I have been thinking about mathematical platonism, and wondering if there is a relationship between asserting the existence of mathematical objects and asserting the existence of mental objects (as in traditional cognitive psychology research). It seems to me that there is an interesting connection between the two, that might help shed light on both sides.
What is a mathematical object? Mathematical platonists (or realists) claim that mathematical objects really exist, as abstract objects. There are various opinions regarding the means by which we can come to believe things about them, and how we can or can’t justify those beliefs.
What is a mental object? Cognitive psychology researchers claim that mental objects really exist. Examples include such things as the mental number line (e.g. Dehaene, Butterworth), the central executive (Baddeley) and the principles of counting (Gelman & Gallistel).
Let’s assume that no-one asserts the existence of objects without good reason. My interpretation of the motivation for mathematical platonism is that it gives us some easy answers to some questions, such as: Why is there such a close relationship between mathematics and the natural sciences? My interpretation of the motivation for the assertion of the existence of mental objects is that they help to explain some observable phenomena, such as children’s developing abilities in solving mathematical problems, and limitations in people’s reasoning abilities. The reification of both mathematical objects and mental objects serves the purpose of a kind of argumental placeholder. There’s no reason to say that they really exist is there?
