Models, Methods and Applications

tools to analyze the structure of psychological models. But they are just abstract tools after all. In any empirical science, the ultimate proof rests on experimental evidence. Nonetheless, perhaps paradoxically, here it is precisely where the full strength of symmetries shows: Not from the models of theories built on symmetry principles but from the intimate connection (through symmetry arguments) between such models and observed phenomena. If we look back to the problems faced by psychological models of associative learning as listed in section 2, we find that they relate to deficiencies that symmetry could be used to resolve. The first shortcoming, that no model accounts for all associative learning phenomena, refers to a lack of explanatory power in such models; the second one, that contradictory rules explain the same phenomena, claims for a normative approach; the third one, that models are partial, relates to the need for unifying principles where different theories that cover disjoint phenomena find common grounds and are made compatible; and the fourth one, that some phenomena remain unaccounted for, identifies a classification problem. It seems, therefore, that symmetries may be useful in solving such problems. First we must find the psychological symmetries. This is the purpose of our research. 6. IN SEARCH OF PSYCHOLOGICAL SYMMETRIES Although there is not a universally accepted ‘law of learning’, all psychological models coincide in assuming that learning takes place when a (relatively permanent) change in behavior happens as a consequence of some experience. Now, we need to know whether such law establishes sufficient symmetry conditions for the occurrence of the observed phenomena –or, in other words, we have to investigate whether the observed phenomena describe necessary conditions for the law to hold (invariantly) true. Unfortunately, a glimpse at the literature suggests it does not: 1. That the sensory and motivational features of the stimuli as well as their novelty and relevance affect learning are well documented facts (Kamin and Schaub, 1963; Pavlov, 1927; Jenkins and Moore, 1973; Randich and LoLordo, 1979; Lubow, 1989; Garcia and Koelling, 1966); 2. Procedurally, the idea that learning is contextspecific is also gaining ground (Bouton, 1993; Bouton and Swartzentruber, 1986; Hall and Mondragón, 1998); also, different results emerge depending on the order in which stimuli are presented during training and on the number (single or compound) and representation (elemental or configural) of the cues themselves (see, e.g., Pearce and Bouton, 2001 for a survey). This first setback may not challenge our search for psychological symmetries though. It could we argued that, after all, we should expect that the parameters in (a) affected the pace of learning (accelerating or decelerating the learning process, i.e., strengthening or weakening the links between nodes/stimuli as time goes), defining, in the extreme, explicit symmetry breaks. Unfortunately, the study of complex phenomena in (b) does not only tell us that the learning rate changes in different experimental conditions. What these results tell us is that the rules of learning themselves fluctuate depending on such factors and, consequently, that they do not reflect any genuine object of invariance. Not surprisingly, a mathematical analysis of the above-mentioned issues reveals that each of them violates one of the conditions for group formation: Associativity. This is rather worrying since associativity is the key condition for symmetry. It tells us that the concatenation of two different operations gives the same result, and that gives us