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• W2592016249 abstract "Grounding Symbol Structures in Space: Formal Notations as Diagrams David Landy (dlandy@indiana.edu) Department of Computer Science Indiana University, Bloomington, IN 47408 USA Robert L. Goldstone (rgoldsto@indiana.edu) Department of Psychological and Brain Sciences Indiana University, Bloomington, IN 47408 USA Abstract than the element representing B – any relation that is inherently asymmetric and transitive. Representations are extrinsic when their inherent structure is arbitrary. They model the represented world by explicitly building the necessary structure so as to conform to the world. Palmer argues that analog representations are intrinsic; correspondences and inferences between represented and representing worlds come for free because of their shared intrinsic structure. Propositional representations, including language, logic, and mathematics, are extrinsic, and hence come to represent objects by explicitly establishing relations with whatever structure is needed. The only intrinsic relation necessary to propositions is the left-right concatenation of basic symbols. Although traditionally understood as extrinsic, it is possible that representations in mathematics and logics nonetheless possess intrinsic and analog properties, and it is this possibility that we empirically pursue here. Specifically, we propose that formal notations are diagrammatic as well as sentential and the property conventionally described as syntactic structure is cognitively mediated, in part, by spatial information. Elements of expressions are “bound” together through perceptual grouping—often induced by simple spatial proximity. Thus, our claim is that mathematical formalizations of syntax are not themselves the direct cognitive mechanisms typically employed in processing that structure. The former really are concatenative, but we propose that people use space and spatial relationships in representational schemas to facilitate the processing of syntax. To be clear: we are not claiming here that the execution of each individual step in a proof or computation is inherently spatial or processed exclusively using sensori- motor mechanisms. We do suggest that spatial reasoning over the physical layout of notational forms is common in reasoning with formal languages, and that spacing practices play a significant role in human reasoning using notations. We have argued previously that a broadly similar interference of metric (non-order-related) spatial properties on syntactic judgments provides evidence that syntax processing typically involves spatial and perceptual grouping processes (Landy & Goldstone, in press; Kirshner & Awtry, 2004). These works conclude that mathematical reasoners are sensitive to non-formal properties of presented equations, and in particular align close proximity with high operator precedence. For instance, in Landy & Goldstone Although a general sense of the magnitude, quantity, or numerosity is common both in untrained people and animals, the abilities to deal exactly with large quantities and to reason precisely in complex but well-specified situations—to behave formally, that is—are skills unique to people trained in symbolic notations. These symbolic notations employ typically complex, hierarchically embedded structures, which all extant analyses assume are the product of concatenative, rule-based processes. The primary goal of this article is to establish, using behavioral measures on naturalistic tasks, that the some of the same cognitive resources involved in representing spatial relations and proximities are also involved in representing symbolic notations. In short, formal notations are used as a kind of diagram. We examine self- generated productions in the domains of handwritten arithmetic expressions and typewritten statements in a formal logic. In both tasks, we find substantial evidence for spatial processes even in these highly symbolic domains. Keywords: symbolic processing, mathematics, embodied cognition, relational reasoning Introduction It is clear that mathematical equations written in modern notation are, in general, visual forms, and furthermore that they share some properties with diagrammatic or imagistic displays. Equations and mathematical expressions are often set off from main text, use non-standard characters and shapes, and deviate substantially from the linear placement of text. Furthermore, evidence indicates that at least some mathematical processing is sensitive to the particular visual form of its presentation notation (Campbell, 1999, McNeil & Alibali, 2004). Nevertheless, notational mathematical representation is typically considered ‘sentential,’ and placed in opposition with diagrammatic representations in fields as diverse as education (Zazkis et al, 1996; Stylianou, 2002), philosophy of science (Galison, 1997), computer science (Iverson, 1980), and cognitive studies of problem- solving (Anderson, 2005; Stenning, 2002). The standard conception of mathematical notation is best understood via Stephen Palmer’s (1978) classic distinction between intrinsic and extrinsic representational schemes. A representation is intrinsic “whenever a representing relation has the same inherent constraints as its represented relation” (p. 271). Line A being shorter than Line B can be intrinsically represented by the representational element that corresponds to A being shorter, taller, brighter, or larger" @default.
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• W2592016249 title "Grounding Symbol Structures in Space: Formal Notations as Diagrams" @default.
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