My Brain Is Open, the mathematical journeys of Paul Erdös: A Book Review

During my stay at the Mittag-Leffler Institute I read this book written by Bruce Schechter. The book is written in accessible English, so that I had no problems reading it though English is not my mother tongue. It tells the life story of Paul Erdös almost chronologically beginning from his childhood in Budapest to the end of his extraordinary career full of traveling from place to place.

In the story telling, one thing I found uncomfortable. The author quite often slides away from his main topic. For instance in the second chapter called Proof the author spends some time telling stories from Friedrich Gauss’ childhood. The story is interesting, no doubt, but it was not really fitting the context, and moreover it is a well known story. In the same chapter the author dives also into the ancient history of mathematics.

What I liked is the really good picture of Paul Erdös. The author cites him a lot and manages somehow to transfer the feeling which was maybe present when Paul was around. The latter is quite impressive since the author never met Erdös. This side of the book is reviewed in the review behind this link. One particular thing in this spectrum, is the Erdös’ attitude to death; merely his own death. He realised at an early age that he will die and after that he never dismissed that fact. He would say “Let us continue our discussion tomorrow. If I live.”. At some stage he started to add a couple of new letters to his signature every five years. When he turned 75 the signature became Paul Erdös PGOMLDADLDCD which means Paul Erdös Poor Great Old Man, Living Dead, Archaeological Discovery, Legally Dead, Counts Dead. These words have some external meaning, for instance “Counts Dead” comes from the fact that at the age of 75 the members of Hungarian Academy of Science were not counted as working members any more.

Let me quote one of the stories I liked. Erdös was around his eighties at that time.

Erdös needed a cornea transplant to restore vision to one of his eyes. As he was leaving Memphis, a donor became available. At first Erdös did not want to delay his travels for the operation, but after a lengthly argument his friends convinced him that his eyesight was more important. Nevertheless, he insisted on taking a pad with him to the operating room to continue his calculations. When the surgeon saw this he said, “You won’t need that. I’ll be working on your eye.”Erdös replied, “I’ll do math with the other eye.”

The author of the book also gives some insights to mathematics in a popular way. As a mathematician I appreciate the effort and consider it important. I am myself a great defender of recreation and popular science. Non-mathematician readers will find not only a story about a mathematician but will also be exposed to some things which might attract them in mathematics and at least explain the burning motivation. Most of the mathematics in the book is nice and one of the puzzles I already explained earlier in this blog. On the other hand, as a mathematician, I have to express some criticism on the following points.

1. One thing I didn’t like too much was saying that mathematics is Platonism and not bringing any other possibilities to the reader. No more about that.

2. In chapter four the author gets into Ramsey theory. He explains the Klein-Szekers problem:

Let k be a natural number. Does there exist a number n such that no matter how n points are placed in the plane, k of them form a convex polygon (assuming no three of them are collinear)? And if exists, then what is the minimal such n?

For k=3 the answer is 3, for k=4 it is 5 and for k=5 it is 9. The complete solution to that problem is still unknown, but the answer to the first question is yes. For each k there exists an n. The proof uses Ramsey theory, more precisely using the Ramsey Theorem for ternary or quaternary relations. The misleading thing in the book is that the author spends some time in explaining the Ramsey Theorem for binary relations and then claims that this is used to prove the existence of the Klein-Szekers n. Well, maybe it is possible, but I do not see nor know any natural way to do it.

And then Ramsey theory gets really misunderstood. The author goes as far as saying that from Ramsey theory it follows that it is inescapable to get some correct predictions about the future just scanning every tenth letter of the Bible. He motivates it by the fact that some rationally poor people have done it, i.e. they found some words such as ROSWELL and UFO by taking every eleventh letter or something and take it seriously. The problem here is that it does not follow from Ramsey theory that it is inescapable. It follows from probability theory that it is very probable (or it is very unlikely not to find any such words), but it is easy to construct a “counter example Bible” which wouldn’t predict anything. (Of course the probability to find a “predicting” word like Clinton in the Bible in that way is the same as to find a “nonsense” word of the same length like Roflmao.)

Another such mistake is the attribution to Ramsey theory the discovery of a human’s face on the surface of Mars by NASA. It is probable that there will be a face in one of the millions of pictures taken by NASA, but unfortunately not inevitable as the author claims.

I would stress that Ramsey theory implies that if a structure is big enough, then no matter how complicated it is, it will inevitably have simple (not complicated) parts. But usually Ramsey theory doesn’t even tell what particular kind of a simple part it is. Sometimes though only one kind is possible as in the case of Klein-Szekers’. But for example the inevitability of some particular word in the Bible would need many extra assumptions and one of them would probably be that the length of the Bible is bigger than can be printed in one billion years by using all existing printers.

About Vadim Kulikov

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