Hirniak would tell Banach, for instance, that there were still some gaps in his proof of Fermat's problem. Then he would add, "The bigger my proof, the smaller the hole. The longer and larger the proof, the smaller the hole." To a mathematician this constitutes an amusing formulation. He would make weird statements about physics. For example, he would say that half the elements in the periodic table are metals and the other half are not. When someone pointed out that this was riot quite correct, he would reply: "Ah, but by definition we can call a few more of them metals!" He had a wonderful way of taking liberties with definitions.
He studied in Göttingen and described how he would drink cups of wine from an automatic dispenser. Once something went wrong in the machine, and the wine continued to flow. Hirniak continued to drink until he found himself lying on the ground surrounded by a group of people. He heard someone ask, "Vielleicht ist etwas los?" (Maybe something went wrong?). He replied, "Vielleicht nicht." (Maybe not.) At this, he was carried home in triumph on the shoulders of the crowd.
Here is the story that entertained von Neumann so much when I told him about Hirniak years later in Princeton: one day Hirniak told Banach, Mazur, and me that he had almost proved Fermat's conjecture and that American reporters would find out about it and would come to Lwów and say: "Where is this genius? Give him one hundred thousand dollars!" And Banach would echo: "Give it to him!" After the war, Johnny said to me one day in Los Alamos, "Remember how we used to laugh at Hirniak's hundred thousand dollars story? Well, he was right, he was the real prophet while we were laughing like fools." What Johnny was referring to, of course, was that representatives of the Defense Department, the Air Force, and the Navy were traveling around the Country at the time bountifully dispensing research contracts to scientists. The average contract amounted to about one hundred thousand dollars. ''Not only was he right," said Johnny, "but he even foresaw the correct amount!"
Sometime around 1933 or 1934, Banach brought into the Scottish Café a large notebook so that we could write statements of new problems and some of the results of our discussions in more durable form. This book was kept there permanently. A waiter would bring it on demand and we would write down problems and comments, after which the waiter would ceremoniously take it back to its secret cache. This notebook was later to become famous as "The Scottish Book."
Many of the problems date from before 1935. They were discussed a great deal by those whose names were included. Most of the questions posed were supposed to have received considerable attention before an "official" inclusion could be considered. In several cases, the problems were solved on the spot and the answers included.
The city of Lwów and the Scottish Book were fated to have a very stormy history within a few years of the book's inception. After the outbreak of World War II, the city was occupied by the Russians. From items toward the end of the book it is evident that some Russian mathematicians must have visited the town. They left several problems and offers of prizes for their solution. The last date appearing in the book is May 31, 1941. Item No. 193 contains a rather cryptic set of numerical results signed by Steinhaus dealing with the distribution of the number of matches in a box! After the start of the war between Germany and Russia, the city was occupied by German troops in the summer of 1941, and the notes ceased. The fate of the book during the remaining years of the war is not known to me. According to Steinhaus, this document was brought to Wroclaw (formerly Breslau) by Banach's son, now a neurosurgeon in Poland.
During my last visit to Lwów in the summer of 1939, a few days before I left I had a conversation with Mazur on the likelihood of war. People were expecting another crisis like Munich and were not prepared for an imminent world war. Mazur said to me, "A world war may break out. What shall we do with the Scottish Book and our joint unpublished papers? You are leaving for the United States and presumably will be safe. In case of a bombardment of the city, I shall put the manuscripts and the book in a case, which I shall bury in the ground." We even decided on a location. It was to be near the goal post of a football field just outside the city. I do not know whether any of this really happened, but apparently the manuscript of the Scottish Book survived in good shape, for Steinhaus sent me a copy of it after the war. I translated it in 1957 and distributed it to many mathematical friends in the United States and abroad.
Of the surviving mathematicians from Lwów many are continuing their work today in Wroclaw. The tradition of the Scottish Book continues. Since 1945 new problems have been posed and recorded and a new volume is in progress.
Chapter 3. Travels Abroad
By 1934 I had become a mathematician rather than an electrical engineer. It was not so much that I was doing mathematics, but rather that mathematics had taken possession of me. Perhaps this is a good place to stop for a moment and ponder what being a mathematician means.
The world of mathematics is a creation of the brain and can be visualized without external help. Mathematicians are able to work on their subject without any of the equipment or props needed by other scientists. Physicists (even theoretical physicists), biologists, and chemists need laboratories — but mathematicians can work without chalk or pencil and paper, and they can continue to think while walking, eating, even talking. This may explain why so many mathematicians appear turned inward or preoccupied while performing other activities. This is quite pronounced and quantitatively different from the behavior of scientists in other fields. Of course, it depends on the individual. Some, like Paul Erdös, have this characteristic in the extreme. His preoccupation with mathematical construction or reasoning occupies a very large percentage of his waking hours, to the exclusion of everything else.
As for myself, ever since I started learning mathematics I would say that I have spent — regardless of any other activity — on the average two to three hours a day thinking and two to three hours reading or conversing about mathematics. Sometimes when I was twenty-three I would think about the same problem with incredible intensity for several hours without using paper or pencil. (By the way, this is infinitely more strenuous than making calculations with symbols to look at and manipulate.)
On the whole, I still find conversation with or listening to other people an easier and pleasanter way of learning than reading. To this day I cannot read "how to" instructions in printed form. Psychologically, these are indigestible for me.
Some people prefer to learn languages by the rules of grammar rather than by ear. This can be said to be true of mathematics — some learn it by "grammar" and others "from the air." I learned my mathematics from the air.
For example, I learned, subconsciously, from Mazur how to control my inborn optimism and how to verify details. I learned to go more slowly over intermediate steps with a skeptical mind and not to let myself be carried away. Temperament, general character, and "hormonal" factors must play a very important role in what is considered to be a purely "mental" activity. "Nervous" characteristics play an enormous role in one's intellectual development. By the age of about twenty, when development is supposed to be fully completed, some of these acquired traits are perhaps essentially frozen and have become a permanent part of our makeup.
Mathematics is supposed to be in essence only a very general and precise language, but perhaps this is only partially true. There are many ways of expressing oneself. A person who starts early has some particular way of organizing his memory or devises his own particular system for arranging impressions. A "subconscious brewing" (or pondering) sometimes produces better results than forced, systematic thinking, as when planning an overall program in contrast to pursuing a specific line of reasoning. Forcing oneself to persist in a logical exploration becomes a habit, after which it ceases to be forcing since it comes automatically (as a subroutine, as computer people like to say). Also, even if one cannot define what we call originality, it might to some extent consist of a methodical way of exploring avenues — an almost automatic sorting of attempts, a certain percentage of which will be successful.