How much of symbolic manipulation is just symbol pushing

How much of symbolic manipulation is just symbol pushing? David H. Landy (dhlandy@gmail.com) Psychology, 603 E. Daniel Street University of Illinois, Urbana-Champaign, IL 61820 Robert L. Goldstone (rgoldsto@indiana.edu) Psychological and Brain Sciences, 1101 E. 10 th Street Indiana University, Bloomington, IN 47405 Abstract This paper explores the hypothesis that schematic abstraction—rule following—is partially implemented through processes and knowledge used to understand motion. Two experiments explore the mechanisms used by reasoners solving simple linear equations with one variable. Participants solved problems displayed against a background that moved rightward or leftward. Solving was facilitated when the background motion moved in the direction of the numeric transposition required to solve for the unknown variable. Previous theorizing has usually assumed that such formal problems are solved through the repeated application of abstract transformation patterns (rules) to equations, replicating the steps produced in typical worked solutions. However, the current results suggest that in addition to such strategies, advanced reasoners often employ a mental motion strategy when manipulating algebraic forms: elements of the problem are “picked up” and “moved” across the equation line. This demonstration supports the suggestion that genuinely schematic reasoning could be implemented in perceptual-motor systems through the simulated transformation of referential (but physical) symbol systems. Keywords: Symbolic reasoning; formal reasoning; high-level cognition; mathematical cognition Introduction Reasoning over abstractions—schemas, rules with variables, or hard to perceive generalities—is a skill which seems on its face to require specialized cognitive structures (Anderson, 2005; Markman & Dietrich, 2000; Sloman, 1996). One common conception of this specialized architecture is that the human mind operates over a set of internal symbols and variables much like external formal languages (Anderson, 2007; Fodor, 1975; Marcus, 2001). On this perspective, mental structures that perform symbolic manipulations are precursors to and ingredients of cognition. A frequently articulated alternative to this view is that schematic rules may be implemented via non-symbolic or incompletely symbolic perceptual manipulations and simulations (Barsalou, 1999; Clark, 1998; Dennett, 1994). Often, such perceptual symbol systems have been conceptualized as simulations of semantic situations picked out by a symbolic form or formalism (Barsalou 2008). Under this conceptualization, symbol systems such as natural or formal languages can play either of two roles. First, they might provide the seeds for a perceptual simulation of the situation referred to in an utterance (e.g., simulating an eagle when reading the word “eagle”, Zwaan et al, 2004). The present paper explores a second possible mechanism for implementing symbolic reasoning in perceptual-motor action that rely on simulating the perceptual-motor environment associated with the physical form of the notation itself. Recently, it has been suggested that formal languages, and mathematical languages in particular, often serve as diagrams whose analog physical structure relates systematically to mathematical or formal truths. Therefore, treating the formal notations as images directly suitable for perceptual-motor processing provides a way to implement abstract cognition in perceptual-motor systems (Dorfler, 2002; Landy & Goldstone, 2007a). Naturally, the results of such processes must then be translated into the appropriate referent situation. Furthermore, symbolic reasoning achieved via the perception and manipulation of physical notations must respect the constraints of systems which may differ substantially from those that would be expected from a Language of Thought (Endress, et al, 2005; Landy and Goldstone, 2007b; Novick and Catley, 2007; Pothos, 2006). In this work, we argue that proofs are often physically designed so that they appeal to processing systems typically used for dynamic events, and we evaluate one possible strategy for solving a standard class of algebra problems (“solve for x”) involving simulating the motion of the elements of the notation used to express these problems. Ways to solve single-variable equations Table 1 displays a standard demonstration that the equation y*3+2=8 has solution y=2. The justification for lines 2 and 4 comes from the Euclidean notion that things done to like things yield like things, and therefore that if an equation X = Y is valid, then the equation X  A = Y  A is also valid, for any operation  or value A . How are such processes carried out by human reasoners? Conceptually, there are (at least) two good strategies for solving such problems. In an algebraic solution, a reasoner constructs the solved equation shown at the bottom of table 1, and then uses straightforward arithmetic to generate the answer. In the unwind strategy, one starts by finding the isolated constant, identifies the next available operation on the variable side (+2 in this case), inverts the operation, and solves the resulting problem (8-2). One then uses this number as the starting point, identifies the next available operations on the left, and repeats.