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When we are solving a mathematical problem, we usually try to find a solution as exact as to the given problem as well as to find a closed form solution. Remind that It is obvious that most realistic problems do not have exact solutions. Hence, we try to simplify the orogonal problem to an approximated problem which probably has an exact or closed form solution. The simplification methods are used as an alternative approach. We will show in this article that the simplification methods have also a number of fundamental problems.
If we have interests only on a closed form solution, we will meet a number of important limitations when solving complex problems.
- First, because of computational complexity it is not easy for the recent approach to find a solution for more than third order polynomial equations or for equations including non-linear functions.
- Second, by simplification in order to reduce the complexity of the given problems we lose accuracy in its solution. For high level complexity problems, we simplify the given problems so as to be less order polynomials or to be linear equations. By doing that, the complex problems change to solvable problems at the expense of accuracy loss.
- Third, we have also difficulty to solve multiple conditional problems especially when the number of conditions are more than two. We can solve the multiple conditional problems using Lagrange multiplication, which has also computational limitation.
We can overcome these limitations by the computer simulation approach. The computer simulation can solve complex problems with high order polynomials, nonlinear equations and many conditions. However, traditional mathematicians do not tend to rely on the computer simulation results because they think it is unproved yet. It is obvious that the computer simulation may provide solutions associated only for a number of given situations because we can run the simulation for all situations. However, it is also wonder whether the closed form solution applicable for all cases without limitation.
Many Japan paper associations are interested in experimental results while western paper associations give less attention to the experimental results without theoretical analysis, i.e., the computer simulation. We now need to rethink which is a better way for the future research between the theoretical approach and experimental approach.
Philosophical Consideration Edit
Based on Plato's cave, many ancient researchers think that the principle can overcome our limited empirical results. They have forgotten that their new principle is designed based on their empirical results, even usually derived from a part of the results. It is true that we can not see real things because of our limited senses. For example, we can not see somebody in a dark night while the beyond red-light equipment enables us to monitor humans in a deep dark night. The principle designed based on the limited empirical results could not represents every facts we have not monitored before.