This is a rather big title for me, maybe for most people even for many professional mathematicians. However, I’d like to write some words here, just for a record for my present understanding.

I began thinking this question a long time ago, when I made up my mind to be a mathematician. To answer this question, we need to know the definition of math. So what is mathematics. Most people begin to learn math in kindergartens, and this process never pause even till university. I didn’t know what is math until I finished my undergraduate study. From the undergraduate education, we can generally define math as a subject that deal with quantity, structure, space and change.

Each of them comes from the process when human beings began to learn the world they live in. Math comes from the concrete objects, but not limited to special objects. Mathematicians always try to include more objects in one mathematical concept by subtracting more general properties from concrete things. Than to study mathematical concepts, we can obtain more general properties for many concrete things. From this point of view, we can see that one kind of good math is to construct new theory or concept in which we can include several old ones. This kind of taste is similar with that of physics. It is ambitious. Hundreds of physicists, including many famous ones like Einstein, have spent more than one hundred years to look for a unified theory to explain our world, but in vain. There seems always a balance in it. Paying attention to generality may sacrify special characters. Taking PDE as an example. After Hormander (who studied linear PDE in a systematic way), no one can find a general theory for nonlinear PDE. Because different equations have different properties, and the method for one equation usually does not work for another. This lead to the fact that nowadays, most of mathematicians in this field usually focus on one type of equations in the whole life time. It seems now in the field of PDE, the top work is to introduce new tools to solve problems.

One view is:

1st class is introducing new concept and building new theory.

2nd class is expressing new ideas and making new tools.

If judging it in this view, PDE is far from the good math.

Another kind of good math is the subject which can connect one branch with the other. One example is Gauss-Bonnet theorem in differential geometry. This is good because it connects geometry and topology. It provides a bridge to understand two different branches as a whole. Another field share the same character may be the geomety analysis, which is using analysis tools (mainly PDE) to study geometry.

To be continued……

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