There is considerable evidence for computationally complex behavior, that is, behavior that appears to require the equivalent of mathematical calculation by the organism. Spatial navigation by path integration is perhaps the best example. The most influential account of such behavior has been Gallistel’s (1990) computational–representational theory, which assumes that organisms represent key environmental variables such as direction and distance traveled as real numbers stored in engrams and are able to perform arithmetic computations on those representations. But how are these computations accomplished? A novel perspective is gained from the historical development of algebra. We propose that computationally complex behavior suggests that the perceptual system represents an algebraic field, which is a mathematical concept that expresses the structure underlying arithmetic. Our field representation hypothesis predicts that the perceptual system computes 2 operations on represented magnitudes, not 1. We review recent research in which human observers were trained to estimate differences and ratios of stimulus pairs in a nonsymbolic task without explicit instruction (Grace, Morton, Ward, Wilson, & Kemp, 2018). Results show that the perceptual system automatically computes two operations when comparing stimulus magnitudes. A field representation offers a resolution to longstanding controversies in psychophysics about which of 2 algebraic operations is fundamental (e.g., the Fechner–Stevens debate), overlooking the possibility that both might be. In terms of neural processes that might support computationally complex behavior, our hypothesis suggests that we should look for evidence of 2 operations and for symmetries corresponding to the additive and multiplicative groups.
With the natural numbers as our starting point, we obtain the arithmetic structure of real (as in R) addition and multiplication without relying on any algebraic tools; in particular, we leverage monotonicity, convexity, continuity, and isomorphism. Natural addition arises by minimizing against monotonicity. Rational addition arises from natural addition by minimizing against convexity. Real addition arises from rational addition via any one of three methods; unique convex extension, unique continuous extension, and unique monotonic extension. Real multiplication arises from real addition via isomorphism. Following these mathematical developments, we argue that each of the leveraged mathematical concepts ---monotonicity, convexity, continuity, and isomorphism --- enjoys, prior to its formal mathematical existence, an intuitive psychological existence. Taken together, these lines of argument suggest a way for psychological representation of algebraic structure to emerge from non-algebraic --- and psychologically plausible --- ingredients.
Human infants have ‘core knowledge systems’ that support basic intuitions about the world including objects and their motion, space, number, and time. What is the origin of these systems, and what is their nature? Although often regarded as separate, domain-specific modules, evidence for similar abilities across many nonhuman species suggests that core systems might be integrated, consistent with views of modularity in evolutionary-developmental biology. Here we propose that core knowledge systems are based on an ability to form representations of the environment with algebraic structure – that is, on implicit computation. Algebraic groups encode symmetries, with computation inherent in the structure – a view that complements an understanding of computation as action or function. Our proposal is related to previous applications of group theory in perception and computational-representational accounts of learning (Gallistel, 1990), but suggests for the first time a common basis for core knowledge across humans and nonhumans. Implicit computation can be studied experimentally with an ‘artificial algebra’ task in which adults learn to respond based on arithmetic combinations of stimulus magnitudes, by feedback and without explicit instruction. Asking why organisms have a capacity for implicit computation suggests two possibilities: Either the geometric invariants of the world have been internalized in perceptual systems by natural selection (Shepard, 1994), or mathematical structure is intrinsic to the mind. Understood more broadly in a framework offered by Penrose (2004), implicit computation is a linchpin with potential to unlock some of the most fundamental questions about relationships between the mind, mathematics, and the world.
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