Taking it Further
by Joseph Rowlands

Recently I've read some books about successful mathematicians and scientists. There's one pattern that has emerged that strikes me as both useful and inspiring. Each of these successful people didn't stop at a single successful achievement. They followed up on it, trying to find out if there was more they could learn from it.

One example of this was with the mathematician Euler and his solution for solving the infinite summation of 1 + ¼ + 1/9 + 1/16 + ... = π2/6. This was an incredible solution and shocking results (since it is the sum of squares, and its result includes π which is related to circles). But if it were left at that, it would simply be the product of genius. He then went on and applied his new method to many other cases, trying to exhaust the capabilities of the method and find new problems to solve.

A different example occurred in science when people started performing experiments to isolate the chemical elements. A scientist named Antoine Lavoisier took one of these methods of producing oxygen, and took it a step further. He carefully measured the weight of the ingredients in the chemical reaction, and found that there was a conservation of mass. While matter changed state, it was all still there at the end.

Time and time again this pattern is repeated. Many of the most accomplished minds built on their previous insights and discoveries, mining them thoroughly for any additional value. And sometimes, the later accomplishments were even more significant.

The lesson I take from this is to follow up on your thoughts and ideas. Maybe you have a thought that might lead to something more interesting if you worked out the details and saw where it took you. The only way to know is to follow through on it. Maybe it'll lead you to a greater understanding. Maybe you'll hit a dead end. But if you don't follow through, you'll have no idea.

Writing these articles is one of the ways I try to put this into practice. I might start off some interesting thought, but not be sure of the implications. So to follow it up, I try to write it down, with details, and as systematic as I can. Sometimes I find that this process leads me to a deeper understanding. Sometimes it helps me to identify tricky cases that need more thinking. Sometimes I manage to integrate the new ideas with things I already knew, shedding additional light in those areas.

There are many areas where this kind of process is useful. Recently I saw a lecture that pointed out a method of factoring polynomials. I hadn't seen that method before, and it struck me that it might be possible to use that method to derive the quadratic equation, which we were taught in school but never learned how it was discovered. Instead of just thinking it might be possible to derive it and leaving it at that, I grabbed a pen and paper and spent a little time going through the exercise. Sure enough, it worked and now I have an understanding of at least one possible way it could be derived. Instead of it existing as a memorized rule that seems to have come from nowhere, I can now understand it as a product of a methodical approach.

These are just examples of natural curiosity with some follow up. There are plenty of examples where you might wonder how something works, or have a half-baked idea that you wonder if it could lead to anything. Instead of just dropping it, follow through with it. Fill in some details. Try to explore the topic. See where it leads you, and see how far you can take it.

Some people find this idea scary or burdensome. They think that if you're going to pursue an idea, you have to be totally serious about it, and be willing to change your life accordingly. If you aren't willing to put 100% into it, then you just aren't serious. So they stifle their natural curiosity.

I think this is the wrong way to view it. When you find something you are interested in, you should pursue it and enjoy the process of learning more and seeing where it goes. Maybe you'll find a topic that you genuinely are interested in, and willing to pursue in much greater detail. Most of the time it will just be satisfying your curiosity and having the positive feelings that come with accomplishing a goal. And as you learn new things, you create even more opportunities to find things interesting. The world is a fascinating place if you're willing to explore.

Not that this approach is only for fun. New knowledge and understanding can benefit you in countless and unexpected ways. You might make sense out of something you always found confusing. You might have a greater appreciation for how people invent things or discover new ideas.

And creating the habit of following through on ideas is useful in other areas of your life, most especially in work. In school we learn mostly through being explicitly taught the information gathered by others. As adults, a significant amount of our learning comes from our own explorations and thinking, and with sharing those thoughts with others. There's always room for more understanding.

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