Modern mathematics with axiomatic systems has been developed to create a complete reasoning system.


This was one of the most exciting mathematical experiments.


However, even after the failure of the experiment, mathematical research is still directed by the vague ideal completeness.


Tight definitions to guarantee logical soundness were good for small toy world, but it could not model complex human knowledge. Perfect prediction of future needs perfect system. By changing the direction from perfection to specific goals, we can build rich world of mathematical systems that can predict the future for given goal. Creation of mathematical concepts needs not be a complex task, but it is one of the most creative task with deep insight for mathematics and real world.