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baroque features or a more “dull” representation, whatever works. Some scientist have been asked to describe how they represent abstract ideas and they invariably seem to entertain some type of visual representation. A beautiful account of this in the case of mathematicians can be found in a marvellous book “XXX” (Hardamard).

By building accurate visual representations of abstract ideas we create a database of knowledge in the unconscious. This collection of ideas forms the basis for what we call intuition. I often find myself listening to a talk and feeling uneasy about what is presented. The reason seems tobe that the abstract idea I am trying to capture from the talk clashed with a similar idea that is already stored. This in turn can be a sign that I either misunderstood the idea before and need to update it, or that there is actually something wrong withwhat is being presented. In a similar way I can easily detect that some idea is a small perturbation of what I already knew (I feel happily bored), or something entirely new (I feel intrigued and slightly frustrated). While the novice is continuously challenged and often feels overwhelmed, the more experienced researcher feels at ease 90% of the time because the “new” idea was already in his/her data-base which therefore needs no and very little updating.

Somehow our unconscious mind can also manipulate existing abstract ideas into new ones. This is what we usually think of as creative thinking. One can stimulate this by seeding the mind with a problem. This is a conscious effort and is usually a combination of detailed mathematical derivations and building an intuitive picture or metaphor for the thing one is trying to understand. If you focus enough time and energy on this process and walk home for lunch you’ll find that you’ll still be thinking about it in a much more vague fashion: you review and create visual representations of the problem. Then you get your mind off the problem altogether and when you walk back to work suddenly parts of the solution surface into consciousness. Somehow, your unconscioustook over and kept working on your problem. The essence is that you created visual representations as the building blocks for the unconscious mind to work with.

In any case, whatever the details of this process are (and I am no psychologist) I suspect thatany good explanation should include both an intuitivepart, including examples, metaphors and visualizations, and a precise mathematical part where every equation and derivation is properly explained. This then is the challenge I have set to myself. It will be your task to insist on understanding the abstract idea that is being conveyed and build your own personalized visual representations. I will try to assist in this process but it is ultimately you who will have to do the hard work.

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Many people may find this somewhat experimental way to introduce students to new topics counter-productive. Undoubtedly for many it will be. If you feel under-challenged and become bored I recommend you move on to the more advanced text-books of which there are many excellent samples on the market (for a list see (books)). But I hope that for most beginning students this_intuitive_style of writing may help to gain a deeper understanding of the ideas that I will present in the following. Above all, have fun!

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Chapter 1