Sometime ago I read somewhere (and I don't remember where it was) that Stefan Banach--a highly creative and great mathematician--did not always write down his ideas.
Allegedly, he did not write his own thesis (but of course, all the mathematics in it came from him). Is that true? And is it known who wrote it then?
Here is a quote from the article by Krzysztof Ciesielski: On Stefan Banach and some of his results. Banach J. Math. Anal. 1 (2007), no. 1, 1–10.
There is a curious story how Banach got his Ph.D. He was being forced to write a Ph.D. paper and take the examinations, as he very quickly obtained many important results, but he kept saying that he was not ready and perhaps he would invent something more interesting. At last the university authorities became nervous. Somebody wrote down Banach’s remarks on some problems, and this was accepted as an excellent Ph.D. dissertation. But an exam was also required. One day Banach was accosted in the corridor and asked to go to a Dean’s room, as “some people have come and they want to know some mathematical details, and you will certainly be able to answer their questions”. Banach willingly answered the questions, not realising that he was just being examined by a special commission that had come to Lvov for this purpose.
It is true that Banach was mainly self-taught as a mathematician, although he attended some lectures by Stanislaw Zaremba at Jagiellonian University. By the way, engineering programs in the former Austro-Hungarian monarchy (including Lvov Polytechnics) required quite an intensive training in mathematics, although of course the latest developments (Lebesgue integral etc.) were not part of the curriculum.
Addendum 0: The above story is also related by Roman Kaluza in his biography of Banach. He heard it from Turowicz, who credits Nikodym as his source (he himself joined the department later, when Banach was already a professor). Well, on one hand, Nikodym was a friend of Banach and his early partner in mathematical discussions, but on the other hand, at the time of Banach's PhD, he was teaching high school in Krakow. (This point was made by Krzysztof Ciesielski in an email exchange with me.)
Addendum 1: Banach's thesis, written in French (which he knew well and used before in publications) can be found here: http://kielich.amu.edu.pl/Stefan_Banach/pdf/oeuvres2/305.pdf It was published in Fundamenta Mathematicae 3 (1922), pp.133-181, and bears only Banach's name. The footnote says that it is a "Thesis presented in June 1920 at the Lvov University for obtaining the degree of the Doctor of Philosophy."
On the first page there is a statement that maybe gives some evidence of Banach's tendency to wait until getting the best version of his results: ``Mr. Wilkosz and I have some results (which we propose to publish later) on operations whose domains are sets of Duhamelian functions(...)". There is no joint work with Wilkosz in the collected works of Banach...
Addendum 2: Some details brought up by other users need correction. First, Steinhaus met Banach and Nikodym in Krakow, where Banach grew up, not in Lvov. This is explicitly recorded in his "Memoirs and Notes", and somewhat less explicitly in the address he gave much later at a session devoted to Banach: http://kielich.amu.edu.pl/Stefan_Banach/steinhaus63.html ("Planty" is a major green belt in the old city of Krakow; in Lvov there were "Waly"). Second, Banach's PhD supervisor (only in the formal sense) was Steinhaus. Antoni Lomnicki held a chair of mathematics at the Lvov Polytechnics (not to be confused with the Lvov University), where Banach got his first position as an assistant (pre-PhD).
Addendum 3: Here is what Steinhaus wrote about working habits of Banach (see the link above):
His lectures were excellent; he never lost himself in particulars, he never covered the blackboard with numerous and complicated symbols. He did not care for verbal perfection; all manner of personal polish was alien for him and, throughout his life he retained, in his speech and manners, some characteristics of a Cracow street urchin. He found it very difficult to formulate his thoughts in writing. He used to write his manuscripts on loose sheets torn out of a notebook; when it was necessary to alter any parts of the text, he would simply cut out the superfluous parts and stick underneath a piece of clean paper, on which he would write the new version. Had it not been for the aid of his friends and assistants, Banach's first studies would have never got to any printing office." And also: "Banach could work at all times and everywhere. He was not used to comfort and he did not want any. A professor's earnings ought to have supplied all his needs amply. But his love of spending his life in cafes and a complete lack of bourgeois thrift and regularity in everyday affairs made him incur debts, and, finally, he found himself in a very difficult situation. In order to get out of it he started writing textbooks.
Addendum 4: This is based on information I received from Danuta Ciesielska, a Polish mathematician and a historian of mathematics (and my classmate from Krakow). The documents from the Lvov University are now split between the Lvov District Archive and Lvov City Archive, http://www.archives.gov.ua/Eng/Archives/ra13.php - Wayback Machine link (the documents of Polytechnics were transported to Wroclaw, Poland after 1945). The catalogs underwent major reorganization, which makes it quite difficult to find particular documents there. Besides the employees' folders, the documentation of PhD and habilitation proceedings is often found in the minutes of faculty meetings. Regarding Banach's PhD, Ciesielska saw a letter from Steinhaus to dean Stanecki (dated September 28, 1920) asking him to set the date for Banach's doctoral exam, to which Stanecki replied that the date cannot be set before Messrs. Steinhaus and Zylinski (the committee members) evaluate the thesis. (Aside: Math Genealogy Project lists Kazimierz Twardowski as one of Banach's advisors. On the surface of it, this makes little sense, as Twardowski was a philosopher and a logician; his expertise was far removed from what Banach worked on. However, as a professor of Lvov University, he was on the committee and signed the papers.)
She also points out that in some institutions (e.g., Jagiellonian University in Krakow), if a PhD thesis was published after the exam, the printed copy/journal offprint replaced the submitted manuscript/typescript. It is not clear if this was the case in Lvov.
When I was a student in Lvov in the 1970s, I heard many legends about Banach, so let me add a few points. Once Steinhaus was walking in a park, and he accidentally heard a conversation of two young people sitting on a bench. The words "Lebesgue integral" were pronounced. At that time very few people in Lvov had heard of the Lebesgue integral. So Steinhaus was curious, and introduced himself... Banach was an engineering student at that time. (The story does not tell who the other person sitting on the bench was.)
According to the legend, Banach worked most of his time in the Scottish café. Students and colleagues joined him for conversation. (One of the results of this was the famous "Scottish book" of unsolved problems. Prizes were offered sometimes and recorded to the book together with the problems. For example, in the 1970s, when Per Enflo solved the "basis problem" from the Scottish book, he won a prize, a live goose, which was delivered by Mazur). He used to write on the table cloth. The owner of the cafe never complained. At the end of the day, he changed the tablecloth for a new one. And he would sell the old one to students.
Banach drank a lot (and there are many stories about this, which I omit). Frequently he was short of money, and had to drink in credit. At some time, the debt grew large, and there was an argument with the owner of the Scottish café. Finally, the owner proposed that Banach writes a calculus textbook to make money to pay for his drinks. (Some version of the legend says this was suggested by students). Indeed, he wrote a calculus textbook :-) But I have never seen his high school textbooks.
The Scottish café still existed in the 1990s, but under a different name, and in the 1970s this was a simple cantina. Then, the rooms passed to some financial institution.
P.S. Wikipedia, https://en.wikipedia.org/wiki/Scottish_Caf%C3%A9, has somewhat different details of doing math in the Scottish café, based on Ulam's recollections.
I also once heard such a story, but I have doubts it is literally true. What is an established fact is that Banach had an unusual start of his career.
He was actually an engineering student (with a personal situation rather on the difficult end) and did math more or less as a hobby. By pure coincidence he met Hugo Steinhaus who was impressed. They worked together and published something together. Then Banach got a position at a university (Lvov) and then a doctorate (under Lomnicki [correction: while he was working for/in the group of Lomnicki, it appears Lomnicki was in no sense the director of his thesis; cf Magaret Friedland's answer]). So he got his doctorate under somewhat unusual circumstances and not following standard rules (though at that time, there were much less rules for doctorates then nowadays anyway).
In that sense, it was likely not so clear when and how he should submit his thesis, and it seems very conceivable that he discussed this matter with various people and/or people close to him pressured/encouraged/helped him to do so. (Added: I see Francesco Polizzi made a comment sort of in this direction.)
Regarding the "laziness":
Not long after the time of his thesis he wrote a lot (including high-school textbooks). So, to attributed this to sheer laziness in a classical sense seems certainly odd. If anything I could imagine a certain uncertainty (and/or occupation with other matters) regarding how to proceed; or how to really write mathematics (not being trained as a mathematician).
Yet, it is also well-documented that he and others worked a lot in cafés. Now, this could to some be taken as a sign of a 'lazy' life-style. But, well, not even this is so clear.
For an overview of Banach's life http://www-history.mcs.st-andrews.ac.uk/Biographies/Banach.html
There is a paper on this topic in pages 1-7 of the September 2021 issue of The Mathematical Intelligencer. The authors are Danuta Ciesielska and Krzystof Ciesielski. If I understand correctly, the aim in their paper is to set the record straight regarding the (infamous) story about the way in which S. Banach obtained his Ph. D.
I am going to share with you the main paragraphs of the Ciesielska - Ciesielski paper below: both the phrases in boldface and the sics are mine.
*** THE STORY ***
"The story goes that Banach could not be bothered with writing a thesis, since he was interested in solving problems not necessarily connected to a possible doctoral dissertation. After some time, the university authorities became impatient. It is said that another university assistant (instructed by Stanisław Ruziewicz) wrote down Banach's theorems and proofs, and those notes were accepted as a superb dissertation. However, an exam was also required, and Banach was unwilling to take it. So one day, Banach was accosted in the corridor by a colleague, who asked him to join him in a meeting with some mathematicians who were visiting the university in order to clarify certain details, since Banach would certainly be able to answer their questions. Banach agreed and eagerly answered the questions, not realizing that he was being examined by a special commission that had arrived from Warsaw for just this purpose. In some sources [11, 19, 20], this event is described only as a possible version of events. Nevertheless, in several (mainly Polish-language) books, it is presented as a fact. There is even a book on the phobias and fears of great Poles that devotes a whole chapter to Banach and this story, claiming to demonstrate that Banach was unable to deal with his own psyche and phobias, although even this story presents Banach simply as someone who did not consider the PhD a very important acquisition."
*** DEBUNKING THE STORY ***
"... good stories aside, the truth about Banach's exam should be known. Nowadays, it is possible to check the facts, since many sources have become more easily available than they were some decades ago. It is enough to look carefully at some dates and university rules to see that the proposed account could not be accurate. Banach moved to Lvov in 1920 to take up his job at the Lvov Polytechnic. On June 24 of that year, he presented his doctoral dissertation to the Philosophy Faculty of Jan Kazimierz University. The time interval of just a couple of months was definitely too short for the university authorities to have become impatient, let alone for someone else to have written a thesis on the basis of Banach's overheard comments. Moreover, in 1920, Banach had already published three research papers. Why would he be reluctant to write a doctoral dissertation, which would be a requirement for him to keep the job?
Now let's have a closer look at the exam. According to the university rules, a PhD dissertation had to be refereed and accepted, and then two exams--in the candidate's main scientific disciplines (in Banach's case they were mathematics and physics) and in pure philosophy--were to be taken by the candidate. It turns out that the records of Banach's PhD exams have survived (they are reproduced in [22] and [26]), and we may read that Banach passed his PhD examinations in mathematics and physics. The examining board consisted of four scientists: the dean of the faculty, Zygmunt Weyberg, wo was a mineralogist; two mathematicians, Eustachy Żyliński and Hugo Steinhaus; and a physicist, Stanisław Loria. None of them was from Warsaw, and Banach knew all of them.
There is another interesting story [sic] concerning Banach's doctoral dissertation. The referees were Żyliński and Steinhaus. In October 1920, Steinhaus, who was mentoring Banach, wrote to the dean to inquire about the date of Banach's doctoral exam, for it had been four months since Banach had delivered his dissertation. The dean replied that everything was ready for the exam, but they were awaiting the referee's report (one of whom was Steinhaus himself!). Indeed, when the joint report from Steinhaus and Żyliński arrived, the exam took place immediately. Banach had submitted his dissertation on June 24, the report is dated October 30, and the exam in mathematics and physics took place on November 3. Bearing in mind that in 1920, October 30 fell on a Saturday, November 3 was therefore a Wednesday, and November 1 (Monday) is a public holiday in Poland, everything must indeed have been prepared for the exam. Banach passed this exam with a unanimous grade of 'excellent' from all four examiners.
On December 11, 1920, Banach passed the exam in philosophy (the examining board consisted of the two philosophers Kazimierz Twardowski and Mścisław Wartenberg and the dean, Zygmunt Weyberg). Banach had now fulfilled all the requirements for being granted the PhD degree, and in many sources (including a CV signed by Banach; see [19]), 1920 is given as the year of Banach's doctorate. However, the precise rules for obtaining a PhD from Austro-Hungarian times had been retained by Poland after regaining its independence (see [14]). According to those rules, the candidate was allowed to call himself a 'doctor' only after the doctoral conferment ceremony, which in the case of Banach took place on January 22, 1921. The official documents state that the academician who conferred the degree on Banach was Kazimierz Twardowski. To a mathematician, that is surprising news indeed. Why Twardowski, who was an eminent Polish philosopher? What was his connection to Banach? Could he have been his dissertation advisor? According to the rules then in force, the conferment of a new doctorate had to be celebrated by a professor from the faculty appointed by the dean, and so there is no reason to regard Twardowski as the supervisor of Banach's thesis. By analogy, one might incorrectly claim that Steinhaus's supervisor in Göttingen in 1911 was the German botanist Gustav Albert Peter, who played the same role as Twardowski in Banach's case (for details, see [9]).
It is frequently said that Banach was not a university graduate, so the fact that he obtained a position at the Polytechnic and a university doctorate was exceptional. This is also slightly misleading. According to the rules that were then in effect in Poland [14], four years of study at the university was enough for one to be eligible for a PhD, but even that requirement could be relaxed. The professors of a faculty could, at their discretion, allow someone with outstanding achievements to apply for a PhD. Moreover, in those years, there was no precise definition of who counted as a university graduate. Banach had studied at the Lvov Polytechnic for precisely four years, which was enough."
*** A KERNEL OF TRUTH? ***
"Let us dig further in an attempt to discover [a kernel of truth underneath the gossip about Banach's doctorate].
This is a good place to recall the illustrious figure of Andrzej Turowicz (1904-1989), a mathematician, priest, and monk active mostly in Kraków, but who also spent some time working in Lvov... Turowicz knew many excellent stories, abounding in colorful detail, about mathematics and mathematicians of his time. It was not unusual for participants in various meetings that he attended to ask him to share some of his anecdotes. Whenever Turowicz had himself been a witness of an event, he recounted it with great accuracy, and one could be sure that things had really happened that way, but there were also stories he had heard from others.
On November 17, 1984, the Jagiellonian University Students' Mathematics Society (see [10]) invited several mathematicians to share their memories during a special meeting. Their reminiscences were taped. Turowicz was one of the guests. He contributed the anecdote about Banach's PhD exam, beginning with the words: 'This is a story I heard from Nikodym, and I am repeating it here at Nikodym's responsibility'. Turowicz recounted this event on several occasions and always credited it to Nikodym. The same attribution is also given in [20].
It was Nikodym whose conversation with Banach was accidentally overheard by Steinhaus in Kraków. Later, Nikodym became a prominent mathematician; after World War II he emigrated to the United States...
And it turns out that it was Nikodym who was reluctant to obtain a PhD. He used to ask: 'Will it make me any wiser?' In 1924, Nikodym (aged 35), still without a PhD, and his wife Stanisława (who was also a mathematician) moved from Kraków to Warsaw. Walerian Piotrowski made a very solid investigation concerning PhDs in mathematics at Warsaw University in the interwar period (see [24, 25]). According to [25], Wacław Sierpiński decided to take the matter of Nikodym's PhD exam into his own hands. He invited Nikodym to a café and began to talk with him. After a while, the dean of the department 'accidentally' appeared in the café and joined the conversation, which quickly drifted toward mathematics. More than an hour later, Sierpiński said to Nikodym: 'Congratulations. You have just passed your PhD exam.'
In our opinion, this is the source of the urban legend about Banach's doctorate. We will never know whether Nikodym gave Turowicz a twisted account of his own PhD exam, changing the main protagonist's name in the process, or whether Turowicz missed something. Our view is that the first explanation is more likely."
These are the references to which D. Ciesielska and K. Ciesielski alluded to in those paragraphs:
[9] D. Ciesielska, L. Maligranda, and J. Zwierzyńska. Doktoraty Polaków w Getyndze. Matematyka. Analecta 28:2 (2019), 73-116.
[10] K. Ciesielski. 100th anniversay of the Jagiellonian University Students' Mathematics Society. Math. Intelligencer 17:4 (1995), 42-46.
[11] K. Ciesielski. Lost legends of Lvov 2: Banach's grave. Math. Intelligencer 10:1 (1988), 50-51.
[14] T. Czeżowski (editor). Zbiór ustaw i rozporządzeń o studiach uniwersyteckich oraz innych przepisów ważnych dla studentów uniwersytetu, ze szczególnym uwzględnieniem Uniwersytetu Stefana Batorego w Wilnie. Wilno, 1926.
[19] E. Jakimowicz and A. Miranowicz (editors). Stefan Banach. Remarkable Life, Brilliant Mathematics. Gdańsk University Press, 2010.
[20] R. Kałuża. Through a Reporter's Eyes: The life of Stefan Banach. Birkhäuser, 1996.
[22] L. Maligranda. 100-lecie doctoratu Stefana Banacha. To appear in Wiad. Mat. 52 (2020).
[24] W. Piotrowski. Doktoraty z matematyki i logiki na Uniwersytecie Warszawskim w latach 1915-1939. In Dzieje Matematyki Polskiej II, edited by W. Więsław, pp. 97-131. Instytut Matematyczny Uniwersytetu Wrocławskiego, 2013.
[25] W. Piotrowski. Jeszcze w sprawie biografii Ottona i Stanisławy Nikodymów. Wiad. Mat. 50 (2014), 69-74.
[26] J. Prytuła. Doktoraty matematyki i logiki na Uniwersytecie Jana Kazimierza we Lwowie w latach 1920-1938. In Dzieje Matematyki Polskiej, edited by W. Więsław, pp. 137-161. Instytut Matematyczny Uniwersytetu Wrocławskiego, 2012.
In the Fall 1988 issue of the Mathematical Intelligencer there is an interview of Andrzej Turowicz who was a contemporary of Banach and Mazur. Here is one of the questions.
Q: Were all the Lvov mathematicians so reluctant to publish their results?
A: No, it was a specialty of Mazur. Banach also left many of his results unpublished, but for a different reason. Banach turned out mathematical ideas so quickly that he should have had three secretaries to compose his papers. That was why Banach published only a small part of the theorems he invented. Not because he did not want to, but because all the time he had new ideas.
I am hesitant to write here, I delayed/procrastinated for long. I am not a historian, I simply was embedded in the Polish mathematical scene for over ten years, and since then I kept in personal touch with several of my Polish mathematical friends.
The notion of Banach assistant is not right. Banach had students and (mathematical) friends, including and especially younger friends. The most important among them was Stanisław Mazur who himself was a fantastically sharp mathematician. Stanisław Mazur truly disliked writing (editing) mathematics because he was doing it so well. For instance, Stanisław Mazur wrote (i.e. edited) the first paper by KS, who was about 30 years younger. However, Prof. Mazur didn't care to publish his own results. Prof. Kuratowski told me that Mazur was happy when someone else rediscovered and published Mazur's results. Mazur would say happily on such occasions: it (the results) had to be good enough if someone bothered to publish it.
Sometime in 1971-72 (or on a next occasion?), Aleksander Pełczyński (Olek) told me, when he had visited me in Ann Arbor (MI) a couple of times, that Banach's classic Theory of Linear Operators was written (i.e. edited) by Mazur.
Stefan Banach didn't care to edit his own research results. However, he did write academic and high-school texts extremely well. At least, this is my opinion based on my studying Banach's 2-volume Calculus monography on my own, when I was a high school student--I'd wake up way before my school day and would read for an hour or two. For a contrast, earlier I had gotten another text--famous--on mathematical analysis by a polytechnic professor. I stopped reading it very soon because it was too boring.
In many places around the world people like to stress how hard they work. It was often the opposite in Poland, especially among many Polish mathematicians. They were particular about being young, brilliant, and lazy. They would not say that they worked hard but that it was nothing, it just came to me at one moment, something like this. Ulam's autobiography illustrates my point. (On the other hand, a close friend of Ulam considers Rota's writing about Ulam as offensive, abusive, dishonest.)
I think this question is very subjective, speculative and gossipy, and I am surprised that it has not yet been criticized as not suitable for MO. Unlike in mathematics, in history it is often enough to raise an unsubstantiated question in order to influence people's beliefs. It is very easy to spread rumours in history, and it is therefore important to provide good evidence for any suggestion that has to do with a historical fact.
What evidence do you have that Banach did not write his thesis, and what makes you think that the word 'lazy' is appropriate here? Would you call Hardy lazy because he only worked a couple of hours a day and spent the rest of his days reading about cricket? Would you call Grothendieck lazy because he did not write up his proof of Grothendieck-Riemann-Roch? Certainly not, because these people, just like Banach, were very prolific and influential mathematicians.
In a similar way, Rota's description of Ulam is historically unhelpful, and only illustrates the fact that Rota sometimes described people in rather arrogant terms (as he also did with Artin in Indiscrete Thoughts).
Please let us stick to the facts and not make MO a forum for speculative historical anecdotes.
In Stanisław Ulam's autobiography Adventures of a Mathematician you can find several references in that sense (mainly in the first Part) about the mathematicians at Lwów in that time, maybe the clearest one is on page 38:
"In general, the Lwów mathematicians were on the whole somewhat reluctant to publish. Was it a sort of pose or a psychological block? I don't know. It especially affected Banach, Mazur, and myself, but not Kuratowski, for example."