Square tries to convince the monarch of the realm of the existence of a second dimension, but to no avail. Following this vision, the Square is visited by a Sphere. The Sphere then levitates figures up and down through Flatland, making the two-dimensional figures expand and contract. Unable to comprehend what is going on (dimension travel), the Square is finally taken by the Sphere to the third dimension via extraction upward, from where he can oversee Flatland and what it really looks like. The novel is a narrative version of Eulerian topology, whereby points (vertices), lines (paths), and figures (with lines and points) are all, literally, figments of the mind which provide a view of each dimensional reality in a constrained way.

Epilogue The fantasia and the inner dialectic it engenders is, as claimed here, the source of mathematics—connecting it to other forms of imagination, as in art and language. The dialectic starts from the experience of something such as seeing significance in the layout of bridges and landmasses and then asking why this is so. At this point, the mathematician’s ingegno gives the experience a mathematical form. The problem is not just an exercise in mental gymnastics, though. It encodes something that exists as a pattern in the world, imagining it is such a way that anyone can understand it. If the Vichian theory discussed here is viable, it is exactly in pinpointing that a model, such as Euler’s graph, originates in the inner dialectic produced by the imagination, eventually giving it concrete form through some visual form, after which the mathematical significance of the network structure becomes increasingly obvious. The ingenious ways of describing some inherent pattern or situation as it occurs in reality comes about after it has been imagined in creative ways. Graph theory is the result of Euler’s imagination translating his experience of something into a creative

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1 Imagination

form of thought, activating his ingenuity to produce new “mathematics.” A colloquial-idiomatic way to characterize the fantasia is to describe it as the faculty of mind that allows us to see with the “mind’s eye” what is not instantly obvious. As Dehaene (1997: 151) puts it, it is what produces the “illuminations” that mathematicians claim to see within their minds: They say that in their most creative moments, which some describe as “illuminations,” they do not reason voluntarily, nor think in words, nor perform long formal calculations. Mathematical truth descends upon them, sometimes even during sleep.

There are many anecdotes in the history of mathematics that support Dehaene’s illumination notion. Henri Poincaré recounted one such anecdote in his book, Science and Method (1908). Poincaré had been puzzling over an intractable mathematical problem, leaving it aside for a while to embark on a geological expedition As he was about to get onto the bus, the crucial insight came to him in a flash of insight (an abduction). He claimed that without it, the solution would have remained buried within him, possibly forever. Consider, as one other illustration of how the mathematical imagination works, the conjecture identified by Poincaré himself in 1904, which was arguably inspired by considering connectivity as defined in Eulerian graph theory. Poincaré noted that by stretching, say, a rubber band around the surface of a billiard ball (which is a three-dimensional sphere), it can be shrunk down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. However, if the same rubber band was somehow stretched in the appropriate direction around a donut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. So, the surface of the billiard ball is “simply connected,” but that the surface of the donut is not. Poincaré, knew that a two-dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three-dimensional sphere. Mathematicians in the twentieth century attempted to prove Poincaré’s conjecture, showing that it was true in some cases, but not in general. The general proof came in 2002 when Grigori Perelman showed its validity (O’Shea 2007). Perelman posted his solution on the Internet. It is over 400 pages. Suffice it to say that his proof involves a blend of poetic and rational logic—analogies, connections, deductions, etc. As Chaitin (2006: 24) observes, “mathematical facts are not isolated, they are woven into a spider’s web of interconnections.” These result from inner flashes of insights (illuminations). Returning to Euler’s solution, it is remarkable how much significance it held for mathematics. One of its insights was truly consequential, leading to topology— namely, that figures can be transformed into other figures and still retain their essential structure. So, a circle is topologically equivalent to a stretched ellipse; and a donut is topologically equivalent to a coffee cup (each has one hole, since the one in the coffee cup is in the handle). Consider the latter example concretely. By stretching and pinching, but not tearing or joining anything, the donut can be molded into a coffee mug, with the hole remaining in the handle of the coffee mug. A drawing of the transformation from donut to coffee mug is the following one by Stephen Barr in his book, Experiments in Topology (1964) (Fig. 1.13):

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Fig. 1.13 The topology of a donut and coffee mug (Barr 1964)

The donut and mug are now said to share the property of invariance, and are thus topologically equivalent. Concepts such as invariance and equivalence are at the core of topology as a modeling device of reality—topological ideas not only model physical phenomena, but also suggest hidden or emergent structural possibilities. So-called manifold topology has become highly useful in geometry and physics, because it makes it possible to describe complicated structures in terms of the simpler local topological properties of Euclidean geometrical spaces. Significantly, manifolds arise as solution sets of systems of equations and as graphs of functions. Now, while all this may seem unrelated to the Ishango Bone, or certainly far removed from it intellectually, the thought process involved is remarkably similar— both are based on iconicity, or the creation of models (tallies or graphs) that convert inner vision into outer vision and thus into memoria, from which more exploration is made possible. The fantasia is an intrinsic attribute of all human beings, no matter where they live, what intelligence they possess, or what they have experienced. Methods such as proof, on the other hand, must be learned and rehearsed—a product of both the ingegno and the memoria. Solving a problem, such as the Königsberg Bridges one, is converted by the ingegno into a model that can be literally seen with the eyes. As Hovanec (1978: 10) has aptly observed, a problem such as this one sparks interest, because it “conceals the answers [that] cry out to be solved,” piquing mathematicians such as Euler to juxtapose “their own ingenuity against [the problem].” The mathematical fantasia thus unfolds much like the imagination of a poet, who also converts inner images into models (poems), making mathematics, as Carl Sagan (1977: 82) so aptly puts it, “as much a [part of] humanity as poetry.”

Chapter 2

Ingenuity

Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. Alan Turing (1912–1954)

Prologue Euler’s demonstration of the impossibility of traversing the Könisgberg network without having to double back on one of its paths (previous chapter) made it possible to flesh out a hidden mathematical principle of connected networks, which laid the foundation for graph theory and topology. Reconstructing his proof allowed us to see how Euler was guided by poetic logic—whereby he initially used his fantasia to envision the geographical map in an image schematic (outline) way, converting his inner vision into a diagrammatic model, via his ingegno, from which he could then use logical reasoning to establish why the network was impossible to traverse as such and, as a result, what this implied more generally. This episode in mathematical history (among many others) brings out how the fantasia is much more than the brain’s ability to generate spontaneous mental imagery from perceptual input—it is a form of insight thinking that interprets the input and then sparks an abduction (a flash of insight), which led Euler to convert the insight into a graph (a model), which highlighted the structural features of the original map, removing extraneous information from it. In other words, the graph is the end product of poetic logic, with Euler’s ingegno leading him to devise something new that enfolded something significant. This chapter deals with this faculty of poetic logic, as it manifests itself in problem-solving, discoveries, inventions, conjectures, and proofs. A classic early example of the workings of the ingegno is found in Book IX (Proposition 20) of Euclid’s Elements (c. 300 BCE) in which he provided a simple, but truly ingenious proof that the prime numbers are infinite. Using background information that composite numbers can be deconstructed into a unique set of prime number factors—the Fundamental Theorem of Arithmetic, also discovered by Euclid (Book IX)—he had an abduction that led to his truly remarkable proof. A

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Danesi, Poetic Logic and the Origins of the Mathematical Imagination, Mathematics in Mind, https://doi.org/10.1007/978-3-031-31582-4_2

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contemporary version of the proof will be used here for the sake of clarity (Ore 1948: 65). We start with the contrary hypothesis—namely, that there is a finite set of primes (P): P = f p1 , p 2 , p 3 , . . . p n g The symbol pn stands for the last (and largest) prime; each of the other symbols stands for a specific prime in sequence: p1 = 2, p2 = 3, and so on. Next, we multiply all the primes in the set, in order to produce a composite number, C, that is thus divisible by any of the primes in P: C = fp1 × p2 × p3 × . . . × pn g At this point, Euclid had his flash of insight, adding 1 to C: C þ 1 = fp1 × p2 × p3 × . . . × pn g þ 1 Let us call the number produced in this way M, rather than C + 1: M = f p1 × p 2 × p 3 × . . . × p n g þ 1 Clearly, M is not divisible by any of the primes in P, because a remainder of 1 would always be left over. So, the number M is either: (1) a prime number that is not in P and thus greater than pn, or (2) a composite number with a prime factor that cannot be found in P, and thus also greater than pn. Either way, there must always be a prime number greater than pn. Once devised, this type of proof “makes sense” in its own truly ingenious way. We continue, in fact, to appreciate to this day Euclid’s flash of insight based on what he knew already, using his mathematical ingenuity to come up with the proof. By revisiting the proof, we become “participants” in observing the mind of a great mathematician at work. Euclid’s actual proof is based on line segments, and need not concern us here, since the reasoning utilized is identical in its overall ingenuity. Now, when such demonstrations are unexpected and seemingly inspired, we then use the term genius to characterize them. Works of genius fundamentally alter our expectations. They are found throughout mathematical history—indeed, they constitute its defining psychological feature (Honsberger 1998; Graham 2012; Simms et al. 2016). The point of departure for the ingegno is the fantasia—Euclid literally had to first imagine how the set of primes can be laid out, which produced a linear image schema that was translated symbolically into an equation (above). From this model of prime numbers came his abduction: What happens if we introduce a “1” into the model? This truly ingenious idea is what allowed him to carry out his proof. Now, like the fantasia, it is impossible to define the ingegno as such (Chap. 1)—all we can do is look at it “in action,” so to speak (i.e., in problems, solutions, and proofs). We can only appreciate it, much like we grasp musical genius by listening to the composer’s

The Mathematical Ingegno

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actual music. Deconstructing or reconstructing mathematical problems and ideas, such as Euclid’s proof is, arguably, the best way to grasp how the ingegno undergirds mathematical thinking during its inventive and creative phases.

The Mathematical Ingegno The question becomes: Was Euclid’s abduction and type of proof a unique episode in mathematical history, or did it reveal something fundamental about mathematical thinking in general? Revisiting the original ideas and proofs of mathematicians in history shows that it is hardly the exception, but the rule. Historical mathematical texts provide abundant evidence of how the ingegno manifests itself from the start in different ways and for different kinds of tasks—that is, for producing novel ideas, solving problems, or in developing proofs. An example of such evidence, predating Euclid, comes from a proof devised by Thales of Miletus, one of the Seven Sages of Greece, considered to be one of the first true geometric proofs—a proof that came to be known as Thales’s Theorem (Boyer 1968). As such, it provides a very early testament of how the ingegno is behind mathematical thinking, before it cedes to deductive logic to complete the process of proving something. It states that if A, B, and C are distinct points on a circle, with AC as its diameter, the angle ABC is a right angle (Fig. 2.1). The theorem is a special case of the inscribed angle theorem mentioned and proved later by Euclid. Proving it requires previously known geometric facts, which Thales clearly possessed—namely, that the sum of the angles in a triangle is equal to 180° and that the base angles of an isosceles triangle are equal. It is this information basis that sparked Thales’s flash of insight, which his ingegno then converted into a diagram that showed concretely how to carry out a proof (Fig. 2.2): 1. Since the radii of a circle are equal, then OA = OB. 2. This produces an isosceles triangle, OAB, in which the base angles are equal, marked on the diagram as α. 3. Similarly, OB = OC, since they are also radii. 4. This produces another isosceles triangle, OBC, in which the base angles are equal, marked on the diagram as β. 5. The angles of a triangle equal 180°. 6. Therefore, in triangle ABC, the sum of its angles 2α + 2β = 180° and, thus, α + β = 90°. 7. Proof concluded. Clearly, the theorem could not be proved with the original diagram (Fig. 2.1), since there was not enough information on it to do so. At this point, Thales used his background knowledge as the spark for his insight, producing the revised diagram (Fig. 2.2). After this abduction, he could then use straightforward deductive logic to

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Ingenuity

Fig. 2.1 Thales’s Theorem

Fig. 2.2 Thales’s proof

complete the proof. Insight thinking and background experience were clearly intrinsically intertwined in sparking the proof process. The mathematical ingegno is thus not identical to reasoning about things, but the spark for reasoning to run its course. Unless there are insights in the first place, there is nothing to reason about. The fact that we can easily reconstruct Euler’s, Euclid’s, or Thales’s thought processes, and understand them on our own terms, is substantive evidence that the ingegno is a kind of universal human quality of genius (Grassi 1980: 91), for which there is no suitable definition, as mentioned. It just shows up when we contemplate something that is unfamiliar or novel. The Latin source of the term ingegno is the verb gignere (“to beget, to give birth to”), which encapsulates what this inherent product of poetic logic allows us to do— to beget something. It is a kind of unconscious creative talent that manifests itself not only in mathematics, but in all areas of human expressive and cognitive activity. But it is in mathematics that it constantly produces the flash of insight, or Aha Moment, as it is called in psychology (Bowdon et al. 2005). Each time something is discovered in mathematics, it can be traced invariably to the ingegno, as it converts the inner vision of the fantasia into something tangible and understandable. Consider another well-known ancient example, namely, the proof of the value of π as 3.14 (Posamentier 2004; Danesi 2020). Historically, π is mentioned in the Old Testament (II Chron. 4:2) where we read: “Also he made a molten sea of ten cubits from brim to brim, round in compass and five cubits the height thereof, and a line of

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Fig. 2.3 Archimedes’ polygon method for π

thirty cubits did compass it about.” This tells us that the Hebrews took the ratio to be 3. The Babylonians also estimated it to be 3, and the Egyptians 3.1604. It was Archimedes who showed that it was 3.14. He did this with a simple, yet ingenious, method of proof. His flash of insight can be stated as follows: If we draw similar polygons outside and inside a circle (touching the circumference), and successively increase their sides, eventually, the polygons would smooth out and become coincident with the circumference (Fig. 2.3): By progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step, eventually, as the number of sides increases, it becomes a more accurate approximation of the circumference of the circle, allowing us to compute π. After polygons of 96 sides each, Archimedes was able to determine that the value of π lay between 3.1408 and 3.1416. Now, the genius of envisioning polygons morphing into a circumference is truly remarkable, since it became the basis for the later notion of infinitesimals, which is at the core of the calculus. Significantly, the same basic image schema that inspired Archimedes goes back even further in time to the Rhind Papyrus (1650 BCE), in which we find a problem that asks us to inscribe a circle in a square of length 9, which is to be divided itself into nine smaller squares (each 3 × 3). Diagonals are then to be drawn in the corner squares, producing an octagon figure, as can be seen in the diagram below (Fig. 2.4): The area of the octagon is, clearly, equal to the areas of the five inner squares (which form the outline of a cross) plus half the areas of the four corner squares. Its area is thus equal to the sum of the areas of seven small squares. The area of one small square is, of course, 3 × 3, or 9 square units. The total area of seven such squares is, therefore, 9 × 7, or 63 square units. With a bit of convenient cheating, it can be assumed that the circle’s area, which is nearly 64, is equal to 82. The value of π can now be computed as follows: Area of circle = πr2 = 64 Diameter = 9 Radius (r) = 9/2 So, r2 = (9/2)2 = 20.25

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Fig. 2.4 Estimation of π in the Rhind Papyrus

Since πr2 = 64 and r2 = 20.25 π = 64/20.25 = 3.1604 This ingenious method was based on turning one figure into another in order to gain control over its structure, prefiguring Archimedes’ transformation of polygons into a circumference. The details are different, but the thought process is similar. The ratio π was at first observed from measuring circles. Establishing a value for it was arguably to save the cognitive effort of having to measure circles over and over. Proofs such as those above turned practical observations into theoretical knowledge via the ingegno. This process is the central feature of mathematical discovery, turning practical needs into systems of knowledge. It should be stressed, however, that the mathematical ingegno works even when there are seemingly no such needs—it is a constant inner creative force that seeks expression even in apparently trivial ways. Consider, as an example, an anecdote that is told about Karl Friedrich Gauss when he was only 9 years old (Bell 1937). In school one day, it is said that his teacher, a certain J. G. Büttner, asked the class to cast the sum of all the numbers from one to one hundred. Gauss raised his hand within seconds, giving the correct response of 5050, astounding both his teacher and the other students who continued to toil over the seemingly lugubrious arithmetical task before them. When his teacher asked Gauss how he was able to come up with the answer so quickly, he is said to have replied somewhat as follows (Bell 1937): There are 49 pairs of numbers in the set of the first 100 numbers that add up to one hundred: 1 + 99 = 100, 2 + 98 = 100, 3 + 97 = 100, and so on. That makes 4,900, of course. The number 50, being in the middle, stands alone, as does 100, being at the end. Adding 50 and 100 to 4,900 gives 5,050.

Impressed, the teacher not only arranged for Gauss’s admittance to a school with a more challenging curriculum, but he also secured a tutor and advanced textbooks for the brilliant child. Let us take a close look at what Gauss arguably saw in his mind. First, he saw the numbers as being laid out in a line, rather than in a conventional addition column of addends, which clearly would pose a huge

The Mathematical Ingegno

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…

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Fig. 2.5 Linear image schema

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Fig. 2.6 Half sequence arrangement

mechanical addition problem. The linear layout is an image schema generated by the fantasia (Fig. 2.5): This inner vision was the spark for Gauss’s ingegno to complete the task at hand—given that the linear sequence itself is a diagram that suggests a segmentation pattern of “half sequences” as follows: (1) from 1 to 49, leaving 50 alone in the middle, and (2) the remaining numbers from 51 to 99, leaving 100 alone at the end (Fig. 2.6): This layout now suggests an addition pattern: Add the first number in the first half and last number in the second half, the second in the first half and the second-last in the second half, the third in the first half and third-last in the second half, and so on. This produces the constant sum of 100: 1 þ 99 = 100 2 þ 98 = 100 3 þ 97 = 100 4 þ 96 = 100 5 þ 95 = 100 ... 49 þ 51 = 100

The pairings end at 49. This means that there are 49 pairs that add up to 100. So, 49 × 100 = 4900. Adding to this the 50 and 100 that Gauss had isolated in the sequence makes 4900 + 50 + 100 = 5050. Now, whether this incident occurred exactly in this way, and while the same ingenious technique is found further back in time in Alcuin’s Propositiones (Hadley and Singmaster 1992; Hayes 2006), it reveals how the ingegno manifests itself in a situation for which, unlike for the calculation of π, there is no apparent need to come up with a proof or theoretical resolution. It appears on the surface as the mind-play of a genius child mathematician or of a skilled teacher of mathematics—Alcuin. But below the surface, the method points to a general pattern involving the summation of a series: {1 + 2 + 3 + . . . + n}, where n is any whole number. The sum of this series is: n(n + 1)/2. Alcuin’s version is found as problem 42 in his Propositiones, paraphrased below: There is a ladder which has 100 steps. One pigeon sat on the first step, two pigeons on the second, three pigeons on the third, four on the fourth, five on the fifth, and so on up to the hundredth step. How many pigeons were there in total on the ladder?

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The total number of pigeons is the sum of the first 100 natural numbers, so it is 5050. There is no evidence that the 9-year-old Gauss was familiar with Alcuin. Even if he was, what matters here is the method itself, which illustrates how the ingegno works in this case, appearing to subserve a different kind of need, from that exemplified by the history of π, namely, to envision shortcuts—ways of doing mathematics efficiently and, at the same time, interestingly and revelatory of hidden structure. Using the language of algebra, the Alcuin-Gauss method reveals (as mentioned) how to sum a series, {1 + 2 + 3 + . . . + n}, where n is any whole number. It also suggests a formal proof. We start by representing the sum of the first n numbers as: S1 = 1 + 2 + 3 + . . . + n. The same sum can be written with the numbers in reverse order. This changes nothing: S2 = n + (n - 1) + . . . + 2 + 1. We then line up these two equivalent versions of the same sum: S 1 = 1 þ 2 þ . . . þ ð n - 1Þ þ n

ð2:1Þ

S2 = n þ ðn - 1Þ þ . . . þ 2 þ 1

ð2:2Þ

By adding each pair of numbers that are lined up (from left to right), we will get the expression (n + 1) each time: First pair : 1 þ n = ðn þ 1Þ Second pair : 2 þ ðn - 1Þ = ðn þ 1Þ Third pair : 3 þ ðn - 2Þ = ðn þ 1Þ ... Last pair : n þ 1 = ðn þ 1Þ The result (n + 1) occurs exactly n times, or n(n + 1). Therefore, the result of adding up (2.1) and (2.2), the two versions of the series, is: S1 þ S2 = nðn þ 1Þ Since S1 = S2, we can simply replace the left side with 2S: 2S = nðn þ 1Þ Dividing both sides by 2, we get S = nðn þ 1Þ=2 There are myriad episodes in the history of mathematics like this one. It seems to be often the case that someone contemplates an ostensibly meaningless, but interesting, pattern, which, later, leads to significant discoveries and, often, forms the foundations of a new branch. It is no coincidence that Gauss went on, by all accounts, to define and study systematically series and their summation (Gauss 1801). So, whether or not there appears to be a practical need for some ingenious idea, the ingegno shows up, no matter what task we give it (Black 1946). Figuring out the solution to any mathematical conundrum involves (1) imagining its outline

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structure via the fantasia, (2) deriving from this inner vision an insight (ingegno), which leads to (3) a modeling of the problem (a diagram, an equation, a layout, etc.), from which (4) further implications can be derived. As one other example of this process, consider the well-known eight queens problem, which again seems to have no practical utility other than engaging us in a ludic form of thought. It asks us to place eight queens on an 8-by-8 chessboard so that no two of them can attack each other. A queen can move in any direction and is not limited as to the span of the move. To phrase the problem more concretely, we are asked to place queens on a chessboard so that no two queens can share the same row, column, or diagonal. There are ninety-two distinct solutions to the problem, although if rotations and reflections of the board are taken into account, then it has twelve unique solutions (from Wolfram Mathworld) (Fig. 2.7): The problem was formulated in 1848 by chess player Max Bezzel in the journal Deutsche Schachzeitung. In 1850, Franz Nauk generalized the problem as follows: How does one place eight queens on an 8 × 8 chessboard, or for general purposes, on an n × n board, so that no queen is attacked by another? In addition, determine the number of such positions.

Nauk purportedly posed the problem to Gauss. Interestingly, as Campbell (1977) has documented historically, this created a situation of misunderstanding as follows (Campbell 1977: 397):

Fig. 2.7 Distinct solutions to the eight queens problem. (Wolfram Mathworld)

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An 1874 article by J. W. L. Glaisher asserted that the eight queens problem of recreational mathematics originated in 1850 with Franz Nauck proposing it to Gauss, who then gave the complete solution. In fact the problem was first proposed two years earlier by Max Bezzel, proposed again by Nauck in a newspaper Gauss happened to read, and only partially solved by Gauss in a casual attempt. Glaisher had access to an accurate account of the history in German but perhaps could not read the language well; the error subsequently spread through the recreational mathematics literature.

Now, the question of what the problem implies for mathematics will come up. As Dahl, Dijkstra, and Hoare (1972) showed, it sparked the required insight for developing a backtracking algorithm for finding solutions to computational problems that incrementally builds potential solutions, and abandons those that do not work out, automatically. In the backtracking approach, the partial solutions are the arrangements of k queens in the first k rows and columns of the board. Any partial solution that contains two mutually attacking queens is abandoned. Immanuel Kant (1781: 278) was among the first to characterize mathematical thinking as the process of “combining and comparing given concepts of magnitudes, which are clear and certain, with a view to establishing what can be inferred from them.” Kant argued further that the whole process becomes reflective and explicit when we examine the “visible signs” that we use to highlight the structural detail inherent in this type of knowledge. For example, a diagram of a triangle compared to that of a square will show where the differentiation occurs “visibly”—one consists of three intersecting lines, while the other has four parallel and equal sides that form a boundary. This type of diagrammatic know-how is based on the brain’s ability to synthesize scattered bits of information into holistic entities that can then be analyzed reflectively. If one examines Kant’s statement in terms of Vichian theory, it can be deduced that he is describing the workings of poetic logic in his own way. In any and all views of the ingegno, no matter how it is labeled, the key aspect is, in fact, that it activates some inner visualization process (paraphrasing Kant), which leads to the Aha insight. This is then given some form (an equation, a diagram, etc.) that suggests a solution or proof. From the solution, a logical form emerges that then implies applications or implications via further abstraction processes. But abstraction cannot occur in the first place, if there is nothing to abstract about. It is the output product of the workings of poetic logic via the ingegno. The whole process, as discussed here, invariably reveals deep connections between different areas of mathematics, or else can suggest conjectures that stimulate further ingenious thinking.

Conjecture The last comment above leads to the topic of conjecture in mathematics, defined as a problem or recurrent pattern that seems to hide a general principle but which resists proof or demonstration. As mathematical history attests, psychologically, it would appear that the state of mind in which a conjecture leaves the mathematician is one of

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an intense need to search for and find a proof. This confirms, indirectly, that the mathematical imagination is programmed to resolve the inner dialectic that besets mathematicians when they confront unexplained pattern. Even proof that a particular problem cannot be solved as described would provide some psychological resolution. What is particularly agonizing is the possibility that proving a conjecture may be undecidable. Some classic conjectures have eventually led to proofs, after much consternation and investigation. Among these are Fermat’s Last Theorem and the Four-Color Problem. Others remain conjectures, including those by Goldbach and Collatz. In all cases, conjectures show how the mathematical ingegno is constantly at work, manifesting itself in how it eventually has devised (or not) a solution or proof. As Vico argued throughout the New Science, the examination of the ingegno cannot be carried out directly, but through the artifacts it has produced, allowing us to work backward, from the artifact to how it could have possibly come about through human ingenuity. Consider first Fermat’s Last Theorem, more accurately a conjecture until it was proved (Aczel 1996; Singh 1997). This is a classic case of how the ingegno will not rest, so to speak, until it finds a resolution to the inner dialectic that the fantasia elicits. As is widely known, around 1637, French mathematician Pierre de Fermat wrote the following in the margin of his copy of Diophantus’s Arithmetica, on the page where Diophantus presents his sum-of-squares problem: It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

In formal notation, Fermat claims to have established that cn = an + bn has integral solutions only if n = 2. Diophantus had included a long discussion of the numbers that satisfy the Pythagorean equation in his Arithmetica (Heath 1910: 144–145). This was the inspiration that clearly sparked Fermat’s inner dialectic to envision his “marvelous proof.” The note in the margin was actually discovered by Fermat’s son, after his father’s death. For almost four subsequent centuries, mathematicians were intrigued by Fermat’s claim, trying to come up with a proof, but always to no avail. For centuries, therefore, it remained a conjecture, constantly kindling the ingegno of mathematicians across the world. In June of 1993, British mathematician Andrew Wiles declared that he had finally proved it. In December of that year, however, some mathematicians found a gap in his argument. In October of 1994, Wiles, together with Richard L. Taylor, filled that gap to virtually everyone’s satisfaction. The Wiles-Taylor proof, which was published in May 1995 in the Annals of Mathematics, was the result of the mathematical ingegno relentlessly searching for, connecting, and modifying previous ideas and formulas to resolve the conjecture. The key idea was a previous conjecture between elliptic curves and modular forms, put forth by Yutaka Taniyama and Goro Shimura—namely, that every elliptic curve is associated with a modular form. Thus, it was soon obvious that a proof of that conjecture would imply a proof of Fermat’s conjecture. That was the flash of insight required to devise the relevant proof. Already in 1985, Gerhard Frey showed that if Fermat’s conjecture was false, then the Taniyama-Shimura conjecture would not hold. Wiles took Frey’s idea and set out to prove the special case of the

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Taniyama-Shimura conjecture, which implied the truth of Fermat’s Last Theorem. The details are highly technical and need not concern us here. The point is that it took centuries for the satisfying (Aha) moment to come, revealing the “tenacity” (so to speak) of the mathematical ingegno to seek a resolution to the inner dialectic. Nonetheless, Fermat’s Last Theorem still haunts some mathematicians, for the simple reason that the Wiles-Taylor proof was certainly not what Fermat could have envisioned, because it depended on mathematical work subsequent to Fermat. In a pure sense, therefore, the Taylor-Wiles proof is not a resolution to Fermat’s Last Theorem. Fermat left behind a true mathematical mystery. What possible “marvelous proof” could he have been thinking of as he read Diophantus’s Arithmetica? Was he mistaken or was his proof so different that it has eluded us to this day? Is there be some ingenious solution that is awaiting discovery? Whatever the case, there is little doubt that the Wiles-Taylor proof is “marvelous” itself, since it is an ingenious demonstration based on integrating previous ideas that did not seem to have any correlation to each other at first. As Babbitt and Kisner (2000: 17) remarked a few years after it was published: Fermat’s Last Theorem deserves a special place in the history of civilization. Because of its simplicity, it has tantalized amateurs and professionals alike, and its remarkable fecundity has led to the development of large areas of mathematics such as, in the last century, algebraic number theory, ring theory, algebraic geometry, and in this century, the theory of elliptic curves, representation theory, Iwasawa theory, formal groups, finite flat group schemes and deformation theory of Galois representations, to mention a few. It is as if some supermind planned it all and over the centuries had been developing diverse streams of thought only to have them fuse in a spectacular synthesis to resolve FLT. No single brain can claim expertise in all of the ideas that have gone into this “marvelous proof.”

The Four-Color Theorem also started out as a conjecture. It states that only four colors are needed to color a map or diagram so that no two areas (or shapes) touching each other have the same color. It was proved in 1976 by Kenneth Appel and Wolfgang Haken (Haken and Appel 1977; Appel and Haken 1986, 2002) using a computer to do so. Consider the simple diagram below which illustrates the kind of pattern that the theorem enfolds. It consists of nine similar curved shapes, which are colored with four distinct tints (Fig. 2.8): Now, for more complex and distorted topological shapes, the question arises if the theorem will still hold? Below is an example of one such complex diagram, which shows that it does indeed seem to hold (Fig. 2.9): It was in 1852 when a young mathematician at University College, London, named Francis Guthrie, realized, as he was coloring maps, that four tints were sufficient to color any map. This sparked an inner dialectic: Is there a way to prove this? Unable to find one himself, he asked his brother Frederick if he knew of any principle or theorem that proved or disproved it. Frederick passed his brother’s query on to Augustus De Morgan who, unaware of any existing proof, immediately grasped the mathematical implications that Guthrie’s question held. In a letter he wrote to William Rowan Hamilton in August 1852, De Morgan asks whether four colors are really enough to color a map. Thus was born the FourColor Conjecture. Although there is evidence that August Möbius had discussed the conjecture in a lecture to his students in 1840, it was Guthrie’s version, as related to

Conjecture Fig. 2.8 Simple example of the Four-Color Theorem. (Wikimedia Commons)

Fig. 2.9 Complex example of the Four-Color Theorem. (Wikimedia Commons)

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De Morgan, that made it widely known (Wilson 2002). In its simplest form, it reads as follows: What is the minimum number of tints needed to color the regions of any map (or diagram) distinctively? (If two regions touch at a single point, the point is not considered a common border).

No map was found that required more than four colors. The challenge was to prove that this is necessarily so, no matter what map or diagram was used—namely, that four colors are sufficient no matter how many regions are involved. After De Morgan, mathematicians started trying to prove it with the traditional methods of proof—showing again the emotional pull of an inner dialectic, spurring on the ingegno to come up with a solution. A proof finally came from Haken and Kenneth, as mentioned, but nearly a century later. At that point, the conjecture had become a theorem. The Haken-Appel proof was seen at first as peculiar and even anomalous—it used an algorithm, which checked to see if a map could be colored by more than four tints. The proof was based on 1936 reducible configurations, which had to be checked one by one by computer. With this method, Haken and Appel concluded that no counterexample exists, because it must be in one of 1936 maps—that is, there would be at least one map with the smallest possible number of regions that required five colors. Such a minimal counterexample does not exist. Initially, the proof was problematic for some mathematicians, because it broke away radically from tradition (Tymoczko 1979). Moreover, as Haken and Appel themselves admitted in 2002: “One can never rule out the chance that a short proof of the Four-Color Theorem might some day be found, perhaps by the proverbial bright high-school student.” If this were in fact to be the case, however, then a novel innovative type of (noncomputer) proof method would have likely emerged. What makes the theorem particularly intriguing is that it entails Eulerian graph theory—it states, in effect, that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color (Wilson 2002). It is thus based on the same Eulerian image schema of paths and vertices in a network—a schema that has been highly productive in mathematics, from Alcuin’s river-crossing problem to the Four-Color Theorem and beyond. As it turns out, the theorem became the basis for further explorations and mathematical generalizations—it applies not only to finite planar graphs, but also to infinite graphs. The proof and the implications need not concern us here. The point is that the initial conjecture initiated a chain of ingenious investigations (literally), some of which were faulty, guided by the urge to resolve the conjecture one way or the other. The two conjectures above were eventually resolved. It is the conjectures that have never yielded to proof (so far) that continue to motivate the mathematical ingegno to find a resolution. This “restlessness” of the brain is encapsulated by Vico as follows (in Bergin and Fisch 1984: 498): Providence gave good guidance to human affairs when it aroused human minds first to topics rather than to criticism, for acquaintance with things must come before judgment of them. Topic has the function of making minds inventive, as criticism has that of making them exact. And in those first times all things necessary to human life had to be invented, and invention is the property of genius. In fact, whoever gives the matter some thought will

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observe that not only the necessaries of life but the useful, comfortable, pleasing and even luxurious and superfluous.

Translating this citation into mathematical language, it says that the mind is “aroused” when some “topic” comes up to the attention (of a mathematician), whereby it becomes “inventive” (which is a “property of genius”). Even the search for the “useful, comfortable, and pleasing” aspects of the topic spurs the inventiveness. Two well-known conjectures that have sparked much inventiveness, but have still not been resolved, resisting the achievement of the “luxurious,” are Goldbach’s and Collatz’s conjectures. The former is traced initially to a letter Christian Goldbach wrote to Leonhard Euler in 1742, in which he mentioned that he could write every even integer greater than 2 as the sum of two primes (Wang 1984; Rassias 2017): 4=2 þ 2 6=3 þ 3 8=3 þ 5 10 = 3 þ 7 ... 2n = pn þ pm ðp = prime numberÞ By exploration, one can always find two primes with which to write an even number. But all we can do is assume this to be true. So far, no one has been able to prove (or disprove) Goldbach’s conjecture, even though it has been shown to hold for larger and larger sets of integers by computer (see, e.g., Wang 1984; Guy 2004). The conjecture above is called Goldbach’s strong conjecture. In the same letter to Euler, Goldbach also conjectured that any number greater than 5 could be written as the sum of three primes: 6=2 þ 2 þ 2 8=2 þ 3 þ 3 7=2 þ 2 þ 3 9=3 þ 3 þ 3 10 = 2 þ 3 þ 5 11 = 3 þ 3 þ 5 ... n = pn þ pm þ pr ðn = any number greater than 5, p = prime numberÞ This is called the weak conjecture. Euler became intrigued by the problem, writing back to Goldbach that the weak conjecture would be implied by Goldbach’s strong conjecture, adding that he was certain that it was true, but that he was unable to prove it. Goldbach’s weak conjecture was apparently proved in 2013 by Harald Helfgott, which has been widely accepted but not definitively so. There have also

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been “implicit” or inductive proofs (e.g., Guiasu 2019, Liang 2018). The point here is that the search for a proof makes manifest how the mathematical ingegno cannot be held back—it constantly seeks to answer the inner dialectic, sometimes at personal cost to the mathematician (as witnessed by the many stories of mathematicians becoming obsessed with find solutions to intractable problems). This is a theme in the history of mathematics, expounded interestingly in a graphic novel, Logicomix: An Epic Search for Truth (2009), by Apostolos Doxiadis and Christos Papadimitriou about the inner quest in mathematics, showing how the personal struggles of some famous mathematicians dovetailed with their almost-obsessive need to resolve issues in the field. Doxiatos had previously written the novel, Uncle Petros and Goldbach’s Conjecture (2000), in which he treats Goldbach’s Conjecture as a revelation provided occasionally by God, as the novel puts it, to mystify human ingenuity, leaving us in a suspended emotional state to linger and thirst for answers that may not even be findable. As one final example, consider the Collatz conjecture, formulated in 1937 by German mathematician Lothar Collatz, who found that we always end up with the number one if we apply the following rule: If a number, n, is even, make it half, or n/ 2; if it is odd, triple it and add one, or 3n + 1. If we keep repeating this rule, we always end up with the number one. Here is a concrete example: Example = 12 First Application of Rule 12/2 = 6 (12 is even so we divide it by 2) Second Application 6/2 = 3 (6 is even so we divide it by 2) Third Application (3)(3) + 1 = 10 (3 is odd so we triple it and add 1) Fourth Application 10/2 = 5 (10 is even so we divide it by 2) Fifth Application (3)(5) + 1 = 16 (5 is odd so we triple it and add 1) Sixth Application 16/2 = 8 (16 is even so we divide it by 2) Seventh Application 8/2 = 4 (8 is even so we divide it by 2) Eighth Application 4/2 = 2 (4 is even so we divide it by 2) Ninth Application 2/2 = 1 (2 is even so we divide it by 2) The question becomes: Is this always the case? Is there a number where oneness is not achieved? Several mathematicians have proved that the conjecture is mainly true, but have not demonstrated that it is always true. Now, in the search for a proof, mathematicians have developed novel methods and found new kinds of information about numbers. It is, significantly, based on the path image schema, which has led to

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Fig. 2.10 Directed graph of the Collatz conjecture

the use of directed graphs (consisting of a set of vertices connected by directed edges called orbits), such as the following one showing the vertices only of odd numbers, and showing that all paths eventually lead to 1 (Fig. 2.10): Taking one of the nodes at random, say, 9, it can be shown how this graph works: (1) the even number nodes are skipped, (2) leaving only the odd ones (shown in boxes below):

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9 → ð3Þð9Þ þ 1 = 28 → 28=2 = 14 → 14=2 = 7 → ð3Þð7Þ þ 1 = 22 → 22=2 = 11 → ð3Þð11Þ þ 1 = 34 → 34=2 = 17 → ð3Þð17Þ þ 1 = 52 → 52=2 = 26 → 26=2 = 13 → ð3Þð13Þ þ 1 = 40 → 40=2 = 20 → 20=2 = 10 → 10=2 = 5 → ð3Þð5Þ þ 1 = 16 → 16=2 = 8 → 8=2 = 4 → 4=2 = 2 → 2=2 = 1 In 1976, Riho Terras demonstrated that, after repeated application of the Collatz rules, almost all numbers eventually wind up lower than where they started, as can be seen in the directed graph above (as one illustrative example). In 2019, Terence Tao proved that for almost all numbers, the Collatz sequence of a number, n, ends up below n. that is, below n/2, below √n, below ln n (the natural log of n), and below every f(n) where f(x) is any function that goes off to infinity. In other words, for almost every number, we can guarantee that its Collatz sequence goes as low as we desire it to go and this is “about as close as one can get to the Collatz conjecture without actually solving it” (Tao 2019). As examples such as this one and others that involve recurring image schemas, the ingegno is psychologically related to the formation of blends of such schemas in the brain (Hubbard et al. 2005), which will be discussed in Chap. 4. The path image schema, for instance, manifests itself in various kinds of problems that are converted into graphs. Interestingly, the same image schema shows up in an ingenious puzzlegame, worth discussing here for this reason—namely, puzzle-maker Sam Loyd’s famous Fourteen/Fifteen game (see Costello 1988). Loyd put 15 consecutively numbered sliding blocks in a square plastic tray large enough to hold 16 such blocks. The blocks were arranged in numerical sequence, except for the last two, 14 and 15, which were installed in reverse order. The object of the game was to arrange the blocks into numerical sequence from 1 to 15, by sliding them, one at a time, into an empty square, without lifting any block out of the frame (Fig. 2.11). We note that when the blocks are in numerical order, each one followed by a block that is exactly one digit higher (1 followed by 2, 2 by 3, and so on), the blocks can be scrambled and the puzzle can be easily solved. This is called the Fifteen Puzzle version. In any other arrangement, some blocks will be followed by blocks Fig. 2.11 The Fourteen/ Fifteen Puzzle. (Wikimedia Commons)

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that are numerically lower (e.g., 2 followed by 1 or 4 followed by 3, etc.). Now, the remarkable thing is that the structure of this puzzle mirrors that of problems such as the Königsberg Bridges one, even though on the surface it does not seem to be related to it. Every instance of a block followed by one that is lower than itself is called an inversion. If the sum of all the inversions in a given arrangement is even, the solution to the puzzle is a possible one. If the sum is odd, it is impossible, just like Euler’s problem. Loyd’s game had only 1 inversion (15 is followed by 14). This is an odd number and thus, it is impossible to arrange the blocks in numerical order. The connection to Euler, Alcuin, Collatz, and others is truly remarkable. Each requires moving from one position to another in determinable ways, whether the movement involves river crossings, interconnected bridges, cell-to-cell shifts, etc. In effect, the image schema is the same one; it simply manifests itself in different guises.

Proof and Decidability Devising a proof that something is the way it is constitutes the core method of mathematics, starting with Thales, Pythagoras, Euclid, and other ancient mathematician in various parts of the world (from Asia to Africa and America). It provides the “closure” to a conjecture raised by the inner dialectic that mathematicians seek to resolve through their ingenious constructs and artifacts. It is a point-of-arrival in the evolution of a problem, based on rational logic (systematic exposure to the consistency of some proposition). However, even in devising a proof by deduction, induction, contradiction, etc., the ingegno never recedes completely; it surfaces in the actual construction of a proof allowing the mathematician to use an amalgam of facts and ideas so as to devise a logical demonstration. Perhaps the main reason why historians see Euclid’s Elements as the founding work in mathematics is because it establishes proof as the core feature of mathematics. As a simple case-in-point, consider his proof of the proposition that the angles formed when two straight lines intersect are equal, found in Book 1, Proposition 15. The idea is to show that such angles are necessarily equal under all circumstances, that is, no matter what their measure in degrees is. The proof unfolded as follows, using contemporary notation and concepts, which are nonetheless identical in ideation to Euclid’s proof. We start by drawing a diagram that shows all the relevant features—two intersecting straight lines and the four vertically opposite angles formed by their intersection (1, 2, 3, 4) (Fig. 2.12): The objective is to demonstrate logically that ∠1 = ∠3 and that ∠2 = ∠4. The proof hinges on the established fact that a straight line is an angle of 180°. That fact was the insight that Euclid used to derive his proof. We start by noting that ∠1 and ∠2 are on one of the lines, and are thus supplementary angles adding up to 180°: ∠1 þ ∠2 = 180 ° We also note that ∠2 and ∠3 are supplementary angles on the other line:

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Fig. 2.12 Intersecting lines

∠2 þ ∠3 = 180 ° These two equations can be rewritten as follows: ∠1 = 180 ° - ∠2

ð2:3Þ

∠3 = 180 ° - ∠2

ð2:4Þ

Now, as per one of Euclid’s axioms, things equal to the same thing are equal to each other, we can now deduce that ∠1 = ∠3, since Eq. (2.3) shows that ∠1 is equal to (180° - ∠2) and Eq. (2.4) that ∠3 is equal to the same expression (180° - ∠2). It is not necessary to figure out what the value of the expression is. Whatever it is, the fact remains that the two angles will be equal to it and thus to each other. Showing that ∠2 = ∠4 is achieved with the exact same reasoning. We can conclude that any two vertically opposite angles produced by the intersection of two straight lines are equal, because we did not assign a specific value to either angle. We have, in effect, proved a proposition for the general case. Proof by deduction works by using previously established information, theorems, axioms, and postulates; nonetheless, for the proof to come to mind in the first place, an insight that relates the information to the task at hand is inevitably required. That insight comes by envisioning the diagram as consisting of lines with supplementary angles on them. This then allows us to envision (literally) how to use this information in a logical way, which constitutes the spark for the proof. It was the Greeks who established the main methods of proof that continue to be used to this day, including proof by contradiction (such as Euclid’s proof of the infinity of primes), by induction (whereby if an observed pattern holds for the nth case and then for the (n + 1)th case it will establish the pattern throughout), by deduction (such as the proof above), and various others. Suffice it to say here they have all one thing in common—demonstrating something as being necessarily so because the various “steps” in the argument follow from each other logically. Stewart (2008: 34) puts it as follows: What is a proof? It is a kind of mathematical story, in which each step is a logical consequence of the previous steps. Every statement has to be justified by referring it back to previous statements and showing that it is a logical consequence of them.

But it should be noted and emphasized that the initial spark for a proof comes not directly from logic, but from a blend of fantasia and ingegno—to greater or lesser degrees. This blend is manifest in all of Euclid’s proofs. He began with accepted mathematical truths (axioms and postulates). From them, he logically demonstrated

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467 propositions, using contradiction, induction, deduction, and other kinds of proof strategies, finishing his proofs with QED, as later translated in Latin. The letters stood for Quod erat demonstrandum (“which was to be demonstrated”)—remaining the symbolic hallmark of what mathematical proof is all about to this day. Non-Euclidean ways of doing mathematics have of course evolved since then, but without the Elements there would be no mathematics as we know it historically. Euclid’s text shows how the mathematical imagination manifests itself in specific logical tasks, connecting ingenuity and logic into a holistic frame of mind. As Kaufman (2001) has observed, there is no separation of the human brain and the mathematics of that brain. Euclid held up one of first mirrors to that brain, showing how it manipulates mathematical objects in ingenious and interconnected ways (Benson 1999). Specifically, the Elements provides substantive philological evidence that the ingegno has always been a spark behind a proof, which constitutes a way of convincing our brain that something is the way it is. And this raises a fundamental question about the human mind: Why do we accept a proof as showing something true or real, since on the surface, it appears to be nothing more than a logical organization of symbols? It was one of Russell and Whitehead’s (1913) central objectives to show that this organization actually provided the external evidence of what was going on in the mind. By stripping down mathematical thinking to its symbols, and how they are combined (syntax) without any reference to real world meaning, we can get a snapshot of what proof is and perhaps why proof is believable. As Russell and Whitehead (1913: 1) asserted in their introduction, their aim was: (1) to analyze the ideas and methods of mathematical logic, minimizing the number of primitive notions and axioms, and inference rules; (2) so as to precisely express mathematical propositions in symbolic logic using the most efficient notation possible; and (3) to solve the paradoxes that plagued logic and mathematics, such as the Liar Paradox of antiquity and Russell’s own Barber Paradox, which involved circularity in thinking. But their Principia did not overcome the so-called Entscheidungsproblem (“decidability problem”)—the problem of determining if it is possible to decide if mathematical statements are true (provable) or not (Davis 2000: 3–20). Russell and Whitehead’s objective was ultimately to devise a system of symbols and rules to construct a mathematical system for proving all possible propositions within it. As discussed (Chap. 1), this supposition was impugned by Gödel (1931) who found the proverbial “fly in such systems.” Nonetheless, mathematicians continued to use all kinds of proof methods after Gödel, including computer-based ones, to carry out the practical task of doing mathematics (Hersh 1997). A few years after Gödel’s demonstrations, Alonzo Church (1936) further showed that there is no computable function, which can be used to decide if two given expressions are equivalent or not. In the same year, Alan Turing (1936) framed the Entscheidungsproblem in computational terms–called the Halting Problem. Given a computer program and an input, will the program finish running or will go into a loop and run forever? Turing proved that no program can be written that can predict whether or not any other program halts after a finite number of steps. Here is a paraphrase of Turing’s proof by contradiction:

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Assume that there is such a program. If so, we could run it on a version of itself, which would halt if it determines that the other program never stops, and runs an infinite loop if it determines that the other program stops. This is a contradiction.

If the halting problem could be solved, many other problems and conjectures could be decided, including the Goldbach and Collatz ones. It is a straightforward task to construct a Turing machine—a hypothetical computing machine capable of using a predefined set of rules to determine a result from a set of input variables— that tests every even natural number greater than 2 on whether it’s the sum of two primes or not (Goldbach Conjecture). If it encounters any counterexample, it would immediately halt, indicating that a counterexample has been found, otherwise it will run forever. If the Halting Problem were decidable, we could then decide whether such a program would halt or not, and thus give an answer to Goldbach’s Conjecture. This type of investigation now falls under the rubric of the P = NP problem. Fortnow (2013) illustrates the core notion behind the problem with the game of Sudoku, which is a useful way to do so. Any proposed solution to the game can be easily verified by a computer algorithm; but the time it takes to check a solution grows slowly (polynomially) as the grid gets larger. Sudoku is thus said to be quickly solvable, but not quickly verifiable. On the other hand, there are problems whose solution can be quickly verified but cannot be quickly solved. In this framework, an “NP-Complete” problem is a problem for which each solution can be verified quickly (in polynomial time) and an algorithm must find a solution by trying all possible solutions. As such, the problem can be used to simulate every other problem for which a solution can be verified quickly that a solution is correct. A classic example of an NP-Complete problem is the so-called Traveling Salesman Problem (TSP), which can be paraphrased as follows (Cook 2014b; Benjamin et al. 2015): A salesman wishes to make a round-trip that visits a certain number of cities. He knows the distance between all pairs of cities. If he is to visit each city exactly once, then what is the minimum total distance of such a round trip?

This is yet another problem in graph theory, requiring that we find the most efficient path, a Hamiltonian cycle, that the salesman can take through each of n cities. No general solution is known, but it can be modeled as an undirected weighted graph, where the cities are the graph’s vertices, the paths the graph’s edges, and a path’s distance the edge’s weight. The problem consists in starting and finishing at a specified vertex after having visited each other vertex exactly once. If the salesman were to visit 5 cities, knowing all the distances, the shortest roundtrip would be determined by checking all possibilities for connecting the vertices standing for the cities. This works for small graphs. If we were to add a new city (vertex), it would need to be tried out in every previous combination. So this method takes factorial time. For 5 cities, the factorial would be 5 × 4 × 3 × 2 × 1. In general, it is n! = n × (n - 1) × (n – 2) × . . . × 1. If a computer has the capacity to solve a 20-city problem in 1 second, then by partitioning the problem into sub-problems (called dynamic programming), it will take about 10 minutes to solve a 30-city problem and about 35,000 years for a 60-city problem. Because of this, the problem is NP-Complete—it has no “quick” solution and the complexity of calculating the best route will increase when more destinations are added to the graph.

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If P were equal to NP then problems that are complex (involving large amounts of data) could be tackled easily as the algorithms become more efficient. The P = NP problem is one of the most important open problems in computer science with deep implications for mathematics and its methods of proof. An early mention of the problem is found in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked Neumann whether theorem-proving could be solved in quadratic or linear time—a central question for automated mathematics (Fortnow 2013). P would consist of all those problems that can be solved on a deterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NP would consist of all those problems whose solutions can be verified in polynomial time given the relevant information, or equivalently, whose solution can be found in polynomial time on a nondeterministic machine. Some of the existing automated proof techniques are not powerful enough to answer the P = NP problem, suggesting that novel technical approaches are required. Some of these will be discussed in Chap. 5. The fact that we are left in a state of suspension when we cannot solve problems or prove something to be necessarily so, such as a conjecture, is indirect evidence of the inquisitive force in us that Vico called the ingegno. Our brains seek answers constantly, even if the answers are of our own making. In antiquity, practical knowledge of the knotting patterns that produced a right triangle was not enough; the ancient mathematicians wanted to understand why these were true in an abstract way. So, they took the next step in establishing mathematics as an explanatory, rather than just descriptive, intellectual tool—developing the methods of proof to show how the patterns fit in with some abstract structure. It is the quest for a proof that shapes mathematical history, as discussed, especially when proof is elusive. One of the most intriguing examples of the latter is related to the primes, known as the Riemann Hypothesis (Derbyshire 2004). In 1859, Bernhard Riemann presented a paper to the Berlin Academy titled “On the Number of Prime Numbers Less Than a Given Quantity” in which he put forth an hypothesis that remains unsolved to this day. It is a proof that is waiting to be discovered or invented, even though it has already led to several significant discoveries in primality theory. On a number line, the primes become scarcer and scarcer as the integers on the line grow larger: twenty-five percent of the numbers between 1 and 100, 17 percent of the numbers between 1 and 1000, and 7 percent of the numbers between 1 and 1,000,000 are primes. Paul Erdös (1934) proved that there is at least one prime number between any number greater than 1 and its double on that line. For example, between 2 and its double 4 there is one prime, 3; between 11 and its double 22 there are three primes, 13, 17, and 19. Riemann argued that the thinning out of primes involves an infinite number of “dips,” called zeroes, on the line, and it is these zeroes that encode the information needed for proving primality and thus for developing a prime number rule. So far, no vagrant zero has been found, but at the same time no proof or disproof of the hypothesis has ever come forward. Riemann showed that at around one million, whose natural logarithm is about 3, every 13th number or so is prime. At one billion, whose natural logarithm is 21, about every 21st number is prime. A pattern seems to jut out from such observations. So, Riemann asked why primes were related to natural logarithms in

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this way. He suspected that he might find a clue to answering his question in a sequence, {1 + 1/2s + 1/3s + 1/4s + . . . 1/ns}, now called the Riemann zeta function. What kind of proof would be involved in connecting these observations? That question is, as argued throughout this book, a product of the ingegno seeking a way to integrate the scattered facts and connect them into a unified vision, much like poets connect words to create a lyrical work. The search for a proof is a powerful motivation, much like the quest for uncovering the laws of nature on the part of physicists, allowing them to seek deep connections between fields of mathematics and between mathematics and reality. The more elusive a proof is, the more it is hunted down, even if it may seem to have no implications above and beyond the proof itself (a feature of the restless ingegno as it has been described colloquially here). As Wells (2012: 140) states: Proofs do far more than logically certify that what you suspect, or conjecture, is actually the case. Proofs need ideas, ideas depend on imagination and imagination needs intuition, so proofs beyond the trivial and routine force you to explore the mathematical world more deeply—and it is what you discover on your exploration that gives proof a far greater value than merely confirming a fact.

Proof and mathematical discoveries in general seem to be located in the same neural circuitry that sustains ordinary language and other cognitive and expressive systems—discussed in Chap. 3. It is this circuitry that allows us to interpret observations, such as those related to unsolved problems or conjectures, as entailing something significant that is hidden within them. A proof would be a way to flesh them out once and for all. Even the formulation of a problem is part of this circuitry. For example, by translating the physical aspects of the traveling salesman problem (distances, cities) into symbolic notions, such as paths and weights, we have thus devised a mathematical model of the problem in terms of graph theory—a model that decomposes all aspects of the problem into its essential parts. For historical accuracy, it should be mentioned that the significance of the problem for logic and mathematics was first mentioned in 1930, evolving over time into one of the most challenging problems in algorithmic optimization ever devised. As Bruno, Genovese, and Improta (2013: 201) have noted: The first formulation of the TSP was delivered by the Austrian mathematician Karl Menger who around 1930 worked at Vienna and Harvard. Menger originally named the problem the messenger problem and set out the difficulties as follows. At this time, computational complexity theory had not yet been developed: “We designate the Messenger Problem (since this problem is encountered by every postal messenger, as well as by many travelers) the task of finding, for a finite number of points whose pairwise distances are known, the shortest path connecting the points. This problem is naturally always solvable by making a finite number of trials. Rules are not known which would reduce the number of trials below the number of permutations of the given points. The rule, that one should first go from the starting point to the point nearest this, etc., does not in general result in the shortest path.

Menger’s challenge has been tackled by computer science and mathematics working in tandem today with the development of the field of combinatorial optimization, one of the offshoots of the problem, starting in 1954 when an integer programming formulation was developed to solve the problem alongside a “cutting-

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plane” method, “which enables the finding of an optimal solution (namely, the shortest Hamiltonian tour) for a TSP involving 49 U.S. state capitals” (Bruno et al. 2013: 202). As such, it became a source for the P = NP problem, since it involves devising methods of how to attack NP-Complete problems in general.

Discovery and Invention Discovery can be divided into three categories, for the sake of argument, each of which implies a dynamic interplay with inventio (invention), the root of the ingegno. First, as we saw with π, some types of discovery in mathematics come at first from observation and experience. A pattern was observed by measuring the size of any circumference with regard to the length of the diameter, resulting in a ratio. As such, however, it remained an observation; only when the relation was proved and formalized by Archimedes did the observation become a fact of geometry. This can be called a “pragmatic” discovery, since it turns practical experience into theoretical knowledge in terms of an invention—an ingenious proof devised by the ancient mathematician. Another type of discovery occurs when a mathematical form itself becomes the basis for further understanding and ideas in mathematics. This can be called “inner-system discovery.” An example is Fermat’s Last Theorem, which resulted from considering a conjecture in itself, leading to the discovery of hidden structures and connections within the system of mathematics itself. Again, it did so through the inventio functions of the ingegno. The real conundrum comes when we discover the presence of a mathematical idea, such as π, in the description of natural phenomena—as, for example, in the spiral of the DNA double helix, the pupil of the eye, in sound and light waves, and on and on. This can be called “outersystem discovery.” It results from applying mathematics to the real world, that is, outside of itself, or in describing the real world via mathematics. The interplay between discovery and inventio has been the source of a debate that goes back to antiquity: Is mathematics discovered or invented? The putative answer to this conundrum is, as suggested by the foregoing discussion, that the two dimensions are connected by poetic logic, which is both the source of discovery, via the fantasia, and of invention, via the ingegno. In the end, the two are virtually impossible to tell apart ontologically. The connection between symbolic mathematics and its meanings and referents is never broken. Like a photo, an invented mathematical symbol or form shows us much more about a “scene” than what it appears to mean at first sight. The dynamic interrelation between discovery and invention suggests a deep epistemic connection between the mathematical imagination and reality, not in any pure “objective” sense, but in an imaginative sense, which is not constrained paradoxically by reality. Recalling Flatland, those living in two-dimensional space perceive, describe, and discover their own world in ways that are constrained by the plane; analogously, living in a three-dimensional world, our discoveries and inventions are constrained by its physical

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structure. Additionally, with our imaginations, we can discover in our own way what a four-dimensional world might “look like” to our minds. Unexpected discoveries in particular, both within and without mathematics, provide evidence of how our poetic logic works at a fundamental level. These now come under the rubric of serendipity (Eco 1998)—a notion that comes initially from an ancient Persian fairy tale, The Three Princes of Serendip, worth summarizing here. Three princes from Ceylon were journeying in a strange land when they came upon a man looking for his lost camel. The princes had never seen the animal, but asked the owner: Was it missing a tooth? Was it blind in one eye? Was it lame? Was it laden with butter on one side and honey on the other? Was a pregnant woman riding it? Incredibly, the answer to all their questions was “yes.” The owner accused the princes of having stolen the animal, since, clearly, they could not have had such precise knowledge. But the princes merely pointed out that they had observed the road, noticing several patterns in it: for example, the grass on either side was uneven, suggesting a lame gait; there were places where food seemed to have been chewed via a gap in the animal’s mouth; there were uneven patterns of footprints, the signs of awkward mounting and dismounting typical of someone who was pregnant; and there were differing accumulations of ants and flies, which congregate around butter and honey. Their questions were really prompted by inferences based on these observations. Knowing the world in which they lived, they were able to make concrete connections between the observations and what happened. English writer Horace Walpole came across the tale and, since Serendip was Ceylon’s ancient name, coined the word serendipity, to designate how we often come across discoveries. Serendipity results because of the operation of poetic logic, which in this case involves inference by analogy, a powerful manifestation of creative-ingenious thinking (Hofstadter 1979; Hofstadter and Sander 2013). Serendipitous discoveries such as the manifestations of π in nature are found throughout the history of mathematics and science; so too are many inner-system discoveries. Consider, as a case-in-point of the latter, going back to Euclid, the so-called amicable and perfect numbers. The numbers 284 and 220, for instance, are called amicable, because the proper divisors of one of them when added together produce the other. The proper divisors of 284 are 1, 2, 4, 71, and 142, and the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Now, we notice that: 220 = 1 þ 2 þ 4 þ 71 þ 142 ðthe proper divisors of 284Þ 284 = 1 þ 2 þ 4 þ 5 þ 10 þ 11 þ 20 þ 22 þ 44 þ 55 þ 110 ðthe proper divisors of 220Þ A perfect number is one that equals the sum of its proper divisors. For example, the proper divisors of the number 6 are 1, 2, and 3. Now, if we add these together, we get the number: 6=1 þ 2 þ 3

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The next perfect number is 28. Its proper divisors are 1, 2, 4, 7, 14, and 1 + 2 + 4 + 7 + 14 = 28. Very few perfect numbers have been discovered. Numbers that are not perfect are called excessive or defective. An excessive number is one whose proper divisors, when added together, produce a result that exceeds its value. The number 12, for example, is excessive, because the sum of its proper divisors, 1, 2, 3, 4, and 6 (1 + 2 + 3 + 4 + 6 = 16) exceeds its value. A defective number is one whose proper divisors, when added together, produce a result that is smaller than its value. One example is 8, since the sum of its proper divisors 1, 2, and 4 (1 + 2 + 4 = 7) is less than its value. Now, one can ask if this is nothing more than “number play.” As it turns out, these numbers have had implications in various sectors of number theory, leading to discoveries within it, and are thus examples of serendipity within mathematics itself. They have also raised questions about infinity: Are there an infinite number of such numbers? An amicable or perfect number did not exist until it was discovered by playing with numerical patterns. This discovery led to a classificatory matrix in which these numbers could be located with respect to other numbers. From this, other discoveries were made—discoveries that would have been literally “unthinkable” without the initial one, since they are connected derivatively to it. An example of a pragmatic discovery, constituting a classic historical example of how the ingegno works at this level, is Eratosthenes’ estimation of the Earth’s circumference. His proof can be paraphrased reductively as follows. Standing during the summer solstice at Alexandria, and knowing that it was due north of the city of Syene, with the distance between the two cities being 500 miles, Eratosthenes used geometrically based symbolic reasoning to calculate the Earth’s circumference— without having to do it physically. In other words, he turned the physical situation into a geometric paper-and-pencil problem. At the summer solstice, he reasoned, the noon sun is shining directly down into a well at Syene, since the sun is directly overhead at that time of day. Eratosthenes drew a diagram with the scale reduced to represent the situation. He represented the direction of the sun with a straight line going directly into the well. Since sun rays are parallel, at the same instant in Alexandria, he drew on his diagram another parallel line. He then measured the angle made by a well near to him, which he found to be 7.2°. From another fact of geometry, he knew that the arc between Alexandria and Syene is equal to 7.2°/360° or 1/50 of a circle (since the Earth is virtually a sphere and therefore almost a 360° angle). Therefore, Eratosthenes concluded, the circumference was 50 times the length of the arc: 50 × 500 = 25,000 miles. This is in close agreement with the actual known value today of approximately 24,859 miles (Fig. 2.13, created by Michael Wakeman, 2015, http://www.classichistory.net/archives/eratosthenes-cir cumference-of-earth): This episode shows how diagrams turn pragmatic knowledge into a theoretical model, compressing information in cognitively manageable ways and then allowing us to discover facts about the real world through them (Miller 2007). But such symbolic methods come not by happenstance, but by experiential hunches or background knowledge. The early geometers of ancient Egypt, Sumer, and Babylon were concerned with such practical problems as measuring the size of fields and

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Fig. 2.13 Eratosthenes’ estimation of the Earth’s circumference

laying out accurate right angles for the corners of buildings. Their empirical work, their meticulous diagrams, and their visual observations produced early pragmatic diagrams that were refined and systematized later into theorems. With the emergence of geometrein as a modeling system of the world, the ancient mathematicians turned pragmatic experience into theoretical knowledge, which, in turn, allowed them to further explore the world through the models themselves, as did Eratosthenes. Stewart (2008: 46) summarizes this remarkable achievement as follows: Using geometry as a tool, the Greeks understood the size and shape of our planet, its relation to the Sun and the Moon, even the complex motions of the remainder of the solar system. They used geometry to dig long tunnels from both ends, meeting in the middle, which cut construction time in half. They built gigantic and powerful machines, based on simple principles like the law of the lever.

The search for better models of description of reality has characterized large stretches of the history of mathematics. Sometimes, this came about by serendipitous discoveries within mathematics itself. As is well known, at least anecdotally, the Pythagoreans serendipitously discovered irrational numbers, and were so surprised and distressed by it that they decided to keep it secret, mainly because the discovery went against their idea of harmony. Like the dissonance ratios they discovered in music, irrational numbers were intellectually dissonant to them. As is well known, these surfaced after the Pythagoreans had proved their famous theorem related to right triangles (Maor 2007). At some point, they noticed that their very own theorem, when applied to an isosceles right triangle with its two sides equal to 1 (the unit

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Fig. 2.14 The discovery of √2

length), produced a very strange number for the length of the hypotenuse, √2 (Fig. 2.14): This was totally unexpected. But there it was. The number √2 cannot be written as a fraction or ratio. It is a repeating decimal (1.414213. . .). So, it could not be explained in existent theoretical ways. For this reason, it came to be called irrational (without a ratio). In effect, a hidden truth—a new type of number—was revealed by a mathematical theorem itself. This whole line of discussion leads us directly to the Platonic-versus-constructivist view of mathematics. Do we discover mathematics or do we invent it and then discover that it works? Was √2 “out there” ready to be discovered, or was it produced inadvertently by the manipulation of symbols? Plato believed that mathematical ideas pre-existed in the world and that we come across them or, perhaps, extract them from the world through logical thought. Just like the sculptor takes a clump of marble and gives it the form of a human body, so too mathematicians take a clump of reality and give it symbolic form. In both “representations,” we discover many more things about the body and about mathematics. The truth is already in the clump; it takes the mathematician to extract it. Many find this perspective difficult to accept today, leaning toward constructivism, or the idea that mathematics is constructed to tell us what we want to know about the world. But, as Berlinski (2013: 13) suggests, the Platonic view is not so easily dismissible: If the Platonic forms are difficult to accept, they are impossible to avoid. There is no escaping them. Mathematicians often draw a distinction between concrete and abstract models of Euclidean geometry. In the abstract models of Euclidean geometry, shapes enjoy a pure Platonic existence. The concrete models are in the physical world.

Without the Platonic models, the concrete ones would have little interest. Moreover, there might be a neurological basis to the Platonic view. As neuroscientist Pierre Changeux (2013: 13) muses, Plato’s trinity of the Good (the aspects of reality that serve human needs), the True (what reality is), and the Beautiful (the aspects of

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reality that we see as pleasing) is actually consistent with the notions of modern-day neuroscience: So, we shall take a neurobiological approach to our discussion of the three universal questions of the natural world, as defined by Plato and by Socrates through him in his Dialogues. He saw the Good, the True, and the Beautiful as independent, celestial essences of Ideas, but so intertwined as to be inseparable. . .within the characteristic features of the human brain’s neuronal organization.

This means that our brain is predisposed to look for reality, but that it might be a reality of our making. Moreover, Plato’s models mean that we never should find faults within our systems of knowledge, for then it would mean that the brain is faulty. This is what Gödel’s (1931) undecidability theorem implies. But then, if mathematics is faulty, why does it lead to demonstrable discoveries, both within and outside of itself? René Thom (1975) referred to discoveries in mathematics as “catastrophes” in the sense of events that subvert or overturn existing knowledge. Thom names the process of discovery as “semiogenesis” or the emergence of forms within symbol systems themselves. This occurs by happenstance through contemplation and manipulation of the forms. As this goes on, every once in a while, a catastrophe occurs that leads to new insights, disrupting the previous system. Now, while this provides a description of what happens—discovery is indeed catastrophic—it does not tell us why the brain produces catastrophes in the first place. Perhaps, the connection among the brain, the body, and the world will always remain a mystery, since the brain cannot really study itself. The connection among truth, discovery, and invention can perhaps be best formulated in terms of Vico’s verum-factum principle, which asserts in essence that poetic logic allows humans to construct their understanding of the world, interpreting it creatively. The term means that “the truth is made”—coming well before the advent of contemporary constructivist models in philosophy and mathematics. As he put it (Bergin and Fisch 1984: 331): “The world of civil society has certainly been made by men, and its principles are therefore to be found within the modifications of our own human mind.” As poetically generated concepts become more and more removed from their imaginative origins, settling into memoria, they generate highly abstract structures on their own. Free from sensory control, these structures gradually come to dominate purposeful thinking. The mind’s cognitive system can then be projected onto the external world of reality to partition it, organize it, classify it, and explain it (in human terms). These projections of the mind have produced our symbolic and cultural systems—our institutions, scientific theories, laws, etc. The structure of these systems, therefore, can be used to investigate the structure of our memoria (next chapter). However, as Bergin and Fisch (1984: xlv) have perceptively pointed out, in being makers of things, Vico believed that human beings were themselves made to do just that: “[Humans] have themselves made this world of nations, but it was not without drafting, it was even without seeing the plan that they did just what the plan called for.”

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Discovery in mathematics could not occur without the invention and use of symbols in the first place to guide it. Invention (a main product of the ingegno) is a construction of our interpretations of reality (truth). It results from experience. And thus it tells us something about experience; for example, fire is a discovery, but rubbing sticks to start a fire is an invention. The general principles of arithmētikē derive from the experience of counting. Naming the counting signs (numerals) allows us to turn these principles into ideas that can be manipulated intellectually and systematically. Analogously, the general principles of geometrein derive from the experience and practice of measuring the size of fields, the angles in the corners of buildings, and so on. To carry out such measurements, diagrams were invented to represent them and names were assigned to the geometric figures employed. Around 2000 BCE, the Egyptians discovered that knotting and stretching a rope into sides of 3, 4, and 5 units in length produced a right triangle, with 5 the longest side (the hypotenuse). The Pythagoreans were aware of this discovery. Their goal was to show that it revealed a general structural pattern. As the historian of science, Jacob Bronowski (1973: 168) has insightfully written, we hardly recognize today how important this demonstration was. It became the blueprint for discovery that reached out into the world: The theorem of Pythagoras remains the most important single theorem in the whole of mathematics. That seems a bold and extraordinary thing to say, yet it is not extravagant; because what Pythagoras established is a fundamental characterization of the space in which we move, and it is the first time that it is translated into numbers. And the exact fit of the numbers describes the exact laws that bind the universe. If space had a different symmetry the theorem would not be true.

All this suggests that we are part of the world in which we live and thus privileged, in a way, to understand it best, on our own terms. As Al-Khalili (2012: 218) puts it, “our very existence determines certain properties of the Universe, because if they were any different we would not be here to question them.” In effect, our brains are designed to discover the universe in its own way, and once discoveries about it are made, we start using them to invent all kinds of ideas derived from them. So, we both invent and discover our worlds. The question becomes why all this is so. We could conceivably live without the Pythagorean theorem. It tells us what we know intuitively—that a diagonal distance is shorter than taking an L-shaped path to a given point. But once we have provided a theory for this intuition, we become equipped to use it for further exploration. There is a commonly held notion among mathematicians that all mathematics is trivial once it is understood (Hardy 1967). To put it another way, theories and proofs make mathematics routine. The followers of Pythagoras are accredited with the discovery of irrational numbers at a time when every number was expected to be the ratio of two integers. This discovery was deemed to be truly catastrophic, so much so that, according to legend, the person responsible for the discovery was thrown overboard by his fellow Pythagoreans while they were at sea (Struik 1987). But, as it turned out, it quickly became a trivial matter of mathematical practice, as irrationals became part of a number system that was expanding all the time, finding their niche within it.

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Epilogue The relationship among invention, discovery, and proof is an intrinsic one, reflecting, as implied in this chapter, the very structure of how the ingegno (the force behind invention) has always played a role in how mathematicians carry out their work. From the Rhind Papyrus to the proofs of Thales, Archimedes, Euclid, Euler, and many others, it is a prelude to logical proof and the source of its inspiration, indicating that poetic logic is the starting point for ideas and methods in mathematics to be devised or discovered (as the case may be). It can be described as a “figuring out” function of the mind that attempts to make connections among an amalgam of facts and observations giving them an order that leads to the relevant Aha insight, which allows us to extrapolate the “big picture,” or salient feature in the connections. When this is not achieved, as in unproven conjectures, it continues to spur mathematicians on to seek the likely hidden connections that they have not as of yet unraveled. As discussed in this chapter, the ingegno is the general term for describing such features of mathematical thought as pattern recognition, often by considering what is already known about some mathematical phenomenon, and using it to extract the relevant pattern. The process involves various stages: detection by the fantasia, which produces an inner vision, translation of the pattern into forms based on such processes as image schematic formation, using comparison and analogies from memory, and finally putting all the abductions into a model of the detected pattern. When Pythagoras united arithmētikē with geometrein, he established mathēmatikē as a means to discover truths about the world. It was more of an art form, than a strictly logical discipline—an art form that subserved, nonetheless, the emerging methods of logic. The word arithmētikē meant, in fact, “the art of counting.” All the ancient civilizations had procedures for carrying out this art, using them in business transactions, in measurement, and in many other practical activities even before they had developed practical and efficient writing systems. When alphabets appeared on the scene around 1000 BCE, the characters were used not only to represent phonemes but to represent the numbers. The order of the Greek alphabet is based on that early practice, where A (alpha) stood for the number 1, B (beta) for the number 2, and so on. In separating this art from the practical world, the Pythagoreans started a true intellectual revolution. They did so by inventing (literally) notational and heuristic symbolic devices as they discovered new facts, often through the intellectual manipulation of their devices. When their inventions revealed new truths, the tool itself (mathematics) was felt to be of divine origin. As Bertrand Russell (1945) aptly stated in his History of Western Philosophy: Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God’s thoughts.

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The ancient mathematicians turned practical and intuitive knowledge into powerful theoretical knowledge via the ingegno, which provided the relevant insights for taking information, facts, and observations and turning them into general concepts. The first theorems introduced into the world the notion that a mathematical proof was not just a recipe of how to do something practical, such as to construct all right triangles; it was a way to convince the brain that it revealed something about the universe. Little did the ancient mathematicians realize how powerful their tool was. The integers that fit into c2 = a2 + b2, for instance, transcended the specific properties of right triangles, with the theorem eventually becoming a vehicle itself for investigating patterns within number systems and the nature of proof itself, ending up with Fermat’s Last Theorem and all the intellectual activities that it has since generated. For Plato, our brains come equipped with mathematics; for Ludwig Wittgenstein (1921), mathematics is an invention that we use to project our minds onto reality. For Vico, the connection between the two is of our own making, but in doing so, we actually discover truth. Plato saw numbers and geometrical forms as ethereal ideals, existing outside of space and time in a realm beyond the reach of humankind. These make their unexpected appearances through human ingenuity and manipulation of the world, much like a Sphere makes its appearance to Flatlanders to generate unexpectedness throughout the two-dimensional realm. We do not have the ability to differentiate ontologically between where one begins and the other ends, since our intellectual activities are products of our brains. As Thom (2010: 494) observes, “quite likely, it is not possible to decide the ontological nature mathematical entities without taking into account the way mathematical constructions may be inserted into the concrete reality of the world surrounding us.” All this suggests that mathematics is a true conundrum. Its forms cannot be tied down to a specific meaning, even if they emerge in a specific context. They can be applied constantly to all kinds of referential domains, known and unknown, as has been the case with graph theory, which shows up almost everywhere in the realm of mathematics, as discussed in this chapter. It is part of an inner vision that continues to guide us to explore “hidden meanings” in the world and mathematics alike. The problem may be that since antiquity philosophers have confused logic with mathematics. Charles Peirce (a logician and mathematician) argues eloquently that the two are different yet intertwined (in Kiryushchenko 2012: 69): The distinction between the two conflicting aims [of logic and mathematics] results from this, that the mathematical demonstrator seeks nothing but the solution of his problem; and, of course, desires to reach that goal in the smallest possible number of steps; while what the logician wishes to ascertain is what are the distinctly different elementary steps into which every necessary reasoning can be broken up. . .In short, the mathematician wants a pair of seven-league boots, so as to get over the ground as expeditiously as possible. The logician has no purpose of getting over the ground: he regards an offered demonstration as a bridge over a canyon, and himself as the inspector who must narrowly examine every element of the truss because the whole is in danger unless every tie and every strut is not only correct in theory, but also flawless in execution. But hold! Where am I going? Metaphors are treacherous—far more so than bridges.

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Metaphor is, in fact, a major manifestation of poetic logic, as will be discussed in Chap. 4. As one of the oldest mathematical texts, the Rhind Papyrus shows how the ingegno appeared from the outset to guide how mathematics is carried out on the basis of the image schemas that undergird metaphor, including the one of infinitely inscribing polygons around and inside a circle—a schema that also guided Archimedes much later. It is arguably the earliest treatise exploring the relation between the internal system of mathematics and the outer world, between the mathematical imagination, the ingegno, and proof, showing that mathematics is a creative response to the world. Mathematics is still on the same wavelength as the papyrus, so to speak, millennia later.

Chapter 3

Memory

Memories of the past are not memories of facts but memories of your imaginings of the facts. Philip Roth (1933–2018)

Prologue The third dimension of poetic logic, along with the fantasia and the ingegno, is the memoria, which has three main functions within a Vichian model of the creative mathematical mind: (1) to allow for a reimagining of established information, both when new mathematics is being discovered or when some task, such as proof, is carried out (as we saw in the previous chapter); (2) to consolidate mathematical discoveries and ideas on the basis of relevant symbolism, which constitutes a compressed memory system of acquired knowledge; and (3) to provide a basis upon which novel ideas and even discoveries can be envisioned. Vico was well aware that his view of memoria was an ancient one, which he adapted, however, in an unprecedented fashion as part of poetic logic. The use of mnemonic devices, for instance, was cultivated by many ancient scholars and teachers, including Plato, Aristotle, Confucius, and others, who were deeply aware that signs (symbols, diagrams, etc.) were memory-preserving and memory-enhancing devices, which could be used to stimulate thought creatively. The evolutionary reason behind the development of numerals was to transform repetitive counting into a symbolic system that could be utilized or consulted at any time to carry out counting in a rapid and economical way. As numeration systems became increasingly important in the everyday affairs of early civilizations, new forms of symbolism emerged, becoming the distinguishing features for establishing mathematics as an autonomous discipline. Without notations, diagrams, equations, and so on, it would require an enormous amount of effort and memory to confront mathematical ideas and tasks anew, since the relevant thoughts would have to be practically discovered or reviewed each time. A notation system, on the other hand, allows us to circumvent this by presenting established information and ideas in a

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Danesi, Poetic Logic and the Origins of the Mathematical Imagination, Mathematics in Mind, https://doi.org/10.1007/978-3-031-31582-4_3

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compact way through symbols and other sign-based artifacts. It is the feature of compression, in fact, that makes notation and other symbolic systems so powerful cognitively, as many episodes in the history of mathematics reveal. Take, for example, the invention of exponential symbolism, which was devised initially to be an abbreviation strategy to facilitate the cumbersomeness of reading repeated multiplications of the same digit and the risks that this would entail, such as introducing errors caused by diminished attention mechanisms: 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = ? The task of just reading the multiplication layout is facilitated with the use of “12” as a superscript symbol, standing for the times the same number 10 is a factor in the multiplication: 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 = 1012 The notation, 1012, makes it instantly obvious how many 10s are involved in the multiplication, thus reducing the time and cognitive effort required to process the given information, as well as saving space to write out the problem. The greater the number to be represented, the greater the need to use such a device. Indeed, the astronomical distances studied by astronomers would remain impracticable to work with, if it were not for exponential notation. The concept of light year, for instance, would be virtually useless as a means to study the universe, since one single light year unit is equivalent to the distance that light travels in 1 year, which is nearly 10 trillion kilometers. But with exponential notation, it can be represented manageably as 9.6 × 1012 kilometers. This utility and significance of exponential notation became saliently obvious right after its invention, providing an efficient way to represent numbers and enhance recall. But the memoria is not just a static recall system; it is a faculty that stimulates further exploration, since it presents information compressed into symbolic forms that can be examined in themselves, beyond their initial meanings. The convention of using a superscript number to represent repeated multiplication did much more than just make this operation less effortful to process and to save physical space in writing out a long multiplication; it allowed for the representation of very large numbers and suggested new ways of doing arithmetic. Although the notion of exponents goes back to antiquity, the first concrete proposals to devise and utilize exponential notation emerged in the Renaissance, culminating in René Descartes’s introduction of superscript symbolism in his treatise, La géométrie (1637). Right after, the symbolism took on a life of its own, so to speak, stimulating the mathematical imagination in ways that would have been literally unthinkable beforehand. Mathematicians started, in fact, to consider exponential numbers in an abstract way, discovering rules for dealing with them and, in the process, unraveling new facts about numbers:

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ðxm Þðxn Þ = xmþn ðxm Þ ÷ ðxn Þ = xm - n ðx ≠ 0Þ ðxm Þn = xmn ðxyÞn = ðxÞn ðyn Þ ðx=yÞn = ðxÞn =ðyÞn ðx - n Þ = 1=ðxÞn ... ð xÞ 0 = 1 Exponential notation likewise became associated with many new ideas and techniques, including logarithms, the inverse function of exponentiation, which also started out as a means of doing arithmetic more efficiently and economically. Logarithms have been used in many areas of mathematics and science, leading to all kinds of discoveries, all of which would have been unthinkable without them. The relevant point here is that a simple notational device invented to make a certain type of arithmetical operation easier to read and to compute became a source prompting on the ingegno to seek and find new ideas and mathematical notions through the symbolism itself. As Alfred North Whitehead (1911: 43–44) so aptly put it: “By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the [human] race.” This chapter discusses the mathematical memoria, which Vico (in Bergin and Fisch 1984: 260) characterizes in general as follows: Imagination, however, is nothing but the springing up again of reminiscences, and ingenuity or invention is nothing but the working over of what is remembered. Now, since the human mind at the time we are considering had not been refined by any art of writing nor spiritualized by any practice of reckoning or reasoning, and had not developed its powers of abstraction by the many abstract terms in which languages now abound, as we said above, it exercised all its force in these three excellent faculties which came to it from the body.

As Verene (1981) has cogently discussed, the fantasia, the ingegno, and the memoria operate simultaneously, with each one sustaining the other, as we attempt to make sense of the world and to encode it in some durable symbolic fashion. One cannot develop a proof of any theorem without the interaction of these three faculties, as illustrated in the previous chapter. The actual form that a proof takes is guided at first by the ingegno, which converts the inner vision produced by the fantasia into an insight (abduction) of how a rational proof can be actually carried out. But before this can occur, past experience and knowledge—the function of the memoria—comes into play, suggesting unconsciously the facts needed to develop the proof. In contemporary psychological terms, the memoria covers both episodic and semantic memory systems (Tulving 1972). Generically speaking, the former is memory of episodes via images elicited by them, and the latter is memory of concepts encoded in symbol systems. Episodic memory pertains to recollections as

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they are imagined, while semantic memory pertains to recollections as they are presented in symbolic forms. Significantly, a study by Tsuda et al. (2008) found that episodic memory and the imagination are linked neurologically, providing a corroborative psychological basis for the Vichian memoria (Tsuda et al. 2008: 99): Because episodic memory includes a time series of events, an underlying dynamics for the formation of episodic memory is considered to employ an association of memories. . .Both episodic memory and imagining things include various common characteristics: imagery, the sense of now, retrieval of semantic information, and narrative structures. Taking into account these findings [it is suggested that] the hippocampus is the common mechanism of episodic memory and imagination.

The Mathematical Memoria The inner visions generated by the fantasia and the creative constructs that the ingegno produces from them are kept “in storage,” so to speak, by the memoria, transforming them into memory-preserving symbolism that becomes the basis for future visions. To put it another way, the memoria conjures up the past so that we can make predictions with it about the future, as Jacob Bronowski (1977: 25) has observed, and which he sees as the feature of the human brain that makes it exceptional among all species: The images play out for us events which are not present to our senses, and thereby guard the past and create the future—a future that does not yet exist, and may never come to exist in that form. By contrast, the lack of symbolic ideas, or their rudimentary poverty, cuts off an animal from the past and the future alike, and imprisons it in the present. Of all the distinctions between man and animal, the characteristic gift which makes us human is the power to work with symbolic images.

So, in this Vichian framework, an object such as the Ishango Bone (Chap. 1) is a symbolic artifact that allows the memoria to replace physical counting with symbolbased numeration, and this in itself produces emergent ideas about the connection between numbers and symbols, becoming the basis for a proto-mathematics. In neuroscientific terms, the mathematical memoria can be characterized as the neural system of imagistic representations that are accumulated in various brain regions from symbolic forms used over and over. The fantasia depends critically on memory to provide the established building blocks for constructing things to be imagined anew or again. Notational symbolism, diagrams, formulas, and the like are thus devised so that the ideas they encode can be used in both routine and new exploratory ways. Diagrams strongly suggest that the mathematical memoria involves a recruitment of the inner visions related to the fantasia, which can then be used by the ingegno to perform mathematics. This tripartite system of mathematical reasoning is evoked every time we solve a mathematical problem. Consider the following puzzle, which is a version of a problem devised by the medieval Persian astronomer and mathematician Abu al-Wafa (Berggren 1986): Draw three identical triangles, and one smaller triangle, similar to them in shape, so that all four can be made into one large triangle.

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Fig. 3.1 Version of Abu Al-Wafa’s puzzle

The way to do this is to lay out the three identical triangles in such a way that when they touch, they will form the outline of a fourth smaller one in the center, as shown in the diagram below. Now, when one specific vertex in each of the four triangles is joined to one specific vertex in the others as shown below, a large triangle emerges (shown in outline form) (Fig. 3.1): The insight comes from imagining arrangements of the triangular form, and thus relies to some extent on previous familiarity with triangles. All such problems in the arrangement of geometric figures are based on the essential structural properties of the figures. Diagrams such as this one are the result of bringing forth knowledge of such properties and applying it to the task at hand in ingenious ways. As such, starting with the ancient mathematicians, such problems were devised to strengthen knowledge of geometry and its possible uses—hence, becoming part of mathematical memoria (Cajori 1928). In effect, we could not solve, or even understand, the problem above without having been exposed to triangles and their structural properties in two-dimensional space. As Vico put it (in Bergin and Fisch 1984: 304): “Memory thus has three different aspects: memory when it remembers things, imagination when it alters or imitates them and invention when it gives them a new turn or puts them into proper arrangement and relationship.” Every diagram or notational device puts ideas into a “proper relationship,” which evokes the mnemonic information that it enfolds. The standard notational conventions that we use today were in fact developed over time and can be divided into evolutionary stages (Smith 1994: 86), starting with the prehistoric tally stage, as discussed in Chap. 1, and then progressing through a rhetorical stage, based on a combination of words with geometric and numeral symbols, and then a syncopated stage, when symbolic abbreviations became increasingly standardized, and finally, a fully symbolic stage, which is traced to the sixteenth century. With the partnership between mathematics and computer science today, new symbolism and notational practices are being developed, merging computer functions with mathematical notions. An example is so-called Big O notation, which was devised at the end of the nineteenth century and early twentieth centuries (Bachmann 1894; Landau 1909) to describe the limiting behavior of a function when it tends toward a particular value or infinity. It has now been adopted to describe the complexity of computer codes using algebraic terms. For example, in O(n2) (Big O squared), the letter “n” represents the input size, and the function “g(n) = n2” gives us an idea of how complex the algorithm is with respect to the input size. Without such notation, even some algorithms cannot be fully explored or manipulated for further development.

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Notation The key feature of mathematical notation is that it compresses information. The Pythagorean equation c2 = a2 + b2, for example, is essentially an economical (compressed) means of saying the same thing as the sentence “the square on the hypotenuse is equal to the sum of the squares on the other two sides.” The equation turns the information and ideas expressed in words into a compact symbolic sentence. In so doing, it takes the finite semantics in the linguistic sentence out, leaving only the structural outline of the information. And it is this feature of notation that makes it cognitively powerful, since we can now find many more meanings and applications for it, in addition to the original geometrical one. For example, one can ask what integers fit the equation and, further, if there are other exponents for which the equation holds in general, cn = an + bn. From this deliberation on the notation itself, detached from its original geometrical meaning, has come much subsequent mathematical contemplation leading to such ideas as Fermat’s Last Theorem. In other words, symbolic notation is suggestive of meanings that likely could not otherwise be contemplated. This illustrates Vico’s notion of memoria as a faculty undergirding rediscovery of the past and its application to present and future situations. In this framework, therefore, notation is itself a source of discovery. Exponential notation not only simplified a complex information-processing task, saving space and reducing the time and cognitive energy required to process the same information, but it guided the ingegno to explore exponential numbers as novel forms of numeration in themselves, thus producing new branches. Without notation and other symbolic practices, there would be only practical counting and measuring, not mathematics. Even a numeral system becomes a source for further mathematics. For example, place value systems have permitted mathematics to progress by allowing for an abstract consideration of numbers via efficient numeration. In contrast to nonpositional systems, the structure of digits in such systems enhances efficiency of representation and allows for the representation of numbers ad infinitum. This is analogous to phonemic systems in language, in which with a small set of phonemes, one can make words ad infinitum—a structural feature called double articulation (Martinet 1955). So, for example, the English word cat is composed of the phonemes /k/, /æ/, and /t/, in that order, which are meaningless in themselves as separate individual phonemes. But when they are combined in a specific way, each one contributes to the semantics of the word. Without this double articulation feature, it would require an enormous amount of effort and memory to construct separate words with different phonemes each time, not a given set of phonemes. This duality of patterning allows for the expression of an infinite number of meaningful words and sentences with a small repertory of phonic resources. The same applies to positional notation systems in mathematics, which also show the property of double articulation. With a small set of numeral symbols (digits), one can construct representations of numbers ad infinitum, by simply considering the values of the digits in their position in a layout.

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Fig. 3.2 Zero on the number line

Now, while a positional system can be seen to increase ease in reading number and in carrying out computations, for it to work, a place-holder symbol is required to indicate that a certain place is to be left “empty” (without value). That symbol is, of course, zero 0 (Seife 2000), which was actually conceptualized as a “void” device in antiquity. For example, the Egyptians and Babylonians left a blank space for it in their numeration practices, as did the Chinese on their counting boards. The Mayans had a symbol for zero by about 250 CE and the Hindus, who developed the symbol we use today, had devised it by the late 800s. The word zero derives from ziphirum, a Latinized form of the Arabic word sifr, which, in turn, is a translation of the Hindu word sunya (void or empty). In 976 CE, the Persian mathematician al-Khwarizmi defined it simply as a place-holder, remarking that if no digit is required in a certain position, then a little circle should be used to indicate it. The zero symbol, along with the Hindu-Arabic decimal system, reached Europe around 1000 CE through the efforts of Pope Sylvester II. But it hardly got noticed at the time. It was reintroduced in a much more practical way a few centuries later by Leonardo Fibonacci with his 1202 book, the Liber Abaci, succeeding in convincing his fellow Italians that the decimal system was far superior to the Roman one in use at the time (Devlin 2011). But Fibonacci realized that a symbol for “nothing” would bring about objections. So he started off his book reassuring readers that zero was simply a sign that allowed for all numbers to be written, without leaving digit slots empty (cited in Posamentier and Lehmann 2007: 11): The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written.

Now, although it was invented as a practical notational symbol, 0 came to be viewed as a number itself, becoming the mid-point on the number line that divides the positive from the negative numbers (Fig. 3.2): Zero is not defined as either negative or positive, assigning it a unique status in number theory. All the real numbers can be located on the number line, including irrational and irrational numbers such as √2, π, and e (ℝ = symbol for real numbers)—showing the extent to which the number line concept has enlarged and even enhanced our understanding of different numbers and their relation to each other (Fig. 3.3). While we now accept such symbolic artifacts as “logical,” originally their conceptualization required a large dose of imagination. And because they are memorypreserving devices (allowing us to categorize elements in specific ways for easy recall and utilization), they constitute a basis for deriving implications, some of which have led to an axiomatic standardization of numbers, culminating in Giuseppe Peano’s (1889) axioms, which start with his definition of 0 as a natural number and a follow-up axiom that natural numbers are successors to 0. It is difficult to see how

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Fig. 3.3 The real number line

Peano’s axioms would have been thinkable in the first place without the number line and the positioning of 0 on it. The line has also allowed for a representation of negative numbers as related vectorially to 0 (namely, to its left). Negative numbers as such are found in a 250 BCE Chinese text titled The Nine Chapters on the Mathematical Art, in which negative numbers are represented with black numeral signs and positive numbers with red signs. These were used, seemingly, to allow for calculations to be carried out. A similar usage of negative numbers appears in the seventh century in the book-keeping practices and astronomical calculations of the Hindus. It was not until the sixteenth century, however, that they were seen as numbers. It is not coincidental that the term negative comes from the Latin negare (“to deny”), perhaps, because the existence of such numbers had been denied for so long, or because it implied philosophically the “denial” of the positive. On the one hand, nothing new was accomplished by introducing the concept of negative number, since it was already used in a practical way for calculation reasons, such as to show loss on an accountant’s ledger. However, only after a suitable notation for negative numbers was introduced (e.g., using the now familiar negative sign) were they perceived to be real numbers, not just convenient heuristic artifacts. Alexander (2012: 28) elaborates the cognitive significance of negative numbers as follows: Using the natural numbers, we made a much bigger set, way too big in fact. So we judiciously collapsed the bigger set down. In this way, we collapse down to our original set of natural numbers, but we also picked up a whole new set of numbers, which we call the negative numbers, along with arithmetic operations, addition, multiplication, subtraction. And there is our payoff. With negative numbers, subtraction is always possible. This is but one example, but in it we can see a larger, and quite important, issue of cognition. . .The numbers, now enlarged to include negative numbers, become an entity with its own identity. The collapse in notation reflects this. One quickly abandons the (minuend, subtrahend) formulation, so that rather than (6, 8) one uses -2.

Once a discovery proves to be useful or interesting, it spreads throughout mathematics, no matter where it is practiced. The decimal system, the zero, the negative numbers all came into being at specific times in different cultures, suggesting the presence in the unconscious mind of what Vico called “imaginative universals” (universali fantastici), which are the set of common inner visions providing the capacity to reflect upon objects and events away from their contexts of occurrence and existence. They correspond largely to image schemas in current cognitive science (Chap. 4). Notation assigns permanence to these schemas—a feature that can be seen even in prehistory, with artifacts like the Ishango Bone (Chap. 1). As Godino et al. (2011: 250) remark, notational systems are practical solutions to the same kind of universal problems:

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As we have freedom to invent symbols and objects as a means to express the cardinality of sets, that is to say, to respond to the question, how many are there?, the collection of possible numeral systems is unlimited. In principle, any limitless collection of objects, whatever its nature may be, could be used as a numeral system: diverse cultures have used sets of little stones, or parts of the human body, etc., as numeral systems to solve this problem.

Some branches of mathematics, such as algebra, are centered on the study of symbols, known specifically as variables, and the rules for manipulating these. As such, algebra is a unifying symbol-based thread of almost all of mathematics, and the basis for advanced mathematics, including group and field theory. The concept of a “general arithmetic” began in ancient Egypt and Babylon. The Babylonians solved algebraic equations by essentially the same procedures that we use today. A Babylonian tablet preserved in the British Museum shows knowledge of quadratic equations, as evidenced by the following problem (Solomon 2008: 11–12): The area of a square added to the side of a square comes to 0.75. What is the side of the square?

The solution is explicated verbally, unfolding as follows, with corresponding contemporary algebraic notation: Babylonian method I have added the area and the side of my square 0.75 You write down 1, the coefficient You break half of 1 You multiply 0.5 and 0.5 You add 0.25 and 0.75 This is the square of 1 Subtract 0.5, which you multiplied 0.5 is the side of the square

Contemporary notation x2 + x = 0.75 Coefficient of x is 1 Half of 1 is 0.5 (0.5)2 = 0.25 0.25 + 0.75 = 1 √1 = 1 1 – 0.5 = 0.5 x = 0.5

This ancient knowledge found a home early in the Arab-Persian world, where it was known as the “science of restoration and balancing.” Indeed, the Arabic word for restoration, al-jabr, is the root of the word algebra. As is well known, it comes from the title of a book written in the ninth century by Mohammed ibn-Musa al-Khwarizmi. It was Descartes, centuries later, who standardized algebraic notational conventions—conventions that we use to this day and which led to his formal unification of algebra and geometry, called coordinate (analytic) geometry. Charles Peirce (1938–1951) often discussed algebraic notation in his work, seeing it as a kind of symbol-based strategy for compressing information (Stjernfelt 2007). To liberally adopt and adapt his views, an equation, such as the Pythagorean one, can be defined as a visual-symbolic portrait of the relations among the variables (originally standing for the sides of the triangle). But, it also tells us that the variables relate to each other in many ways other than geometrically. Expressed in language, we would literally not be able to see the possibilities that the equation presents us. To use Susanne Langer’s (1948) concept of discursive-versus-presentational form, the equation tells us much more than the discursive statement (“the square on the hypotenuse is equal to the sum of the squares on the other two sides”) because it

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literally “presents” the structure inherent in the linguistic version symbolically, fleshing it out as an abstract form. We do not read the equation as consisting of unrelated individual elements, but presentationally, as a totality which is inscribed in the form of the equation itself. Taking any element out of the equation will make it crumble intellectually. To cite Bellos (2010: 123): Replacing words with letters and symbols was more than convenient shorthand. The symbol x may have started as an abbreviation for “unknown quantity,” but once invented, it became a powerful tool for thought. A word or an abbreviation cannot be subjected to mathematical operation in the way that a symbol like x can be. Numbers made counting possible; but letter symbols took mathematics into a domain far beyond language.

Algebra has made exploration in mathematics and science a constant possibility. As Crilly (2011: 104) observes, the “desire to find a formula is a driving force in science and mathematics.” It has also made possible the formulation and derivation of general abstract principles of numbers and mathematical concepts that would have been unthinkable without it. To cite just one example, the Fundamental Theorem of Algebra states, in its simplest form, that an equation of degree n has n roots—a linear equation will have one root, a quadratic equation will have two roots, a cubic equation three, a quartic equation four, and so on: Linear x-5=0 One root : x = f5g

Quadratic x2 - 4 = 0 Two roots : x = fþ2, - 2g

Cubic x3 - 2x2 - 5x þ 6 = 0 Three roots : x = f1, - 2, 3g From this theorem, many features of equations have subsequently been discovered. Some equations do not have real roots, but complex ones, and others, have only complex roots. This theorem was implicit in the work of Descartes and Albert Girard. The first attempt to prove it is traced to French mathematician Jean d’Alembert in 1746. But his proof contained some infelicities, which were corrected by Karl Friedrich Gauss in 1799. Gauss published a complete proof in 1816, building on ideas established previously by Leonhard Euler. Logan and Pruska-Oldenhof (2022) have cogently argued that mathematical notation and writing emerged at around the same time to serve similar kinds of group social functions, and are thus based on the same semiotic tendencies,

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Fig. 3.4 Babylonian numerals. (Wikimedia Commons)

becoming the basis for preserving ideas. Some of the earliest writing shapes so far discovered were unearthed in western Asia, dating from the Neolithic era. They were apparently made by means of elemental clay tokens that were probably used as image-making forms or molds (Schmandt-Besserat 1978, 1992). At around the same time, tally sticks and bones are found, which indicate that writing and mathematical notation share semiotic properties, starting with iconicity as the central cognitive force in ideation (whereby signs resemble their referents or the concepts they encode). It is thus little wonder that the early writing and numeral systems, such as those used by the ancient Sumerian and Babylonian cultures, form a unitary representation system. Called cuneiform, because it consisted of wedge-shaped picture symbols, the Sumerians recorded their representations on clay tablets, making it a very expensive and impracticable means of communication. For this reason, cuneiform was developed, learned, and used primarily by rulers and clerics. As such, it showed that numerals and writing characters were devised with the same kinds of signs. Below are the Babylonian numerals (Fig. 3.4): This system first appeared around 2000 BCE. An obvious function of such systems was to preserve knowledge, and it is this memory-preserving function that has allowed mathematicians to develop new ideas and branches. It is remarkable indeed to consider that symbolism itself generates ideas. As Cellucci (2020: 1397) aptly observes, the interplay between notation and exploration in mathematics is truly remarkable. It is evidence that human consciousness is not only attentive to physical patterns (color, shape, etc.), resulting in iconic representational activities (such as tally marks), and cause and effect patterns (contingent on time and space constraints), resulting in indexical representational activities (including vectorial sign forms such as the number line), but also to abstract pattern in itself.

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Diagrams Proofs in antiquity were carried out initially with geometric diagrams, which provide, to this day, a philological key for grasping the visual-cognitive nature of the mathematical imagination. In this framework, a diagram is anything that shows some structural feature, and includes not only the typical geometric diagrams, but also layouts that suggest a geometric form, such as the diagonal demonstration by Cantor (to be discussed subsequently). Consider the Pythagorean theorem, as a case-inpoint, which, in contemporary terms, is represented algebraically as c2 = a2 + b2. The Pythagoreans did not leave any evidence of a proof, so we can only surmise how they carried it out on the basis of secondary sources. Proof of the theorem appeared also in several important mathematical works in China—The Arithmetic Classic of the Gnomon and the Circular Paths of Heaven, dated to around the third century BCE and the Nine Chapters on the Mathematical Art, around the second or third century CE, but traced back much further in its original form. Historians surmise that the Pythagoreans used a dissection proof, such as the following one, which is marked up with letters here for the sake of clarity and convenience. We first draw a right-angled triangle with sides {a, b} and hypotenuse {c}. Then, we construct a square with length (a + b), the sum of the lengths of the two legs of the triangle. This is equivalent to joining four copies of the triangle together in the way shown below (Fig. 3.5): The diagram shows us literally how to elaborate a proof as follows. The area of the internal square that emerges though the union of the triangles is c2 and the area of the large square is (a + b)2 = (a2 + 2ab + b2). The area of any one of the triangles is ½ab, and since there are four of them, the overall area covered by the four triangles is 4(½ab) = 2ab. If we subtract this from the area of the square, (a2 + 2ab + b2) - 2ab, we get (a2 + b2). This corresponds to the area of the internal square, which is equal to c2. So, by the axiom of equality, c2 = a2 + b2. Diagrams are, at one level, visual compressions of information; at another, their visual form and structure suggest how to carry out a proof. As such, they stimulate the fantasia and the ingegno to literally visualize what to do. In the version of Abu Al-Wafa’s puzzle above the stimulation is of a, different kind of ingenuity, but nonetheless based on the same poetic logic faculties as the above e proof.

Fig. 3.5 Diagrammatic proof of the Pythagorean theorem

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Fig. 3.6 Argand diagram

Diagrams also function to represent novel ideas and discoveries, so as to stabilize them mnemonically. When imaginary numbers were discovered serendipitously, it was not clear how they fit into the extant number system or how they could be represented on the Cartesian plane. This conundrum led to the ingenious invention of a diagram, called the Argand diagram—after Jean-Robert Argand (1806)—which made it possible to show the relation of imaginary numbers to real ones, utilizing previously established notions from coordinate geometry and trigonometry (memoria). Below is one version of the diagram (Fig. 3.6): The diagram allows for a plotting of complex numbers, x + iy, as points in the plane using the x-axis as the real-number axis and y-axis as the imaginary axis. In the diagram above, the circle represents the absolute value (modulus) of a point and the angle θ its complex argument (the angle between the positive real x-axis and the line joining the origin to the complex point). In effect, the diagram gives a visual interpretation to an abstract idea, thus assigning it to the memoria for further use, whereby it becomes a source of new mathematics. It shows, for example, that complex numbers can be added like vectors and can be multiplied in terms of polar coordinates with the product of the two moduli (absolute values). Multiplication by a complex number is a rotation—a discovery that has been incorporated into the theory of complex numbers. As discussed previously, graph theory emerged from a consideration of a practical situation, which Euler turned into a diagram, and which subsequently became the basis of graph theory (Chap. 1). The whole field depends on devising relevant diagrams (such as the network one used to solve the Königsberg Bridges Problem) to portray situations or concepts so as to be able to examine possibilities and meanings hidden within them. Euler (1760–1763, reprinted in 1833) was also the one who introduced the type of diagram that eventually became indispensable to the development of set theory. Below are a few examples of his “circles” as they came to be called (Hammer and Shin 1996, 1998) (Fig. 3.7):

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Fig. 3.7 Euler circles

These consist of simple circular shapes in a two-dimensional plane depicting a set or category. Each one divides the plane into two regions—the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. Separated circles represent disjoint sets, indicating that they have no elements in common (No A is B), whereas intersecting circles show common elements (Some A is B, Some A is not B). A circle completely within the interior of another circle is a subset of it (All A are B). It was John Venn (1881) who refined and expanded Euler’s circle system. The main difference between the two is that Euler diagrams tend to be more simplified, and generally limited to two circles overlapping, while Venn diagrams can have more overlapping circles. For example, the elements of sets A = {1, 2, 5}, B = {1, 6}, and C = {4, 7} would be represented differentially as shown below (Fig. 3.8, see Wikimedia Commons): As Lakoff and Núñez (2000) have argued, such diagrams are semiotic expressions of the image schema of the container (Chap. 4), whereby elements such as numbers (and other symbols) are imagined as objects in a container, which has a circular shape in this case, and which is traced neurologically to a visual “filling-in” mechanism, whereby neural circuitry spreads from outside to inside in a region of the visual field, which is the basis for providing the topological properties of the container schema (Feldman 2006). As Lakoff and Núñez (2000: 33) elaborate, the containment and its correlative boundary concept (such as the division of the plane into insideness and outsideness) are crucial to all of mathematics: Closed sets of points are conceptualized as containers, as are bounded intervals, geometric figures, and so on. The concept of orientation is equally central. It is used in notions like angles, direction of change (tangents to a curve), rotations, and so on. The concepts of containment and orientation are not special to mathematics but are used in thought and language generally. Like any other concepts, these arise only via neural mechanisms in the right kind of neural circuitry. It is of special interest that the neural circuitry we have evolved for other purposes is an inherent part of mathematics, which suggests that embodied mathematics does not exist independently of other embodied concepts used in everyday life. Instead, mathematics makes use of our adaptive capacities—our ability to adapt other cognitive mechanisms for mathematical purposes.

Grasping the importance of diagrams for understanding the mathematical mind, Charles Peirce (1931–1958) developed a powerful theory based on what he called “Existential Graphs.” Like Euler, Peirce saw diagrams as producing the outline of mental ideas showing how the thought process unfolded (Stjernfelt 2007; Queiroz

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Fig. 3.8 Euler versus Venn diagrams

Fig. 3.9 Peircean existential graphs

and Stjernfelt 2011). Existential Graphs thus mirror on paper the very process of thinking in an iconic-schematic way. For example, in the graphs below, Peirce showed how we likely envision the two statements “something good is ugly” and “everything good is ugly” (Fig. 3.9): The difference between the two concepts is shown with a container schema that encloses the statement involving the notion of “everything is good” of which “is ugly” forms a subset; whereas the notion of “something” which is generic, does not involve a containment idea, as such.

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Fig. 3.10 Legendre’s theorem

Diagrams clearly enlist the imagination directly, becoming models after they have been established conventionally within mathematics itself—and hence settling into the mathematical memoria for further usage and exploration, as we saw with the Argand diagram, which embedded ideas within it that led to insights for developing complex number theory. Diagrams have also been used to show relations between dimensions, such as the hypercube. A diagram showing the relation between a triangle on the plane and one on a sphere became the basis for Legendre’s (1805) theorem on spherical triangles, since it literally showed the relation (Fig. 3.10): ABC is a spherical triangle with sides a, b, c. Let A′B′C′ be the planar triangle with the same sides. Then the angles of the spherical triangle exceed the corresponding angles of the planar triangle by approximately one third of the spherical excess (the spherical excess is the amount by which the sum of the three angles exceeds π):

To reiterate one more time, without diagrams, some branches, like graph theory, topology, and n-dimensional geometry, would never have come to mind (literally). Topology concerns itself with determining spatial phenomena such as insideness or outsideness. A point outside a circle cannot be connected to a point inside it by a continuous path in the plane without crossing the circle’s circumference. If the plane is deformed, it may no longer be flat or smooth, and the circle may become a crinkly curve, but it will continue to divide the surface into an inside and an outside. That is its defining structural feature. Topology is diagrammatology. Richeson (2008: 155) puts it as follows: The fruitful dialogue about Eulerian and non-Eulerian polyhedra in the first half of the nineteenth century set the stage for the field that would become topology. These ideas were explored further by others, culminating in Poincaré’s marvelous generalization of Euler’s formula at the end of the nineteenth century.

Interestingly, from topological theory have come insights into knots as models of natural phenomena, showing again the principle that mathematical forms often contain within them hidden (outer-reaching) information. It appears, for example, that the DNA and the universe itself may have topological structure of a certain kind. In effect, the mathematician’s diagrams often become the scientist’s theories. As Poincaré pointed out, he came to his own discoveries by considering the simple

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diagrams of the two Newtonian curve models of orbits: “When one tries to depict the figure formed by these two curves and their infinity of intersections, each of which corresponds to a doubly asymptotic solution, these intersections form a kind of net, web or infinitely tight mesh. One is struck by the complexity of this figure that I am not attempting to draw” (cited in Stewart 2012: 64). In sum, diagrams not only model certain phenomena, but also suggest emergent new meanings and possibilities. Once established, a certain type of diagram enters the memoria from whence it is further utilized for carrying out tasks and for exploring novel ideas. In a way, it is the faculty that subserves and encodes generalizations. When Euler tackled the Königsberg Bridges Problem, he clearly had a larger problem in mind— to determine whether the unitary walk would be possible for any number of bridges, not just the seven in Königsberg. To answer this question, Euler devised the notion of graphs from which he developed a whole branch of mathematics. At first glance, Euler’s approach seemed to be a trivial matter, but, as it turned out, his conclusions led to a significant discovery (graph theory) that continues to have significance in mathematics and computer science. As Paoletti (2011) aptly comments: When reading Euler’s original proof, one discovers a relatively simple and easily understandable work of mathematics; however, it is not the actual proof but the intermediate steps that make this problem famous. Euler’s great innovation was in viewing the Königsberg bridge problem abstractly, by using lines and letters to represent the larger situation of landmasses and bridges. He used capital letters to represent landmasses, and lowercase letters to represent bridges. This was a completely new type of thinking for the time, and in his paper, Euler accidentally sparked a new branch of mathematics called graph theory, where a graph is simply a collection of vertices and edges. Today a path in a graph, which contains each edge of the graph once and only once, is called an Eulerian path, because of this problem. From the time Euler solved this problem to today, graph theory has become an important branch of mathematics, which guides the basis of our thinking about networks.

Looping Structure The accumulation and preservation of established ideas, encoded in notations and diagrammatic practices, allows for the institutionalization of mathematics as a system of knowledge, that is, as a collective system of memoria, which can be utilized for future imaginative work. Every equation, geometric diagram, etc., alludes to some previous mathematical meaning, even if devised for novel purposes, as was the case with the Argand diagram, which extended the use of coordinate geometry to the domain of complex number representation. The process involved can be characterized as one that connects the memoria and fantasia as if they were in a loop, and the ingegno as the connecting link in the process of finding a solution, a procedure, a notation, a proof, or some other outcome (if there is one), as it goes back and forth between the fantasia and memoria. The looping structure of mathematical understanding is consistent with Vico’s cyclical model of the mind as a projection of the past into the present and vice versa (Fig. 3.11):

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Ingegno

Memory

Fantasia

Fig. 3.11 Looping structure of poetic logic

Fig. 3.12 Diagram of Dudeney’s puzzle. (From Dudeney 1907)

The back-and-forth modality is constantly at work, and can even be seen in how we solve problems and puzzles (Fig. 3.12). As a case-in-point, consider a wellknown puzzle composed by British puzzle-maker Henry Dudeney (1907: 92): Inside a rectangular room, measuring 30 feet in length and 12 feet in width and height, a spider is at a point on the middle of one of the end walls, which is 1 foot from the ceiling as at A; and a fly is on the opposite wall, 1 foot from the floor in the centre, as shown at B. What is the shortest distance that the spider must crawl in order to reach the fly, which remains stationary? Of course the spider never drops or uses its web, but crawls fairly.

As we observe the three-dimensional figure, the fantasia might come into play by envisioning different possibilities for the layout, when an insight from the ingegno, based on geometric memoria, suggests that if we “flatten the room,” we would transform the problem into a more manageable one in plane geometry (Fig. 3.13): The room has dimensions 30 × 12 × 12. The spider is in the middle of one of the 12 × 12 walls, 1 foot from the ceiling, while the fly is in the middle of the opposite wall, 1 foot from the floor. The fly remains stationary, so the spider must crawl along the walls, the ceiling, and the floor in order to capture the fly on the hypotenuse of the right triangle shown above. The answer is 40, given that √(242 + 322) = 40. In effect, what we know about two- and three-dimensional geometry and our ability to envision layouts that connect the two dimensions, similar to Flatland, was the likely spark of the insight. Needless to say, there may be other ways to explain the solution, but the one based on poetic logic would still hold as highly plausible.

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Fig. 3.13 Flattened diagram. (Wolfram Mathworld)

Interestingly, this exact type of problem may have even been the source of Descartes’s invention of coordinate geometry. Legend has it that Descartes himself recounted an episode whereby he was watching a fly on the ceiling from his bed, wondering how best to describe the fly’s location and decided that one of the corners of the ceiling could be used as a reference point (Forbes 1977). Whether or not the anecdote is verifiable historiographically, and whether or not Dudeney may have been familiar with the anecdote, the point is that without having experienced the history of geometry directly, we can easily reconstruct it with our familiarity with its principles and structures. In effect, we use our fantasia in similar ways to envision the same problem. Needless to say, the tripartite looping system is also putatively at the cognitive base of discovery, as the discovery of imaginary numbers reveals, which became part of mathematical memoria after they were connected to the real numbers via a simple diagram. So too, the invention of the calculus as a modeling system of change and flux in the world, comes from a long memoria that goes back to Zeno’s paradoxes (to be discussed in Chap. 4). As another example, consider the binomial theorem. The likely hunch behind the theorem is traced to Pascal’s triangle—a triangular arrangement of numbers wherein each number is the sum of two numbers directly above it (except for the edges, which are all “1”). Below is a small section of the triangle (Fig. 3.14): The triangle is named after Blaise Pascal (as is well known) who studied it in his 1655 work, Traité du triangle arithmétique, although the same figure appears centuries before him in India, Persia, China, and various parts of Europe (Coolidge 1949; Kennedy 1958). It was partially Pascal’s work on binomial coefficients, inspired by the triangle, that led to the binomial theorem, or the expansion of (a + b)n:

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Fig. 3.14 Pascal’s triangle

ða þ bÞ0 = 1 ða þ bÞ1 = a þ b ða þ bÞ2 = a2 þ 2ab þ b2 ða þ bÞ3 = a3 þ 3a2 b þ 3ab2 þ b3 ða þ bÞ4 = a4 þ 4a3 b þ 6a2 b2 þ 4ab3 þ b4 ða þ bÞ5 = a5 þ 5a4 b þ 10a3 b2 þ 10a2 b3 þ 5ab4 þ b5 ða þ bÞ6 = a6 þ 6a5 b þ 15a4 b2 þ 20a3 b3 þ 15a2 b4 þ 6ab5 þ b6 ða þ bÞ7 = a7 þ 7a6 b þ 21a5 b2 þ 35a4 b3 þ 35a3 b4 þ 21a2 b5 þ 7ab6 þ b7 ða þ bÞ8 = a8 þ 8a7 b þ 28a6 b2 þ 56a5 b3 þ 70a4 b4 þ 56a3 b5 þ 28a2 b6 þ 8ab7 þ b8

In the top row, the numerical coefficient is 1; in the row below the numerical coefficients of a and b are both 1; in the next row below the numerical coefficients are 1, 2, and 1; and so on. Extracting these coefficients and leaving them in their respective rows the outline of a triangle with the above properties emerges. In effect, the expansion generates a triangular image schema, which suggests a method for computing (a + b)n, since the numbers on each row are binomial coefficients (Kline 1972: 273). So, in the expansion of (a + b)4, which is a4 + 4a3b + 6a2b2 + 4ab3 + b4, the numerical coefficients (1, 4, 6, 4, 1) coincide with the fourth row of numbers in the triangle (counting down from (1, 1) as the first one; similarly, the coefficients in the expansion of (a + b)5 coincide with the fifth row; those in the expansion of (a + b)6 with the sixth row; and so on. In other words, the exponent is an index of row height in the triangle. The triangle has been found to have many other interesting properties and hidden features, including the following: • The numbers on the second diagonal (in either orientation) form the counting numbers (1, 2, 3, 4, 5,. . .); the numbers on the third diagonal (either way) form the triangular numbers (1, 3, 6, 10, 15,. . .); the numbers on the fourth diagonal the tetrahedral numbers (1, 4, 10, 20, 35,. . .); and so on • The sum of the numbers in each row is increasing by powers of two (2n). For example, the sum of the numbers in the first four rows is as follows:

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Fig. 3.15 Pascal’s triangle and the Fibonacci numbers

First row: 1 = 1 = 20 Second row: 1 + 1 = 2 = 21 Third row: 1 + 2 + 1 = 4 = 22 Fourth row: 1 + 3 + 3 + 1 = 8 = 23 Fifth row: 1 + 4 + 6 + 4 + 1 = 16 = 24 Sixth row: 1 + 5 + 10 + 10 + 5 + 1 = 32 = 25 And so on • Diagonal lines in the triangle produce the Fibonacci sequence {1, 1, 2, 3, 5, 8, . . .}, which has the structure Fn = Fn - 1 + Fn - 2 (any number in the sequence is the sum of the previous two numbers) (Fig. 3.15): The formula for the expansion is the following one. It is the final output of the cognitive movement back-and-forth between the imagination and what is known about algebraic expressions (memoria): n

ð a þ bÞ n = k=0

n k

a n - k bk

The Greek letter sigma is the summation notation. The vertical symbols in the parenthesis compute the binomial coefficient, representing the combination formula n!/(n - k)! k!, which is explained with combinatory theory and method using factorial notation. To see how this formula captures the binomial expansion, consider the case when n = 3, and k = 0, 1, 2, 3. 3

3

k=0

k

ða þ bÞ3 = =

3 0

a3 - k bk

a 3 - 0 b0 þ

3 1

a 3 - 1 b1 þ

3 2

a 3 - 2 b2 þ

= 1  a3 b 0 þ 3  a2 b 1 þ 3  a 1 b 2 þ 1  a0 b 3 = a3 þ 3a2 b þ 3ab2 þ b3

3 3

a 3 - 3 b3

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Fig. 3.16 The golden ratio

The foregoing discussion implies that for something to become memoria, imaginative-ingenious experimentation is required, including experimentation that seems, at first, to be nothing more than that—a kind of ludic mathematics in itself but which, as it turns out, can lead to general ideas. Another example can be seen in the history of the so-called golden section or ratio (Dunlap 1997; Livio 2002). The ratio results from a division of a line segment in such a way that the ratio of the whole segment to the larger part is equal to the ratio of the larger part to the smaller part. The ratio, represented with the Greek letter phi (ϕ) is approximately 1.61803 to 1. (Fig. 3.16): As is well known, ϕ is a number mentioned at the beginning of Book VI of Euclid’s Elements. Since antiquity philosophers, artists, and mathematicians have been intrigued by this ratio, which Renaissance writers called the divine proportion, after Luca Pacioli (1509). Remarkably, the ratio of two successive terms in the Fibonacci sequence converges on the golden ratio: for example, 5/8 = .625, 8/13 = .615, 13/21 = .619, and so on. It has also been found in nature, such as in the curve of a nautilus shell. One cannot help but be struck by such serendipities. The point here is that without the notion of ratio as part of ancient mathematical memoria, which allowed for a specific type of cognitive experimentation, the mathematicians would not likely have discovered the golden ratio—hence the looping pattern in mathematical ideation. As one final example, consider the emergence of fractal geometry. This branch is concerned with complex shapes called fractals, which consist of small-scale and large-scale structures that resemble one another. The term fractal was coined by Benoit Mandelbrot, after observing that random fluctuations in nature and in human affairs formed geometrical patterns that could be reduced to smaller elements. Mandelbrot was interested in self-similarity, a property of geometrical figures that have a similar appearance when viewed at different scales. In his paper “How Long is the Coast of Britain?” published in Science magazine in 1967, he discussed the phenomenon of self-similarity in coastlines. In 1975, he coined the term fractal to describe any repeating self-similar pattern. A fractal is thus defined as any form that is altered by the application of a transformational rule to it ad infinitum. Actually, the main ideas undergirding fractal geometry are prefigured in a paper presented by Karl Weierstrass to the Royal Prussian Academy of Sciences in 1872 (Edgar 2004: 7–11). In it, Weierstrass provided the first definition of a function with a graph that would today be considered a fractal, since it was a function that was everywhere continuous but nowhere differentiable. Shortly after, in 1883, Georg Cantor, published

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Fig. 3.17 Sierpinski triangle

examples of subsets of the real line known as Cantor sets, which had properties that are now recognized as fractals (Edgar 2004: 11–24). By the last part of the nineteenth century, Felix Klein and Henri Poincaré started developing the domain of such study considerably, culminating in the work of Mandelbrot (1977). Below is a classic and well-known example of a fractal form, known as the Sierpinski triangle, named after the Polish mathematician Vaclav Sierpinski (Barnsley 1988). As can be seen, it is produced by repeating the triangle form over and over in a patterned way (Fig. 3.17): Fractal shapes were known to the human imagination, long before fractal geometry provided a theory for them. They are found in the ancient world, especially in the arts. A Sierpinski figure crops up in a thirteenth-century pulpit in the Ravello Cathedral in Italy, designed by Nicola di Bartolomeo of Foggia. In Mahayana Buddhism, the fractal nature of reality is captured in the Avatamsaka Sutra by the god Indra’s net, a vast network of precious gems hanging over Indra’s palace, arranged in such a way that all the gems are reflected in each one. In more recent times, artists Dalí, Pollock, and Escher have exploited the fractal technique of creating a new shape out of repeated copies of another to great aesthetic effect (Barnsley 1988). In sum, the binomial theorem, the golden ratio, and fractals, among an infinitude of inventions and discoveries, start with the fantasia, which impels mathematicians to experiment with constructs such as figures and diagrams with their ingenuity (ingegno). When these lead to general ideas, they become part of the architecture of mathematics (memoria). Without this looping (cyclical) structure, the mathematical mind would hardly have “thought” what it has thought historically, leaving mathematics at the level of instinctive counting and tallying. The foregoing discussion implies that the imagination depends critically on memory, which provides the building blocks for things to be imagined. Moreover, it suggests that memory may itself be (actually) a part of imagination. Significantly, this process has been corroborated by neuroimaging studies, which have come forth to provide an empirical basis to the Vichian model of the mind. They show, essentially, that memories are imaginative reconstructions of past events. As Gaesser (2013) has aptly observed, in reviewing the studies, there is strong “evidence of a relationship between memory, imagination, and empathy. . .[showing that] imagination influences the perceived and actual likelihood an event occurs. . .and shares a neural basis with memory and empathy.”

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Epilogue The invention of exponential notation, as mentioned at the start of this chapter, is one of myriad examples of how symbolism subserves the mathematical memoria, compressing information for further utilization. During Descartes’s era, exponents presented the possibility of novel meanings of numbers, becoming intrinsic to algebra, coordinate geometry, and number theory, among many other branches. But ancient mathematicians were aware of the concept itself; even though they did not assign any symbolic status to it, relating it typically to geometric shapes. For instance, the Pythagorean theorem, which in modern notation involves the use of the exponent “2,” c2 = a2 + c2, the relation of the three integers, {a, b, c}, was envisioned as related to the lengths of the sides of geometric squares—a name that this particular exponent bears to this day. The Babylonians had actually discovered Pythagorean triples, before the theorem was introduced into mathematics much later, as corroborated by one of the most famous tablets of mathematics ever discovered—Plimpton 322—believed to have been created about 1800 BCE. The tablet shows a table of four columns and fifteen rows of numbers written in the cuneiform script of the period. It was Neugebauer and Sachs who noticed in 1945 that in each row, the square of the number in the third column, minus the square of the number in the second column, is itself a square number, reflecting the Pythagorean theorem—{3, 4, 5}, {5, 12, 13}, {6, 8, 10}, {9, 12, 15}, and so on (Fig. 3.18): Plimpton 322 was likely used as a memory device, for carrying out practical matters related to measurement. But it also enfolded an exponential relation of the numbers that became the basis for many subsequent discoveries in mathematics. The tablet simply lacked a symbolic encoding method, which came much later. As Robert Smith (2017) has emphasized, such artifacts are memory records of how the mathematical imagination seeks to uncover core concepts (Smith 2017): Fig. 3.18 Plimpton 322. (Wikimedia Commons)

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Human thought could be recorded, shared among generations, and thus concepts and ideas could develop and evolve. The Babylonians, at the start of human civilization, harnessed this power as much as possible: they were not only avid record keepers, but they developed lists in order to keep track of acreages of wheat and quantities of livestock. They also kept track of taxes and legal disputes. And from the basis of these very humble foundations, using soft clay tablets, early ideas in mathematics could be pursued.

All symbolic forms, imagined and invented, become fixed properties of the mind through the workings of the memoria, which makes them available constantly for inventive and creative uses, as well as for reasons of envisioning how to perform proofs logically (Chap. 2). The need for symbols in mathematics goes back to its origins as a distinct discipline. For instance, in the Rhind Papyrus, which is traced to Egypt a few centuries before 1650 BCE, the year in which it was transcribed (Gillings 1972: 246–247), all kinds of notational techniques are found, confirming that mathematics had evolved into a symbolic language all its own around 5000 years ago. Without this language, we would have no means of accessing Egyptian mathematics of that era, and reconstructing how they viewed the world via mathematics. Interestingly, the papyrus even had notation for representing fractions, which may be one of the first times in history that this was done. Multiplications and divisions were carried out by a succession of doubling operations, based on the fact that any number can be represented as a sum of powers of 2, thus anticipating the notion of exponentiation millennia before it became intrinsic to mathematics. Symbols allow us to think about their referents away from their original contexts of occurrence. They constitute an inner modeling system that envisions mathematical concepts as access points into abstractions. This system was called the Bauplan by the biologist Jakob von Uexküll (1909) at the turn of the twentieth century, establishing a point of contact between the mainstream scientific approach to the study of organisms, biology, and semiotics, the science of signs and symbols. For Uexküll organisms lived different inward and outward realities. Animals with widely divergent modeling systems do not live in the same kind of experiential world as other animals. Each species possesses a particular kind of Bauplan—the ability to grasp the world in a biologically determined way, selecting from it what is essential for survival. The human Bauplan, however, adds interpretation to the selection; that is, it impels humans to seek meaning in the information beyond its context of occurrence. It is this initial meaning that a symbol is designed to preserve, but which can be reimagined and interpreted subsequently in different ways. The implies that the human brain is highly autopoietic—a term introduced by Maturana and Varela in their famous 1973 book, Autopoiesis and Cognition, where they claimed that an organism participates in its own evolution, since it has the ability to produce, or at least shape, its various biochemical agents and structures, thus ensuring their efficient and economical operation. In the case of the human mind, autopoiesis seems to know no bounds. Mathematical ideas and the symbolic systems of thought that encode them are autopoietic artifacts. To exemplify the difference between the human and nonhuman Bauplan, consider a well-known episode in mathematical history, which concerns the Alexandrian geometer Pappus, who was apparently contemplating the following problem:

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What is the most efficient way to tile a floor? There are three ways to do so with regular polygons—with equilateral triangles, equal four-sided figures, or regular hexagons (Flood and Wilson 2011: 36). The hexagonal pattern has the most area coverage. Now, Pappus remarked that bees used the hexagon pattern for their honeycombs, having the ability for holding more honey than other geometric shapes. He saw this as a truly intriguing fact, which continues to raise a whole series of questions related to the human and apiary Bauplans. Clearly, the internal modeling system of bees is well adapted to acting upon the external world in a specific way— honeycomb construction. But this remains instinctive knowledge, since bees do not have the evolutionary prerequisites for developing their instincts into geometry (Banks 1999: 19). The anecdote shows that each species is tuned into the structure of the world according to its own particular Bauplan. In the case of humans, the ability to convert this structure into abstract symbolism makes the species unique. This propensity to model something in the abstract is, in fact, the basis of human knowledge, and may even form a kind of unique “cognitive instinct” so to speak (Sebeok and Danesi 2000). This instinct has even impelled humans to devise machines based on symbol manipulation, so as to facilitate computation and other tasks (discussed further in Chap. 5). The mathematical programmer starts with a simple question: How can we best model phenomenon X? Instructions are then devised to produce the required output. Significantly, this mechanical process has produced unexpected discoveries. A well-known anecdote bears this out. In 1656, John Wallis showed that π can be calculated in terms of an infinite series consisting of the product of consecutive terms with the following structure: π=2 = ð2=1 × 2=3Þ × ð4=3 × 4=5Þ × ð6=5 × 6=7Þ . . . Remarkably, mathematician Tamar Friedmann and physicist Carl Hagen found this formula unexpectedly in the structure of the hydrogen atom. The discovery was made in 2015, while Hagen was teaching a class in quantum mechanics on the variation principle—used to approximate the energy states of a hydrogen atom. As he was illustrating the conventional calculations, he noticed an unexpected pattern in the ratios. So, he asked Friedmann to help him figure out what was going on. Together, they identified the pattern as a manifestation of the Wallis formula for π, which was the first time it had appeared in quantum physics. Since 1656, when Wallis proposed it, there had been many proofs of the formula, but all had come from the world of mathematics. The fact that it emerged in the field of quantum mechanics of the hydrogen atom is astonishing. Schmidhuber (2010) has argued that is it perhaps best to think of mathematical creativity as the result from any optimizing performance, which would explain how a computer might hit upon a pattern that it comes across during optimization procedures. But, as the above anecdote illustrates, a computer would have remained unaware of the significance of its discovery; it takes humans to realize it. It is suggestive to note that Google has developed its own AI theorem-proving program, which can prove, essentially unaided by humans, many basic theorems of mathematics. In 2019, two members of Facebook’s AI research group, Guillaume Lample

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and François Charton, even developed a neural network system capable of solving symbolic mathematical problems (a topic discussed in Chap. 5). Despite such truly remarkable feats, automated mathematics does not have a fantasia, which is based on the body converting experience into inner visions. Artificial systems are programmed with the relevant image schemas and then allowed to explore all possibilities with their organization into vast networks. From this exploratory navigation, they will indeed come up with innovative ideas. But if asked why they are significant, then there is no program that can be devised to provide the answer in human terms. When mathematicians talk about something coming into existence as the result of some proof or serendipitous discovery, they are talking about something that only they can truly interpret and understand. As Ian Stewart (2013: 313) observes, the problem of existence is hardly a trivial one: The deep question here is the meaning of “exist” in mathematics. In the real world, something exists if you can observe it, or, failing that, infer its necessary presence from things that can be observed. We know that gravity exists because we can observe its effects, even though no one can see gravity. However, the number two is not like that. It is not a thing, but a conceptual construct.

As Black (1962) pointed out at the start of AI, the idea of trying to discover how a computer has been programmed in order to extrapolate how the mathematical mind works was bound to become a guiding principle in both mathematics and AI research. But there is a caveat here, expressed best by physicist Roger Penrose (1989), who argued that computers can never truly be creative in the human sense, because the laws of nature will not allow it. Nevertheless, computer modeling can serve many functions, from outlining the structure of a problem to indicating what processes are involved in solving it, as will be discussed in Chap. 5. Vico pointed out throughout his writings that the mind cannot be studied objectively as a separate entity, as can a physical object or a natural phenomenon. It cannot be taken out of the body for observation or inspection. It results from the interactions between the body and brain. It is, in other words, an epiphenomenal by-product of physiological and neural activities working in tandem. However, the essential nature of mind could be unraveled by reconstructing its “modifications” (his term for “creations”), that is, by examining the uniquely human artifacts that it has allowed humans to produce. Vico saw the imagination as the crux to understanding human mentality, and the memoria as a kind of reconstructive imagination that provides the basis for further exploration and invention. As such, the human brain presents a paradox—it is both a biological product of the physical world and an organ reflecting on it, creating symbolic artifacts that allow it to understand reality on its own terms—the verum-factum principle. As Michael Frayn (2006) has remarked, with his own version of this principle: It’s this simple paradox. The Universe is very old and very large. Humankind, by comparison, is only a tiny disturbance in one small corner of it—and a very recent one. Yet the Universe is only very large and very old because we are here to say it is. And yet, of course, we all know perfectly well that it is what it is whether we are here or not.

Chapter 4

Metaphor

The greatest thing by far is to have a command of metaphor. This alone cannot be imparted by another; it is the mark of genius, for to make good metaphors implies an eye for resemblances. Aristotle (384–322 BCE)

Prologue As suggested in Chap. 1, the origins of conscious mathematical ideas can be plausibly traced to the fantasia taking in information and transforming it into inner visions that become the basis for ingenious mental activities (the ingegno), which, in turn, lead to the invention and discovery of novel ideas and methods, after which they are incorporated into the memoria in the form of symbolic artifacts that can be used subsequently for further creative mind work. This tripartite model is encapsulated in the Vichian notion of poetic logic, as discussed throughout the previous three chapters. The question now becomes: Once an idea is formed through poetic logic, what actual (imagistic) shape does it take in the mind? Specifically: What does a mathematical idea look like in the imagination? This very question was addressed cogently by Lakoff and Núñez (2000) in a way that clearly echoed Vico—namely, as metaphor. The purpose of this chapter is to discuss the role of the Vichian metafora, which he developed it in his New Science, and to map it against the work of Lakoff and Núñez and related research in the cognitive science of mathematics. Vico saw metaphor as the master trope, the primary indicator of how thoughts are formed, with metonymy, synecdoche, and irony subserving cognition subsequently to metaphor. His “theory of mind,” as such, was a rhetorical one, which, as with many of his other notions, can be traced to antiquity. As Fernyhough (2006) has pointed out, in fact, one of the oldest rhetorical theories of mind was put forth by Plato on the basis of what is now called the container metaphor, whereby thoughts are perceived to be objects that are stored in the brain (the container), at greater or lesser levels of accessibility. Metaphor has consistently undergirded theories of mind, as Vico claimed, because the only access to the mind is to observe how

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Danesi, Poetic Logic and the Origins of the Mathematical Imagination, Mathematics in Mind, https://doi.org/10.1007/978-3-031-31582-4_4

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thoughts are formed metaphorically. Philosophers and psychologists have consistently used metaphors as the basis of their theories. Thomas Hobbes (1651), for instance, used the metaphor of a stream, whereby thoughts are described as flowing in a stream—a metaphor that became the basis (either coincidentally or derivatively) for William James’s (1890), stream of consciousness theory—a psychological metaphor that is traced even before James to Alexander Bain in his 1855 book, The Senses and the Intellect. For Vico, as mentioned, metaphor was the primary way we can actually understand the mind, since it reveals how we form ideas. He also developed a theory of history divided into four stages based on evolutionary rhetorical forms of thought inscribed in the language spoken in each stage—from metaphor to metonymy and synecdoche and finally irony (White 1973, 1978). For the present purposes, the main aspect of the Vichian model of specific interest is that mathematical ideas manifest metaphorical structure in their origins. Although they make no reference to Vico, Lakoff and Núñez (2000) put forth a proposal that is substantively Vichian, whereby they see mathematical ideas as emanating from the same neural processes that undergird verbal metaphor. The relevant research on their own model has actually produced intriguing findings and insights that have shed light, putatively, on what might happen in the brain as people do, learn, and discover mathematics (Danesi 2016). Critiques have, needless to say, emerged to challenge this model; but it remains a viable one to this day, claiming in a nutshell that mathematics, like language, emerges and develops from every day mechanisms of experience that assume metaphorical shape in the mind. In the Lakoff-Núñez approach, metaphor is not a mere figure of speech, but rather the result of a cognitive process of creative association that manifests itself in verbal and nonverbal ways. The historical record in mathematics actually appears to support this claim (Schlimm 2013). A classic example is the origination of the concept of the number line, which was suggested by a metaphor, as its inventor, John Wallis, points out in his Treatise of Algebra (1685: 265) in which he describes addition and subtraction as akin to someone walking forward and backward on a linear path. The linear path, as discussed in previous chapters, is an example of an image schema, which is the source of the line metaphor. Interestingly, Immanuel Kant (1781: 278) defined mathematical thinking as the process of “combining and comparing given concepts of magnitudes, which are clear and certain, with a view to establishing what can be inferred from them.” The inference becomes explicit through the “visible signs” that are used to highlight the structural detail inherent in the comparison—that is, through the metaphorical forms that undergird artifacts such as number lines. Charles Peirce (1882) extended the Kantian view by suggesting that metaphorical thought unfolds in the form of what he called Existential Graphs, which are outlines of thought that are akin to the image schemas described by Lakoff and Núñez.

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The Vichian Metafora Vico’s notion of metaphor (metafora) as a primary output of poetic logic, allowing us to make sense of things, constitutes what a mathematical idea might look like as it is formed in the mind, either as a novel idea or as an exploratory guide for doing mathematics. Although the Vichian notion of metafora has never been applied to mathematics, as far as can be told, Lakoff and Núñez’s view that mathematical concepts stem from conceptual metaphors certainly resonates with the Vichian approach. As the two cognitive scientists emphasize, metaphor arises from the brain’s tendency to map experiences against each other, sensing verisimilitude within them. In this framework, mathematics is an offshoot of the same conceptual mapping system that leads to the creation of language and other human skills and faculties. Prefiguring the work of Lakoff and Núñez, Vico also saw metaphor as a strategy for grasping abstractions on the basis of sensory, perceptual, and physical reactions to things in the world. It results from an association of sense between what is unknown and what is familiar: “It is another property of the human mind that whenever men can form no idea of distant and unknown things, they judge them by what is familiar and at hand” (Bergin and Fisch 1984: 122). The example Vico gave to illustrate what he meant was naming the base of a mountain its foot, given that it is a lower extremity like the bodily foot. By mapping biology onto nature, humans are engaging in metaphorical cognition, which allows them to connect the facts of reality in a human-based way. This type of mapping is at the core of mathematical reasoning. The number line concept, for instance, is derived from mapping the physical characteristics that a path entails and the possibilities that it allows us to do—moving forward and backward—onto a layout of numbers, suggesting that one can move conceptually in a similar way on the line, forward and backward, so that rightoriented motion involves gaining territory, and left-oriented motion involves losing territory. This then becomes the source of negative numbers as concepts. The association between paths and number lines is hardly an act of comparison—it is the result of imagining one as equivalent conceptually to the other. Vico thus saw metafora as giving abstract thought an understandable concrete form (in Bergin and Fisch 1984: 116): All the first tropes are corollaries of this poetic logic. The most luminous and therefore the most necessary and frequent is metaphor. It is most praised when it gives sense and passion to insensate things, in accordance with the metaphysics above discussed, by which the first poets attributed to bodies the being of animate substances, with capacities measured by their own, namely sense and passion, and in this way made fables of them. Thus every metaphor so formed is a fable in brief. This gives a basis for judging the time when metaphors made their appearance in the languages. All the metaphors conveyed by likenesses taken from bodies to signify the operations of abstract minds must date from times when philosophies were taking shape. The proof of this is that in every language the terms needed for the refined arts and recondite sciences are of rustic origin.

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The two main points are as follows: (1) metaphors are compact “fables” of ideas and (2) that they are imaginative projections of human sense-making that constituted the first “operations of abstract minds.” This is a remarkable statement given that, as claimed by various contemporary cognitive scientists, abstract concepts frequently have metaphorical structure, constituting “fables” constructed by humans themselves. The view of metaphor as a “fable-making” capacity generated by poetic logic makes it the cognitive source of early myths, abstract concepts, and even names for things (Verene 1981: 60–100). For example, Vico claimed that Jove was the name created by early humans as they became aware of and feared the thundering sky, imagining a god to be the “thunder-maker” (in Bergin and Fisch 1984: 374–384). Once the word Jove came into existence, all other experiences of the same phenomenon—the fear associated with the thunder god—were “found again” in this name. Jove was thus a metaphor that came to stand for all kinds of abstractions, including the sky, our sense of metaphysical reality, the human world, and so on. From such metaphorical names, the first conscious humans learned to make sense together and communicate meaningfully beyond the instincts. Research in various branches of cognitive science has come forth to confirm Vico’s basic blueprint for the source of metaphorical ideas, as a creative mapping of experience onto abstraction. Metaphor has been found to play a key role in the childhood learning of mathematical ideas and operations, recalling Vico’s idea that metaphor is the vehicle through which our first thoughts are formed and through which we learn about the world (Semino and Demjén 2020). Elementary school teachers, for instance, use the image schema of the container to teach notions of quantity and operations such as addition. The operation of adding, say, two numbers, such as 2 and 3, is imparted experientially by exposing children to two boxes containing respectively 2 and 3 balls and then putting them together in a third box. Asking the child how many balls there are together in the third box is tantamount to eliciting from them their background knowledge of containers (Bloomfield and Rips 2003). In current conceptual metaphor theory, as it is called (Lakoff and Núñez 2000), poetic logic can be interpreted as the neural mechanism that converts experience into abstraction via metaphor, allowing, as Verene (1981: 101) has put it, “to know from the inside” by extending “what is made to appear from sensation beyond the unit of its appearance and to have it enter into connection with all else that is made by the mind from sensation.” The example of using the container schema to teach elementary arithmetic implies that poetic logic is a rather insightful term for the neural mechanisms involved, allowing us to construct our abstractions on the basis of our sense of things. Vico saw metaphor as the evolutionary source of language and consciousness, elaborating a cyclical theory of historical evolution whereby a society (and the individual mind) progresses through three stages—the “divine,” the “heroic,” and the age of “equals.” He portrayed each one as manifesting its own particular kind of language and mind-frame. So, in the “age of the gods,” the mind grasps reality via myths, which are based on root metaphors; in the subsequent “age of heroes,” people

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Fig. 4.1 Vico’s stages

Now Fig. 4.2 Placement of “Now” on a line

come to understand truths via legends revolving around certain powerful figures, or cultural heroes. The language used is metonymic, with heroic themes becoming symbols for cultural values. Finally, in the “age of equals,” history is devised as rational and sequential, based on prose forms of language (Fig. 4.1): This three-stage development is recapitulated in condensed chronological stages during childhood—constituting one of the first formulations of the theory on record of ontogeny recapitulating phylogeny, which gained ground in the nineteenth century. So, the earliest stage of understanding depends on metaphor, and cannot be avoided as such; only subsequently do other aspects of understanding come into play, including rationality (Fletcher 1991: 149). As Rudolf Arnheim (1969: 233) has observed, in line with the notion of poetic logic, the categories of thought have experiential and perceptual origins. In effect, the power of poetic logic lies in the fact that it allows people to come up with a common understanding of similar concepts via metaphor (Núñez 2017; Danesi 2017). Research based on conceptual metaphor theory, which started with Lakoff (e.g., Lakoff 1979; Lakoff and Johnson 1980), has made it obvious that metaphor is not an option, but often the only way that something can be understood, especially when novel ideas are presented, as in the case of elementary school mathematics. Moreover, as Vico understood, once a metaphor is introduced into life as a thought pattern or habit, it colors and affects cognition from that point onward. Let us look more closely at the concept of the number line, since it is clearly related to a writing system in which the words are laid out from left to right. Users of such a system tend to perceive past time as a “left-oriented” phenomenon and future time as a “right-oriented” one. Consider the line below which has a point on it labeled Now (Fig. 4.2): Where would a speaker of English likely put the verbal labels Before and After on that line? The speaker would almost certainly put Before to the left and After to the right, because English writing flows from the left side of a page to the right. So, something that has been written “before” is left-based in the visual space and something that is written “after” is right-based in the same visual space (Fig. 4.3):

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Aer

Fig. 4.3 “Before” and “After” on the line

Those who are accustomed to writing and reading from right to left will tend to reverse the order of the labels, for the same reason. The allocation of the three words on the line also reflects English verb tenses as being marked for present, past, and future. These too are perceived to lie on a timeline, with past placed before (to the left of) the present, and future placed after (to the right). This kind of conceptual linearity dovetails with the conceptualization of the number line. It is thus no coincidence that the one who first suggested the number line was a speaker of English, arguably mapping his perception of visual space as based on a left-to-right movement onto the linear structure of numbers. Now, once introduced into mathematics, that same metaphorical construct becomes the basis for understanding numbers and arithmetic in new ways, via new metaphors. If a certain number is farther to the right on the number line than is another number, then the first number is greater than the second (or the second is less than the first). So, the concept of greater-versus-lesser is now formed in the mind as an image schema of distance between the numbers, not just quantity, as in the container schema. It now follows that the distance between two numbers is the magnitude of their difference. The operations can now also be described in terms of movements forward and backward along the line and in terms of distances between numbers. Significantly, the metaphorical chain of reasoning becomes the source of invention and discovery, as well as new ways to represent numbers, including the following: 1. Logarithmic scale: the representation of numbers, such that the distance of two points on a line is represented by multiplying the previous value by the same amount. Logarithmic scales are used for representing many physical phenomena. 2. Number line combinations: a line drawn through the zero point at right angles to the real number line, which can be used to generate the Cartesian plane and to represent the imaginary numbers in the same plane. 3. Linear continua: the number line is a continuum, whereby the numbers between two particular units (integers, fractions, irrationals, etc.) constitute an uninterrupted continuum under the same distance metaphor (forward and backward). This has led to the formulation of various theorems based on the continuum concept, including the notion of differentiable infinities. 4. Metric space: the real number line forms a metric space, with the distance function given by absolute difference, which has also been the source of theorems and real-world applications. 5. Topological space: as a topological space, the real number line is homeomorphic to the open interval (0, 1); that is, the real number line is topologically pathconnected, though it can be disconnected by removing any one point.

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6. Vector space: the number line provides a vector space over the field of real numbers (i.e., over itself). In other words, it provides a set of elements, called vectors, which can be added or multiplied by scalars. 7. Real algebra: is concerned with the real numbers on the line within algebra. It is truly remarkable to consider how a seemingly simple metaphorical mapping—the number line as based on the experience of paths—became the source of so many novel ideas, branches of mathematics, and new systems of numerical representation. This gives credence to Vico’s idea that at the source of all ideation is initially poetic logic, which produces metaphorical mind-worlds that keep on suggesting extensions and expansions of ideas in themselves. The even more remarkable aspect is that these metaphorical constructs have had applications to all kinds of sciences, from physics to computer science. Explaining the connection between mind-worlds and real-worlds remains an enigma nonetheless (discussed further in the next chapter). There are a few caveats that must be taken into account vis-à-vis this whole line of explication. First, whether or not all abstract mathematical concepts are formed metaphorically is an open question. When we generalize adjectives into nouns— great → greatness, beautiful → beauty—the abstractive process may not easily fit into the conceptual metaphor framework, although it could be argued that these too are mappings from a concrete qualitative (adjectival) domain onto an abstract (nominal) domain. But this might be a somewhat artificial explanation. Despite such cautions, the current research within conceptual metaphor theory has made it obvious that the Vichian metafora is a central feature of mathematical ideation.

Conceptual Metaphors in Mathematics Conceptual metaphors are “cognitive formulas,” so to speak, that have crystallized from the repeated mappings of a class of related experiences (called the source domain) onto a specific abstract topic (called the target domain). So, for instance, if different geometrical figures and relations (points, lines, etc.) are mapped onto the target domain of ideas, a conceptual metaphor crystallizes: ideas are geometrical figures and relations—“Those ideas are circular;” “I don’t see the point of your idea;” “Her ideas are central to the discussion;” etc. When we come across an expression such as “Her ideas are central,” which is called a specific linguistic metaphor, we can trace it to a mapping of the source domain of geometry onto the target domain of ideas, which is the conceptual metaphor. The mappings themselves are guided by image schemas, which are produced in the visual regions of the brain where sensory experiences of locations, movements, shapes, substances, etc., are located—containers, paths, distances, etc. In mathematical and scientific representational practices, the meaning of a verticality image schema can be gleaned, for instance, from the ways in which forms in coordinate geometric diagrams are graphed—lines that are oriented in an upward direction indicate a growth or an

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increase of some kind, while those that are slanted in a downward direction indicate a decline or decrease. Conceptual metaphor theory is prefigured in Vico, as briefly suggested above (for example, in Bergin and Fisch 1984: 156), especially when he defines metafora as the model of reality “drawn from natural objects according to their natural properties or sensible effects.” This essentially describes the general notion of an image schema as based on the “natural properties” of objects. There are other ways of “knowing,” of course. But these come subsequent to metafora, as discussed, including metonymy, which is a part-whole, whole-part form of thinking, and can be seen in mathematical induction, as well as in cases where a set of procedures implies a general pattern. But metaphor is the form of thought, which results from the inner vision of the fantasia, rendering visible within the mind those things we can never see with our eye— atoms, sound waves, gravitational forces, magnetic fields, etc., K. C. Cole (1984: 156) puts it as follows: The words we use are metaphors; they are models fashioned from familiar ingredients and nurtured with the help of fertile imaginations. “When a physicist says an electron is like a particle”, writes physics professor Douglas Giancoli, “he is making a metaphorical comparison like the poet who says “love is like a rose”. In both images a concrete object, a rose or a particle, is used to illuminate an abstract idea, love or electron.

As Robert Jones (1982: 4) has also pointed out, for the scientist, metaphor serves as “an evocation of the inner connection among things.” Interestingly, Fernand Hallyn (1990) has identified the goal of science in a Vichian fashion as that of giving the world a “poetic structure.” Overall, conceptual metaphors are the basis of specific abstractions, such as the number line, which is the result of mapping the image schema of a path onto a conceptualization of numbers as part of a linear layout. As we saw, this then becomes the source of further metaphors, suggesting that complex thinking in mathematics is the product of layers of metaphorical mappings. Consider the Cartesian plane. Conceptually, it is the result of imagining what would happen if two number lines were to cross at right angles at the zero point, constituting a new metaphorical layer of thought. The result is four quadrants where points can be located as coordinates. From this, we can now name the parts of the plane, starting with the x-axis and the y-axis and the origin. Now, with this mind experiment (Descartes 1637), a new way of describing geometric figures algebraically has crystallized, allowing mathematicians to build an entire branch of mathematics— coordinate geometry (Fig. 4.4): The coordinate system allows us to describe geometric figures with equations, thus uniting geometry and algebra. For example, by plotting the points described by the equation x2 + y2 = 4, we get a circle whose diameter is four units—two to the left of the origin and two to the right (Fig. 4.5): As Lakoff and Núñez (2000: 38–39) maintain, all this was made possible by extending the path image schema involved in conceptualizing the number line into a source-path-goal schema, which specifies that the metaphorical concept involves physical movement from place to place, consisting of a starting point, a goal, and a series of intermediate points:

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Fig. 4.4 The Cartesian plane

Fig. 4.5 A circle in the Cartesian plane

The Source-Path-Goal schema is ubiquitous in mathematical thought. The very notion of a directed graph, for example, is an instance of the Source-Path-Goal schema. Functions in the Cartesian plane are often conceptualized in terms of motion along a path—as when a function is described as “going up,” “reaching” a maximum, and “going down” again.

As Lakoff and Núñez (2000: 43–44) go on to explain, spatial sources (“from”), goals (“toward”) and paths intermediate between them (“along,” “through,” “across”) do not occur as isolated notions, but rather, as part of the source-pathgoal schema, which implies the following:

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Fig. 4.6 Some members of the source-path-goal family. (Wikimedia Commons, from Hedblom 2020)

1. 2. 3. 4. 5. 6. 7. 8.

A trajector that moves A source location (the starting point) A goal, or the intended destination of the trajectory A route from the source to the goal The actual trajectory of motion The position of the trajector at a given time The direction of the trajector at that time The actual final location of the trajector, which may or may not be the intended destination

It is relevant to note that this very schema has been productive in the area of computer science, suggesting that conceptual metaphor theory is hardly just speculation (e.g., Hedblom et al. 2017; Hedblom 2020). Hedblom (2020) breaks down the source-path-goal schema for computer science as follows, which is worth reproducing here, since it shows the various components of the schema in a comprehensive way, as it involves conceptual paths and loops (Fig. 4.6): Without going into details here, the theory of image schemas, as adopted in artificial intelligence and cognitive robotics to help solve issues with natural language comprehension, provides support for the plausibility of the theory.

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Q (a, b)

Secant line Dy P (x, y)

Dx Fig. 4.7 Representing the derivative

The chain of image schemas that a diagram, such as the one constructed to produce the Cartesian plane, becomes the source of further exploration and insight, allowing the ingegno to contemplate different ways to use the new diagrammatic systems. One of these is the calculus, which would not have been possible, arguably, without the coordinate system, at least in its formal version. For instance, the notion of the derivative of a function can be represented as the slope made by a tangent at a point on a curve plotted in the system. Consider the curve below. The derivative is calculated by choosing two points on the curve, P(x, y) and Q(a, b), and connecting them with a secant line (Fig. 4.7): We can now determine the slope of the secant line PQ as follows—the slope is the increase in its rise, denoted by Δy, over the increase in its run, denoted by Δx. We can now make the following substitutions, according to the diagram: • Since P = (x, y) and Q = (a, b), and since a y-coordinate defines a function in terms of the values of the x-coordinate, P = (x, y) = (x, f(x)) and Q = (a, b) = (a, f(a)). • The difference between the x-coordinate of P(= x) and the x-coordinate of Q(= a) is Δx, so a - x = Δx, and thus a = x + Δx. • Since the y-coordinate of Q is f(a), and a = x + Δx, it can be represented as f(x + Δx). • Now, the coordinates of point Q can be represented as follows: Q(x + Δx, f(x + Δx)). • Thus, the difference between the y-coordinate of Q(= f(x + Δx)) and P(= f(x)), is f(x + Δx) - f(x), which equals Δy. The slope of the secant line is the rise/run: Δy/Δx. Substituting, we get f ðx þ ΔxÞ - f ðxÞ Δx

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Now, let us consider the interval Δx. The smaller we make this interval, the more it approaches the slope of the secant line—that is, as Δx decreases, the secant line approaches the tangent. At 0, it would become the tangent. This is represented as the limit of Δx as it approaches 0: lim

Δx → 0

So, by adding this to the formula above, we get the formula for the slope of the tangent‚ or the derivative: f ðx þ ΔxÞ - f ðxÞ df : = lim Δx dx Δx → 0 If we carry out differentiation on various curves, such as 5x2 or 7x3, a pattern emerges that can be symbolized as follows: xn = nxn-1. So, df/dx of 5x2 = 2(5) x1 = 10x, and of 7x3 = 3 (7) x2 = 21x2. Now, none of this would have been literally “thinkable” without the coordinate system as a constructed image schema, which, when blended with the image schema of limits—to be discussed below—has led to the calculus. As Davis (2015: 15) has observed, because of this simple diagrammatic model, Descartes “brought together arithmetic, analysis, geometry, and logic through the masterstroke of linking number and shape through a coordinate system.” As Lakoff and Núñez 2000: 44) comment: What will eventually become an abstract concept is formed as a metaphor; further concepts are derived from the primary metaphors and by other tropical processes. The first abstract concepts were formed in an identical way—it is in origination concepts that the same kinds of image schemas are found archeologically, such as the path schema. Each metaphorical layer carries inferential structure systematically from source domains to target domains— systematic structure that gets lost in the layers unless they are revealed by detailed metaphorical analysis.

As Vico understood long before the current age of cognitive mathematics, once a metaphorical layer is formed, it becomes the basis for ingenuity, activating the poetic logic looping structure (Chap. 3). Only when something is unfamiliar, do we resort once again to our corporeal imagination (in Bergin and Fisch 1984: 140): It is noteworthy that in all languages the greater part of the expressions relating to inanimate things are formed by metaphor from the human body and its parts and from the human senses and passions. . .All of which is a consequence of our axiom. . .that man in his ignorance makes himself the rule of the universe, for in the examples cited he has made of himself an entire world. So that, as rational metaphysics teaches that man becomes all things by understanding them. . .this imaginative metaphysics shows that man becomes all things by not understanding them. . .and perhaps the latter proposition is truer than the former, for when man understands he extends his mind and takes in the things, but when he does not understand he makes the things out of himself and becomes them by transforming himself into them.

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The study of metaphor has shown that its forms are iconic—that is, they are based on resemblance. The number line resembles a path in the imagination, and is thus an iconic metaphorical product of the fantasia. As the psychologist Fincher (1976: 25) states, this may be traced to “the brain’s ability to manufacture mental images was originally more important than its ability to produce language.” And this may be the reason why iconic pedagogy, such as using containers to teach arithmetic (above), is so widespread in elementary education (Presmeg 2005; Yee 2017). By putting objects into containers of varying sizes (larger versus smaller), and a numerical name applied to each container, corresponding to the quantity of objects in it, the concept of numeration emerges (at least in principle). This type of pedagogy is based on the association of numerical size to the experience of putting objects in containers of various sizes.

Image Schemas As Lakoff and Núñez (2000: 33) emphasize, we are hardly aware of the extent to which an image schema, such as the container one, is at the root of a large portion of mathematics: Given the spatial logic of Container schemas, the Categories Are Containers metaphor yields an everyday version of what we might call folk Boolean logic, with intersections and unions. That is why the Venn diagrams of Boolean logic look so natural to us, although there are differences between folk Boolean logic and technical Boolean logic. . .Folk Boolean logic, which is conceptual, arises from a perceptual mechanism—the capacity for perceiving the world in terms of contained structures.

As the neural mechanisms that transform bodily experiences into mind-forms, becoming the basis for the conceptual shape of a metaphor, image schemas are key to understanding poetic logic itself and how mathematics emerges as a visual-iconic system of thought (Lakoff 1987; Johnson 1987; Lakoff and Johnson 1980). Image schemas give what the fantasia has visualized a specific form, namely, metafora. As we saw, the verticality (or orientation) image schema guides the construction and interpretation of many graphs. This is derived from experiencing orientation in the real world, which then guides the conceptualization of a host of concepts that are felt to implicate it in some imaginary (metaphorical) way. The range of image schemas is typically (or normally) constrained by actual physical and affective experiences and everyday events. However, the fantasia can, and often does, transcend the constraints. Consider a well-known episode in mathematical history—namely, Euclid’s fifth postulate, also known as the parallel postulate, which is based on the image schema of nonintersecting linear paths extending to infinity. One way of representing the postulate in diagrammatic form is to draw two parallel lines on a sheet of paper, m and l, and drawing through a point P on line m, various lines to show that only one line can be drawn parallel to the given line l, namely m itself, since all the other lines will intersect with l if they are extended (Fig. 4.8):

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Fig. 4.8 A representation of Euclid’s fifth postulate Fig. 4.9 The Beltrami model

During Euclid’s time, and for centuries thereafter, mathematicians attempted to prove that the parallel postulate could be derived from Euclid’s axioms. The problem was that it constituted an image schema of path linearity in two-dimensional space (the plane), falling outside the Euclidean axioms. In the 1800s, mathematicians finally demonstrated that the parallel postulate cannot be proved from Euclid’s other axioms. This discovery led to the ideation of geometric systems in which the parallel postulate no longer held. In one of these, called Lobachevskyan geometry (after Nikolai Lobachevsky), the parallel postulate is replaced by the following: through a point not on a given line, more than one line may be drawn parallel to the given line. This required literally a “re-imagining” of lines, surfaces, spaces, and the like. In one such model, called the Beltrami model, after Eugenio Beltrami (1868) who put it forth, the plane is defined as a set of points that lie in the interior of a circle. In the diagram below, therefore, the lines going through point P are all parallel, even though they all pass through the same point, P. The reason for this is, of course, that those lines are inside the circle, and end at its circumference (Fig. 4.9):

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Fig. 4.10 Clustering ICM for “Number”

etc.

objects container

orientaon NUMBER

paths

rotaon placement geometry

points

Such new ideas were truly “imaginative,” since they required thinking outside of pre-established visual models. Since then it has become obvious that the geometry used to describe the physical world will vary. Some situations are better described in non-Euclidean terms, such as aspects of the theory of relativity. Other situations, such as those related to building, engineering, and surveying, seem best described still by Euclidean geometry. The notion of image schema has, clearly, far-reaching implications for grasping the origin of mathematical ideas. Substantive research has come forward to show how the image schemas underlying the layering of conceptual metaphors coalesce into a system of meanings that define complex or advanced mathematics. Lakoff and Johnson (1980) saw this coalescence as producing what are now called idealized cognitive models (ICMs), defined as over-arching models that result from the repeated mapping of certain image schemas onto specific target domains. For example, the notion of number includes the following image schemas: objects (distinct quantitative units), containers (as elements in containers), paths (as being laid out in a path such as the number line), points (in reference to the points on a coordinate plane), geometrical forms (as in triangular numbers, square numbers, etc.), placement (as in decimal and binary numerals), rotation (as in sexagesimal systems), orientation (as in negative numbers represented as numbers to the left of zero on a number line), etc. The clustering of these image schemas around the target domain of number is an ICM, which allows us to understand the concept of number in experiential ways in mathematics. The clustering structure of this ICM can be shown as follows (Fig. 4.10): Not all ICMs manifest a clustering structure. Another type involves different target domains being connected by the same image schema. This process can be called radiation, since it can be envisioned as a single source domain “radiating outward” to deliver different target domains (Danesi 2022). For example, the path image schema not only allows us to conceptualize number, but also such other

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infinity

number line

PATH graphs

linearity

topology

circularity

Fig. 4.11 A radiation ICM based on the “Path” image schema

concepts as coordinates (intersecting number paths), infinity (paths that never end), linearity (paths that are straight), circularity (circular paths), topology (paths that can take any shape), graphs (which show how certain paths are constrained by their network structure, as in the Königsberg Bridges Problem), and number line (which shows a linear path as consisting of equally-spaced number points), among many others. The radiation structure of this ICM can be shown as follows (Fig. 4.11): Radiation can be defined as the tendency to envisage abstract concepts as implicating each other through a specific associative frame of reference (a single image schema). Relevant research indicates that some conceptual metaphors are based on radiation, while others are more diverse (clustering). Suffice it to say here, that they are the cognitive elements that connect the different parts (branches, systems, etc.) of mathematics around conceptual cores. ICMs exemplify what Lakoff (1979) calls the principle of invariance: namely, the fact that metaphorical clusters (or radiants) preserve the image-schema structure of the initial source domain, in a way that is consistent with the inherent structure of the target domain—so, once a path schema is enlisted as a source domain, its features are preserved, no matter what derived metaphorical layers are built up subsequently: What the Invariance Principle does is guarantee that, for container schemas, interiors will be mapped onto interiors, exteriors onto exteriors, and boundaries onto boundaries; for pathschemas, sources will be mapped onto sources, goals onto goals, trajectories onto trajectories; and so on. . .If one looks at the existing correspondences, one will see that the

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Invariance Principle holds: source domain interiors correspond to target domain interiors; source domain exteriors correspond to target domain exteriors; etc. As a consequence it will turn out that the image-schematic structure of the target domain cannot be violated: One cannot find cases where a source domain interior is mapped onto a target domain exterior, or where a source domain exterior is mapped onto a target domain path. This simply does not happen.

An early image schema that can be found as far back in texts such as the Rhind Papyrus is the partition one, which is the conceptual base for the notion of fraction. Consider the following ingenious problem, devised by Renaissance mathematician, Niccolò Tartaglia, which plays on the everyday experience of partitioning wholes into parts (Petkovic 2009): A father dies, leaving 17 camels to be divided among his three sons, in the proportions 1/2, 1/3, 1/9. How can this be done?

Dividing up the camels in the manner decreed by the father would entail having to split up one of the camels, which would, of course, kill it. So, Tartaglia suggested “borrowing an extra camel,” for the sake of argument. With 18 camels, we arrive at a practical solution: one son was given ½ (of 18), or 9, another 1=3 (of 18), or 6, and the last one 1/9 (of 18), or 2. The 9 + 6 + 2 camels apportioned in this way, add up to the original seventeen. The extra camel could then be returned to its owner. Whatever the interpretation of this solution in real legal terms, as a puzzle it shows that the concept of fractions is derived from common partitioning experiences such as dividing an inheritance into (real) parts, and what this implies in mathematical terms. Now, the fascinating thing is that the puzzle itself generated further mathematical reasoning, namely, a solution based on n camels, which can be summarized as follows. If there are n camels, three heirs (a, b, c), and proportions 1/a, 1/b, 1/c, then the relevant equation is: n=ðn þ 1Þ = 1=a þ 1=b þ 1=c → ðn þ 1Þ Solutions: n = 7 ða = 2, b = 4, c = 8Þ n = 11 ða = 2, b = 4, c = 6Þ n = 11 ða = 2, b = 3, c = 12Þ n = 17 ða = 2, b = 3, c = 9Þ n = 19 ða = 2, b = 4, c = 5Þ n = 23 ða = 2, b = 3, c = 8Þ n = 41 ða = 2, b = 3, c = 7Þ Consider the first one. It says that 7 camels are to be divided among the three heirs in the proportions 1/2, 1/4, and 1/8. So (n + 1) = 8, and thus the camels can be divided as shown:

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First heir : 1=2 × 8 = 4 Second heir : 1=4 × 8 = 2 Third heir : 1=8 × 8 = 1 Together : 4 þ 2 þ 1 = 7 ðthe original number of camelsÞ Let us put these values in the general equation, and we will get an identity (as expected): n=ðn þ 1Þ = 1=a þ 1=b þ 1=c 7=8 = 1=2 þ 1=4 þ 1=8 = 4=8 þ 2=8 þ 1=8 Note that when n = 17, we get Tartaglia’s original puzzle. The question becomes: Is this really a solution to the division of inheritance? Or is it just a clever mathematical maneuver? Leaving aside this consideration, the key aspect of the puzzle is that it highlights the partitioning schema as the crux of the concept of fraction. This schema has had significant consequences, historically. A rational number is a number that can be made by dividing two integers, p/q (q ≠ 0). If q = 1, that is, p/1, then it represents an integer as well. This amplification would never have been possible without the notion of the fraction schema, showing, remarkably, that fractions are more common than integers, which are a subset of fractions. Also, it has led to the discovery of irrational numbers, which cannot be represented with p/q. This too has led to a further amplification of mathematical knowledge—the rational and irrational numbers are classified together as real numbers. Further explorations led to the discovery of complex numbers, which indicates that the real numbers are subsets of the complex numbers, defined as numbers, a + bi, where a and b are real numbers and i is √-1. Every real number can thus be represented as a complex number, by simply letting the imaginary part be 0. Now, the discovery process did not stop there. Consider π and √2, which are irrational numbers, because they cannot be written as a ratio, p/q. The difference between the two can now be examined. As can be seen, √2 is a root (solution) of the following equation: x2 - 2 = 0 x2 = 2 x = ± √2 For this reason, it is called algebraic. As it turns out, π is not algebraic; that is, there is no equation in which it occurs as a root. For this reason, it is called a transcendental number (Baker 1990). As it has turned out there are an infinite number of transcendental numbers. So, π is hardly an exception; it is part of a rule. Clearly, one simple image schema, partitioning, has been conceptually productive, even though it is often difficult to envision its presence in advanced mathematics. But, as Lakoff and Núñez have argued, as for all image schemas, “it is there.”

Conceptual Blending

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Conceptual Blending The notion of conceptual blending as part of conceptual metaphor theory makes explicit what happens in the mind as layers of metaphorical concepts are formed on the basis of clustering and radiating image-schematic connectivities and continuities. To cite Lakoff and Núñez (2000: 48): A conceptual blend is the conceptual combination of two distinct cognitive structures with fixed correspondences between them. . .When the fixed correspondences in a conceptual blend are given by a metaphor, we call it a metaphorical blend. An example is the NumberLine Blend, which uses the correspondences established by the metaphor Numbers Are Points on a Line. In the blend, new entities are created—namely, number-points, entities that are at once numbers and points on a line.

One of the first mentions of conceptual blending is found in Mark Turner’s book, The Literary Mind (1997: 93), in which he states that “Conceptual blending is a fundamental instrument of the everyday mind, used in our basic construal of all our realities, from the social to the scientific.” Blending theory was elaborated in an in-depth manner by Fauconnier and Turner (2002). In this framework, mathematical concepts are formed from inputs in different regions of the brain, which share a common experiential basis (e.g., the path or container schema), and are then amalgamated to form a new concept, with new properties and ideas emerging from the blend, that did not exist in the original inputs (Turner 2014). This might also explain Vico’s notion of the ingegno neuropsychologically, as the faculty that employs blending to create mathematical insights. One never knows when and to whom the insight will come. Consider an anecdote that Henri Poincaré himself recounted in his book, Science and Method (mentioned in Chap. 1). Poincaré had been puzzling over an intractable mathematical problem, leaving it aside for a little while to embark on a geological expedition. As he was about to get onto a bus at one point, the crucial idea came to him in a flash of insight. He claimed that without it, the solution would have remained buried somewhere in his mind, possibly forever. Now, as he elaborates, the insight was the result of linking (blending) Fuchsian functions with non-Euclidean geometry (Poincaré 1908: 23): Just at this time I left Caen, where I then lived, to take part in a geologic excursion organized by the École des Mines. The circumstances of the journey made me forget my mathematical work; arrived at Coutances we boarded an omnibus for I don’t know what journey. At the moment when I put my foot on the step the idea came to me, without anything in my previous thoughts having prepared me for it; that the transformations I had made use of to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify this, I did not have time for it, since scarcely had I sat down in the bus than I resumed the conversation already begun, but I was entirely certain at once. On returning to Caen I verified the result at leisure to salve my conscience.

Blending creates connective and continuous thinking about something. Reading a mathematical theorem in a book might lead some individual mathematician to devise another one or to use it as part of some new idea, based on the individual’s experiences and background knowledge related to the theorem. When others take it on to develop it, the idea becomes a shared one. This implies that a blend, once

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completed, is available for subsequent or additional blending. The modus operandi of mathematicians is, in fact, to build upon ideas created before them. In this way, they construct entire edifices of ideas to solidify their objectives, producing the Vichian memoria, in terms of new “fables” or metaphorical narratives, as Vico called them. To quote Turner (2005): As long as mathematical conceptions are based in small stories at human scale, that is, fitting the kinds of scenes for which human cognition is evolved, mathematics can seem straightforward, even natural. The same is true of physics. If mathematics and physics stayed within these familiar story worlds, they might as disciplines have the cultural status of something like carpentry: very complicated and clever, and useful, too, but fitting human understanding. The problem comes when mathematical work runs up against structures that do not fit our basic stories. In that case, the way we think begins to fail to grasp the mathematical structures. The mathematician is someone who is trained to use conceptual blending to achieve new blends that bring what is not at human scale, not natural for human stories, back into human scale, so it can be grasped.

For even a simple concept such as “7 is larger than 4,” we hardly realize that it involves the blending of a source domain that involves concepts of size with the target domain of numbers—namely, numbers are collections of objects of differing sizes. Similarly, the more is up, less is down image schema underlies the representations of functions on the Cartesian plane. The linear scales are paths conceptual metaphor manifests itself in concepts such as rational numbers are far more numerous than integers, and infinity is way beyond any collection of finite sets. Lakoff (1979) explains the path metaphor as follows: The metaphor maps the starting point of the path onto the bottom of the scale and maps distance traveled onto quantity in general. What is particularly interesting is that the logic of paths maps onto the logic of linear scales. Path inference: If you are going from A to C, and you are now at in intermediate point B, then you have been at all points between A and B and not at any points between B and C. Example: If you are going from San Francisco to NY along route 80, and you are now at Chicago, then you have been to Denver but not to Pittsburgh. Linear scale inference: If you have exactly $50 in your bank account, then you have $40, $30, and so on, but not $60, $70, or any larger amount. The form of these inferences is the same. The path inference is a consequence of the cognitive topology of paths. It will be true of any path image-schema.

Blending theory has various precursors. For instance, both Richards (1936) and Black (1962) envisioned a metaphor as an attempt to establish a conceptual link between what is known (the vehicle source domain) and what needs to be known (the topic target domain). The linkage assumes that the two domains share a common experiential ground, which is elicited in the metaphorical meaning (called the ground, itself an image schema). Soskice (1985) suggests that the two domains can be said to “animate” each other. Consider the concept of the number line again as a derivative of the path metaphor. It is a diagrammatic model of how we actually count and organize counting in a sequence from small to large to infinity. It both mirrors and then subsequently structures the cognitive tasks we perform when we count. Now, the number line became a source of further mathematics, after Wallis, leading to more complex and advanced mathematics, as discussed above. Without the number line, it is unlikely that such inventions or discoveries would have been possible in the first place.

Mathematical Infinity

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Mathematical Infinity A path schema that became the conceptual basis for mathematical infinity is traced to antiquity. It describes a path that never ends, or a path that becomes infinitely smaller when segmented into intervals. Among the first to portray infinity in this way was Zeno of Elea with his famous paradoxes (Salmon 1970). Consider his Achilles and the Tortoise Paradox, which he formulated in support of the doctrine of his teacher, Parmenides, that motion is an illusion. The paradox is recounted in Aristotle’s Physics (VI: Chap. 9). Below is a paraphrase: Achilles decides to race against a tortoise. To make the race fairer, he allows the tortoise to start at half the distance away from the finish line. In this way, Achilles will never surpass the tortoise. Why?

In order for Achilles to surpass the tortoise, he must first reach the halfway point, which is the tortoise’s starting point. But when he does, the tortoise will have moved forward a little bit. Achilles must then reach the tortoise’s new point before attempting to surpass it. When he does, however, the tortoise has again moved a little bit forward, which Achilles must also reach again, and so on ad infinitum. In other words, although the distances between Achilles and the tortoise will become smaller and smaller (in fact, infinitesimally so), Achilles will never surpass the tortoise. Of course, in reality, Achilles will do so, but explaining why he does in terms of the paradox soon after became a major debate in philosophy, science, and mathematics, leading over time to the theory of limits and the calculus. It also embedded the image schema of infinity as a linear path that involves segmentation on it that never ends. Zeno’s Arrow Paradox utilizes the same schema in a different way—namely, to show the logical impossibility of motion. Zeno asks us to imagine an arrow in flight. At any instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In effect, at every instant of time, there is no motion occurring. Another way to put it is as follows: An arrow in flight is in a specific position at a given instant of time. At that instant, it is indistinguishable from a motionless arrow in the same position. Below is its formulation as reported by Aristotle in his Physics (VI: 9): If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time.

A retort to this paradox is that the arrow does not “stop” along its path, but is constantly moving. However, if we see its “instances” as “time frames” like in a comic strip, then the paradox remains. Suffice it so say that this paradox, like the others put forth by Zeno, led to much debate in mathematics, and subsequently to new ideas and branches.

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Consider one other paradox, namely, the Dichotomy Paradox. Zeno argued that if we use logic, then we must conclude that a runner will never reach the end of a race course, even though the runner will actually do so if we just look at the race with our eyes, thus differentiating between inner and outer vision, in line with the Vichian notion of the fantasia. He argued that the runner must first complete half of the course, then half of the remaining distance, and so on infinitely, which means that he will never cross the finish line. If the length of the race course is represented by a line, with unit length, the successive stages of the runner’s location form an infinite series with each term in it half of the previous one: f1=2 þ 1=4 þ 1=8 þ 1=16 þ 1=32 þ . . .g Adding up the terms, in fact, shows that the sum will come close to 1, but never reach it. The implication is that when dealing with infinite summation, there is no end to the process, thus identifying mathematical infinity as behaving differently from mathematical finiteness. Lakoff and Núñez (2000) explain the difference in terms of what they call a Basic Metaphor of Infinity (BMI). The BMI is based on the image schema of “adding one more” to any collection or to any path. As an aside, it is the schema that guides proof by induction, which implies that if some condition holds for the (n + 1)th case, given n, then it holds infinitely, because we can add the (n + 2)nd case, the (n + 3)rd case, and so on—one case at a time, until we decide to stop, which Lakoff and Núñez call completion. We can count as high as we want, but as soon as we feel that we have reached the highest number we possibly can reach, we can always add one more to it and make it bigger. This image schema is at the core of the notion of infinite set. So, the series {1/2 + 1/4 + 1/8 + 1/16, . . .} will go on forever because of the BMI, but, even as it continues to infinity, it has a summation limit, namely 1—a schema that led to its formalization in the invention of the calculus. So, the paradoxes of Zeno were hardly mind games they eventually became the inspiration for developing the implications of mathematical infinity. They still present challenges to this day for explanations of space, time, and motion, however—challenges that the Greeks raised but were not able to explain in any viable way. The peculiar theoretically features of the mathematics of infinity, in contrast to the mathematics of finiteness, were eventually unpackaged Georg Cantor (1874), elaborating a seemingly paradoxical observation by Galileo in his book, Dialogue Concerning Two New Sciences (1638), in which he noted that each member in the set of square integers can be compared, one-by-one, with all whole numbers (positive integers). The conundrum was that there were, apparently, as many square integers as there are numbers (even though the squares are themselves only a part of the set of integers) (Fig. 4.12): 1 b 1

2 b 4

3 b 8

4 b 16

5 b 25

6 b 36

7 b 49

Fig. 4.12 Galileo’s one-to-one correspondence

8 b 64

9 b 81

10 b 100

11 b 121

12 b 144

… …

Mathematical Infinity

1 b 1n

2 b 2n

3 b 3n

117

4 b 4n

5 b 5n

6 b 6n

7 b 7n

8 b 8n

9 b 9n

10 b 10n

11 b 11n

12 b 12n

… …

Fig. 4.13 Cantor’s elaboration

Fig. 4.14 Cantor’s sieve

Cantor showed that this kind of technique, as simple as it was, revealed something intrinsic about sets, noting that Galileo’s finding can be generalized to encompass any exponential value (Fig. 4.13): Cantor came up with many ingenious demonstrations of this kind related to mathematical infinity. Consider the set of rational numbers, which, to reiterate, are numbers that can be written in the form p/q where p and q are integers (and q ≠ 0). The cardinal numbers (integers) are, themselves, a subset of the rationals—since every integer p can be written in the form p/1. Terminating decimal numbers are also rational, because a number such as 3.579 can be written in p/q form as 3579/1000. Finally, all repeating decimal numbers are rational, although the proof of this is beyond the scope of the present discussion. For example, 0.3333333. . . can be written as 1/3. Amazingly, Cantor demonstrated that the rational numbers and the integers have the same number of elements, if the set of all rational numbers are arranged in an array as shown below, since then called “Cantor’s sieve” (Fig. 4.14): In each row, the successive denominators (q) are the integers {1, 2, 3, 4, 5, 6, . . .}. The numerator ( p) of all the numbers in the first row is 1, of all those in the second row 2, of all those in the third row 3, and so on. In this way, all numbers of the form p/q are covered in the above array. He highlighted every fraction in which the numerator and the denominator have a common factor. If these fractions are deleted, then every rational number appears once and only once. Then, Cantor set up a oneto-one correspondence between the integers and the numbers in the array in a zigzag

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1 b 1/1

2 b 2/1

3 b 1/2

4 b 1/3

5 b 3/1

6 b 4/1

7 b 3/2

8 b 2/3

9 b 1/4

10 b 1/5

Metaphor

… …

Fig. 4.15 Cantor’s cardinality demonstration

fashion as follows: he let the cardinal number 1 correspond to 1/1 at the top left-hand corner of the array; following the arrow, 2 to the number below (2/1); 3 to 1/2; 4 to 1/3; and so on through the array ad infinitum. The zigzag path thus allowed him to set up a one-to-one correspondence between the cardinal numbers and all the rational numbers (eliminating the repeated numbers such as 2/2 = 1). This implies that there are as many rational numbers as there are integers—that is, they have the same cardinality (Fig. 4.15): Cantor then classified those numbers with the same cardinality as belonging to the set “aleph null,” or ℵ0, calling it a transfinite number. He then demonstrated that there are other transfinite numbers, and indeed an infinitude of them—that is, sets with a greater cardinality than the integers. He labeled each successively larger transfinite number with increasing subscripts {ℵ0, ℵ1, ℵ2, . . .}. Cantor’s proof is remarkable for its simplicity and ingenuity, highlighting the power of the ingegno to elaborate a vision of numbers that could only be sparked by the fantasia. Suppose we take all the possible numbers that exist between 0 and 1 on the number line and lay them out in decimal form. Let us label each number {N1, N2,. . .}: N 1 = :4225896 . . . N 2 = :7166932 . . . N 3 = :7796419 . . . ... Cantor showed how to construct a number that is not on that list. Let us call it C. To create it, we do the following: (1) for its first digit after the decimal point we choose a number that is greater by one than the first digit in the first place of N1; (2) for its second digit we choose a number that is greater by one than the second number in the second place of N2; (3) for its third digit we choose a number that is greater by one than the third number in the third place of N3; and so on: N 1 = 4225896 . . . The constructed number, C, would start with 5 rather than 4 after the decimal: C = :5 . . . N 2 = :7166932 . . .

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Fig. 4.16 Cantor’s array with binary numbers

s1 = s2 = s3 = s4 = s5 = s6 = s7 = ...

(0, (1, (0, (1, (1, (0, (1,

0, 1, 1, 0, 1, 0, 0,

0, 1, 0, 1, 0, 1, 0,

0, 1, 1, 0, 1, 1, 0,

0, 1, 0, 1, 0, 0, 1,

0, 1, 1, 0, 1, 1, 0,

0, 1, 0, 1, 1, 1, 0,

...) ...) ...) ...) ...) ...) ...)

The constructed number would have 2 rather than 1: C = :52 . . . N 3 = :7796419 . . . The constructed number would have 0 rather than 9 C = :520 . . . ... Now, using this diagonal method, we have produced the number C = .520. . . , which is different from N1, N2, N3, . . . because its first digit is different from the first digit in N1; its second digit is different from the second digit in N2; its third digit is different from the third digit in N3, and so on ad infinitum. We have in fact just constructed a different transfinite number than ℵ0. It appears nowhere in the list above. With such demonstrations, Cantor laid the foundations for the systematic study of both sets and mathematical infinity. He accomplished this under the guidance of a primordial image schema that underlies the process of counting itself, where objects (stones, sticks, etc.), are put into a one-to-one correspondence to solve the problem of counting in an unlimited way. Now, it is remarkable to note that it was Cantor’s diagonal method above that became the basis of Gödel’s proof. Using binary digits, rather than decimal digits, where s1 = N1, s2 = N2, and so on above, Cantor’s proof can be grasped as follows (Fig. 4.16): Next, a sequence s is constructed (as above) by choosing the first digit as complementary to the first digit of s1 (exchanging 0s for 1s and vice versa), the second digit as complementary to the second digit of s2, the third digit as complementary to the third digit of s3, and so on (Fig. 4.17). As can be seen, the numbers in the diagonal do not occur anywhere in the infinite array. It was this diagrammatic image schema that inspired the first of Gödel’s theorems, as Lakoff argued, in a lecture at the Field’s Institute in 2011 (Danesi

120 Fig. 4.17 Cantor’s diagonal proof

s1 s2 s3 s4 s5 s6 s7 ...

= = = = = = =

(0, (1, (0, (1, (1, (0, (1,

0, 1, 1, 0, 1, 0, 0,

0, 1, 0, 1, 0, 1, 0,

0, 1, 1, 0, 1, 1, 0,

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Metaphor

0, 1, 0, 1, 0, 0, 1,

0, 1, 1, 0, 1, 1, 0,

0, 1, 0, 1, 1, 1, 0,

...) ...) ...) ...) ...) ...) ...)

2011). Lakoff explained how Gödel’s proof had adopted this schema, guided by the BMI, as it manifests itself diagonally. As Rafael Núñez (2005: 1717) explained in an article he wrote several years after the publication of Where Mathematics Comes From, in such proofs, the BMI can be adjusted to what he calls the Basic Mapping of Infinity, since this describes concretely what Cantor and Gödel actually did with their proofs: [Cantor’s] analysis is based on the Basic Metaphor of Infinity (BMI). The BMI is a human everyday conceptual mechanism, originally outside of mathematics, hypothesized to be responsible for the creation of all kinds of mathematical actual infinities, from points at infinity in projective geometry to infinite sets, to infinitesimal numbers, to least upper bounds Under this view “BMI” becomes the Basic Mapping of Infinity.

The ancient Greeks grappled constantly with the fact that certain things could not be proved within their system of demonstration. Why, for example, was it seemingly impossible to trisect an angle with compass and ruler, given that bisection was such a simple procedure? For centuries after, mathematicians attempted trisection with compass and ruler, but always to no avail. The demonstration that it was impossible had to await the development and spread of Descartes’ method of converting every problem in geometry into a problem in algebra. The proof came in the nineteenth century after mathematicians had established that the equation, which corresponds to trisection, must be of degree 3—that is, it must be an equation in which one of its variables is to the power of 3. A construction carried out with compass and ruler translates, on the other hand, into an equation to the second degree. Thus, trisection with compass and ruler is impossible. The formal proof was published by mathematician Pierre Laurent Wantzel in 1837, which was based on a blending of algebra and geometry, showing relationships among ideas and facts that were previously considered to be separate or unrelated but, which, in effect, turned out to have a common problematic—namely, that they did not fit in with any established imageschematic system, including change is motion, sets are collections in containers, continuity is gapless, functions are sets of ordered pairs, geometric figures are objects in space, numbers are object collections, recurrence is circular, etc. (Lakoff and Núñez 2000). Each of these underlies a specific mathematical conceptualization

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such as the calculus (change is motion), infinity (continuity is gapless), set theory (numbers are object collections), and so on. In this framework, metonymy (the part for the whole) is seen as the mechanism that allows for generalizations from particular instances to emerge. The ontological difference between metaphor and metonymy can be reduced to a simple paraphrase: metaphor amalgamates information, metonymy condenses it. So, metonymy is operative in how symbols arise to compress ideas; metaphor is operative in how different experiential inputs are amalgamated to produce the ideas. Both processes reflect blending in general, taking different inputs and putting them together in image-schematic ways. The BMI is also behind the notion of infinitesimals (numbers relating to, or involving, a small change in the value of a variable that approaches zero as a limit). The early calculus was often critiqued because it was thought to be an inconsistent mathematical theory, given its use of bizarre notions such as the infinitesimals. These were defined as changing numbers as they approached zero. The problem was that in some cases they behaved like real numbers close to zero, but in others they behaved paradoxically like zero. Take, as an example, the differentiation of the polynomial f(x) = ax2 + bx + c (Colyvan 2012: 121): f 0 ð xÞ = f 0 ð xÞ =

f ðx þ δÞ–f ðxÞ δ

a ðx þ δÞ2 þ b ðx þ δÞ þ c–ðax2 þ bx þ cÞ δ f 0 ð xÞ =

2axδ þ δ2 þ bδ δ

f 0 ðxÞ = 2ax þ b þ δ f 0 ðxÞ = 2ax þ b Colyvan (2012: 122) comments insightfully on the solution as follows: Here we see that at lines one to three the infinitesimal δ is treated as non-zero, for otherwise we could not divide by it. But just one line later we find that 2ax + b + δ = 2ax + b, which implies that δ = 0. The dual nature of such infinitesimals can lead to trouble, at least if care is not exercised.

Perhaps one of the most significant findings emerging from the examination of metafora in mathematics is the omnipresence of specific image schemas—the container, the conduit, phases, and so on. This entails that increasingly complex cognition is the result of the higher-order blends of source image schemas. Immanuel Kant (1781: 278) saw language as connected to the brain’s ability to synthesize scattered bits of information into holistic entities that can then be analyzed reflectively. Kant’s ideas found their implicit elaboration and amplification in blending theory, which mirrors the notion of poetic logic, as discussed throughout.

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Epilogue Since Lakoff and Núñez’s book, the gist of the relevant research in a large sector of cognitive science suggests, overall, that the crux to understanding mathematics involves examining its conceptual metaphors (Danesi 2023). A diagram or an equation, for example, is an external representation of a metaphorical thought, as Wallis clearly understood with his characterization of the number line as a path of numbers moving forward and backward (above), whereby he established a formal link between positive and negative numbers in terms of this image schema. He illustrated his concept with the following diagram (Wallis 1685)—we note that rather than digits, Wallis used letters to indicate what he described as advancing and retreating from point A (now represented as zero on the line) (Fig. 4.18): The presence of a linear path image schema as a product of the mathematical imagination goes right back to the origins of mathematics. It can be seen in Zeno’s paradoxes as an unconscious inner vision of linear motion, as well as in other ancient metaphors, some of which even prefigured coordinate geometry as a system of numerical representation based on the intersection of two lines at right angles. For example, Apollonius of Perga dealt ingeniously with the question of finding points on a line that were in a ratio to the other points, thus coming close to developing a proto-coordinate geometry. But he did not take into account negative magnitudes, which is the crux of both the number line and coordinate systems (Boyer 1968). Descartes’s ideas in La géometrie (1637) prefigured Wallis’s metaphor. However, he did not articulate it explicitly as did Wallis, because he did not use specific numbers mapped onto lines, only abstract quantities (Núñez 2018). Incidentally, it is well known that coordinate geometry was independently discovered by Pierre de Fermat, even though he did not publish his version. Both Descartes and Fermat used a single axis and variable lengths measured in reference to the axis. The concept of using a pair of axes was introduced later, after Descartes’ work was translated into Latin in 1649 by Frans van Schooten. The point here is that the development and utilization of metaphorical models in mathematics constitute its core methodology. Without them, ideas and entire branches could not have evolved as they have. As Max Black (1962) emphasized, there is no formal science or mathematics without them, since they constitute creative attempts to render visible those things we can never see with the physical eye, as mentioned—that is, phenomena such as atoms, gravitational forces, magnetic fields, and so on. The trace to this inner vision is metaphor. In physics it was at first speculated that atomic structure mirrored the solar system—a metaphor that went back to the ancient Greek concept of the cosmos as having the same structure at all Fig. 4.18 Wallis’s illustration of the number line

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its levels, from the microcosmic (the atom) to the macrocosmic (the universe). It is truly extraordinary to think that the ancient metaphor led subsequently to real knowledge about atoms. Physicists have never seen the inside of an atom with their eyes. So, they used their inner eye to produce a metaphorical hunch, initially, that the atom has the same kind of structure that the solar system has. This model of atomic structure as a miniature solar system, as Black pointed out, led to a whole spate of experimental research and discoveries that eventually led to quantum physics. Physicists no longer use the simple cosmic metaphor, but without it, they never would have come up with the ideas that crystallized afterward. Through a metaphorical analysis of basic concepts, from simple counting to the infinitesimal calculus, Lakoff and Núñez put forward the claim that mathematics is guided by layers and layers of metaphors that accrue over time into systems of thought. As Marcus (2012: 124) has observed, even the words mathematicians have coined reflect their origin in metaphorical cognition, including union, inclusion, border, frontier, distance, bounded, open, closed, imaginary number, rational/ irrational number, and so on, whereby each one reflects a specific image schema, such as an impediment schema, the container schema, and so on. Mathematical models, in this view, are layered metaphors. Marcus (2012: 184) writes on this notion insightfully as follows: Starting as a cognitive model or metaphor for a definite, specific situation, mathematics acquires an autonomous status and it is open to become a model or a metaphor for another, sometimes completely different situation. Mathematics may acquire some interpretation, but it can also abandon it, to acquire another one. No mathematical construction can be constrained to have a unique interpretation, its semantic freedom is infinite, because it belongs to a fictional universe: mathematics. Mathematics has a strong impact on real life and the real world has a strong impact on mathematics, but all these need a mediation process: the replacement of the real universe by a fictional one.

As Vico claimed throughout the New Science, metaphors reveal how the brain converts outer visions into inner visions, rendering them visible in language, mathematics, and all other types of human symbolic expression. As Cicero (55 BCE: 274) so aptly put it in antiquity: All metaphors, at least such of them that are best chosen, are applied to the senses, especially the seeing, which of all senses is the most exquisite. Thus when we say, the tincture of politeness, the softness of good-breeding, the murmur of waters, and sweetness of language; these metaphors are all taken from the other senses. But the metaphors taken from the sense of seeing are much more striking, because they place in the eye of the imagination objects otherwise impossible for us to see or comprehend. For there is nothing in nature but what we may adapt its name to signify something else; and every object from which a likeness may be raised, as it may from all objects, if metaphorically applied.

The distinguishing feature of the human fantasia is that it allows us to imagine the world beyond our senses. Without image-schematic thought, there would be no theoretical-formal mathematics, just instinctive counting and intuitive spatial estimations.

Chapter 5

Logic

Logic will get you from A to B. Imagination will take you everywhere. Albert Einstein (1879–1955)

Prologue The terms logic and mathematics are often paired in the historiography of the latter. But, as discussed in this book, logic is not a monolithic process, since it manifests itself in various ways, spanning a cognitive continuum that starts from the highly poetic to the highly rational. Starting with philosophers such as Heraclitus, Parmenides, and Aristotle, rational logic came to be viewed as equivalent to reasoning, a view that became the core of Euclidean geometrical method, which started from a small set of self-evident axioms upon which it built a comprehensive system based on proofs using these axioms. But to construct the system, Euclid showed that proof involved more than one form of logic, as discussed in previous chapters. The association of rational (deductive) logic with mathematics was further embraced by Descartes, who even maintained that such logic was the only practical way to solve human troubles, caused by the emotions and the passions (Descartes 1633). In their book, Descartes’ Dream, Davis and Hersh (1986: 7) encapsulated Descartes’s vision as “the dream of a universal method whereby all human problems, whether of science, law, or politics, could be worked out rationally, systematically, by logical computation.” This universal method was elaborated subsequently by Leibniz (1646), which he called a characteristica universalis, and which he claimed could reduce errors in thinking to errors in logic and thus easily fixed. He described his universal system as follows (Leibniz 1646: 30–31): And although learned men have long since thought of some kind of language or universal characteristic by which all concepts and things can be put into beautiful order, and with whose help different nations might communicate their thoughts and each read in his own language what another has written in his, yet no one has attempted a language or characteristic which includes at once both the arts of discovery and judgement, that is, one whose signs and characters serve the same purpose that arithmetical signs serve for numbers, and

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Danesi, Poetic Logic and the Origins of the Mathematical Imagination, Mathematics in Mind, https://doi.org/10.1007/978-3-031-31582-4_5

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algebraic signs for quantities taken abstractly. Yet it does seem that since God has bestowed these two sciences on mankind, he has sought to notify us that a far greater secret lies hidden in our understanding, of which these are but the shadows.

By alluding to the use of signs in arithmetic and algebra, Leibniz was clearly prefiguring the kind of symbolic language that formal mathematics adopted centuries later, as well as the system of binary logic developed by George Boole (1854), based on the binary arithmetical system that Leibniz himself had helped to develop. The purpose of this final chapter is threefold: (1) to discuss the role of logic in mathematics; (2) to consider the relation between poetic logic and the rationalsymbolic logic used in formal mathematics; and (3) to revisit the origins question discussed in Chap. 1 in a cohesive way. The theme that will be threaded throughout is that poetic logic is at the core of mathematical thought in its origins, while other kinds of logic are subsequent to it, phylogenetically and ontologically. Charles Peirce (1931–1958) also differentiated between two general kinds of logic—logica utens and logica docens. The former is a rudimentary intuitive logic that everyone possesses without being able to specify what it is; the latter is a sophisticated and tutored logic as practiced by mathematicians, scientists, detectives, and medical doctors (Sebeok and Umiker-Sebeok 1983: 40–41). Because everyone possesses logica utens, no special training is required to understand rudimentary mathematics; as such, it dovetails with the notion of poetic logic. Understanding and proving formal mathematical structures or theories, and the problems associated with them, requires logica docens instead, which is equivalent to rational logic as defined here. The connecting mental mechanism between the two forms of logic is metaphor, as discussed in the previous chapter. Vico saw the kind of rational logic used in proving theorems as an evolutionary achievement, not a point-of-departure, but as a point-of-arrival, phylogenetically and ontogenetically. He put it as follows (in Bergin and Fisch 1984: 162): The metaphysics of the philosophers, fulfills its first task, that of clarifying the human mind, which needs logic so that with clear and distinct ideas it may form its conclusions, and descend there with to cleanse the heart of man with morality. . .The idea was of course not formed by reasoning. . .but by the senses, which, however false in the matter, were true enough in form which was the logic conformable to such natures as theirs.

In the contemporary era, set theory surfaced as the model for the relevant cognitive architecture for expanding mathematical logic. As such, Cantor’s proofs of infinity and the existence of transfinite numbers (Chap. 4) defied the classical forms of logical demonstration. They were based, instead, on an ingenious idea— namely, pairing symbols in terms of the number line ad infinitum. It is little wonder that Cantor’s “logic” encountered resistance from some of his mathematical contemporaries, being even subjected to caustic attacks. However, by 1904, most of the critiques subsided, whence David Hilbert (1926: 170) declared: “No one shall expel us from the paradise that Cantor has created.” The use and development of “logic machines” to explore the structure of mathematics is now part of mathematical inquiry (Samuel 1959). Starting in the mid-1950s, such machines were used to both simulate and test the validity of the kind of propositional logic elaborated by Gottlob Frege (1879) and later by Russell

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and Whitehead in their Principia Mathematica (1913). Russell and Whitehead thought they could derive all mathematics using a core set of axioms and rules of formal logic. The first machine learning system based on this kind of logic (called first-order logic) was developed by Martin Davis in 1954. It was capable of proving only a very small set of theorems. As Davis quipped later on about his early system (cited in Davis 2000: 3): “Its great triumph was to prove that the sum of two even numbers is even.” In 1956, Newell, Simon, and Shaw developed a Logic Theory Machine, based on the Principia Mathematica, which had the ability to elaborate a larger set of first-order proofs—namely, 38 of the first 52 theorems of the Principia. This approach was called “heuristic,” because the machine attempted to mimic human mathematicians, but it could not ensure that a proof could be carried out for every valid theorem. Since then, more sophisticated programs have been designed, which, in theory, could carry out any proof based on first-order logic. Remarkably, machine logic has helped prove a number of hard theorems. One of these was the Robbins conjecture, which had eluded human mathematicians, until William McCune devised a computer program that proved it (McCune 1997). The conjecture asserted that for elements a, b, and c, if a _ (b _ c) = (a _ b) _ c, and a _ b = b _ a, then Ø (Ø (a _ b) _ Ø (a _ Øb)) = a (in which _ is a single binary operation and Ø a unary operation. This is known as a Robbins algebra, after Herbert Robbins, who formulated it on premises expounded by Edward Huntington in 1933. For many years, it was conjectured, but unproven, that all Robbins algebras were Boolean algebras. This was proved by McCune using his automated theorem prover, so that the term Robbins algebra became thereafter simply a synonym for Boolean algebra. The event was one of the first to raise the possibility that computers could actually do mathematics, perhaps even better than humans. McCune’s proof was checked independently by other mathematicians. The question it raises is the following: Does the computer possess the capacity for poetic logic? The problem with answering this question is that the notion of the fantasia is an intuitive one, like logic utens, and thus impossible to define formally, which is a requisite of all formal logics (McCormack and D’Inverno 2012). As Newell, Shaw, and Simon pointed out in 1963, creative artifacts, produced by either humans or machines, can only be evaluated as being novel or interesting. And this can only be accomplished by human interpretation, not by any logical system of formal symbolism, since it is the same kind of interpretive process that assigns meaning to poetry and metaphor, and other faculties of human expression.

Logic and Mathematics The Greek term lógos implied a mode of reasoning for arguing or demonstrating that something is the way it is, and could not be anything else. It was Euclid who showed how this meaning of logic, which took on different forms (deduction, induction, contradiction, etc.), could be applied to establish mathematical truths. His approach

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instituted mathematics as an autonomous discipline, based on logic. Ever since, it has been considered to be the core of mathematical thinking and method, as Berlinski (2013: 12) aptly observes: Euclid required a double insight before he could strike for immortality. The first: that the various propositions of geometry could be organized into a single structure; and the second: that the principle of organization binding geometric propositions must be logical.

The term logic, as mentioned, is not ontologically monolithic, since it is applied to different kinds of proofs, including Euclid’s ingenious proof of the infinity of primes, which reveals the operation of poetic logic in tandem with rational forms of logic (as discussed). This blend of the two forms, in fact, is what makes advanced mathematics highly productive, called “bi-part” logic by Edgar Allan Poe, discussed below (Canada 2001; Cook 2014a, Cook 2014b). The question of whether the logical structure of mathematical proofs and the truths they uncovered could be formalized as derivative of a symbolic language in the brain, a characteristica universalis, to use Leibniz’s term, became a major goal in the late nineteenth century with the work of George Boole (1854), Gottlob Frege (1879), and Giuseppe Peano (1889). The search for such a language was motivated by the fact that the proofs of mathematics made sense to anyone who were taught them, and thus hid within them a common language of logic. Leibniz (1668) deconstructed the forms and categories of this language as displaying the following features: 1. A set of universal logical symbols, which he called an alphabet, and which could be used to encode concepts in numerical form 2. A dictionary of logic, so that complex concepts could be translated into numerical language in terms of a code (a dictionary) 3. A syntax of logic, which governed the ways in which inferences could be made, as they are combined or organized in some way This provided a kind of implicit blueprint for discussing and developing formal logic—a universal blueprint of thought based on the same laws of structure, which could be described as a single system that would unravel the source of ideas in mathematics. Ludovico Richeri (1760) introduced the notion of classifying statements and their provability, and Johann Lambert (1782) reduced the laws of logic to the laws of arithmetic, echoing Hobbes (1656) before him. Lambert represented statements and inferences with numbers and arithmetical operations. More than a half century later, Augustus De Morgan (1847) introduced the idea of a “formal logic,” following up on Lambert’s method and mirroring Leibniz’s notions of universal and particular logic. Below is an excerpt from his book showing how he intended to do so: The order of a proposition has relation to the choice of subject and predicate. Thus ‘Every X is Y’ and ‘Every Y is X’ though both establish a universal affirmative relation between X and Y, yet are in fact two different propositions. They are called converse forms. When the subject and predicate are of the same sort of quantity, both universal or both particular, the converse forms give the same proposition. Thus ‘No X is Y’ and ‘No Y is X’ are the same; neither has any meaning, except perhaps of emphasis, which the other has not.

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Less than a decade later, George Boole (1854) came forward to present his own version of the laws of logic as connected with thought in general. Elaborating Leibniz’s method of binary arithmetic, Boole represented the truth values of variables with the digits 1 and 0, and showed how they are connected logically with operators such as conjunction (and) denoted as ^, disjunction (or) denoted as _, and negation (not) denoted as Ø. Presumably, with two symbols, one can reduce logical operations to their different combinations. And since computers work on this principle, Boolean logic became highly significant in the development of electrical engineering and computer science, starting at the end of the nineteenth century. As Wolff (1963: 242) puts it: George Boole, in his Laws of Thought, shows us that there is yet another branch of learning that is part of mathematics. Here we have logic being treated as part of mathematics. To be sure, it is strange mathematics, with strange laws and strange propositions. Later developments have shown that Boole was on the right track. At present, it is a question whether logic ought to be considered a branch of mathematics, or mathematics a branch of logic. Indeed, the best way to solve this problem may be to say that logic and mathematics are one.

Given that arithmetic and logic were being seen increasingly as synchronous, Giuseppe Peano (1889) came forth to break down arithmetic into basic axioms, in logical order, starting with the first natural number, zero, from which the other axioms can be derived in order (as De Morgan suggested): 1. 2. 3. 4. 5. 6. 7. 8. 9.

0 is a natural number. For every natural number x, x = x. For all natural numbers x and y, if x = y then y = x. For all natural numbers x, y, and z, if x = y and y = z, then x = z. For all a and b, if a is a natural number and a = b, then b is also a natural number. That is, the set of natural numbers is closed under the previous axioms. For every natural number n, S(n) is a natural number: S(n) is the successor to n. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection. If K is a set such that 0 is in K, and for every natural number n, if n is in K, then S(n) is in K, then K contains every natural number.

While these may seem like self-evident arithmetical concepts, it is in formalizing them that they can be connected to some universal system of logical thought. At the First International Congress of Mathematicians of the twentieth century in Paris, David Hilbert asked if all science cannot be broken down into similar groups of fundamental axioms (Hilbert 1902). The question is still an open one. Overall, Hilbert sought to define mathematics logically using the method of formal systems, based on a set of axioms. The method of axiomatic logic in mathematics was further developed by Bertrand Russell together with Alfred North Whitehead (1913), which would presumably be impervious to the challenges posed by paradoxes such as those by Zeno (Chap. 4) and, more specifically, by self-referential paradoxes, such as the famous Liar Paradox, which is worth revisiting schematically here, given its deep

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implications for developing a formal logical system for mathematics (Shurz 2013). In their introduction, Russell and Whitehead (1913: 1) indicated that they aimed to achieve three goals: (1) to analyze the ideas and methods of mathematical logic, minimizing the number of primitive notions and axioms, and inference rules; (2) to precisely express mathematical propositions in symbolic logic using the most efficient notation possible; and (3) to solve the paradoxes that plagued logic and set theory, such as the Liar Paradox of antiquity, which involved circularity in logic because of self-reference. The paradox was known to various Greek philosophers, including Protagoras, Gorgias, Eubulides of Miletus, and Epimenides. The Epimenides version, which is the most widely cited one, is given below: Epimenides, a Cretan makes the following statement: “All Cretans are liars.” Did Epimenides speak the truth?

Let us assume that Epimenides did indeed speak the truth. Thus, his statement that “All Cretans are liars” is a true statement. However, from this, we must deduce that Epimenides, being a Cretan, is also liar, and thus, that his statement cannot be true. Obviously, we must discard our assumption. Let us assume the opposite, namely, that Epimenides is in fact a liar. But, then, if he is a liar, the statement he just made—“All Cretans are liars”—is true. But this is a contradiction—liars do not make true statements. Obviously, we are confronted with a circularity. For some reason, circular statements such as this paradox have a bizarre appeal, as the famous British puzzle-maker, Henry E. Dudeney (1907), perceptively observed: A child asked, “Can God do everything?” On receiving an affirmative reply, she at once said: “Then can He make a stone so heavy that He can’t lift it?”

The child’s question is similar to a philosophical conundrum: “What would happen if an irresistible moving body came into contact with an immovable body?” As Dudeney went on to observe, such bizarre paradoxes arise only because we take delight in inventing them. In actual fact, “if there existed such a thing as an immovable body, there could not at the same time exist a moving body that nothing could resist.” In their 1986 book, The Liar, mathematician Jon Barwise and philosopher John Etchemendy dismissed the Liar Paradox, because, they asserted, it arises only when a statement is not tied to real-life contexts. So, for instance, when Epimenides says “All Cretans are liars,” he may be doing so simply to confound his interlocutors. His statement may also be the result of a slip of the tongue. Whatever the case, the intent of Epimenides’ statement can only be determined by assessing the context in which it was uttered along with Epimenides’ reasons for saying it. Once such factors are determined, no paradox arises. But in the framework of the logic of mathematical systems, the paradox cannot be eliminated by simply contextualizing it as a discourse phenomenon—it remains a selfreferential paradox that requires resolution, if one exists. As it turned out, a resolution does not exist, as Gödel showed (1931).

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Interestingly, the paradox has broad appeal for its “logical mischief,” so to speak. It was once even discussed by St. Jerome (c. 342–420) in a sermon (in Ewald 1964: 294): I said in my alarm, Every man is a liar! Is David telling the truth or is he lying? If it is true that every man is a liar, and David’s statement, “Every man is a liar” is true, then David also is lying; he, too, is a man. But if he, too, is lying, his statement that “Every man is a liar”, consequently is not true. Whatever way you turn the proposition, the conclusion is a contradiction. Since David himself is a man, it follows that he also is lying; but if he is lying because every man is a liar, his lying is of a different sort.

The same paradox is found in the work of Indian grammarian-philosopher Bhartrhari (fifth century CE), which he formulated as “everything I am saying is false,” exploring the boundary between statements that are unproblematic in daily life and logical paradoxes (Houben 1995). Clearly, the paradox is independent of language—it is not just true in English, but in any language that allows for a sentence to make a claim about itself. As such, it is a problem in logic, and for this reason, it was the likely spark for Gödel’s first theorem, which set the logical limitations for mathematics (as discussed). Interestingly, a version of the paradox, called the Barber Paradox, is attributed to Russell himself, albeit maybe incorrectly, given that he himself denies having formulated it (Gardner 1979). Whatever the case, the paradox goes as follows, highlighting the nature of self-reference more precisely: The village barber shaves all and only those villagers who do not shave themselves. So, shall he shave himself?

Let us assume that the barber decides to shave himself. He would end up being shaved, of course, but the person he would have shaved is himself. And that contravenes the requirement that the village barber should shave “all and only those villagers who do not shave themselves.” The barber has, in effect, just shaved someone who shaves himself. So, let us assume that the barber decides not to shave himself. But, then, he would end up being an unshaved villager. Again, this goes contrary to the stipulation that he, the barber, must shave “all and only those villagers who do not shave themselves”—including himself. It is not possible, therefore, for the barber to decide whether or not to shave himself. It is said that this paradox was suggested to Russell as an alternative form of his own paradox—known as Russell’s Paradox—which he had devised to show that set theory as it was used by Georg Cantor and Gottlob Frege contained various unresolved issues (Gardner 1979). Russell’s Paradox concerned the definition of the set of all sets, and that such a set should not contain itself, or else the paradox arises. His comment on the Barber Paradox is a relevant one (Russell 1918): That contradiction is extremely interesting. You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as “one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning.

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For his first theorem, Gödel used a modified version of the paradox, replacing a statement such as “this sentence is false” with “this sentence is not provable,” called the “Gödel sentence G.” His proof showed that for any theory T, G is true, but not provable in T. Suppose G is provable, then what it says of itself, namely, that it is not provable, is a contradiction. Suppose that G is not provable, then what it says of itself is true. Therefore, there is no proof to demonstrate that G is provable, and there is also no proof that it is unprovable. Hence, T is incomplete, because it cannot prove all statements within it. Gödel represented statements by numbers and questions about the provability of statements as propositions about the properties of numbers (as discussed previously), which would be decidable by the theory if it were complete. The Gödel sentence, G, states that no natural number exists that would encode a proof of the inconsistency of the theory, T. If there were such a number, then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number. Gödel showed, in effect, that logical systems are human inventions, and like all things human are inevitably faulty. Thus, a central goal of a Vichian approach in the mathematical imagination is to examine the role of rational logic in mathematics, not to equate it with mathematics. The following citation indicates that Vico also saw logic as critical, but only as a point-of-arrival, whereby poetic logic (a blend of fantasia, ingegno, and memoria) remains the originating force of ideas and sensemaking (Vico in Bergin and Fisch 1984: 114): That which is metaphysics insofar as it contemplates things in all the forms of their being, is logic insofar as it considers things in all the forms by which they may be signified. Accordingly, as poetry has been considered by us above as a poetic metaphysics in which the theological poets [early humans] imagined bodies to be for the most part divine substances, so now that same poetry is considered as poetic logic, by which it signifies them. The word logic comes from logos, whose first and proper meaning was fabula, “fable,” carried over into Italian as favella, “speech”. . .whence logos means both “word” and “idea.”

As mentioned, Peirce (1931–1958) differentiated between logica utens (a practical logic) and logica docens (a theoretical or learned logic). Because everyone possesses logica utens, no special training is required to understand what most logical statements are about or what to do in order to solve problems of everyday logic. If we say that John is taller than Mary and that Mary is taller than Bob, then we have no problems concluding that Bob is the shortest of the three. However, understanding formal logical structures or theories, such as the RussellWhitehead one, requires familiarity with logical traditions, theories, and developments. It requires training of some sort—hence logica docens (Pietarinen 2005). In other words, the utens form is universal, while the docens form is based on principles of logic developed by logicians. It was Aristotle who introduced logica docens, with his categorical syllogism, which he believed would show how logical thinking unfolds in the mind. Aristotle claimed that the syllogism is based on common sense, and only gives it a general propositional form (Aristotle in Prior Analytics, 350 BCE). Aristotle started by establishing the kinds of statements that would characterize logical ideas in general form:

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All A are B. No A are B. Some A are B. Some A are not B. The letters A and B, or any other symbols that might be used, are terms that represent various classes of things, such as numbers, people, objects, sounds, animals, and so on. The connectivity of specific premises is what determines the logical validity of a particular syllogism: All mammals are warm blooded. All cats are mammals. Therefore, all cats are warm blooded. The form of this syllogism in its skeletal symbolic logical structure is as follows: All A are B. All C are A. Therefore, all C are B. One does not need to use syllogistic reasoning to accept this as true. Common sense tells us that this is so. However, common sense does not show us the structure of the logic behind it. Moreover, the syllogism may show us why something is anomalous: No cats are planets. Some satellites are not planets. Therefore, some satellites are not cats. This syllogism has the following form: No A are B. Some C are not B. Therefore, some C are not A. Without going into the logical details, this syllogism fails to meet the requirement that the conclusion must be true if the premises are true. It does so by associating entirety (All) with partiality (Some), producing a semantic anomaly, even if it appears to be consistent. In effect, syllogisms enable us to test the validity and meaningfulness of some types of argument. Lewis Carroll became fascinated by the notion of the syllogism. Aware that the pitfalls within it might be due to language itself, he put forth what is perhaps the first diagrammatic system of syllogistic proof in his 1896 book, Symbolic Logic. Below is one of his logic diagrams from the book (Fig. 5.1): Carroll’s aim of converting Aristotelian syllogistic logic into a diagrammatic form lends evidence to Vico’s notion that poetic logic aids the mind in visualizing ideas, including syllogistic ones. Perhaps inspired by Carroll (Wagner 2012), Wittgenstein (1921) also saw visual symbols as a means to ensure that the form of a syllogism could be examined in itself for logical consistency, separate from any content to which it could be applied. If the form held up to logical scrutiny, then that

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Fig. 5.1 Carroll’s Diagrammatic Logic (1896)

was the end of the matter. The problem, Wittgenstein affirmed, was that we expect logic to interpret reality for us. But that is expecting way too much from it. Eventually, Wittgenstein had serious misgivings about his own theory of logic, as he delved more and more in Carrollian logic games (Wagner 2012), which showed that logical arguments never end, thus becoming impracticable in themselves, a conclusion that is imprinted in the words of the character Tweedledee in Carroll’s Through the Looking-Glass (1871): “if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.” In his posthumously published Philosophical Investigations (1953), Wittgenstein became perplexed by the fact that language could do much more than just construct propositions about the world. So, he introduced the idea of “language games,” by which he claimed that there existed a variety of linguistic functions (describing, reporting, guessing riddles, making jokes, and so on) that went beyond simple syllogistic logic. Carroll’s approach to logic remains significant to this day, since it shows that ultimately one has to keep on proving propositions as true ad infinitum in order to keep the system cohesive, making it impracticable. Gardner (1996: 71–73) summarizes this as follows: In logic and mathematics you cannot prove a theorem except within a formal system based on a set of posits or assumptions. But are the assumptions true? To prove them you have to make additional assumptions, and to prove those assumptions requires still further posits.

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You thus seem to be trapped in an infinite regress. Deductions can never reach absolute certainty. You are forced to stop at some point and accept a set of posits as true by an act of faith.

As Carroll certainly knew, Aristotelian logic is nonetheless still useful in everyday life (Carroll 1896): Once master of the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real use to you in any subject you may take up. It will give you clearness of thought—the ability to see your way through a puzzle— the habit of arranging your ideas in an orderly and get-at-able form—and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art.

In the end, logic and mathematics are not one and the same. Moreover, as Smolin (2013: 46) suggests, mathematics may not be able to capture reality because of the limitations of the human mind: Logic and mathematics capture aspects of nature, but never the whole of nature. There are aspects of the real universe that will never be representable in mathematics. One of them is that in the real world it is always some particular moment.

Forms of Logic The Greek philosophers and mathematicians claimed that mathematical truths could be proved in several fundamental ways, such as by induction, deduction, or contradiction (also known as reductio ad absurdum). These were established by Euclid as part of the methodology of mathematics. Induction involves reaching a general conclusion from observing a recurring pattern; deduction entails reasoning about the consistency of a recurring pattern as applied to something new; and reductio ad absurdum involves showing that if some contrary hypothesis would lead to absurdity or contradiction, then its opposite (the original hypothesis) must be true. All these play a significant role in the proof of many mathematical truths. But they hardly explain how the proofs were envisioned in the first place, via hunches and guesses along with their translation into logical constructs of some kind. In effect, any form of proof involves poetic logic at some point, as the initial form of thought that attempts to literally “figure out” what to do or what to think in a specific situation. It also keeps us “on our toes,” so to speak, in assessing the end products of the processes of logical method. Consider the logic behind induction, which can be represented as follows: If a = x If b = x If c = x ...

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Then {a, b, c, . . .} = x Now, consider the following arithmetical computations: Multiplication 2×2=4 3/2 × 3 = 4 ½ 4/3 × 4 = 5 1=3 5/4 × 5 = 6 ¼

Addition 2+2=4 3/2 + 3 = 4 ½ 4/3 + 4 = 5 1=3 5/4 + 5 = 6 ¼

From these examples, we might be led to conclude by induction that multiplying numbers always produces the same result as adding them. But, of course, that is not true. Induction involves enlisting the BMI, as discussed in Chap. 4, whereby a certain imagined scenario is perceived to continue ad infinitum by an “adding one more” procedure. So, in the list above, the metaphor impels us, almost instinctively, to assume that the next observation will be “one more” example confirming the previous set of observations. The BMI crystallizes by observing some recurrence, as we seek an explanation of that recurrence, even if it leads to a falsehood. But the question that our inner dialectic begs, in contrast, is: Is this always the case? Recurrence that does seem to conceal a pattern, on the other hand, is based on what Lakoff and Núñez (2000: 35) call an aspect schema, which involves the general structuring of events: Everything that we perceive or think of as an action or event is conceptualized as having that structure. We reason about events and actions in general using such a structure. And languages throughout the world all have means of encoding such a structure in their grammars. What [research] tells us is that the same neural structure used in the control of complex motor schemas can also be used to reason about events and actions.

The aspect schema is the source of many serendipitous discoveries, leading to poetic abductions, which, as we have seen, often lead to serendipitous discoveries. Consider complex numbers—numbers with a real and an imaginary component. Now, if i is an imaginary number, why not add more sets of similar kinds of numbers, such as j and k, to produce complex numbers with four parts. That idea, as a flash of insight, actually occurred to William Rowan Hamilton (1843), leading to his notion of quaternions, which have the form: w + xi + yj + zk. In this formula, w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions. They are governed by the usual rules of operation. An example is the multiplication table below (Fig. 5.2): Hamilton defined a quaternion as the quotient of two directed lines in a threedimensional space, or, equivalently, as the quotient of two vectors. Now, one may ask: What is the point of such an ingenious invention? As it turns out, quaternions have led to the discovery of many previously unknown number patterns and have had significant applications in physics and other sciences. The point is that the invention of quaternions was due to an ingegno moment, as it has been called throughout this book. Group theory emerged in a similar “poetic”

Forms of Logic Fig. 5.2 A quaternion table

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way, on the basis of a hunch: How can we specify the members of a group so as to be able to combine them to make more members? An example of a group is the integers under the operation of addition, which has the following property: any integer n plus another integer m always produces another integer. However, under division the integers do not form a group, because it is not true that an integer divided by another integer always produces a third integer (Cayley 1854). Group theory originated from the fact that two mathematicians, Neils Henrik Abel and Évariste Galois in the nineteenth century, were contemplating the solutions to polynomial equations (Mackenzie 2012: 118–119). They were looking at quintic polynomials that have no solution. The insight came when they adapted the concept of symmetry from geometry, as the correspondence of the parts of an object in size, shape, and position after certain operations are carried out. This was the imaginative abduction that they needed to develop an overarching theory. The details need not concern us here. Now, while all this may prove to be very interesting, is it nothing more than a thought experiment within mathematics? Does group theory have any discernible meaning? As it has turned out it, it provides an accurate language for many natural phenomena, as Mackenzie (2012: 121) indicates: Chemists now use group theory to describe the symmetries of a crystal. Physicists use it to describe the symmetries of subatomic particles. In 1961, when Murray Gell-Mann proposed his Nobel Prize-winning theory of quarks, the most important mathematical ingredient was an eight-dimensional group called SU(3), which determines how many subatomic particles have spin ½ (like the neutron and proton). He whimsically called his theory “The Eightfold Way.” But it is no joke to say that when theoretical physicists want to write down a new field theory, they start by writing down its group of symmetries.

The foregoing discussion implies that the forms of logic are themselves guided by imaginative processes that are based in image schematic forms. Sometimes, a schema shows up in the actual structure of a proof. For example, the containment and circularity schemas (Lakoff and Núñez 2000) can be enlisted to characterize the cognitive frame of mind that undergirds proof by contradiction, or reductio ad absurdum. We imagine inserting something, like √2, into the collection of rational numbers. By so doing, we can assay if it leads to a type of circularity in logic that leads us to conclude that it does not belong there. This type of proof has become highly important, since it undergirds many computer proofs, such as the one developed for the Four-Color Conjecture, discussed previously.

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As we saw, this form of proof was used by Euclid to demonstrate the infinity of primes. Consider the irrationality of √2, proved as well by Euclid. His proof can be paraphrased (with modern notation and algebraic concepts) as follows. We start by noting that the general form of a rational number is p/q (q ≠ 0). So, if √2 cannot be written in the form p/q, then we could conclude logically that it is not rational—in image schematic terms, this means that it does not fit into the container of rational numbers. Euclid assumed that it could be written in that form, √2 = p/q, and then showed that this would lead to a contradiction. We begin by eliminating the square root sign, by squaring both sides of the equation: √2 = p=q ðassumptionÞ √2

2

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Therefore: 2 = p2 =q2 We then multiply both sides by q2, thus eliminating the denominator on the right side of the equation: 2q2 = p2 Now, 2q2 is an even number, because any number multiplied by 2 is even. So, p2, which represents the same value of the even number on the right side, must of course be even. The general form of an even number is 2n. So, we let p = 2n: 2q2 = p2 Since p = 2n: 2q2 = ð2nÞ2 = 4n2 : Therefore: 2q2 = 4n2 This equation can be simplified by dividing both sides by 2: q2 = 2n2 This shows that q2 is an even number, and thus that q itself is an even number and can be written as 2m (to distinguish it from 2n): q = 2m. Now, let us go back to our original assumption—namely that √2 is a rational number:

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√2 = p=q In this equation, we make the substitutions that we have proved above: p = 2n and q = 2m: √2 = 2n=2m √2 = n=m Now, the problem is that we find ourselves back to where we started—which implies the circularity image schema, whereby we end up with what we started with, rather than establishing something as necessarily so; that is, we have simply ended up replacing p/q with n/m. We could, clearly, continue on in this way, always coming up with a ratio with different letters for numerators and denominators: √2 = n/m, a/b, x/y, and so on. This has been caused by the fact that √2 was assumed to have the form p/q of the rationals, which it obviously does not. Now, proofs by computer have taken proof by contradiction one step further—they let the computer decide if something crops up as a contradiction or not. At this point, it should be mentioned that programming computers to do mathematics has become, actually, a separate branch of its own, called Experimental Mathematics (EM), which has produced truly remarkable results. For example, in 2017, an AI system called AphaGo, beat the world’s Go champion with a creative move that was previously unknown, surprising Go experts, implying that if an AI system can be devised to come up with a truly intelligent move in the game of Go, previously unbeknownst to humans, then the question arises: Can AI create new mathematics? That is the key question in EM, whose primary objective is to carry out “experiments” with algorithms to see what they yield mathematically. These include creating computer programs with the ability to solve mathematical problems, prove theorems, and unravel patterns in classic conjectures. Although its origins go back to computer experiments on theorem proving in the 1950s, the first true advancements in this field are the ones by Donald Knuth, which go back to the late 1960s (Knuth 1968). EM has achieved a number of impressive results, from discovering a formula for the binary digits of π to finding the smallest counterexample to the sum of powers conjecture by Euler. As Borwein and Bailey (2004: vii) aptly point out, the importance of EM inheres in providing a concrete understanding of mathematical properties, by confirming or confronting conjectures, thus making mathematics “more tangible, lively and fun for both the professional researcher and the novice.” As just mentioned, the first practical outcome of the mathematics-computer alliance came in the mid-1950s with automated theorem proving (ATP) (Urban and Vyskočil 2013), which was at first based on the kind of logic discussed above—namely, as formalized by Russell and Whitehead. It was called a firstorder logical proof system, developed initially by Martin Davis in 1954, as also pointed out (see Davis 1958). It was capable of solving a small set of logical theorems and grasping elementary mathematical properties. As Davis quipped about this early system (cited in Davis 2000: 3), as mentioned, but worth

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repeating here: “Its great triumph was to prove that the sum of two even numbers is even.” In 1956, Newell, Simon and Shaw, developed a Logic Theory Machine, reflecting logical symbolism developed in the Principia Mathematica (Newell and Simon 1956), which was able to prove various theorems of the Principia. Work on ATPs has since helped prove a number of hard theorems, such as the Robbins conjecture, which had previously eluded human mathematicians, as mentioned at the start of this chapter (McCune 1997). To distance himself from the ATPs of the past, McCune called his project an experiment in “automated reasoning,” rather than “automated theorem proving.” The core problem in developing automated logic machines is, however, the Entscheidungsproblem (Chap. 2), which originated with Leibniz, who wanted to build a machine that could determine the truth values of all mathematical statements, realizing that this would entail developing a consistent formal language (Davis 2000: 3–20). The problem was elaborated explicitly by David Hilbert (1928), who believed that there was no such thing as an undecidable or unsolvable problem—a belief that was put to rest by Gödel (as discussed throughout). A few years after Gödel’s proofs, Alan Turing (1936) examined the Entscheidungsproblem in terms of his “Turing machine”—a mathematical model of a hypothetical computing machine, which can use a set of predefined rules to determine a result from a set of input variables: Can an algorithm capable of deciding whether a given statement is provable from the axioms using the rules of first-order logic be devised? Turing showed that this was impossible, calling it the Halting Problem (as discussed previously). Given a computer program and an input, will the program finish running or will go into a loop and run forever? Turing argued that no algorithm for solving this problem can exist logically. Here is a paraphrase of Turing’s proof by contradiction (discussed already, but repeated here for convenience): Assume that there is such a program. If so, we could run it on a version of itself, which would halt if it determines that the other program never stops, and runs an infinite loop if it determines that the other program stops. This is a contradiction.

As Lakoff and Núñez (2000) have shown, mathematics makes sense when it encodes concepts that fit our experiences of the world. It is unclear how AI can experience input in the same way, turning it into a system of logical organization with no inconsistencies. And if it did, what would a computer make of the system that it produced? It would still take human intervention to interpret it. Nonetheless, as discussed in Chap. 4, image schema theory has now been adopted to improve computer logic in mathematics to expand its range conceptually. Neural network theory and deep learning systems utilize a version of blending theory that have allowed computers to connect aspects of a situation that simulate how the human brain works in blending different regions. One type of program, called DeepMind, is a very useful one for examining how amalgamating different components of a problem can lead to solutions. It receives as input a series of equations along with their solutions, without any explanation of how those solutions can be reached. In recent versions, DeepMind was devised to “intuit” how to solve equations without any instructions or input structure, based solely on examining a limited number of

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completed examples. As opposed to other AI systems, DeepMind is not pre-programmed, learning by induction, that is, by pattern-extraction from data that is introduced into the system by humans. In a comprehensive article discussing the nature of logic as programmed into computers, Davies et al. (2020) conclude that the computer is essentially a heuristic device that aids mathematicians by allowing them to sift through huge swaths of data and identify patterns within it that have not been spotted before. As such, computers allow for assessing conjectures and proofs for unproven ideas in mathematics in novel ways. They do so through human guidance, in most cases, coming up serendipitously with a proof via their capability of amalgamating seemingly disparate facts into coherent systems. One machine learning technique, called a supervised learning model, has been used to detect previously undiscovered relationships between different kinds of mathematical ideas that have been established already. In effect, AI can find connections that the human mind might not always easily spot because of the enormity of the task. As Black (1962) pointed out at the start of AI, the idea of trying to discover how a computer has been programmed in order to extrapolate how the mind works was bound to become a guiding principle in AI research for the simple reason that algorithms are so understandable in human terms and so powerful in producing unexpected outputs. But there is a caveat here, as expressed by physicist Roger Penrose (1989), that computers can never truly be intelligent or creative like humans because the laws of nature will not allow it.

The Bi-Part Soul Consider, as an example of how poetic and rational logic might form a blended system in the discovery and conduct of mathematics, namely, Ibn Khallikan’s famous chessboard grain puzzle, which he devised in 1256 (Bardi 2019). The original insight for the puzzle, as can be assumed from the formulation of the puzzle itself, came from a flash of insight about putting grains of wheat on a chessboard (Pickover 2009: 102): How many grains of wheat are needed on the last square of a 64-square chessboard if 1 grain is to be put on the first square of the board, 2 on the second, 4 on the third, 8 on the fourth, and so on in this fashion?

Ibn Khallikan’s fantasia was thus the initial spark for the puzzle, as corroborated latently by the fact that today the puzzle still impels us to imagine the collocation of grains at the start, not deduce it logically. Only after, does the solution involve logic in the rational (systematic) sense, albeit in ingenious ways. If one grain (= 20) of wheat is put on the first square, two grains (= 21) on the second, four on the third (= 22), eight on the fourth (= 23), and so on, it is obvious that 263 grains will have to be placed on the sixty-fourth square. Since the first term is 20, and the term 263 is the sixty-fourth term in the sequence, it can be represented as 2n-1. Now, the value of

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263 is so large that no chessboard could ever hold that many grains—it would contain about 1.84 × 1019 grains. This figure amounts to some 3 × 1013 bushels, which is several times the world’s current annual crop of wheat. The problem conceals some truly intriguing patterns that were discovered subsequently. For example, if a second chessboard is placed next to the first, then the pile on the last square (= 128th square) of the second board contains 2127 grains. If we subtract the number 1 from this, (2127–1) we get the following result: 170,141,183,460,231,731,687,303,715,884,105,727. Incredibly, this is a prime number. If we take away one grain from each square, the result would be 2n-1. As it turns out, this formula can be used to test the primality of each square. For example, the fourth square has 23 or 8 grains on it. If we take one away from it, 23–1, we get 7, which is a prime number. These are Mersenne primes, as is widely known, which involve the formula 2n-1 as a test of primality. Applied to Khallikan’s chessboard, the formula produces prime numbers on the squares shaded below (Fig. 5.3): The two formulas 2n-1 and 2n-1 go back to Euclid, but the fact that they show up in a simple puzzle with implications for prime number theory is mind-boggling. It is an example of discovery by happenstance—a phenomenon that has always characterized discovery in mathematics and science, because of the exploratory nature of poetic logic as the starting point; that is, all this came about from an initial imaginative thought, developed through rational logic, and ending up with serendipitous discoveries via the ingenuity of mathematicians. The puzzle exemplifies in

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microcosm how mathematical thinking has always unfolded and what it has allowed us to achieve. In sum, to belabor the obvious, there is no one form of logic in mathematics, which can be formalized in terms of some system of symbols. It can be inductive, deductive, abductive, oppositional, combinatorial (as in group theory), spatial (as in graph theory), analogical, metaphorical, metonymical, probabilistic, iterative (as in fractal geometry), algorithmic, computational, topological, paradoxical, selfreferential, and so on. Any attempt to answer what mathematics is by equating it with logic is a moot point. In his story, Murders in the Rue Morgue, Edgar Allan Poe portrays the method of solving a crime exhibited by his detective character, C. Auguste Dupin, as a blend of rational logic with imagination (creative thinking), which Poe calls a “bi-part” soul (Poe 1841): At such times I could not help remarking and admiring (although from his rich ideality I had been prepared to expect it) a peculiar analytic ability in Dupin. He seemed, too, to take an eager delight in its exercise—if not exactly in its display—and did not hesitate to confess the pleasure thus derived. . .Observing him in these moods, I often dwelt meditatively upon the old philosophy of the Bi-Part Soul, and amused myself with the fancy of a double Dupin— the creative and the resolvent.

Just before writing his story, Poe had apparently developed a keen interest in cryptography, believing that breaking ciphers was required in order to stimulate the imagination together with rational logic—hence the term “bi-part.” In the December 1839 issue of Alexander’s Weekly Messenger, he egged readers on to write to him using substitution ciphers rather than typical written letters, believing that his readers shared his enthusiasm. Interestingly, Poe’s ideas stimulated scientific and professional interest in cryptography. It inspired William F. Friedman, a famous military cryptanalyst, who read Poe’s “The Gold Bug” as a child (which contained a cryptographic mystery). Freidman became renowned for cracking Japan’s so-called PURPLE code during World War II. As an example of how the bi-part soul manifests itself in actual ways, consider the Kissing Number Problem, which asks us to determine the number of spheres that can be arranged so that they touch a given sphere. Below are six spheres that touch the central given sphere. The kissing number in this case is 6 (Fig. 5.4): Clearly, this genre of puzzle involves a blend of poetic logic—imagining how to pack objects—and rational logic, showing how a certain configuration is unique. Indeed, if we arrange the spheres above in different ways, we do not get the solution. From this, we can now examine all kinds of packing problems, where the goal can be: (1) to pack a single container as densely as possible, (2) to pack objects using a minimum number of containers, or (3) to pack them together in a spatially-minimal way. For example, we might be asked to pack ten circles optimally in a circular arrangement. Below is one answer (Fig. 5.5): Without going into details here, packing puzzles have been the source of many mathematical discoveries. Like all other types of problems, they represent situations in a simple yet insightful (imaginative) way. They allow us to eliminate physical intervention through representations of the real world in condensed outline (diagrammatic) forms, which, in turn, allow us to experiment mentally with that very

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Fig. 5.4 Kissing Number Problem

Fig. 5.5 Circle packing problem

world to see what they yield. From them, we can derive implications of a mathematical or scientific nature. Even a Sudoku puzzle requires that the solver imagine, or envision, specific number placements, combinations of numbers in a row or column, for example, and the like, before employing a placement logic systematically. When a particular state of the grid does not concede a definitive placement pattern, then the imagination suggests that we use hypothetical placements and make appropriate deductions from them. Thus, logic and imagination are blended to produce the relevant insight. Actually, it can be said that the operation of one type of logic or other is a matter of gradation. If it is in fact valid that mathematical thinking involves both imagination and strict logic (such as deductive logic), then we can perhaps represent it as

The Origins Question Redux

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varying according to a scale, ranging from I to L, where I is pure imagination and L pure logical deduction (Fig. 5.6): So, any particular mathematical idea, proof, technique, etc., can be represented as falling on this hypothetical scale. The closer it is to I, the more it requires activating the imagination; the closer it is to L, the more routine systematic thinking is involved. Needless to say, this is a relative assessment, and will vary significantly among individual mathematicians and in terms of the kinds of mathematics involved. Nevertheless, it is a useful way to summarize the discussion in this chapter. With the rise of mathematics as a discipline, the need for ordered classificatory thinking, as opposed to inferential guessing, came to assume a privileged place in philosophy and mathematics. But, as we have seen at various points in this book, the fantasia has always played a silent role in novel ideas and in discovery. Imagination and logic are inextricably intertwined—they are products of interacting parts of the brain, distributed throughout the cortex. Signs in mathematics, from annotations to diagrams, give expression to this inextricability and, thus, invariably shed light on snippets of mathematical thinking. The problem has always been devising an overall picture of mathematics. Signs lead to piecemeal models of things, not overarching theories of the world. Anthropologist Bronislaw Malinowski (1948) believed, similarly, that early symbol systems reflected the ability to intuit and reason at once. Claude Lévi-Strauss (1978) saw symbolism as a form of connective, imaginative logic. People used symbols to express, for the most part unconsciously, their deeply held ideas about themselves and their relationship to the world around them. In a fundamental sense, advanced discoveries in mathematics are the result of using the symbolism developed through a process that is activated by poetic logic. So, the number signs of early cultures provide the first, metaphorical symbolism swiping how we initially come to grips with reality; as cultures developed, the signs evolved into symbolic artifacts and to increasingly larger symbolic narratives, including early treatises in science and mathematics, such as the Rhind Papyrus.

The Origins Question Redux The last point brings us back to the origins question (Chap. 1): How did mathematics originate? Like the early myths, there is little doubt that it too started in primordial metaphorical cognition, which is the primary product of the fantasia: “the mythologies, as their name indicates, must have been the proper languages of the fables; the fables being imaginative class-concepts” (Vico in Bergin and Fisch 1984: 139).

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In terms of the origins question, therefore, it is useful to take a rapid look at some relevant research: 1. Some animals (primates, rodents, felines) seem to have a rudimentary sense of number (Dehaene 1997). The experiments utilized to determine this have included training a rat to press a bar a number of times to obtain a food reward. In another type of experiment, hidden speakers were used to transmit from 1 to 5 artificial male lion mating calls. If a lioness heard three calls, she would leave, presumably feeling outnumbered. If she was with four other lionesses, however, the five together would go and seek out the mating caller (McComb et al. 1994). In both experiments, the ability to distinguish number, or numerosity, seems to be at play. There is no evidence, as far as can be told, of other species converting numerosity into numeration, as do humans. 2. Psychological studies have shown that infants display a similar kind of rudimentary number sense. When presented with arrays of 4 to 8 dots on various surfaces or objects, infants respond in a patterned way: If they had become habituated to, say, 8 dots, they would stop looking at novel displays involving that number of dots. But by presenting them with a display of a different number of dots for the first time, they once again became intrigued. This shows that even during infancy, there is a capacity for association number sense with numeration. 3. Neuroimaging studies have made it obvious that the parietal lobe and the inferior parietal lobule are activated in subjects asked to carry out calculation tasks. It is no coincidence, therefore, that the left parietal lobe controls the movement of fingers, constituting a neurological clue to the evolution of our number sense, explaining why we count on our fingers. This is a significant finding, because the same regions are involved in language and gesture. 4. Studies have shown that mathematical thinking might overlap with spatial cognition—a topic treated extensively by Walter Whiteley (2012). Without going into specifics here, suffice it to say that, perhaps, the most significant finding in this domain is that spatial cognition overlaps with conceptual metaphorization. Brian Butterworth (1999) has argued cogently that numerosity constitutes a separate and unique kind of intelligence with its own brain module, located in the left parietal lobe, as mentioned. So, the reason a person falters at mathematics is not because of a “wrong gene” or “engine part” in the left parietal lobe, but because the individual has not fully developed the number sense with which they are born, and the reason is due to environmental and personal psychological factors. Butterworth presents findings that neonates can add and subtract even after a few weeks, and that people afflicted with Alzheimer’s have unexpected numerical abilities. Overall, the research is suggestive of how language and mathematics may have originated with respect to each other. Interestingly, work in evolutionary anthropology has brought forth findings, which suggest that mathematics and language are united by several key evolutionary factors (Cartmill et al. 1986). Both are a consequence of four critical evolutionary events—bipedalism, a brain enlargement unparalleled among species, an extraordinary capacity for tool-making, and the advent of

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the tribe as the main form of human collective life. Bipedalism liberated the fingers to do several things—count and gesture. The former is the basis for developing numerosity symbolically, the latter for language. Both likely occurred in tandem, thus seemingly negating any uniqueness to finger-use for the math ability, since it was also needed for verbal gesturing. Although other species, including some nonprimate ones, are capable of tool use, only in the human species did complete bipedalism free the hand sufficiently to allow it to become a supremely sensitive and precise manipulator and grasper, thus permitting proficient tool making and tool use in the species. Shortly after becoming bipedal, the evidence suggests that the human species underwent rapid brain expansion. In the course of human evolution, the size of the brain has more than tripled. The size of the brain, however, does not determine the degree of intelligence of the individual; this appears to be determined instead by the number and type of functioning neurons and how they are structurally connected with one another. And since neuronal connections are conditioned by environmental input, the most likely hypothesis is that any form of intelligence, however it is defined, is most likely shaped by upbringing. Like most species, humans have always lived in groups. Group life enhances survivability by providing a collective form of shelter and a context for the partitioning of labor. But at some point in their evolutionary history—probably around 100,000 years ago—bipedal hominids had become so adept at tool-making, communicating, and thinking in symbols that they became consciously aware of the advantages of a group life based on a common system of representational activities. By around 30,000 to 40,000 years ago, the archeological evidence suggests, in fact, that hominid groups became increasingly characterized by communal customs, language, and the transmission of technological knowledge to subsequent generations. The early tribal collectivities have left evidence that gesture (as inscribed on surfaces through pictography) and counting skills, as imprinted in tally bones and sticks, occurred in tandem. The early tally symbols then evolved into more abstract symbols, from the first sets of cuneiform numerals (chosen for the ease with which they could be carved into wooden sticks), to modern numerals and numeral systems. Aware of the limitations of human knowledge, Vico, as argued throughout this book, claimed that the essential nature of mind could be unraveled by considering what its “modifications” revealed, that is, by examining the symbols that the mind has made possible and which have come to constitute how human sapience is expressed and remembered. The modifications made to mathematics over time have lead to modifications in how we understand physical reality. The meaning of the term science in Vico’s New Science implies an evolutionary view of how symbols and ideas develop in tandem. In Western culture, this term has always been synonymous traditionally with the “objective” knowledge of “facts” of the natural world, gained and verified by exact observation, experiment, and ordered thinking. Vico, like quantum physicists today, broke away from this tradition. For Vico, we can know only what we ourselves have made, including mathematics and physics. As Bergin and Fisch (1984: xxxi) have aptly put it, for Vico we “can have scienza in mathematics, because we are deducing the consequences of our own definitions and postulates; and we can have scienza in physics to the extent of our

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capacity for experiment.” For Vico, the “laws” of Euclidean geometry, for instance, are not present in the universe in the precise ways that this form of geometry has specified them. Rather, they constitute a cognitive strategy for organizing and rationalizing our visual perceptions of space. The laws of quantum physics actually bear this out, for something can be a wave according to one model, and a particle according to another, and even a wave-particle according to a third. Our mathematical constructs capture reality in a fleeting instant as it morphs into something else, often a paradox of sorts. So, what is the connection between mathematics and reality? The connection between the imagination, ingenuity, mnemonic symbolism, and mathematics is a dynamic one, with one guiding the other. From the archeological record, it can be asserted that numerosity may be instinctive, but when humans started envisioning counting objects as generalized ideas, they started producing symbolic constructs, such as the Ishango Bone. The first sign of mathematics as an abstraction system resulting from the imagination and the ingenuity of people to produce forms of meaning is in early numeration systems, such as the one exemplified by this bone. Perhaps, the most relevant of all questions is whether we really need mathematics. Since counting is instinctive (numerosity) and since we could carry out practical affairs with intuitive arithmetic and geometry, we could potentially do without mathematics. But then, the same comment could be made about language. We could survive without it, because of our innate corporeal signaling systems. But language and mathematics go far beyond subserving instinctive survival. So, the question of why mathematics is the same as why language, art, music, and all the other creative products of the human brain. Perhaps, we can never explain scientifically why such expressive-representational forms have evolved. We seem to have within us a kind of “poetic instinct,” an instinct to understand things in a conceptual way, rather than just practical. In his truly insightful treatise, the Novum Organum (1620), Francis Bacon praised the ancient Greeks for making abstraction (going from particular facts to general ideas that can be extrapolated from them) as the key achievement of philosophy and mathematics. A little later, in 1644 (the date of the first edition of La scienza nuova), Vico also praised the ancients for showing how abstraction was the greatest achievement of the human mind; but unlike Bacon who saw rational logic as the mental force behind abstractions, Vico saw the imagination as the “creative fuel” initially behind it. There is no way to explain the imagination, however, because it takes the imagination to do so—thus creating an endless loop of speculation. As he emphasized, “The criterion and rule of the true is to have made it.” And for this reason, the mind can never know itself. So, answering what the mathematical imagination is and how it originated can never be truly grasped in any truly understandable way; only observed via its “luminous” products, including numeration systems, proofs, and other modes of abstraction. And yet, mathematics allows us to gain knowledge about the very world that is outside of our minds. In his book Il saggiatore (1623), Galileo puts it as follows (cited in Popkin 1966: 65):

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Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth.

As modern quantum physics attempts to develop a theory of everything on the basis of increasingly abstract mathematics, it does not seem far-fetched to assert that our brains may well be designed to create such things as mathematics to better understand both the outside world, which von Uexküll (1909) called the Umwelt, and our connections to it, called the Innenwelt. Arguably, mathematics originated from the dynamic interplay between the Umwelt and the Innenwelt. Pythagoras and Plato regarded the world as a whole with interconnected parts, each one of which could be studied mathematically. The basic forms of reality are thus immanent in the universe, and these forms are subject to mathematical laws. The problem is that every once in a while, there emerges a fault in the mathematics, which may have its source in the mind, as Gödel made “logically clear” to that mind, using logic itself to deconstruct it.

Epilogue According to a legend, the Greek philosopher Parmenides invented logic as he sat on a cliff meditating about the world. Whether true or not, the point is that it characterizes the connection made between logic and thought, which, as David Gallop (1991: 3) has aptly observed, has shaped the view of mathematics in Western traditions as a manifestation of “explicit and self-conscious argumentation.” Thomas Hobbes (1656) even claimed that rational logic, such as the type displayed when we do arithmetic, was the defining attribute of the human mind. But the history of mathematics shows that logic is only a point-of-arrival in mathematical thinking, which, as claimed throughout this book, starts out instead as poetic logic—a form of creative thinking that was used by Cantor, Gödel, Turing, and others to show how logical systems themselves are constituted through human ingenuity. As Rucker (1987: 218) has aptly observed, the “great dream of rationalism has always been to find some ultimate theory that can explain everything,” but Gödel showed that this is not achievable, because human logical systems are ultimately imperfect, made by imperfect logicians. This does not deny the need for a logica docens for carrying out mathematical tasks and for stabilizing mathematical knowledge systematically, which, starting with Euclid, has provided an enormous mathematical memoria on which to build for future explorations. In Carroll’s Alice’s Adventures in Wonderland (Carroll 1865: 71–72), Alice asks the Cheshire cat: “Would you tell me please, which way I ought to go from here?” The cat’s answer is ironical, yet revealing: “That depends a good deal on where you want to get to,” to which Alice responds with “I don’t much care where.” The shrewd cat’s rejoinder to Alice’s response is Vichian in nature: “It doesn’t matter which way we go.” In effect, the cat is alluding to the fact that the (mathematical) fantasia is

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constantly roaming inside the mind, seeking new paths and attempting to unveil new vistas (literally). This quest for meaning was actually ritualized at the origins of mathematics as a distinctive discipline, as can be seen from the Pythagorean practice of giving sacrifice to the gods for mathematical discoveries and the seventeenthcentury practice on the part of Japanese mathematicians of giving sangaku (the Japanese word for “mathematical tablet”) to the spirits for allowing them to discover ideas and proofs. The ancients even thought that mathematics was the code that allowed humans to unlock all the secrets of the universe. Pythagoras is reported as saying in this regard: “There is geometry in the humming of the strings, there is music in the spacing of the spheres” (James 1995). This view still holds great intuitive appeal, as Arthur Koestler insightfully remarked in his book The Sleepwalkers (1959): These geometrical ratios are the pure harmonies which guided God in the work of Creation; the sensory harmony which we perceive by listening to musical consonances is merely an echo of it. But that inborn instinct in man which makes his soul resonate to music, provides him with a clue to the nature of the mathematical harmonies which are at its source. The Pythagoreans had discovered that the octave originates in the ratio 1:2 between the length of the two vibrating strings, the fifth in the ratio of 2:3, the fourth in 3:4, and so on. But they went wrong, says Kepler, when they sought for an explanation of this marvellous fact in occult number-lore. The explanation why the ratio 3:5, for instance, gives a concord, but 3:7 a discord, must be sought not in arithmetical, but in geometrical considerations.

The search for hidden Order is what poetic logic impels us to constantly pursue. It turns instinctive curiosity into a quest for meaning. Vico’s notion is not a precise model of how the mathematical imagination originated or even works, as mentioned several times, because as Vico maintained it cannot study itself; but it does provide a lexicon that it highly synchronized with how we do mathematics and with current research on mathematical cognition. Ideas come into view (literally) within the mind as we contemplate a problem in geometry, arithmetic, or even logic. This inner vision would remain dormant, however, if not for our inbuilt ingenuity to turn it into something practicable (a diagram, an equation, and so on), by amalgamating known facts that allow us to devise a logical sequence of statements or maneuvers that provide a solution. The whole process, which goes from poetic to rational logic, is recapitulated every time novel information and ideas confront mathematicians, as argued throughout this book. Aristotle considered phantasia as the capacity for eliciting mental images (Chap. 1), which led to the notion of the “mind’s eye” put forth by Cicero (55 BCE). It is this inner eye of the mind, metaphorically speaking, that the fantasia activates constantly, wherever its neural substrate is. As Vico (in Bergin and Fisch 1984: 143–144) put it, alluding to mathematics indirectly, the start of “theorems and mathemata” is through “the bodily eyes,” which turn inward to produce inner vision. However, we must always be aware that, sometimes, the images produced by the mind’s eye are skewed, as can be gleaned from the one of lessons inherent in Flatland, given that outer vision is constrained by the type of space in which we live. And this is the reason why often mathematical ideas go awry or astray. The same kind of argument was made by Plato in his Republic (2021a), in which he presents his famous Allegory of the Cave, whereby a prisoner, bound and incapable

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of moving, sits in a cave with his back to a fire watching the shadows cast on the wall before him by people carrying objects behind his back. The shadows represent the fragment of reality that we perceive through our senses, while the objects represent the true forms of objects that we can only perceive through reason. But what Plato seems to have ignored is that we can “reason” at first through fantasia, which is not constrained by the physics of vision. The mental images that come from perception are in fact limited by a visual system adapted to three dimensions from evolution. But when these are acted upon by the fantasia (as distinct from simple imagination), they allow for experimentation with perception itself, becoming the source of ideas and novel systems of thought. In a way, Lewis Carroll was Vico’s direct intellectual descendant, who saw with his inner eye that below the logical forms of mathematics, there is a deep meaning that our fantasia continues to explore. In Alice in Wonderland, we see Alice entering the rabbit hole, because she was feeling bored and drowsy, noticing a White Rabbit who was muttering “Oh dear, Oh dear, I shall be late” before entering the hole. She follows the rabbit down when suddenly she falls into a strange world. It is the world of the imagination, which she describes rather aptly as “curioser and curioser.” There she discovers mathematics in the form of paradoxes, puzzles, and riddles, which is perhaps the only way to decipher what the mathematical imagination is all about, as Kasner and Newman (1940) clearly understood. To conclude, it is actually useful to return one more time to Kasner and Newman’s book, Mathematics and the Imagination, in which the authors use an apt metaphor to characterize the mathematical imagination, calling it a “mathescope,” which, like a telescope, allows the mind to see beyond the constraints of real space and find new meaning in reality (Kasner and Newman 1940: 26): The mathescope is not a physical instrument; it is a purely intellectual instrument, the everincreasing insight which mathematics gives into the fairyland which lies beyond intuition and beyond imagination. Mathematicians, unlike philosophers, say nothing about ultimate truth, but patiently, like the makers of the great microscopes, and the great telescopes, they grind their lenses. . .[through which] strange sights [appear].

In a substantive portion of their book, Kasner and Newman (1940: 220–212) illustrate what the mathescope can and cannot do, using optical illusions to show how inner vision and outer vision may sometimes clash. Consider, for example, Schröder’s Reversible Staircase, a picture in which there seem to be two different staircases at the same time: one going up from right to left, the other turned upside down, named after the German scientist Heinrich G. F. Schröder, who devised it in 1858 (Fig. 5.7): When concentrating on A, one staircase appears to jut out; however, when looking at B the upside down one suddenly appears to our perception. This illusion projects the mind toward the “I” end of the mathematical thinking scale (above), since rational logic cannot be easily applied to explaining the illusion (if at all). Another relevant optical illusion is the one discussed by Penrose and Penrose (1958), showing a staircase, which can be climbed or descended forever without getting any higher or lower (Fig. 5.8): Again, this baffles us, as we might seek to find an answer to the illusion that would make sense. But the answer does not seem to come in any logical way. It

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Fig. 5.7 Schröder’s Reversible Staircase. (Wikimedia Commons)

Fig. 5.8 Impossible Staircase (Penrose and Penrose 1958)

suggests, perhaps, that perception and logic may sometimes be disconnected entities of mind. As is well known, the master artist who used optical illusions consistently in his art works was Maurits Escher (Hofstadter 1979), who saw the implications of the relations among art, perception, and the mathematical imagination. As Shepard (1990: 168) has aptly put it, our “visual experience evidently is the product of highly sophisticated and deeply entrenched inferential principles that operate at a level of our visual system that is quite inaccessible to conscious introspection or voluntary control.” In effect, these illusions show how artistic traditions affect perception—an implication that is highly relevant to mathematical history, as Kasner and Newman argue. Drawing pictures of the world, filtered and guided by the fantasia’s inner visions, has conferred upon the human species the ability to both encode perceived reality and to play with it ingeniously. In the end, mathematics is, as Kasner and Newman (1940: 358) assert, echoing Vico, “a creation of the mind, both mystic and pragmatic in appeal. . .wrought by imagination and logic out of a handful of childish rules.” The aim of this book has been to provide a Vichian framework for understanding, or at the very least, describing that handful of childish rules, for in the end, as Kasner and Newman (1940: 358) emphasize, the question “What is mathematics? must go unanswered.”

Glossary

abduction Charles Peirce’s term for inferential thinking based on connecting bits of experience together in such a way that an insight emerges. Basic Metaphor of Infinity Term introduced by George Lakoff and Rafael Núñez to explain the mathematical concept of infinity as an image schema in which there is always an “add on” in the configuration of the schema. The simplest example is that of the linear layout of integers, which never ends because of the “add-on” feature. bi-part soul Edgar Allan Poe’s notion that the mind has two dimensions, imagination and logic, that it uses constantly in figuring out things. characteristica universalis Leibniz’s idea of a universal symbolic language able to express mathematical, scientific, and philosophical concepts unambiguously. conceptual blending Notion in cognitive science by which abstract concepts are formed in the brain by the amalgamation of different regions to produce single abstractions. conceptual metaphor In cognitive linguistics, a generalized metaphorical idea that undergirds large segments of cognition and language. For example, the notion that numerical increase is upward and decrease is downward is a conceptual metaphor that can be seen not only in language but in diagrammatic representations such as graphs. eidos Ancient Greek term referring to the distinctive expressions associated with the cognitive or intellectual character of a culture or an individual. eikon Ancient Greek term referring to the image that the mind elicits in specific ways, and to the forms (words, symbols, etc.) that represent something via imagery. Entscheidungsproblem The so-called decidability problem in mathematics and computer science, which considers if any problem (or input) is decidable or if it is valid logically; that is, given all the axioms of mathematics it asks if there is an algorithm that can tell if a proposition is provable.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Danesi, Poetic Logic and the Origins of the Mathematical Imagination, Mathematics in Mind, https://doi.org/10.1007/978-3-031-31582-4

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fantasia Italian for “imagination” and “fantasy” as a singular thought form, which, according to Giambattista Vico, is the universal state of mind for discovery and invention. iconicity Notion in semiotics whereby a sign stands for it referent by resemblance; for example, in the Roman numeral system, the first three numerals, I, II, III, stand iconically for the notions of “oneness,” “twoness,” and “threeness,” since the form of the numeral delineates visually the quantitative nature of the referent. idealized cognitive model Notion in cognitive linguistics that image schemas converge around a core concept; for example, the concept of path in mathematics is delivered by a convergence of referential domains such as line, movement, and orientation image schema Notion in cognitive linguistics, which claims that abstractions are the result of recurring structures of experience that are imagined in outline form; for example, the image schema of a container undergirds operations such as numerical values, since the latter is perceived as consisting of containers of different sizes. indexicality Semiotic notion referring to the process of representing something in terms of its existential relation to something else: for example, an arrow pointing to the left or right on a page as representing location or representing a vector notion. ingegno Italian word for “ingenuity,” used by Vico to refer to the ability of the human mind to convert imagined constructs into actual expressive artifacts; for example, Euler’s graph of the Königsberg Bridges situation is an ingenious representation that came from the fantasia, or the creative imagining of the situation. Innenwelt Semiotic term referring to the brain’s modeling system, whereby information is turned into models (signs and other structures) allowing it to interpret it in a particular way. logica docens Charles Peirce’s term for logic that is learned formally, such as Aristotle’s syllogistic logic. logica utens Charles Peirce’s term for the kind of innate practical logic that allows humans to instinctively figure out how things in the world work or fit together. memoria Italian for “memory,” used by Vico in reference to how the mind can conjure up what it knows in new and imaginative ways. metafora Italian for “metaphor,” used by Vico in reference to the first words and thoughts as resulting from experiential associations, as for example, naming the bottom part of a mountain as a “foot.” poetic logic Vico’s notion that thinking involves the coordination of imagination, ingenuity, and memory in such a way that it allows us to understand and discover things constantly. poetic wisdom Vico’s notion that we are all born with good judgment about the nature of things, turning it into expressive forms. scienza Italian for “science,” used by Vico in its original sense of “knowledge.”

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semiosis Semiotic term referring to the innate ability to produce and understand signs. source domain The set of concrete concepts that undergird a conceptual metaphor; for example, the human body could be the source domain for understanding abstract concepts such as superiority (she is the head of mathematics at that university). target domain The actual topic of a conceptual metaphor; for example, to grasp the notion of infinity—the target of understanding—the metaphor of a path might be used, which can go on forever. Umwelt Semiotic term referring to the world of experience that is filtered by the human brain and turned into signs and sign systems. universali fantastici Italian for “imaginative universals,” used by Vico in reference to the images that humans the world over share to understand reality; an example would be tally marks to represent numerical quantities. verum-factum Latin for “the truth is made,” used by Vico in reference to the creative artifact, such as number systems, that humans make in order to grasp truths on their own terms.

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Index

A Abacus, 21, 22 Abbott, E.A., 4–7 Abduction, 19, 20, 27, 28, 30, 33–35, 64, 69, 136, 137 Abu Al-Wafa, 70, 71, 78 Achilles and the Tortoise Paradox, 115 Age of equals, 99 Age of heroes, 98 Age of the gods, 98 Alcuin of York, 13 al-Khwarizmi, Mohammed ibn-Musa, 21, 73, 75 Appel, K., 44, 46 Archimedes, 24, 37, 38, 57, 64 Argand diagram, 79, 82, 83 Aristotle, 2, 11, 18, 67, 95, 115, 125, 132, 150 Arrow Paradox, 115 Artificial intelligence (AI), 92, 93, 104, 139–141 Aspect schema, 136 Automated theorem proving (ATP), 139, 140 Axioms, 7, 52, 53, 73, 74, 78, 106, 108, 125, 127, 129, 130, 140

B Babylonian numerals, 77 Bach, J.S., 12 Bacon, F., 148 Barber Paradox, 53, 131 Basic Metaphor of Infinity (BMI), 116, 120, 121, 136

Bauplan, 91, 92 Beltrami model, 108 Bimodality, 25 Binary number, 23, 119 Bi-part soul, 141–145 Boole, G., 126, 128, 129 Butterworth, B., 146

C Cantor, G., 9, 10, 78, 89, 116–120, 126, 131, 149 Cantor’s sieve, 117 Cardinality, 75, 118 Carroll, L., 17, 18, 133–135, 149, 151 Cartesian plane, 79, 102, 103, 105, 114 Characteristica universalis, 125, 128 Church, A., 53 Circle Packing Problem, 144 Clustering structure, 109 Collatz, L., 43, 48, 50, 51, 54 Collatz’s conjecture, 43, 47–50 Complex number, 79, 82, 83, 100, 112, 136 Compression, 68, 78 Computer science, 55, 56, 71, 101, 104, 129 Conceptual blending, 113–114 Conceptual metaphor, 97–99, 101–107, 109, 110, 113, 146 Conjecture, 6, 30, 33, 42–51, 54–57, 64, 127, 137, 139–141 Coordinate geometry, 79, 85, 90, 102, 122

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Danesi, Poetic Logic and the Origins of the Mathematical Imagination, Mathematics in Mind, https://doi.org/10.1007/978-3-031-31582-4

169

170 D Decidability, 51–57, 62 Decimal system, 73, 74 Deduction, 15, 18, 30, 51–53, 127, 135, 144 Dehaene, S., 25, 30, 146 De Morgan, A., 44, 46, 128, 129 Derivative, 59, 96, 105, 106, 114, 128 Descartes, R., 8, 68, 75, 76, 85, 90, 102, 106, 120, 122, 125 Diagonalization, 9 Diagonal proof, 9, 10, 120 Diagram, 10, 15–17, 25, 28, 35–37, 39, 41, 42, 44, 46, 51, 52, 59, 60, 63, 67, 70, 71, 78– 85, 89, 101, 105, 107, 108, 119, 122, 133, 145, 150 Diagrammatology, 82 Dichotomy Paradox, 116 Diophantus, 43, 44 Discovery, 4, 7, 11, 12, 19, 28, 29, 33, 38, 40, 44, 55–64, 67, 69, 72, 74, 79, 82, 83, 85, 89, 90, 92, 93, 95, 100, 108, 112, 114, 115, 123, 125, 136, 141, 142, 145, 150 Double articulation, 72 Doxiadis, A., 48 Dudeney, H., 84, 85, 130

E Eight queens problem, 41, 42 Entscheidungsproblem, 53, 140 Episodic memory, 69, 70 Eratosthenes, 59, 60 Euclid, 17, 18, 33–36, 51–53, 58, 64, 88, 107, 108, 125, 127, 128, 135, 138, 142, 149 Euclidean geometry, 15, 16, 18, 31, 61, 109, 113, 125, 148 Euclid’s Elements, 33, 51, 88 Euler, L., 26–31, 33, 36, 47, 51, 64, 76, 79–83, 139 Euler circle, 80 Existential Graphs, 80, 81 Experimental Mathematics (EM), 139 Exponent, 68, 72, 86, 90 Exponential notation, 68, 69, 72, 90 Exponentiation, 69, 91

F Fantasia, 2, 4–11, 13–15, 19–26, 28, 29, 31, 33, 34, 36, 39, 41, 43, 52, 57, 64, 67, 69, 70, 78, 83–85, 89, 93, 95, 102, 107, 116, 118, 123, 127, 132, 141, 145, 149–152 Fechner, G., 7

Index Fermat, P. de, 43, 44, 122 Fermat’s Last Theorem, 43, 44, 57, 65, 72 Fibonacci, L., 21, 73, 87, 88 Fifth postulate, 107, 108 Flatland, 4–8, 29, 57, 65, 84, 150 Flatland: A Romance of Many Dimensions, 4 Formal logic, 127, 128, 130, 132 Four-Color Problem, 43 Fourteen/Fifteen game, 50 Fractal, 88, 89, 143 Frey, G., 43 Fundamental Theorem of Algebra, 76 Fundamental Theorem of Arithmetic, 33

G Galileo, 116, 117, 148 Gardner, M., 16–18, 131, 134 Gauss, C.F., 7, 23, 38–42, 76 Gödel, K., 9, 55, 120 Gödel sentence, 132 Goldbach, C., 43, 47, 48, 54 Goldbach’s Conjecture, 47, 48, 54 Golden ratio, 88, 89 Googol, 24 Googolplex, 24 Graph, 14, 26–31, 33, 46, 49, 50, 54, 56, 65, 79–83, 88, 96, 103, 107, 110 Graph theory, 14, 28–30, 33, 46, 54, 56, 65, 79, 82, 83, 143 Group theory, 136, 137, 143 Guthrie, F., 44

H Haken, W., 44, 46 Haken-Appel proof, 46 Halting problem, 53, 54, 140 Hamiltonian cycle, 54 Helmholtz, H. von, 7 Hexagonal pattern, 92 Hilbert, D., 126, 129, 140

I Ibn Khallikan’s chessboard, 141, 142 I Ching, 23 Iconicity, 20, 21, 31, 77 Image schema, 9–11, 14, 22–24, 28, 34, 37, 39, 46, 48, 50, 51, 66, 74, 80, 86, 93, 96, 98, 100–102, 104–112, 114–116, 119, 121– 123, 139, 140

Index Imagination, 1–31, 43, 53, 56–58, 66, 68–71, 73, 78, 82, 87–90, 93, 95, 102, 106, 107, 122, 123, 132, 143–145, 148, 150–152 Imaginative universal, 14, 74 Impossible Staircase, 152 Incompleteness, 9, 10 Indexicality, 21 Induction, 18, 51–53, 102, 116, 127, 135, 136, 141 Infinite series, 92, 116 Infinity, 18, 35, 50, 52, 59, 71, 83, 100, 107, 110, 114–121, 126, 128, 138 Ingegno, 2, 7, 8, 10, 12–16, 18, 21, 22, 24–26, 31, 33–44, 46, 48, 50, 51, 53, 55–57, 59, 63–67, 69, 70, 72, 78, 83, 84, 89, 95, 105, 113, 118, 132, 136 Innenwelt, 149 Invention, 3, 12, 21, 33, 46, 57–65, 68, 69, 71, 79, 85, 89, 90, 93, 95, 100, 114–116, 132, 136 Irony, 95, 96 Irrational number, 60, 63, 73, 112, 123 Ishango Bone, 3, 20, 23, 31, 70, 74, 148

J Johnson, M., 10, 99, 107, 109

K Kant, I., 42, 96, 121 Kasner, E., 2, 4, 24, 151, 152 Kissing Number Problem, 143 Knuth, D.E., 139 Königsberg Bridges Problem, 26, 79, 83, 110

L Lakoff, G., 9, 10, 24, 80, 95, 96, 99, 102, 103, 106, 107, 112, 113, 116, 119, 120, 122, 123, 136, 153 Law of Octaves, 12 Legendre’s theorem, 82 Leibniz, G.W., 23, 28, 125, 126, 128, 129, 140 Liar Paradox, 53, 129, 130, 132 Lobachevskyan geometry, 108 Logarithm, 55, 69, 100 Logic, 7–10, 13–19, 23, 25, 26, 28, 30, 33, 36, 42, 53, 56–58, 62, 64, 67, 78, 84, 95, 97–99, 101, 106, 107, 114, 116, 125–152 Logica docens, 126, 132, 149 Logica utens, 126, 132

171 Looping structure, 83–89, 106 Loyd, S., 50, 51

M Machine logic, 127 Mandelbrot, B., 88, 89 Mathematical imagination, 3, 4, 9, 13, 17, 19, 30, 43, 53, 68, 78, 88, 90, 132, 148, 150, 151 Mathematics and the Imagination, 24, 151 Memoria, 7, 10, 13, 14, 19, 22, 25, 31, 62, 67–77, 79, 82–85, 88–91, 93, 95, 114, 132, 149 Metafora, 95, 97–102, 107, 121 Metaphor, 2, 11, 15, 65, 66, 95–123, 126, 127, 136, 151 Metonymy, 95, 121 Möbius, A., 44 Musica universalis, 23 Myths, 98, 145

N N-dimensions, 6 Negative number, 73, 74, 97, 109, 122 Network, 27, 28, 33, 46, 79, 83, 89, 93, 110, 140 Neumann, J. von, 55 Newlands, J.A.R., 12 Newman, J., 2, 4, 24, 151, 152 Notation, 43, 51, 53, 67–77, 83, 87, 90, 91, 130, 138 Novum Organum, 148 Number line, 55, 73, 74, 77, 96, 97, 99–102, 107, 109, 110, 114, 118, 122, 126 Number system, 22, 63, 65, 79 Numeration, 2, 3, 10, 20–24, 67, 70, 72, 73, 107, 146, 148 Numerosity, 146, 148 Núñez, R., 9, 24, 80, 95–99, 102, 103, 106, 107, 113, 116, 120, 136, 140

O Optical illusion, 151, 152 Origins of mathematics, 122

P Papadimitriou, C., 48 Pappus, 91, 92 Parmenides, 115, 149

172 Pascal, B., 85–87 Pascal’s Triangle, 85–87 Peano, G., 73, 128, 129 Peirce, C., 1, 4, 19, 21, 28, 65, 75, 80, 81, 96, 126, 132 Periodicity, 22 Periodic Law, 12 Phoneme, 64, 72 Pi (π), 37–40, 57, 58, 73, 82, 92, 112, 139 Piaget, J., 22 Place-value, 21, 22 Plato, 8, 61, 62, 65, 67, 95, 149, 150 Plimpton 322, 90 P = NP, 54, 55, 57 Poe, E.A., 128, 143 Poetic logic, 2, 7–10, 12–20, 23, 26, 28, 33, 36, 42, 57, 58, 62, 64, 66, 67, 78, 84, 95, 97–99, 101, 107, 121, 126, 127, 132, 133, 135, 142, 143, 149, 150 Poetic wisdom, 7, 8, 11–14, 26 Poetry, 2–4, 11, 31, 127, 132 Poincaré, H., 30, 82, 89, 113 Prehistoric bones, 3, 19 Prime number, 33, 34, 47, 55, 142 Principia Mathematica, 127, 140 Proof, 7, 9, 10, 15, 16, 18, 28, 30, 31, 33–40, 42–44, 46, 48, 49, 51–57, 59, 63–65, 69, 76, 78, 83, 91–93, 116–120, 125–128, 132, 133, 135, 137, 139–141, 145, 148, 150 Proof by contradiction, 52, 53, 137, 139, 140 Propositiones ad acuendos juvenem, 13 Prose, 99 Pythagoras, 3, 4, 12, 13, 23, 51, 63, 64, 149, 150 Pythagorean theorem, 8, 63, 78, 90

Q Quaternions, 136, 137 Quod erat demonstrandum (QED), 53

R Radiation structure, 110 Rationality, 99, 138 Rational number, 112, 117, 137, 138 Recurrence, 22, 120, 136 Reductio ad absurdum, 18, 135 Rhind Papyrus, 37, 38, 64, 66, 91, 145 Riemann, B., 55 Riemann Hypothesis, 55 Riemann zeta function, 56

Index Robbins algebra, 127 Russell, B., 4, 53, 64, 126, 129–132, 139

S Sand Reckoner, The, 24 Schröder’s Reversible Staircase, 151 Scienza nuova, 4 Semantic memory, 69, 70 Semiogenesis, 62 Semiosis, 1 Serendipity, 58, 59 Series, 40, 70, 92, 102, 116, 140 Sierpinski, V., 89 Sierpinski triangle, 89 Spatiality, 25, 28, 82, 103, 123, 143 Sphereland, 4–6 Stream of consciousness, 96 Sudoku, 54, 144 Syllogism, 7, 17, 18, 132, 133 Symbols, 10, 20, 34, 53, 61, 62, 67, 68, 70–72, 75–77, 80, 87, 91, 99, 121, 126, 128, 129, 133, 143, 145, 147, 153 Synecdoche, 95, 96

T Taniyama-Shimura conjecture, 44 Tartaglia, N., 111, 112 Thales of Miletus, 35 Thales’s Theorem, 35, 36 Thom, R., 62, 65 Topology, 7, 28–31, 33, 82, 110, 114 Traveling Salesman Problem (TSP), 54, 56 Turing, A., 53, 54, 140, 149

U Uexküll, J. von, 91, 149 Umwelt, 149

V Venn diagram, 80, 81, 107 Verene, D.P., 4, 8, 69 Verum-factum, 62, 93 Vichian ages, 2, 10, 11, 21, 25, 28, 29, 42, 67, 70, 89, 95–102, 114, 116, 132, 149, 152 Vico, G., 2, 4, 6, 7, 10–14, 43, 46, 55, 62, 65, 67, 69, 71, 72, 74, 83, 93, 95–99, 102, 106, 113, 114, 123, 126, 128, 132, 133, 145, 147, 148, 150–152

Index W Wallis, J., 92, 96, 114, 122 Walpole, H., 58 Whitehead, A.N., 53, 69, 127, 129, 130, 132, 139 Wiles-Taylor proof, 43, 44

173 Z Zeno of Elea, 115 Zeno’s paradoxes, 85, 115, 116, 122 Zero, 24, 55, 73, 74, 109, 121, 122, 129