Two Doors to Open

Mathematicians do not care to speculate in print about the future of their subject. There are several good reasons for this. One is that significant advances in mathematics nearly always involve the injection of unpredictable new ideas. A few rash mathematicians have published predictions for their own fields; these predictions were generally based on the stock of ideas already in the field, and with hindsight they became steadily more embarrassing as the subject moved onwards to higher things. Nor is it wise to publish bold speculations until you have checked that they work – and then they are no longer bold speculations. For a mathematician, problem lists are a safer bet than predictions. A description of a problem leaves it open whether one can solve the problem routinely with known methods, or whether something radically new is needed. The best problem lists have come from relatively young researchers working at or near the crest

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