Imagine a society in which the citizens are encouraged, indeed compelled up to a certain age, to read (and sometimes write) musical scores. All quite admirable. However, this society also has a very curious--few remembered how it all started--and disturbing law:  Music must never be listened to or performed !

   Though its importance is universally acknowledged, for some reason music is not widely appreciated in this society. To be sure, professors still excitedly pore over the great works of Bach, Wagner, and the rest, and they do their utmost to communciate to their students the beautiful meaning of what they find there, but they still become tongue-tied when brashly asked the question, "What's the point of all this?!" 

   In this parable, it was patently unfair and irrational to have a law forbidding would-be music students from experiencing and understanding the subject directly through "sonic intuition." But in our society of mathematicians we have such a law. It is not a written law, and those who flout it may yet prosper, but it says, Mathematics must not be visualized !

   More likely than not, when one opens a random modern mathematics text on a random subject, one is confronted by abstract symbolic reasoning that is divorced from one's sensory experience of the world, despite the fact that the very phenomena one is studying were often discovered by appealing to geometric (and perhaps physical) intuition. 

   This reflects the fact that steadily over the last hundred years the honour of visual reasoning in mathematics has been besmirched. Although the great mathematicians have always been oblivious to such fashions, it is only recently that the "mathematician in the street" has picked up the gauntlet on behalf of geometry. 

   The present book openly challenges the current dominance of purely symbolic logical reasoning by using new, visually accessible arguments to explain the truths of elementary complex analysis. 

The text above appears on the first page of the preface of Tristan Needham's book, Visual Complex Analysis, Oxford University Press, 1997. Needham's insights bring to mind how in my own educational experience reliance on visualization was taboo, and how as a point of honor "real mathematicians don't do (use) pictures." 

I am interested in various questions regarding the status of diagrams and visual reasoning including: 

Q1. Aren't there compelling biological reasons why we must "use our eyes" if we wish to reduce the (computational) complexity and increase the efficiency of certain reasoning tasks involved in problem-solving and learning? 

Q2. Given a positive answer to Q1. won't any constructivist theory-of-learning/cognitive-model have to seriously address mental constructions that are essentially visual, and the role they play in the construction of mathematical concepts, both visual and nonvisual? 

Q3. Under what circumstances is a diagram/picture a 1st class object of mathematical construction and reasoning (e.g. as in Tristan Needham's book or ...)?