Manipulative imagination: how to move things around in mathematics

COSMIC SPIRIT, 2020

Only the linguistic expression of genuine mathematics is typical, since, as being a language, it is a human construction of symbolic/point patterns which express the earthly dimensional (Euclidean) space-time environment. These extreme spot symbols of mathematical expression are nothing else but the extreme sections of the net (intangible) mathematical universe. Namely, they are material thickenings that take place due to the function of the human brain (1). An example may analyze the above: as electromagnetism is material (electricity) and “power” (“something else”) altogether, the mathematical universe is language (material) and thought (“something else” – “energy”) together. The language should not be coincided with thought (2). On the other hand, neither the thought should be coincided with the written language. Both the oral and the written language are nothing more than a form of the net structure of thought – logic. Logic in turn, should not be coincided with mathematics (neither as being a mathematical logic), because mathematics are an immaterial – invisible “language” which is expressed as a visible material through mathematical symbols – spots. Ultimately, Mathematics themselves cannot be categorized, as we showed, in rationalism or in empiricism (“sensualism”), or in intuitionism (“intuitive mathematics”) either.

2000

A metaphor is an alteration of a woorde from the proper and naturall meanynge, to that which is not proper, and yet agreeth thereunto, by some lykenes that appeareth to be in it. ... —Thomas Wilson, The Arte of Rhetorique ...

Educational Studies in Mathematics, 2009

The goal of this paper is to explore qualities of mathematical imagination in light of a classroom episode. It is based on the analysis of a classroom interaction in a high school Algebra class. We examine a sequence of nine utterances enacted by one of the students whom we call Carlene. Through these utterances Carlene illustrates, in our view, two phenomena: (1) juxtaposing displacements, and (2) articulating necessary cases. The discussion elaborates on the significance of these phenomena and draws relationships with the perspectives of embodied cognition and intersubjectivity.

The Routledge Handbook of Philosophy of Imagination, 2016

The eminent mathematician Felix Klein wrote, in his intimate history of nineteenth-century mathematics, that "mathematics is not merely a matter of understanding but quite essentially a matter of imagination" (cf. Klein [ ], p.). Klein was responding to the turbulent trajectory of imagination's role in mathematics during the nineteenth century, which began with Gaspard Monge teaching new ways of representing three-dimensional figures in the plane to engineers at the École Polytechnique in Paris, and ended with Moritz Pasch fulminating against visualization in geometry. In that light Klein's remark concerns imagination as representation of the visual. But there is another sense of imagination that mathematicians frequently employ: imagination as the ability to think of novel solutions to problems; in other words, ingenuity. This twofold usage of the term "imagination" was characterized by Voltaire, in his entry on imagination in the Encyclopédie, as distinguishing between what he called the "passive" and "active" faculties of imagination. Both are faculties of every "sensible being" by which one is able "to represent to one's mind sensible things" (cf. Diderot and d'Alembert [ ], p.). The passive imagination consists in the ability "to retain a simple impression of objects", while the active imagination consists in the ability "to arrange these received objects, and combine them in a thousand ways". The active imagination is thus the faculty of invention, and is linked with genius, in particular in mathematics: "there is an astonishing imagination in mathematical practice, and Archimedes had at least as much imagination as Homer" (Ibid., p.). This distinction between imagination as faculty of representation and as faculty of reconstruction, however dubious it may be as cognitive science (on this, cf. Currie and Ravenscroft [ ]), organizes well the two primary ways that mathematicians and philosophers have understood the nature and role of imagination in mathematics. This essay will focus on just the first of these ways, since the second would make for a study of discovery in mathematics, demanding consideration of a rather di erent set of issues than those called for by the first (cf. Hadamard [ ] for such a study for mathematics, as well as Stokes, "Imagination and Creativity", this volume). It will consider imagination in mathematics in the first sense from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.. I Aristotle [ ] characterized imagination as "that in virtue of which we say that an image occurs to us" (a-), and this seems to have been the basic functional view of imagination in antiquity: a mental faculty capable of receiving and reproducing presentations of the senses. Aristotle also held that imagination can also produce images when no sense perception has taken place, as it does in dreams (a), and is thus capable of augmenting thought by unseen images Date: May ,. Thanks to Juliette Kennedy and Sébastien Maronne for comments on drafts of this article. This article is forthcoming in the Routledge Handbook on the Philosophy of Imagination, edited by Amy Kind.

2001

In this paper we propose a theory of cognitive construction in mathematics that gives a unified explanation of the power and difficulty of cognitive development in a wide range of contexts. It is based on an analysis of how operations on embodied objects may be seen in two distinct ways: as embodied configurations given by the operations, and as refined symbolism that dually represents processes to do mathematics and concepts to think about it. An example is the embodied configuration of five fingers, the process of counting five and the concept of the number five. Another is the embodied notion of a locally straight curve, the process of differentiation and the concept of derivative. Our approach relates ideas in the embodied theory of Lakoff, van Hiele's theory of developing sophistication in geometry, and the processobject theories of Dubinsky and Sfard. It not only offers the benefit of comparing strengths and weaknesses of a variety of differing theoretical positions, it also reveals subtle similarities between widely occurring difficulties in mathematical growth.

Frontiers in Psychology

Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue that embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it.

Neoplatonism in Late Antiquity

Chapter 8 considers the role of the imagination as it appears in Proclus’ commentary on Euclid, where mathematical or geometrical objects are taken to mediate, both ontologically and cognitively, between thinkable and physical things. With the former, mathematical things share the permanence and consistency of their properties; with the latter, they share divisibility and the possibility of being multiplied. Hence, a geometrical figure exists simultaneously on four different levels: as a noetic concept in the intellect; as a logical definition, or logos, in discursive reasoning; as an imaginary perfect figure in the imagination; and as a physical imitation or representation in sense-perception. Imagination, then, can be equated with the intelligible or geometrical matter that constitutes the medium in which a geometrical object can be constructed, represented, and studied.

2007

Objectives The objective of this study is to contribute to research on mathematical cognition by illuminating implicit processes of embodied reasoning in situated problem solving. I argue that situated mathematical reasoning transpires as an embodied negotiation between material/perceptual affordances of phenomena and evolved cultural–historical cognitive artifacts that include physical utensils, symbolical forms, and figures of speech.

18 Unconventional Essays on the Nature of Mathematics

Robotics, artificial intelligence and, in general, any activity involving computer simulation and engineering relies, in a fundamental way, on mathematics. These fields constitute excellent examples of how mathematics can be applied to some area of investigation with enormous success. T his, of course, includes embodied oriented approaches in these fields, such as Embodied Artificial Intelligence and Cognitive Robotics. In this chapter, while fully endorsing an embodied oriented approach to cognition, I will address the question of the nature of mathematics itself, that is, mathematics not as an application to some area of investigation, but as a human conceptual system with a precise inferential organization that can be investigated in detail in cognitive science. The main goal of this pi ece is to show, using techniques in cognitive science such as cognitive semantics and gestures studies, that concepts and human abstraction in general (as it is exemplified in a sublime form by mathematics) is ult imately embodied in nature.

These Are Situationist Times, 2019

Educational Studies in …, 1999

In this paper we analyze, from the perspective of 'Embodied Cognition', why learning and cognition are situated and context-dependent. We argue that the nature of situated learning and cognition cannot be fully understood by focusing only on social, cultural and contextual factors. These factors are themselves further situated and made comprehensible by the shared biology and fundamental bodily experiences of human beings. Thus cognition itself is embodied, and the bodily-grounded nature of cognition provides a foundation for social situatedness, entails a reconceptualization of cognition and mathematics itself, and has important consequences for mathematics education. After framing some theoretical notions of embodied cognition in the perspective of modern cognitive science, we analyze a case study -continuity of functions. We use conceptual metaphor theory to show how embodied cognition, while providing grounding for situatedness, also gives fruitful results in analyzing the cognitive difficulties underlying the understanding of continuity.

2007

Conclusions These exercises appear to be rewarding in the sense that they liberate the intellect of the student and place it in closer contact with the realm of feelings and the unconscious indicating the fundamental role of topological concepts in common perception and the assignment of meaning.

Complicity: An International Journal of Complexity and Education, 2010

Working from the premise that mathematics knowledge can be described as a complex unity, we develop the suggestion that network theory provides a useful frame for informing understandings of disciplinary knowledge and content learning for schooling. Specifically, we use network theory to analyze associations among mathematical concepts, focusing on their embodied nature and their reliance on metaphor. After describing some of the basic suppositions, we examine the structure of the network of metaphors that underlies embodied mathematics, the dynamics of this network, and the effect of these dynamics on mathematical understanding. Finally, implications for classroom teaching and curriculum are discussed. We conjecture that it is both instructive and important to use the network structure of mathematical knowledge to shed light on both cognition in mathematics and on mathematics education.

… of the 25th Conference of the …, 2001

We consider Gray & Tall's (2001) idea of embodied objects in mathematics from a semiotic perspective. We explore two main issues arising from Gray & Tall's notion of embodied objects in mathematics. The first is that the simplest embodied objects of a mathematical ...