Cognitive Units, Connections and Mathematical Proof

Mathematical proof seems attractive to some, yet impenetrable to others. Inthis paper a theory is suggested involving “cognitive units” which can be theconscious focus of attention at a given time and connections in the individual’scognitive structure that allow deductive proof to be formulated. Whilstelementary mathematics often involves sequential algorithms where each stepcues the next, proof also requires a selection and synthesis of alternative pathsto make a deduction. The theory is illustrated by considering the standardproof of the irrationality of √2 and its generalisation to the irrationality of √3.Cognitive units and connectionsThe logic of proof is handled by the biological structure of the human brain. As a multi-processing system, complex decision-making is reduced to manageable levels bysuppressing inessential detail and focusing attention on important information. A pieceof cognitive structure that can be held in the focus of attention all at one time will becalled a cognitive unit. This might be a symbol, a specific fact such as “3+4 is 7”, ageneral fact such as “the sum of two even numbers is even”, a relationship, a step in anargument, a theorem such as “a continuous function on a closed interval is bounded andattains its bounds”, and so on. What is a cognitive unit for one individual may not be acognitive unit for another; the ability to conceive and manipulate cognitive units is a vitalfacility for mathematical thinking. We hypothesise that two complementary factors areimportant in building a powerful thinking structure:1) the ability to compress information to fit into cognitive units,2) the ability to make connections between cognitive units so that relevantinformation can be pulled in and out of the focus of attention at will.Compression is performed in various ways, including the use of words and symbols astokens for complex ideas (“signifiers” for something “signified”). These may sometimesbe “chunked” by grouping into sub-units using internal connections. A more powerfulmethod uses symbols such as 2+3 as a pivot to cue either a mental process (in this caseaddition) or a concept (the sum). This has become a seminal construct in process-objecttheories (Dubinsky, 1991; Sfard, 1991). Gray & Tall (1994) coined the term procept forthe amalgam of process, concept and symbol which could evoke either. However, thenotion of procept is not the only instance of compression in mathematics: