The amount of concentration, of time spent (by mathematicians) in concentration on mathematical issues is very big. The unconscious effects need the previous concentration time.
Concentration vs. communication: is there any influence to communication during concentration? Is there a difference between being alone concentrated on something on one hand and discussing this thing with others (or presenting/performing it for others) or the concentrated perception of a lecture or talk? There is an obvious analogy to sports and performing arts or to the active part of non-performing arts.
Concentration is linked to attraction (positively) and to distraction (negatively). It is linked also to ‘absorption’ – ‘to be absorbed with something’ as well as ‘to be focussed’ on something, to focus one’s mind.
Is there something that might be called the transcendence of concentration? Sometimes we see this in performing arts when the artist becomes ‘one’ with the performance. There is no special notion for this ‘unity’. It seems similar to the proficiency in using a language – one stops concentrating on the language but uses it ‘fluently’ in order to express something with it. And moreover – not just to express something with it but also to identify oneself with it. Can this be the case in mathematics as well, if we consider it a form of art, or at least having artistic aspects? Is this linked to the unconscious, to insight? Is art the expression of insight? There are very few testimonials about this state of ‘unity’. The pianist Friedrich Gulda has described such moments by saying that the feeling is somehow that he is no more playing by himself but ‘it’ is playing. It would be extremely interesting to link such experiences made by artists with similar experiences made by mathematicians. By the way, it does not mean that in such moments the artist does not control the situation. It is something like a controlled state/flow of trance – so concentration is still there. There are amazing (but very few) depictions of such situations on video during music performances e.g. by Glenn Gould, Thelonious Monk or Wilhelm Furtwängler.
What is the emotional aspect of concentration? Is it linked to attraction, to passion, to devotion, to dedication? Are these the true criteria for good vs. bad mathematicians (hence mathematics)? Curiosity might also play a role but it seems secondary to me compared to the just mentioned criteria.
This item is linked to training and to commitment.