Paul Erdös: review of Paul Hoffmann's The man who loved only numbers.

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Paul Erdös was the second most prolific mathematician in history, after the Swiss, Leonhard Euler. He is the most prolific collaborator with other mathematicians. Hence, an ordinal number system named after him, which grades how closely any mathematician came to work with him. If you have an 'Erdös number one', that means you actually did a mathematical paper with this prodigy. 'Erdös number two' means you have done a joint paper with a mathematician who did a paper with Erdös.

I know someone whose bigger brother is an Erdös number two. She wonders if having her homework done for her, by said brother, makes her an Erdös number three. Sadly not.
At our book club, we gave little talks about a book we had read. She enjoyed my talk about Paul Hoffmann's title, some time ago. She said I was right: he did bring his mother along with him. Indeed, that much I had faithfully gleaned from Hoffmann. ( Unfortunately, I can't remember exactly what I said. The following account differs somewhat. )

Paul Erdös was of Jewish Hungarian extraction. The book says the name Erdös is pronounced 'air-dish'. But I'm told the Hungarian pronunciation is actually 'err-desh'. That is err, as in 'to err is human, to forgive is divine', and desh, as in Bangladesh. ( Scots would give 'err' its traditional and phonetic pronunciation, which does sound like the normal pronouncing of 'air', but I mean the more typical pronunciation of 'err', as the unstressed vowel sound. )

The physics nobel prize winner, Abdus Salam also was a mathematical prodigy, which was how he came to be discovered in his home country of Bangladesh. He set up a foundation in Italy to help others like himself in under-privileged countries. Evidently, they put much back in their homelands, and do not constitute a 'brain drain'.

You may think it is good to tell I am not a mathematician, because they do not work on irrational associations. The conclusion is correct but the inference false. Yes, I'm not a mathematician. No, they do. A versifying colleague ( half ) rhymed his name with 'Kurdish', as about the only people not to benefit from one of his math papers. Erdös was enthused to submit a paper to a Kurdish journal of mathematics. But he found there wasnt one.
( Not being a mathematician, apologies for not knowing other mathematicians' names from Hoffmann's book -- Erdös' own name being all I could master. )

Paul Erdös' parents were both mathematicians. But Paul, himself, never married, as the title of Hoffman's biography suggests. His genetic heritage went no further. He called children 'epsilons', the Greek letter mathematicians use for small quantities. He loved children and was good in their company. Earlier fotos show him beaming in their company.

Paul never really grew up. He was always a mother's boy. War and conquest deprived him, as it has many, of a father's influence. He was not allowed to tie his shoelaces, till his age was into double figures. His mother was so possessive, she once appeared out of an upstairs window to ask her son, in the street, what was he doing with that girl. She was the girl-friend accompanying Paul's friend.

By all accounts, his mother was likable, as well as dominating. Five years after she died, Paul was gloomily crossing some campus. A colleague asked him what was the matter. He replied: He missed his mother. Reminded this was five years ago, he said, he knew.
After her death, he threw himself even deeper into his vocation. More than one foto of him, in later life, show him asleep at formal group takes or just at dinner. Those in mathematical conversation with him would think he hadnt heard. He was like a dolphin that sleeps with one half of his brain, while the other half stays alert. The mathematics went on even while he was dozing.

A mathematical collaboration with Erdös was a somewhat taxing seventeen-hour day. He would arrive at the door-step. If it was Christmas, he would say something like: Happy Christmas. Let n be the number...
At about four o'clock in the morning, the guest would start rummaging noisily about the kitchen, in his undomesticated way -- he was used to his mother having done everything for him and expected everyone else to. Moreover, this was the hint it was about time his colleague and host got up to do mathematics.

Erdös had no home. He lived from a suit-case, on a perpetual tour to tap other mathematicians' brains. 'Property is a nuisance' is one of his sayings. He kept himself solvent with earnings from mathematical journals and prizes. He never kept more than he needed to meet colleagues. He was constantly giving money away to the current needy cause, wherever in the world it was.
He might have seemed a sad case, had he not had this professional talent, because no-one might have known the extent of his goodness of character. Perhaps that is the moral of his life. He only needed mathematics but others had a most pressing need of money. He knew this and cared without fuss and without their asking.

One married couple of mathematicians built an extension for him to stay periods. He could be difficult to work with. He could drive the woman, of these hosting partners, to vow she would never work with him again, in her frustration at his bad conversational manners.
He would ask her to explain some maths and then interrupt her to try and re-formulate the problem in his own terms, which naturally stopped her in her tracks.

Yet Erdös was an incomparable ambassador of mathematics. He would set people questions with rewards, starting with a five dollar question, grading the prize according to the difficulty of the answer. He could judge ability, so he knew just what level of problem to set.
At the top end of ability, he once advised a graduate against a particular thesis. It was too difficult. The young man had cause to be grateful. The problem still wasnt solved by the mathematical world twenty or thirty years later.

Erdös prefered tackling problems that didnt need a lot of specialist knowledge. He best liked solutions 'straight from the book' -- God's manual of creation, as it were -- that carried immediate conviction.
In perhaps forty pages or so, two mathematicians, to their intense pride. had written a proof of a theorem. Erdös happened to notice it on a black-board. Asking the meaning of the notation, from a field of math he didnt know, he wrote straight down, in a couple of lines or so, a new proof -- straight from the book.

He wasnt in the slightest interested in the practical value of his findings. He would be satisfied if no applications were found for another five hundred years. This was not just the doctrine of pure science but the freedom to enjoy mathematics for mathematics' sake.

Paul Hoffmann's book is of interest for also discussing some recent mathematical milestones, such as the proof to Fermat's last theorem. The solver happened to know the right branches of study and worked long alone to win the prize on offer. This way is in complete contrast to the co-operative Erdös. Ironically, a hole was found in the closet solver's proof and he was driven to seek help from another mathematician, to plug the leak in his proof.

However, I must admit to finding the cited problems, in pure math, neither practical nor interesting. There was one old brain-teaser I found appealing and remembered for a little talk to a non-mathematical group. A well-known tv hostess with a super IQ caused a storm of controversy with it:

Suppose you are on a quiz show. You may choose to open one of three doors. One has a prize behind it, an other a booby prize. Suppose you choose one door but before you open it, the show hostess, opens another door which reveals no prize. The hostess then allows you to stay with your first choice or to choose, instead, the other unopened door.
The question is: which is the best strategy?

The tv woman with the genius IQ said: change your choice. Letters at a rate of nine to one disagreed with her, including some academics, on the degenerate influence of tv, saying things like: you really blew it, this time! Their argument was that the move from one door to the other shouldnt make any difference, because there was an equal probability that the prize would be behind either still unopened door.

Of the little group, this reviewer talked to, some guessed right some wrong. None really know. I had thought like the ignorant ninety per cent. The interesting thing is that the most prolific mathematician of the twentieth century couldnt understand, either. Like a green student, he pestered his host and colleague for an explanation.

Erdös was shown a computer simulation of the quiz show given a large number of trials. On average, the probability of winning the prize was one-third, if one stayed at the door of one's first choice. If one changed one's choice, the probability of winning became two-thirds.
Erdös accepted the result but he still wanted a transparent explanation 'straight from the book'.
Erdös' friend and colleague put it this way. You, as the quiz contestant, know you are going to be given the chance of making two choices for the prize.

Richard Lung.

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