What can I tell you from my decades of mathematical experience?
Real progress in mathematics is only possible with unusual ideas that result from intensive consideration. The fewest problems are solved with luck. We need a mathematical pool from which we can use the mathematical tools for our problems. Success only comes with a healthy dose of tenacity, with which we can pursue a problem for decades if necessary, before we can solve it.
If we think deeply enough, it may require making whole theories obsolete. Therefore, it is necessary to challenge the so-called self-evident and to develop better solutions if we are not satisfied with the state of research. We have to develop a high sensitivity for the feasible and we must not be content with what we have achieved: we have to criticise ourselves.
It is not just the big and complicated ideas that solve a problem in the best way possible, but require long proofs. It is often the small, simple steps that bring success, as elaborate transformations have unfavourable complexity. We should also train the powers of observation and detach ourselves from existing literature, avoiding having to reinvent the wheel or miss important impulses.
If we cannot solve a mathematical problem iteratively, we should use automated computer-aided (proof) methods or be satisfied with a good approximation, even if the beauty and conciseness of the exact and closed representation is lost and the effort for the implementation is considerable if the solution to the problem is important enough or the required resources are available.
Often, the specification of error terms or numerical solutions (e.g., using interval arithmetic) is sufficient because high precision is not required. While certain statements are of value only in their exact form, it is more important to evaluate the problems according to their practicality and use, then to prioritise and finally satisfactorily resolve them.
There will always be numbers that cannot be represented by short sums or products of few (algebraic or transcendental) numbers, even if we have to (additionally) distinguish some of the numbers and to use them under a symbol. There is a good reason that we do not have the ability of the gods to survey entire levels of infinity at once and to filter out of them concise results.
© 2009-2018 by Boris Haase