Thinking Better

The shortcut, the shorter way to reach your target, is what this book is about. The shortest path from A to B is the straight line. That is the natural way that elephant paths emerge in parks if no precautions are taken. But this only holds in Euclidean geometry, and shortcuts are not always that obvious. The most obvious path followed by our brain to solve a problem is not always the shortest. A classical stereotypical example, legend or not, is how the young Gauss added the numbers from 1 to 100. Adding 1+100, 2+99, 3+98, . . . gives 50 times 101 or 5050. His teacher's intention was that the class should add 1+2+3+. . . to keep them busy for a while, but Gauss's answer came almost immediately. This is the kind of shortcut that Marcus du Sautoy is after in this book. His message is that these less obvious shortcuts are the way to success, they form the engine for the progress made in science, and they are the essence of mathematics. It is not a coincidence that the summing example appears in the beginning of the book because it is obvious that Gauss is one of du Sautoy's mathematical idols, since Gauss returns as a hero or at least as a sidekick in practically every chapter.

 

This book made me realize that since we were children, we have constantly learned how to do things better and more efficiently and that nearly everything we do involves a learning process that shows us the shortcuts to do things better the next time we have to solve the problem. Actually, it is not only a matter of nurture. When considering the long-term evolution, then in a way, nature itself also has learned to find shortcuts for the best way to evolve.

 

Although the titles of the ten chapters refer to mathematical subjects and the main story is indeed about mathematics, there are many excursions to broader social aspects and everyday life. These are most explicit in the somewhat shorter "pit stops" that are attached to every chapter. These pit stops are usually the result of an interview or a meeting with a person who represents or has explicit ideas, related to the subject discussed in the preceding chapter but in a quite different social context.

 

In the first chapter, the way to detect a shortcut is by recognizing some pattern in the problem tissue. A repeating pattern allows us to predict the solution, the outcome, or the future. That holds for astronomical observations as well as for summation formulas for number sequences and for the analysis of economic time series. The associated pit stop examines music patterns and the ability to recognize a pattern versus the amount of training one has to put into a subject in order to master it.

 

Other chapters discuss our way to denote numbers and formulas to simplify computations. Computers help to extract the wisdom of the crowd contained in massive amounts of data, but they can be a disaster when they follow blindly inaccurate models for the stock market. As for humans, thinking outside predefined roads is a way to find a shortcut. That is how complex numbers emerged in mathematics and how a start-up business can become a billion-dollar company.

 

At first, I was somewhat surprised to see calculus announced as a major way to achieve a shortcut. But if you think of finding the "best" solution to a problem measured by some criterion that has to be optimized, then, under some conditions, it reduces to finding a zero of a derivative, which is way simpler than finding an optimal solution by trial and error. It is essential for architects, engineers, and scientists to do what they do and to realize what would not be possible without calculus in a reasonable time or within a reasonable budget.

 

There is also a chapter on networks, which are the shortcut to analyze all the connections boosted by the big internet business and are creating the big data to be analyzed. While Euler initiated graph theory when solving the problem of walking the seven bridges of Köningsberg connecting four areas, the mathematical analysis of graphs is now a discipline for analyzing our social networks, automatic grading systems, and simulating the connections in the human brain.

 

In the final chapter, we meet the, yet unsolved, hypotheses and conjectures like the P vs NP problem, of which the traveling salesman problem is a typical example. But it also illustrates how long-standing conjectures that were eventually solved have led to new branches of mathematics that would perhaps never have been explored or not that soon. The shortcut is clearly not always an obvious step to take and it requires you to find your shortcut as a path through a labyrinth as is beautifully illustrated on the cover of the book. If we have not found a way to solve the problem yet, it is not a reason to despair. Du Sautoy points to new computer models like quantum computing or biological computers as a way to find the road through the labyrinth.

 

Marcus du Sautoy is the Simonyi Professor for the Public Understanding of Science, and he has once more illustrated in this book why he was appointed to this position. Although most of its contents are mathematical, there are practically no formulas or complicated technical discussions. There is history as well as current applications and excursions to many non-mathematical topics. Everything is told in an entertaining way, not trying to be funny in an absurd or unnatural way. Each chapter could be a public lecture to entertain a general audience. Some stories are well known among mathematicians but there is a pinch to be learned that cannot be found elsewhere.

 

Will this book teach you how to function better in life as the subtitle claims? I doubt that it contains a non-mathematical prescription for that. We can certainly learn from mathematics, since mathematics is everywhere, as well as from these pit stops describing people who found successes in life by applying mathematical ideas. Remember that we may learn from examples, seeing a pattern, and perhaps, putting the pieces together, finding a shortcut for our own situation. Thinking outside of the box is not a guaranteed success, but sharing ideas with experts from different disciplines has been successful in many cases. Each chapter also starts with a short puzzle that the reader should think about. It usually does not have an easy solution, but the solution becomes obvious in the end after the general principle has been explained and the shortcut lets the aha!-solution drop out immediately. I can warmly recommend mathematicians and non-mathematicians alike to read this very entertaining reflection on many different disguises of the shortcut.

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Adhemar Bultheel is emeritus professor at the Department of Computer Science of the KU Leuven (Belgium).  He has been teaching mainly undergraduate courses in analysis, algebra, and numerical mathematics.  More information can be found on his home page.