Some time 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?

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    $\begingroup$ Is there also a claim that he didn't write his book either (which appeared two years later)? Seems a little suspect. The charge of laziness was also leveled against his compatriot Ulam, particularly in reminiscences of Rota in his Indiscrete Thoughts. $\endgroup$ – Todd Trimble Nov 7 '12 at 13:33
  • $\begingroup$ I heard this story, too. The version I know is that one of the professors in Lvov University asked one of his assistent to help Banach in writing down his mathematical ideas. The name of this assistent, as far as I know, is unknown. But maybe the whole story is only a legend... $\endgroup$ – Francesco Polizzi Nov 7 '12 at 13:47
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    $\begingroup$ Who ia in Grant's tomb? $\endgroup$ – Lunasaurus Rex Nov 7 '12 at 20:54
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    $\begingroup$ tea.mathoverflow.net/discussion/1464/… $\endgroup$ – Margaret Friedland Nov 9 '12 at 17:51

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.

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    $\begingroup$ Thank you for the quote, but I still find it hard to believe this literally happened like this. $\endgroup$ – user9072 Nov 7 '12 at 15:52
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    $\begingroup$ I do believe it, since I heard a similar story about the PhD exam (in 1950's) of Henryk Markiewicz, a Polish literary historian and theorist, which he told himself in a public lecture I attended sometime in 1990's (there is also an audio file in Polish here, under the number 46, archiwum.uj.edu.pl/henryk-markiewicz). Maybe the professors in Krakow got inspired by the earlier event in Lvov :) (plausible, since some of them taught in Lvov before WWII) $\endgroup$ – Margaret Friedland Nov 7 '12 at 16:56
  • $\begingroup$ Thank you for the link to the thesis. A question: do you know if there does in addition to this journal version also (still) exist an 'original' version of the thesis (in the Lvov library, a national library or alike), or was this not common anyway. $\endgroup$ – user9072 Nov 10 '12 at 11:56
  • $\begingroup$ This is something I would like to find out. I can ask Ciesielski or other people dealing with history of Polish mathematics, they may know. Definitely there must have been a hard copy submitted before the exams (as it was practiced then, and long thereafter), but given the turbulent historical times in between, one cannot be sure it survived. $\endgroup$ – Margaret Friedland Nov 10 '12 at 20:02
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    $\begingroup$ Thank you for the interesting updates! I only changed some quotatiin-marks, as some "backward" ones caused minor trouble due to markdown intepreting them as instructions. $\endgroup$ – user9072 Nov 14 '12 at 20:15

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

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    $\begingroup$ Re. working in cafes. I visited Lvov once and was keen to find the `Scottish Cafe' where Banach and his contemporaries were reputed to have done a lot of great work. It took a deal of finding and when I got there it had turned into..... a bank. Big anticlimax! $\endgroup$ – Nick Gill Nov 7 '12 at 14:17
  • $\begingroup$ If you are referring to the order of getting his position and getting a doctorate as unusual, I think it was quite common during that days. I read from an interview with Selberg that it was a general practice to write at least a few papers published before writing your thesis. $\endgroup$ – timur Nov 7 '12 at 14:37
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    $\begingroup$ I made this CW as it contains a bit much speculation, and not much original information. $\endgroup$ – user9072 Nov 7 '12 at 14:38
  • $\begingroup$ @timur: No, mainly I refer to the fact that he was not educated as mathematician, but essentially self-taught. Likely he hardly ever followed any courses in mathematics. He finished some engineering studies in 1914, then in 1916 he met Steinhaus and they started to work together, then in 1920 he got a position and submitted his thesis. $\endgroup$ – user9072 Nov 7 '12 at 14:48
  • $\begingroup$ Thanks for your edits and for prompting me to find out as many details as possible. After doing this, I can summarize the situation as "Ignoramus et ignorabimus"... $\endgroup$ – Margaret Friedland Nov 15 '12 at 1:54

When I was a student in Lvov in 1970-s, I've 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. (Who was the other person on the bench, the story does not tell).

According to the legend, Banach worked most of his time in the Scottish cafe. 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 1970-s, 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 the old one he sold to students.

Banach drunk 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 cafe. 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 cafe still existed in 1990-s, but under a different name, and in 1970-s 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 cafe, based on Ulam's recollections.

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    $\begingroup$ Steinhaus included the story about the meeting in the park in his "Memoirs and Notes". It is also repeated in Ciesielski's article quoted below. The other person was Witold Wilkosz, Banach's fellow student, later a logician and a linguist, and a professor at the Jagiellonian University. $\endgroup$ – Margaret Friedland Nov 7 '12 at 23:46
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    $\begingroup$ Yes, although the professor's salary was quite high then, Banach wrote texts to support his lifestyle. The high school textbooks he wrote are available here: kielich.amu.edu.pl/Stefan_Banach/podreczniki.html $\endgroup$ – Margaret Friedland Nov 8 '12 at 0:04
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    $\begingroup$ @Margaret: Quote from Steinhaus: "During one such walk I overheard the words "Lebesgue measure". I approached the park bench and introduced myself to the two young apprentices of mathematics. They told me they had another companion by the name of Witold Wilkosz, whom they extravagantly praised. The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and ... we decided to establish a mathematical society." (www-history.mcs.st-and.ac.uk/Biographies/Steinhaus.html) $\endgroup$ – Harun Šiljak Nov 8 '12 at 5:31
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    $\begingroup$ @Harun: Thanks for the quote, my memory did not serve me too well, and I did not have the copy of Steinhaus's memoirs at hand. Otto Nikodym is certainly better known than Wilkosz, yet (perhaps) Wilkosz's permanent association with Krakow (where I studied) made me remember him better. $\endgroup$ – Margaret Friedland Nov 8 '12 at 15:26
  • $\begingroup$ And I made a mental shortcut by calling Wilkosz a "logician and a linguist". He did hold the chair of logic at Jagiellonian University and published in set theory, but he also dealt with real analysis, mathematical physics, radio technology and Oriental languages. $\endgroup$ – Margaret Friedland Nov 9 '12 at 18:04

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.


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."


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.

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    $\begingroup$ Well, what to say. First, the positive thing, in an abstract sense I can see some merit in your opinion and share it to a certain extent. Second, a procedural thing: your "answer" is unrelated to the goal of answering the question, as such it is completely misplaced as an answer (it would be fine on meta though; just sign up there, there is not rep limit or anything, it is automatic). Third, OP did not raise any unsubstantaited question but by contrast asked for confirmation or refutation and an additional detail of a well-know thing; it turns out it is official published. Fourth,... $\endgroup$ – user9072 Nov 9 '12 at 12:11
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    $\begingroup$ As to quid's second point, it's mitigated by the fact that Bok doesn't have enough points to leave a comment (and possibly wasn't aware of meta). But Bok's comment does strike me as a little bit harsh, since the OP is precisely looking for hard evidence of some sort (of something which wasn't well-known to me). He or she is probably right that the question would be improved by leaving off the bit about 'laziness', which is indeed subjective. (And I agree with him about Rota's book, which exasperates me on so many levels.) $\endgroup$ – Todd Trimble Nov 9 '12 at 12:19
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    $\begingroup$ Dang it -- substitute "agree with him or her" in my last sentence. $\endgroup$ – Todd Trimble Nov 9 '12 at 12:20
  • $\begingroup$ perhaps do not project your own(?) or least some value system to much on everybody. To some extent I prefer somebody knowing my work considers me as lazy over considering me as working hard. And, to some academics (present company ambivalently included) to be told they work hard is basically an insult. For example, I am virtually certain Hardy had no interest whatsoever (rather the opposite) to be considered as working all the time. $\endgroup$ – user9072 Nov 9 '12 at 12:28
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    $\begingroup$ tea.mathoverflow.net/discussion/1464/… $\endgroup$ – Margaret Friedland Nov 9 '12 at 17:58

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 got 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.)


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