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My work in the future must be devoted entirely to pure mathematics in its abstract meaning. I shall apply all my strength to bring more light into the tremendous obscurity which one unquestionably finds in analysis. It lacks so completely all plan and system that it is peculiar that so many have studied it. The worst of it is, it has never been treated stringently. There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general, and it is extremely peculiar that such a procedure has led to do few of the so-called paradoxes. It is really interesting to seek the cause.
In analysis, one is largely occupied by functions which can be expressed as powers. As soon as other powers enter—this, however, is not often the case—then it does not work any more and a number of connected, incorrect theorems arise from false conclusions. I have examined several of them, and been so fortunate as to make this clear. ...I have had to be extremely cautious, for the presumed theorems without strict proof... had taken such a stronghold in me, that I was continually in danger of using them without detailed verification. (en) |
