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If your interest is future pedagogic utility in academia (funny etymology for pedagogue: 'leader of children') that's one thing. If your interest is in what can

If your interest is future pedagogic utility in academia (funny etymology for pedagogue: 'leader of children') that's one thing. If your interest is in what can be called today the cognitive dynamics of creativity as experienced by highly creative individuals then who better than Albert Einstein to report on his experiences. In mathematician Jacques Hadamard's 'The Psychology of Invention in the Mathematical Field' (Dover (1954),ISBN 0486201074, pp. 142-3) Einstein reports his creative cognitive processes in response to a questionnaire titled 'An inquiry into the working methods of mathematicians' as follows: A) The words or language as they are written or spoken do not seem to play any role in my mechanism of thought. The psychical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be "voluntarily" reproduced and combined... B) The above mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associated play is sufficiently established and can be reproduced at will. Or what might be called today visual/tactile qualial imagery manipulated in cognitive workspace (Baars). So yeah it would seem, 'a visual medium (can) be used to do serious mathematical exploration', at least as a lead-in to transcription from conceptual imagery to reporting.

OisinMoran 9y ago
This is an incredibly interesting comment! I think you've hit the nail on the head that this is better suited for pedagogy than for exploration. The extracts from the book are especially insightful and really something I had not considered before so thank you for that--I may just have to read the whole book (and it's by the same Hadamard of the Hadamard gate! It seems that anyone even slightly related to quantum computing has an interest in exploring and enhancing creative thought--that's 3 just in this thread). I hope I didn't come across as suggesting that a visual medium couldn't be used to do serious mathematical exploration--I definitely think it can (and Strilanc's comment proves that it can). To clarify, I was more suggesting that it could be the case that not every mathematical concept is conducive to being explained or taught effectively using a visual medium, or even more loosely that in some cases a symbolic approach could be more effective than a visual one. However, I agree that for most concepts visual trumps symbolic and I currently have no concrete examples of the opposite.

OisinMoran 9y ago

This is an incredibly interesting comment! I think you've hit the nail on the head that this is better suited for pedagogy than for exploration. The extracts from the book are especially insightful and really something I had not considered before so thank you for that--I may just have to read the whole book (and it's by the same Hadamard of the Hadamard gate! It seems that anyone even slightly related to quantum computing has an interest in exploring and enhancing creative thought--that's 3 just in this thread). I hope I didn't come across as suggesting that a visual medium couldn't be used to do serious mathematical exploration--I definitely think it can (and Strilanc's comment proves that it can). To clarify, I was more suggesting that it could be the case that not every mathematical concept is conducive to being explained or taught effectively using a visual medium, or even more loosely that in some cases a symbolic approach could be more effective than a visual one. However, I agree that for most concepts visual trumps symbolic and I currently have no concrete examples of the opposite.