Complexity and simplicity

Complexity and simplicity is a cliche that is sometimes referred to. But what does it mean and how can a higher complexity sometimes make life simpler? Let’s take a classic example from the history of mathematics.

Carl Friedrich Gauss was one of the giants within mathematics, amongst other things known for the normal distribution curve or the bell curve, got as a 10-year-old schoolboy the task of summing all the numbers from 1 to 100, i.e. to calculate 1+2+3+4+5+ … +96+97+98+99+100. After just a few minutes Gauss could give the correct answer: 5050.

But how did he do? Was he extremely fast at calculating?

No, he discovered a symmetry and used the trick of dividing the list of numbers into two parts of equal length, 1-50 and 51-100. He turned the latter part, and summed them element by element: 1+100, 2+99, 3+98, etc..

When you add the first and last elements, second and penultimate and so forth, you end up with the same result, 101. Since all the numbers are to be added and since both lists are 50 elements long, the result is given by 101 * 50 = 5050.

Have you seen this trick once, it’s possible to derive an expression for the sum S of all the integers from 1 to n, which is called an arithmetic sum, which gives

S = n (n +1) / 2

The question we now could ask ourselves is the following: Did Gauss make the problem simpler or more complex? Simplicity or complexity? Or both?

What was the complexity of the problem that Gauss got according to MHC? He was supposed to add a lot of numbers, 1+2+3+ etc.. The task of adding two numbers is order 7 primary. And to do that many times is still order 7 primary, though with a much higher degree of horizontal complexity. Gauss’ problem had thus a high horizontal but low vertical complexity. In principle very easy to carry out, but very time consuming.

But Gauss transformed the problem into one that is more vertically complex, to be specific, order 10 formal (according to a discussion we had on yahoo tech group adult development)! And to generalize the result to the formula with n instead of 100 is yet another order, 11 systematic.

The conclusion here is that Gauss instead of solving a problem with high horizontal complexity, he transformed it into a problem with high vertical complexity. He makes the problem simpler in that it requires fewer operations, simplicity, yet more difficult because it requires a deeper mathematical understanding, complexity!

This is a typical example of how a new order of complexity can emerge, by having a large horizontal complexity of the previous order. This is typically how you plan the mathematics teaching, consciously or unconsciously. The student is made to solve a lot of similar problems until they think something like “Now it’s the same routine again, there seems to be a pattern here!? What if this can be systematized? That would make it easier!

Often there is a reluctance and resistance to systematize and go up to the next level or order, but when the horizontal load on the working memory gets too big, it appears like the price to take the leap to the next level is worth paying. The following complex level coordinates and organizes the previous so that it becomes easier to manage. This may apply to individuals but perhaps also for entire communities.

• To domesticate the soil is more complex than hunting and gathering.
• To come up with a written language is more complex than to pass on information orally.
• To computerize administrative operations is more complex than do the work with paper and pencil.
• For companies to use social media is more complex than using the one-way communication.

In all cases, new problems are created that are vertically more complex but are still worthwhile because it saves a lot of time and work, at least in the long term. How many have cursed over the new computer application that is not compatible with the current OS? How many have not been annoyed over negative comments on the company’s Facebook page? Or cursed crop failure? Yet we seem to be willing to pay that price. We have otherwise been required to keep up with the competition.

Life has become more complex but simpler.