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Are Mathematicians, Physicists and Biologists Irrational? Intransitivity Studies vs. the Transitivity Axiom
The status of the axioms of transitivity of dominance (“if x dominates y and y dominates z, then x dominates z” and “if a person prefers A to B and B to C, then that person should prefer A to C”) as key components of rationality is discussed. The discussion is conducted in the context of modern mathematical and natural science studies of systems in objectively intransitive (rock—paper—scissors) relations contradictory to the transitivity axioms. Experiential studies of solving problems about various intransitive objects and relations are described.
From an epistemological point of view, it seems reasonable to distinguish between 4 types of situations.
(1) Relations are objectively transitive, and problem solvers: (1a) make correct conclusions about them; or (1b) wrongly consider the relations as intransitive. Most cognitive studies are conducted in this paradigm.
(2) Relations are objectively intransitive, and problem solvers: (2a) make correct conclusions about them; or (2b) wrongly consider the relations as transitive (e.g., because of taking the transitivity axiom for granted). This has been minimally studied in cognitive psychology.
In general, for objectively transitive options, intransitive choices are fallacies. For objectively intransitive options, fallacies are transitive choices of the intransitive options.
Topological and graph theoretic approaches are more appropriate to present objectively intransitive objects and systems than metric approaches. Models of intransitive relations in biological studies are worthy of special interest in this context. Teaching intransitivity problem solving should be based on studies how abilities develop in this area and broader educational studies.