The well-designed young mathematician

This paper complements McCarthy’s “The well designed child”, in part by putting it in a broader context, the space of possible well designed progeny, and in part by relating design features to development of mathematical competence. I first moved into AI in an attempt to understand myself, especially hoping to understand how I could do mathematics. Over the ensuing four decades, my interactions with AI and other disciplines led to: design-based, cross-disciplinary investigations of requirements, especially those arising from interactions with a complex environment; a draft partial ontology for describing spaces of possible architectures, especially virtual machine architectures, for behaving systems (including our precursors); investigations of varied forms of representation and how they are suited to different functions; analysis of biological nature/nurture tradeoffs and their relevance to future machines; studies of control issues in a complex architecture; and showing how the states and processes possible in such an architecture relate to our (simplified) intuitive concepts of motivation, feeling, preferences, emotions, attitudes, values, moods, consciousness, etc. In 1971 I thought working models of human vision could lead to models of visual/spatial reasoning that would help to support Kant’s view of mathematics, against Hume’s. This has not yet happened, but I am still exploring requirements for such models, partly motivated by the hypothesis that human mathematical abilities are a natural extension of abilities produced by biological evolution that are not yet properly understood, and have barely been noticed by psychologists and neuroscientists. Some aspects of our ability to interact with complex 3-D structures and processes extend Gibson’s ideas concerning action affordances, to include proto-affordances, epistemic affordances and deliberative affordances. Some of what a child learns about structures and processes starts as empirical then as a result of reflective processes can be transformed to the status of necessary (e.g. mathematical) truths. These processes normally develop unnoticed in young children, but provide the basis for much creativity in behaviour, as well as leading, in some, to development of an interest in mathematics. We still need to understand what sort of (possibly self-extending) architecture, and what forms of representation, are required to make this possible. This paper does not presuppose that all mathematical learners can do logic, though some fairly general form of reasoning seems to be required.