Memories from the Scottish Café

In 1969 Ulam published Memories from the Scottish Café for the Polish Mathematical Society:

https://www.lwow.com.pl/banach/ulam-wspomnienia.html

A translation of this is below.

Throughout history, the development of mathematics has received impulses either from certain environments or from certain groups of people. These environments, both large and small, were formed around one or sometimes several individuals, and sometimes they were the result of the research work of a number of people who constituted an even group working simultaneously and developing mathematical activities. Such a group is more than just a community of specific interests; it has a completely specific mood and character both in the selection of problems and in the methods of thinking. At first glance, this may seem strange, because a mathematical achievement, whether a new, content-laden definition or a complex proof that settles a question, seems to be the result of a completely individual effort, almost like a musical composition in which it is not easy to understand how could have been written by more than one individual.

However, when it comes to a group of individual mathematicians, the choice of certain problems or methods is repeated many times, which is the result of common interests. Such choices are often influenced by a set of questions and answers that a single mathematician could probably pose and solve on his own, but which arise naturally as a result of the work of several minds. In this way, the great mathematical centres of the 19th century, such as Göttingen, Cambridge in England, Paris and Russian centres, had a special and specific influence on the development of mathematics. A significant part of the achievements of mathematicians in Poland in the interwar period constitutes an important stage in creating the foundations of modern world mathematics. They influence not only the subject matter, but also the tone of contemporary research.

Since Cantor's time, the spirit of set theory has increasingly permeated mathematics; We have recently witnessed a renaissance of interest in this theory and unexpected advances. I mean not only set theory in its most abstract form, but also its immediate applications, topology in its most general approach, the most general presentation of algebraic ideas. All this was given direction and impulse by the Polish school. A significant part of this contribution is due to mathematicians from Lwów. Here, the interests were not focused solely on set theory, but on a new approach to classical problems, which can be called functional analysis in a geometric and algebraic spirit. If we wanted to give a very simplified description of the sources of this activity, we could say that research based on the work of Cantor, logicians of the German school, French mathematicians Baire, Borel, Lebesgue and others has become established in Poland. These studies, together with the problems of analysis formulated by Hilbert and others in Germany, led to simple, general constructions of infinitely high-dimensional function spaces. At the same time, in America, and somewhat independently, the works of E H Moore, O Veblen and others, stimulated by general tendencies, led to the rapprochement of different ways of seeing and the unification of mathematical intuition.

An important feature of modern mathematics, which was fully developed in Lwów, is cooperation between different individuals and even entire mathematical schools. Despite the growing diversity and specialisation, or even hyperspecialization, of mathematical research, research directions and threads from diverse and independent sources often converge.

I will not try to give a historical or genetic description or philosophical explanation of this excellent Lwów environment. I will only give my personal impressions, both as a student and a participant, about the spirit and nature of the work of a group of employees of the University and Technical University of Lwów.

I am writing these memories from the period between the two world wars after thirty years spent in the United States. At that time I had only sporadic contact with Polish mathematicians, except for a short period just before World War II, when I visited Lwów during the holidays.

The kaleidoscope of types of Lwów mathematicians presented a great variety of mathematical individuals, not only in terms of interests and education, but also in the types of intuition and mathematical habits. The main driving force of the original research work was set-theoretic fields: the basics of set theory, set topology, and then - under the influence of Banach and Steinhaus - functional analysis with applications to classical analysis. Schauder, who was an assistant professor at the university, worked on partial differential equations. His methods and results have become classic today and constitute one of the most powerful tools for proving existence theorems. Banach, Mazur and Schauder are the creators of the now popular method of treating analysis problems using geometric methods of function spaces.

If I wanted to define the main characteristic of this school, I would first mention the interest in the foundations of various theories. By this I mean that if we consider mathematics as a tree, the Lwów group devoted itself to studying the roots and trunks, perhaps even the main branches, and was less interested in side shoots, leaves and flowers.

Deep research into the construction of classical mathematics led the Lwów group to consider more general concepts that could serve as a basis for other possible definitions. From this set-theoretic and axiomatic point of view, the nature of general spaces was studied rather than any particular example; the general meaning of continuity rather than, for example, the continuity of a function of one variable; the nature of more general sets of points in Euclidean space rather than just classical geometric figures; rather, general functions of one or more real variables, general functional spaces, more general concepts of curve length, area and volume, i.e. the concept of measure and the formulation of the concept of probability. Known mathematical structures were studied and compared, and common structural features were abstracted from them. The overall results could be interpreted in each specific example without re-examining the evidence in each individual case. For example, many well-known mathematical spaces satisfy the axioms of what later became commonly known as Banach spaces.

Looking back, it seems strange to me that algebraic ideas were not considered there against an equally general background. It is clear that the Lwów group was numerically modest and the development of algebra in a modern spirit had to wait for the establishment of other centres in other countries. It is equally strange that the study of the foundations of physics, and in particular the study of space-time, has not been undertaken anywhere in this spirit to this day.

With such a general approach, it is not surprising that new and strange mathematical objects appear parallel to general classical ideas. For example, in topology, along with ordinary geometric figures, there are strange continua of points of the plane and three-dimensional space. In the study of functions of a real variable, it turned out that among continuous functions, non-differentiable functions constitute the "majority". In the study of infinitely multidimensional vector spaces, it turned out that a whole series of such spaces has the same importance as the Hilbert space. Analysis of various properties of functions, their differentiability or types continuity showed that each of these concepts leads to a certain infinitely multidimensional vector space, sometimes as interesting as the Hilbert space. The properties of sequences of real numbers, their convergence or summability, were considered using vector spaces of such sequences, i.e. axiomatic research the formulation of probability theory required the study of very general measures and the construction of new spaces of complex "events", which were constructed starting from given spaces.

The excitement of finding such a variety of new objects that could be manipulated by a few general methods was so great that the frequency of discussion and teamwork during these years was truly exceptional. The only time I encountered a similar community of interests and intensity of intellectual coexistence was during the war years when I was researching a then new issue - nuclear energy.
Much of our mathematical conversations took place in cafes near the university. The first one was called "Roma". After a year or two, Banach decided that our sessions should be moved to the "Scottish Café", located on the opposite side of the street. Our sessions continued in small restaurants where mathematicians ate. I now think that the food was average, but there were plenty of drinks. The café tables were covered with marble slabs that could be written on with a pencil, more importantly, quickly erased. In our mathematical conversations, often a word or a gesture, without any additional explanation, was enough to understand the meaning words thrown out during long periods of thought. A viewer sitting at another table might notice sudden short bursts of conversation, the writing of a few lines on the table, the occasional laugh from one of the people sitting there, followed by long periods of silence during which we just drank coffee and watched. unconsciously at ourselves. The habit of perseverance and concentration created in this way, sometimes lasting for hours, became for us one of the most important elements of real mathematical work.

I first saw Banach when, as a student in my final years of junior high school, I attended a series of lectures on various aspects of mathematics intended for a wider audience. Banach was then about thirty-five years old. Unlike the impression young people usually get when they are about fifteen years older, he seemed very young to me. He was tall, with blond hair, blue eyes, and a rather heavy posture. His way of speaking already struck me with its directness, power, and sometimes even oversimplification; a feature which - as I later found out - was in a sense consciously pushed by him. When later, as a student of the polytechnic, I had the opportunity to observe him in the Department of Mathematics while talking to others, these impressions were confirmed. His expression usually reflected good humor, combined with a certain skeptical attitude. In conversations, he generally avoided expressing strong opposition; however, when he did not agree with the interlocutor's opinion, he manifested it by asking questions. In mathematical discussions, in which he was involved very willingly, even enthusiastically, one immediately felt the power of his mind. Whether in a university office or in a café, you could sit with Banach for hours, discussing a mathematical problem. He drank coffee and smoked cigarettes almost constantly. These types of sessions with Banach, and more often with Banach and Mazur, made the atmosphere in Lwów something unique. Such intimate collaboration was probably something completely new in mathematical life, or at least on such a scale and intensity.

Mazur in particular was the one who taught me how to control my innate optimism - not to rush to conclusions from a sketch of a proof without thoroughly examining it. Banach once confessed to me that from his early youth he had been particularly interested in proving hypotheses. Indeed, his genius for finding hidden and unexpected paths uniquely characterised him as an insightful and original mathematician! Mazur's strength was what he himself called making observations and "observations". They usually contained, in an extremely concise and precise form, certain properties of concepts that, once noticed, were perhaps not so difficult to check, but were generally within "in an area" not easily visible to most mathematicians. It often happened that these comments were decisive in finding a proof or counterexample.

Usually, after a math session in a café, one could expect that Banach would appear the next day with a few loose pieces of paper on which he sketched the evidence he had found in the meantime. Sometimes it happened that they were not actually complete or even correct in the form given by him, and Mazur was the one who managed to bring them into a truly satisfactory form.

It is probably unnecessary to add that our mathematical discussions were interspersed with conversations about politics, about the country and our city, about university politics, as well as, to a large extent, about science in general, and physics and astronomy in particular.

Let me quote some of the discussed ideas, which later became the subject of many mathematical works. During a café conversation, Mazur gave the first example of an infinite mathematical game. I remember - and it was in 1929 or 1930 - that Mazur raised the issue of the existence of automata that would be able to give an answer to themselves, given a certain amount of inert material in their environment. We discussed this in a very abstract way, and some of our thoughts, which were not written down anywhere, were actually pioneering theories such as von Neumann's theory of abstract automata. We speculated a lot about the capabilities of computers performing calculations and even how formal algebra works.

It seems to me that in 1933 or 1934 we decided to give our current formulations of problems and the results of discussions a more permanent form. Banach purchased a large notebook in which problems were to be written down, with each author's name and date listed. This notebook was kept in the café and the waiter brought it upon request. We would write in the problem and the waiter would ceremoniously take it back to where it was buried. This document later became famous under the name "The Scottish Book", after the name of the café. The original of this document is in the possession of Dr S Banach, Jr, son of Stefan, a mathematician. I translated the copy sent to me by Steinhaus and sent it to many friends in the United States and Europe.

Kuratowski and Steinhaus represented, each in their own way, elegance, precision and mathematical intelligence. Kuratowski was in fact a representative of the Warsaw school, which flourished almost explosively at the end of the First World War. He came to Lwów in 1927. He was preceded by the reputation of his works on pure set theory and the axiomatic topology of general spaces. However, his work contained important results about the properties of general continua in Euclidean spaces. He and his Warsaw collaborator - Knaster - gave examples of paradoxical flat sets that go beyond Brouwer's approach. As editor of Fundamenta Mathematicae, he organised and gave direction to many of the studies published in this famous journal. His mathematics was characterised by what I would call Latin clarity. Among the multiplying variety of definitions and mathematical problems, even more surprising today than then, Kuratowski's measured selection of problems he had a certain property that is difficult to define - common sense in abstraction.

Steinhaus's approach to problems of analysis, real functions, function theory, and orthogonal series was marked by a deep understanding of the historical development and continuity of ideas in mathematics. His book What is and What is not Mathematics had a great influence on me. Perhaps because he had less interest and feeling for the very abstract areas of mathematics, he tried to steer our new ideas towards practical applications, almost to everyday life. He had a penchant and talent for inventing geometric problems that could be treated combinatorially, capturing what could pose a visual, almost tangible challenge to mathematical formulation. He had a special sense of language. At times he was almost pedantic in his insistence on speaking or writing about mathematics or mathematical sciences in precise language.

The number of professors, both at the University and the Polytechnic, was extremely modest and their salaries were low. In order to earn a living, people like Schauder had to work as middle school teachers to supplement the modest income of an assistant professor or assistant. Zbigniew Łomnicki worked as an expert in probability theory at the National Institute of Statistics and Insurance. Still, most mathematicians found time to frequent coffee shops.

Stożek, who was a professor and dean of the General Faculty of the Lwów Polytechnic University, came to the cafe every day, like many others. Stożek, short, round, completely bald and jovial, was playing chess with Nikliborc, who was both a docent and a senior assistant. They spent most mornings drinking coffee and playing chess, surrounded by other cheering mathematicians.

Auerbach, who was also a good chess player, often visited there. He was a senior university assistant and later an associate professor. He was shy and silent, with occasional flashes of sarcastic wit. His mathematical works were elegant, and from him I learned some of the more classical areas of mathematics, such as group theory and Lie algebras, which were not particularly studied in Lwów. Both Mazur and I later published joint works with Auerbach. Kaczmarz, tall, very slim, Nikliborc's friend, appeared from time to time. Orlicz, who was also a university assistant and a friend of Mazur's, showed up less frequently.

The shadow of the impending tragedy weighed over us, although at first only subconsciously - Hitler in Germany and the premonition of world war. Stożek's presence and conversations with him helped to lighten the mood. Sometime around 1930, I told Mazur that the sight of Stożek eating sausages with mustard could dispel the most melancholic mood.

Meetings of the Polish Mathematical Society were held at the University almost every Saturday evening. Typically, three or four short announcements were made per hour, after which many of the participants retired to the café. I was nineteen or twenty years old when Stożek asked me to become the secretary of the Lwów Branch of the Polish Mathematical Society, which job consisted mainly of sending out notices of meetings and preparing short summaries of messages for the Society's bulletin.

Of course, our branch maintained correspondence with other branches. At that time, the issue of moving the Society's headquarters from Krakow to Warsaw was controversial. Needless to say, this involved a lot of manoeuvering, discussion and politicking. One day, a letter arrived from the Krakow centre, which - like the Warsaw Branch - was trying to gain the vote of the Lwów Branch. I notified Stożek, who was the president of the Branch, about this; I said, "An important letter arrived today." Stożek replied, "Hide it away so that no human eye can see it again." I was shocked by this in my youthful naivety.

Ruziewicz, a university professor, often came to the café, as did the aristocratic-looking Antoni Łomnicki. He was a professor of mathematics at the University of Technology, an expert in probability theory and cartography, and the author of a good textbook. His nephew, Zbigniew Łomnicki, became my close friend and collaborator. My teacher Kuratowski, as well as Steinhaus, appeared only occasionally at these café meetings. They usually went to a fancier confectionery shop located near our café.

Of the younger, most active and most productive people at that time were Schreier, a university student, and me. A number of works we published together at that time testify to our almost daily cooperation. We met for the first time at a Steinhaus lecture and talked about a problem I was working on at the time. Almost immediately, it turned out that we had so many common interests that we started meeting regularly. Our research, although determined by the methods used in Lwów at that time, touched upon a then new field, the abstract theory of infinite groups and topological groups. It seems to me that our works are among the first to show the application of modern methods and a more algebraic point of view to a wide range of mathematical objects. I don't know Schreier's fate. He was murdered by the Germans at an unknown time and place.

Mark Kac, who was four or five years younger than Schreier, was Steinhaus's student at university. Already as a beginner student he showed exceptional talent and obtained important results, at first together with Steinhaus, in probability theory and Fourier series theory. My contacts with Kac have developed. occurred during my summer visits to Lwów shortly before World War II.

I remember professors of other subjects: chemistry, physics and others who frequented cafes or pastry shops and often sat with us. Małachowski, a very witty chemistry professor, sometimes played chess with Auerbach and Stożek. He was a good player and sometimes got irritated when, while supporting his fellow mathematicians, I suggested certain moves to them. The philosopher Ajdukiewicz came regularly. Żyliński, who looked more like a cavalry officer than a university professor, appeared from time to time for a short visit. Some of the physicists would rather go to a confectionery shop, which was said to have the best biscuits in Poland, which, according to the owner, were sent by plane from Lwów to Warsaw every day. Writers, musicians and others also gathered in nearby cafes. As a result, almost every academic at the university or polytechnic showed up at the café from time to time.

The interaction between the Lwów centre and others, especially the one in Warsaw, was intense. Sierpiński and Mazurkiewicz came from Warsaw, as did Knaster and Tarski, who was a friend and almost the same age as Kuratowski. During their visits, they gave short lectures at Saturday meetings of the Polish Mathematical Society. In 1930, Mazurkiewicz spent the entire trimester in Lwów, giving lectures.

Like Knaster in topology, Mazurkiewicz was a master of finding counterexamples in analysis. Moreover, Mazurkiewicz's strong point was the evidence of various constructions in analysis, sometimes very complicated, but always inventive and elegant. From Sierpiński came a steady, steady and continuous stream of results from abstract set theory and set topology. He eagerly learned about new problems, considering them seriously and intensively. After returning to Warsaw, he often wrote solutions to the problems we discussed during our café sessions. Zygmund, who was originally from the Warsaw school, also came to us from Vilnius several times.

We also had younger guests from Warsaw. Borsuk visited us several times, once for a longer stay. Our cooperation began from the first arrivals. He introduced me to more visual and tangible procedures and methods of thology, as a result of which we published a number of works in Polish and foreign journals. We have defined the concept of epsilon homeomorphisms, approximating transformations, and invariants of some topological properties with respect to more general transformations. Lichtenstein, a professor at the University of Leipzig, of Polish origin, also visited for one trimester. He lectured on differential-integral equations and hydrodynamics.

Hurewicz, one of the most inventive topologists, was born in Poland and studied abroad. He visited Lwów several times and gave lectures at the Society's meetings. A number of French mathematicians - Lebesgue, Borel, Montel - also visited us and held discussions during our meetings.

Von Neumann visited Lwów twice. I learned about his first visit in junior high school from Zawirski, my logic and philosophy teacher. Von Neumann was little more than twenty years old when he came to Lwów in 1927 to the Congress of Polish Mathematicians. The fame of his discoveries in the foundations of mathematics and set theory preceded him, and I remember how eager I was to attend some of the meetings of the Congress. Unfortunately, at that time I was preparing for my high school final exams and I was unable to be present during the discussion.

I met von Neumann in person only in 1935, after exchanging correspondence, when he had already settled permanently in the United States. At that time, he and several other American mathematicians (including Stone, Garret and Birkhoff) were passing through Poland on their way to Moscow for the Topological Symposium. Our friendship and collaboration began shortly thereafter. In 1938, at von Neumann's invitation, I came to the Institute for Advanced Study in Princeton. In 1938, we were traveling from the United States to Europe together, and I invited von Neumann to visit Lwów again. He spent a few days with our group, gave a lecture to the Society, and entered some problems into the Scottish Book.

My personal contact with Lwów continued after I moved to the United States. In 1936, 1937, 1938 and 1939 I spent summer holidays in Lwów, for three months each time. Most of the mathematicians remained in the city at that time, so I continued to cooperate with Banach and Mazur at café meetings. Once or twice, when Banach was spending his holidays in some Carpathian summer resort, I went to visit him. He wrote textbooks there at the time. We spent many hours in country inns, continuing mathematical conversations.

The last time I saw Banach was in the late summer of 1939 at the Scottish Café. We discussed the possibility of war with Germany. Despite the situation at that time, we still talked about mathematics and wrote a few problems into the Scottish Book. Mazur, who was with us at the time and was much more certain than we were that war would break out, suddenly said to me: 'We have never published many of our joint results; I intend to hide our manuscripts in a box that I will bury somewhere, for example near the goals of a football field. You're going to the United States, maybe you'll find it after the war, when it's all over.

Last Updated April 2024