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. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

中文翻译：

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关于计算复杂行为的起源。

有大量证据表明计算复杂的行为，即似乎需要有机体进行数学计算的行为。通过路径集成进行空间导航也许是最好的例子。这种行为的最有影响力的解释是加利斯特尔（Gallistel）（1990）的计算表示理论，该理论假设有机体将关键的环境变量（例如行进的方向和距离）表示为以字母表示的实数存储，并且能够对这些表示进行算术计算。但是这些计算如何完成？从代数的历史发展中获得了一个新颖的观点。我们认为计算复杂的行为表明，感知系统代表代数场，这是一个数学概念，表示算术基础结构。我们的场表示假设预测知觉系统在表示的幅度上计算2次运算，而不是1次。我们回顾了最近的研究，其中对人类的观察者进行了训练，可以在没有明确指示的情况下（非格雷斯，莫顿，沃德）估计刺激对的差异和比率。 ，Wilson和Kemp，2018年）。结果表明，当比较刺激幅度时，感知系统会自动计算两个操作。场域表示法解决了长期以来在心理物理学中有关两个代数运算中的哪一个是基本运算的争论（例如，费希纳-史蒂文斯辩论），而忽略了两者可能存在的可能性。在可能支持计算复杂行为的神经过程方面，我们的假设表明，我们应该寻找2个运算的证据，以及与加和乘性组相对应的对称性。（PsycINFO数据库记录（c）2019 APA，保留所有权利）。