Informal Rigor and Mathematical Understanding

These two introspective cognitive experiments have shown how the rational imagination can be a satisfactory mental arena for the conduct of rigorous mathematical thinking. The diagrams and other representative imagery used in such thinking need not be external and public, but are capable of being privately constructed, viewed, and analyzed “in the mind's eye”. The conclusion I draw from this and other similar introspective cognitive experiments with. actual proof scenarios is that such examples of rigorous but unformalized mathematical thinking provide a rich source of material for the development of a more authentic model of mathematical reasoning and communicating results than the more limited model offered by the rigid formal systems of traditional mathematical logic. p]It seems clear that in any such realistic model there will have role for the thinking subject. That is to say, we must explicitly include features in the model which capture the way the intuition and imagination deal directly (without the mediation of formal definitions) with spatial and temporal relationships, with order relations, with comparison and combination of quantities, and with elementary manipulations of symbols. p]Understanding the rules and conventions involved in the engineering and operation of formal systems of deduction and computation is itself an example of intuitive and informal mathematics, just as Hilbert long ago stressed in explaining the ideas behind his metamathematics. p]In the end, we are left with the question: how does the ma actually function? To answer, we must watch real minds in action, rather than guessing how idealized minds would work if they existed.