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Are mathematicians, physicists and biologists irrational? Mathematical and natural science studies vs. the transitivity axiom
An important and interesting phenomenon of the last few decades is the increasing number of mathematical studies of so-called intransitive dice with non-standard numbers on their faces and the popularization of them. The dice beat one another like in the rock-paper-scissors game. They violate the transitivity law (or axiom): “if it were true that whenever x dominates y and y dominates z, then also x dominates z”. Physicists and biologists study intransitivity in their areas too. Yet many texts on rational decision-making contain statements that a key component of rationality is the transitivity axiom and that violation of transitivity is irrational. Are all the physicists, biologists and mathematicians, authors of popular books and math lovers—fans of the intransitive dice irrational? It is not the case. The main difference of most cognitive studies of intransitivity of preferences from intransitivity studies in mathematics and biology is that the cognitivists work with transitive options. In this case, intransitive options are fallacies. In the case of objectively intransitive options, fallacies are transitive choices of the intransitive options. Not only an Euclidean metric but also topological and graph theoretic approaches to rational cognition of objectively intransitive objects is a possible way to overcome some cognitivists’ belief in the universality of the transitivity axiom. One can say about transitivity-oriented and intransitivity-oriented paradigms in different scientific areas. Supporters of the transitivity-oriented paradigm consider transitivity as one of axioms of rational decision making and state that its violations is a fallacy. Supporters of the intransitivity-oriented paradigm consider intransitivity as a part of objective relations in the real world which should be revealed and comprehended. From the point of view of cognition of complex systems containing intransitive relations, it seems reasonable to distinguish between four types of situations. (1) Relations are objectively transitive and problem posers and (or) solvers make correct conclusions about their transitivity. (2) Relations are objectively transitive, but problem posers and (or) solvers wrongly consider them as intransitive. (3) Relations are objectively intransitive and problem posers and (or) solvers make correct conclusions about their intransitivity. (4) Relations are objectively intransitive, but problem posers and (or) solvers wrongly consider them as transitive (e.g. because of taking the transitivity axiom for granted). This type has been minimally studied in cognitive psychology. Models of intransitive relations in biological studies are worthy of cognitivists’ potential interest. Making rational ecological decisions should take intransitive competition of species into account.
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English