The geometry of cognitive biases and fallacies

 

Categorizations of objects are certainly a useful, even necessary, part of human cognitive development. We cannot underestimate the interest of so-called Venn diagrams which while going back to Leibniz arguably are strongly yet somewhat implicitly present in Aristotle's Organon. Venn diagrams can describe human categorical-cognitive architecture but with the caveat that we should be aware of the extreme geometric complexity of the sets involved as well as how a vast array of psychological tendencies and schemata are at work in shaping our perception and reasoning about such sets. 

Here is one example of cognitive fallacy (linked to geometric inadequacy) which doubtlessly occurs in political propaganda and statistically flavored pseudoscience. We have two categories of human beings A and B (the universe U can represent a given human population). A can be a desirable or desirable trait while B can represent a self-identified human group. The situation in the diagram above represents a situation in which for a standard notion of probability we can say both that 'most As are Bs' and yet  'most B's are not A's. A typical fallacy (perhaps linked to a tendency for the mind to generate a symmetrical geometric scenario) is to assume that if  'most As are Bs' then most 'Bs are As'.   We can however say that B is over-represented in category A and seek a sociological explanation. In some instance a category may appear to be defined precisely by its intersection geometry with other categories.

The above considerations show that while categories are useful cognitive constructs one must be aware of  fallacies and biases - some linked to a poor grasp of geometry - and never lose sight of what is cognitively of far greater importance: the individual.