Dhariwal Manuj, "An experimental and theoretical tool for studying the language of geometric concepts" master's thesis, Massachusetts Institute of Technology, 2018, https://dspace.mit.edu/handle/1721.1/120899

Abstract

In this thesis, I propose concretizing the Piagetian view of children as 'gifted learners' to children as 'gifted language builders', who construct and learn many languages to reduce their uncertainty about the world. These include languages such as, the language of geometry, the language of music & rhythm, even a child playing with blocks (eg: LEGO) is actually learning or rather building a language for themselves. As a specific case, I introduce an experimental paradigm and tool, Finding GoDot, for studying the cognitive language of geometry. Using the above lens, I model constructive actions as a language, specifically looking at the task of drawing shapes. Next, majority of this thesis deals with the problem of calculating the entropy and redundancy of such a language for which there is no readily available language data. For this, I utilize Shannon's insight of accessing our implicit statistical knowledge of the structure of a language by converting it to a reduced text form, through a prediction experiment. I generalize Shannon's experiment design to make it applicable for a wide variety of languages, beyond just text-based, especially those lacking existing language data. Finally, I compute entropy (average information per letter) values for individual shapes to show evidence of subjects using a rich forward model to mentally simulate incomplete shapes, thus gaining information about the underlying shape more than is visible. I also share results on bounds for the entropy and redundancy of the proposed language of actions for generating shape drawings.