Shurik, a friend of mine I used to share student dorm with, was a mathematician. Algebraist, to be precise. We talked a lot discussing multitude of topics, not necessarily mathematical ones. In those days hot water was customarily shut off across campus during summer season so students could prepare for exams without such a distraction as hot showers. That fact prompted me to comment that our lifestyle, while notably different, still somewhat resembles what a lifestyle in Upper Paleolithic might look like. Shurik was digesting my remark for a few moments with the stamp of intense thinking on his face (his Calculus test was next day), then said excitedly:
- Dude, you know why there was no hot water in Upper Paleolithic? That's because the water pressure was not strong enough in the Lower Paleolithic!
That insignificant episode from my student years characterizes true mathematicians very eloquently. They are quite unusual breed of humankind with extraordinary abilities to locate not very obvious properties and relations in seemingly regular objects and notions. Having been exposed to interaction with mathematicians for sometime I, by the time the book of Mr. Schechter was read through, felt I knew Paul Erdos almost personally. Very light and elegant writing style of the author was a contributing factor as well.
Mathematicians rarely can be aggressive. Usually, they are very sensitive and kind people. In this regard the portrait of Paul Erdos by Mr. Schechter goes along quite naturally with my experience of dealing with them. At the same time that portrait leaves a very sad impression of the true inner nature of Erdos - depressingly lonely person, with no family and no home. The deep tragedy of the Erdos family with Paul's siblings gone by disease, father's suffering in Russian exile, terrible WWII ordeals - all that makes you wonder how Paul and his parents can continue "to prove and conjecture" so successfully under such horrendous circumstances? Author partly explains this phenomenon very brightly describing the scientific and especially educational traditions in Hungary before the war. Indeed, the density of incredible talents generated in this small central European country somewhat shocking. It underscores how important the role of truly good teacher in elementary school can be. Taking into account all that and also the fact that both parents of Erdos were superior math teachers in high school themselves a reader can see the roots of the enormous productivity of Erdos, who published more math papers in multiple branches of it than any other scientist in history. But it also can be a city of Budapest whose streets, as per Mr. Schechter, are very inviting for any kind or scientific reasoning - although not a scientist myself, I did experience the same when I was roaming with friends along Duna shores in Buda one summer.
The mathematical content of the book is very engaging for non-mathematicians. It is explained almost with no formulas but Mr. Schechter manages to convey the depth of the mathematical ideas very well without them. It is especially applicable to the chapter about prime numbers. The primes, although endless in the set of integers, do have very strange properties. Take the theorem proved by Chebychev first and re-proved by Erdos by elementary means - between N and 2N there is always a prime. At the same time we know that the intervals without primes can be as long as one would wish. At first glance two facts seems to contradict to each other but they do not. Facts like that are abundant in the Numbers Theory with most enigmatic one as a problem of primes distribution and Riemann function. Mr. Schechter does a good job providing historical background of the Numbers Theory, its evolution, contributions of Paul Erdos and controversy of Erdos and Selberg.
I have to admit the author did a brilliant homework researching all kinds of details pertinent to mathematics and its origins. I did enjoy pages about clay table Plimpton 322 with its incredible content of Pythagorean triplets as well as multitude of other stories like most bizarre "application" of Numbers Theory when close collaborator of Erdos avoided deportation to Gulag just because he happened to have his publication on the subject in Russian mathematical magazine with him. In this regard, the book of Mr. Schechter can be considered as not so much as biography of Paul Erdos but as biography of mathematics as a scientific discipline. Humor, albeit sometimes very dark (for example, about math students, who were "studying" Jordan theorem being confined to "inner area", id est being imprisoned) sparking the text regularly and appropriately.
Mathematics is somewhat similar to soccer. While everybody can perceive the beauty of ball handling by say Riquelme or Robinho, very few of us can do the same on the soccer field. In math, formulation of the conjecture can be deceptively simple and elegant, and most of us can understand it well. At the same time, it is very different story once you start thinking about trying to prove that conjecture. In many cases it might require years of learning and tons of exercises. But even that no guarantee to success. The inclination to a special way of thinking is required. In this regard, magic of Riquelme on the stadium is direct equivalent of wizardry of Erdos in Numbers Theory. The books similar to Mr. Schechter facilitate our comprehension of the conjecture beyond mere formulation, opening the curtain after which the proof is hidden.
On the other note, I can't stop thinking of what kind of future European science might have should its development was not brutally aborted by sad realities of Second World War. True, many of bright Hungarian (and other) minds escaped from the inferno of warfare and extermination campaigns; true, many of them intensified their research in military related directions and achieved significant results. Still so many perished needlessly making a good number of famous European scientific centers empty and forgotten for a very long time. It seems incredible that one person's paranoia can mercilessly terminate so much in such a short period of time. Let's us hope the future Erdoses will never be forced to travel so intensively against their wills even with theirs brains open so widely.
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Jonathan
A superb introduction to why one man thought.
Reviewed in the United Kingdom on June 27, 2021
