Gaps Between Human and Artificial Mathematics

The Turing-inspired Meta-morphogenesis project begun in 2011 was partly motivated by deep gaps in our understanding of mathematical cognition and other aspects of human and non-human intelligence and our inability to model them. The project attempts to identify previously unnoticed evolutionary transitions in biological information processing related to gaps in our current understanding of cognition. Analysis of such transitions may also shed light on gaps in current AI. This is very different from attempts to study human mathematical cognition directly, e.g. via observation, experiment, neural imaging, etc. Fashionable ideas about “embodied cognition”, “enactivism”, and “situated cognition”, focus on shallow products of evolution, ignoring pressures to evolve increasingly disembodied forms of cognition to meet increasingly complex and varied challenges produced by articulated physical forms, multiple sensory capabilities, geographical and temporal spread of important information and other resources, and “other-related meta-cognition” concerning mental states, processes and capabilities of other individuals. Computers are normally thought of as good at mathematics: they perform logical, arithmetical and statistical calculations and manipulate formulas, at enormous speeds, but still lack abilities in humans and other animals to perceive and understand geometrical and topological possibilities and constraints that (a) are required for perception and use of affordances, and (b) play roles in mathematical, and proto-mathematical, discoveries made by ancient mathematicians, human toddlers and other intelligent animals. Neurally inspired, statistics-based (e.g.“deep learning”) models cannot explain recognition and understanding of mathematical necessity or impossibility. A partial (neo-Kantian) analysis of types of evolved biological information processing capability still missing from our models may inspire new kinds of research helping to fill the gaps. Had Turing lived long enough to develop his ideas on morphogenesis, he might have done this.