The well-designed young mathematician
This paper complements McCarthy's "The well designed child", in part by putting it in a broader context, a space of sets of requirements and a space of designs, and in part by relating design features to development of mathematical competences. I moved into AI hoping 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, especial those arising from interactions with a complex environment; a draft partial ontology for describing spaces of possible architectures, especially virtual machine architectures; investigations of how different forms of representation relate to different functions; analysis of biological nature/nurture tradeoffs and their relevance to machines; studies of control issues in a complex architecture; and showing how what can occur in such an architecture relates to our intuitive concepts of motivation, feeling, preferences, emotions, attitudes, values, moods, consciousness, etc. I conjecture that working models of human vision can lead to models of spatial reasoning that would help to support Kant's view of mathematics by showing 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 requirements for such models, are described, including aspects of our ability to interact with complex 3-D structures and processes that 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 recognised as 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 self-monitoring and 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. (C) 2008 Elsevier B.V. All rights reserved.
mathematics development architectures forms of representation kant evolution self-monitoring altricial vs. precocial species and competences empirical vs. necessary geometry hume nature-nurture tradeoffs philosophy of mathematics