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A language of thought for the mental representation of geometric shapes

A language of thought for the mental representation of geometric shapes

Why do geometric shapes such as lines, circles, zig-zags, or spirals appear in all human cultures, but are never produced by other animals? Mathias Sablé-Meyer et al. formalize and test the hypothesis that all humans possess a compositional language of thought that can produce line drawings as recursive combinations of a minimal set of geometric primitives. By presenting a programming language that combines discrete numbers and continuous integration in higher-level structures based on repetition, concatenation, and embedding, they show that the simplest programs in this language generate the fundamental geometric shapes observed in human cultures.

We could never know the geometric triangle through the one we see traced on paper, if our mind had not had the idea of it elsewhere

René Descartes

Evidence for abstract concepts of geometry, including rectilinearity, parallelism, perpendicularity, and symmetries, is widespread throughout prehistory. Zig-zag carved patterns, equal angels, parallelism, and bifaces – archeological evidence that suggests an aesthetic drive for symmetry, that was already present in ancient humans.

Geometric shapes in human cultural history

According to a recent comparison study, all humans, regardless of age, culture, or education, exhibited a striking effect of shape regularity: squares and rectangles were processed better than other, more irregular quadrilaterals, and there was a continuous ordering of complexity, from squares and rectangles to parallelograms, trapezoids, and fully irregular shapes. Surprisingly, baboons did not show this geometric regularity effect.

Our core hypothesis is that perceiving a shape, in humans, consists in finding the shortest program that suffices to reproduce it. Our proposal thus connects shape perception to the problem of program induction, i.e. the identification of a program that produces a certain output.

Mathias Sablé-Meyer

A crucial aspect of the proposal is that humans encode a shape mentally by inferring a simple program that could generate it. Thus, the perception of a simple shape is an act of “program induction”. While program induction remains a difficult challenge for computer science, the study deployed a state-of-the-art program induction technique, the DreamCoder algorithm. This algorithm is given programming problems via examples of the desired behavior and searches for the simplest program that performs the task.

Sample shapes generated by the enumeration of all programs in the proposed language.
Testing the DreamCoder algorithm for program induction.

First, we show that our language predicts which shapes are judged simple. Second, we show that any such language has to satisfy a set of additive relationships for repeated, concatenated or embedded shapes, and that those universal laws can be experimentally validated.

Mathias Sablé-Meyer

The DreamCoder approach opens up a number of perspectives on how human cognition could efficiently address the problem of program induction. First, it naturally accounts for cultural drifts. Second, it may explain how simple geometric shapes may be efficiently recognized and used by young children in the absence of much or any training (poverty of the stimulus argument).

The bottom-up neural network in a future version of DreamCoder might be repurposed to immediately obtain the most likely program for a particular shape, perhaps offering a mechanism for people to swiftly detect simple shapes.

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Simple arguments show that the suggested language can generate the vast majority of the shapes that people consider simple and that has been documented in both human cultures and the history of geometry.

The prospect of building reusable abstractions or program templates is an attractive component of this approach that has yet to be extensively explored. While the square is not primitive in the original language, a square-drawing program schema may become abstracted over time, allowing the participant to understand ideas like “a square of circles” or “a square twice as huge as the last one.”

A language of thought for the mental representation of geometric shapes, Mathias Sablé-Meyer, Kevin Ellis, Joshua Tenenbaum, Stanislas Dehaene

Published: December 2021

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