Recently, Backman (2006) has brought to our attention the mistake that occurs when mathematical models - convenient calculation devices - are ascribed physical reality, being perforce thrust into the realm of physics. He however suggests that a mathematical model may be ascribed physical reality when it allows testable predictions on physical phenomena.
Building on this idea, this letter takes the discussion a step forward, arguing that the fact that a mathematical model helps us predict various physical phenomena does not grant it physical reality; the implications for cognitive science are explored.
Let us consider the following mathematical model: F: N-N; F(x)=2x. This function helps us: (1) explain why a specific codomain value (e.g., 4) is given a specific domain value (e.g., 2); (2) predict a specific codomain value (e.g., 6) given a specific domain value (e.g., 3); (3) describe the entire range of domaincodomain relations (e.g., 0-0; 1-2); (4) summarize by a mathematical model all the possible domain-codomain relations. If the input and the output refer to physical quantities, this mathematical model allows testable predictions in physical reality, although few would argue that the model itself [F(x)=2x)] is part of physical reality. Such a mathematical model is useful if we resist the temptation of its reification (i.e., transforming a mental model into a physical reality). By populating physical reality with such "ghost phenomena", confusions appear between "what is real" and "what is illusory", with a negative impact on research. A mathematically described physical concept can be accepted as a part of our physical reality only if we have a clear-cut definition of the concept and a qualitative description of the physical phenomenon, both of which can be fully grasped by the human mind;
In a cognitive system, a set of inputs is associated with a set of outputs by several cognitive models (some of them described mathematically). But are they any more "real" than the "F(x)=2x" formula, such as other constructs like "gene" or "mass" are? …