The author is indebted to Wioletta Miskiewicz and Philippe Nabonnand for their important suggestions.
- 1 Lviv (romanized Львів in Ukrainian) has different names in different languages: Leopolis (in Latin (...)
1The Lwów1 School of Mathematics, which acted between two world wars, is, in a certain sense, the implementation of Zygmunt Janiszewski’s program for the development of mathematics in Poland at the beginning of the 20th century [see, e.g., Murawski 2010, for description of the program]. The school is a phenomenon in the history of mathematics, in particular, it had its own style of activity, based on permanent meetings and discussions of its participants. During a relatively short historical period, the school’s participants obtained important scientific results that had a significant impact on the development of mathematics in the world. It is customary to date the beginning of the school from a chance meeting in 1916 of Hugo Steinhaus with Stefan Banach.
2One of the important points of Janiszewski’s program was the concentration of research directions in relatively new, modern areas. The interests of the representatives of the Lwów School of Mathematics were mainly focused on functional analysis, a part of mathematics which studies linear spaces, in particular, spaces of functions, and linear maps between these spaces. It can be said that the foundations of functional analysis are laid in the Lwów school. One of the basic concepts of functional analysis, that of Banach space, is named after Stefan Banach, the leader of the school. In Lwów, the following results that are considered the principles of functional analysis were established: the Banach-Steinhaus theorem (also known as the principle of uniform boundedness), the Hahn-Banach theorem on extending functionals from subspace of a normed linear space to the whole space, Banach’s open map theorem and closed graph theorem.
3In addition to functional analysis, topology, topological algebra, real analysis and other sections of mathematics, mainly related to analysis, were developed in Lwów at that time. It is remarked in [Domoradzki & Zarichnyi 2015] that the achievements of the Lwów School of Mathematics in the field of topology and topological algebra are comparable with those in functional analysis.
4S. Domoradzki published the monograph [Domoradzki 2011] in which he interpreted mathematics from the standpoint of culture and analyzed its development in the region during the period of Polish autonomy. The content of the book is an attempt at a historical explanation of the unexpected appearance of a powerful scientific school in a city that, at first glance, did not have significant scientific traditions.
5Józef Puzyna (1856-1919) is considered as a precursor of the Lwów School of Mathematics. While in Krakow in 1917, Puzyna invited Hugo Steinhaus to the Lwów University. This invitation determined the fate not only of H. Steinhaus himself, but also of all mathematicians in Lwów in the 20s and 30s years of the 20th century.
6Puzyna worked mostly in complex analysis. His monograph [Puzyna 1898, 1900] on the theory of analytic functions in two volumes was the first Polish mathematical book that used the language of set theory.
7Famous mathematicians Stanisław Saks and Antony Zygmund highly praised Puzyna’s monograph, considering it a real encyclopedia of analysis, which, in addition to the contemporary exposition of the theory of analytic functions, contained information from the field of set theory and set-theoretic topology, group theory, algebra, differential equations, and harmonic functions.
8Actually, Puzyna’s book was the first attempt to teach a course in the theory of analytic functions based on set theory. One of the initial chapters of the book was devoted to the foundations of this theory, as well as some fundamental concepts of set-theoretic topology (accumulation point, derivative, compactness, connectedness, etc.). Examples of subsets of the set of real numbers with predetermined properties of the transfinite derivative (necessary for countable ordinal numbers) were presented.
9However, later in the book, Puzyna changes the style of exposition of the material: the chapters on the topology of surfaces are actually written without the use of language of set theory. The author mostly relies on an intuitive-visual argument.
10In fact, in Puzyna’s book we observe a certain eclecticism, a combination of both the set-theoretical and the visual-intuitive approach. The latter corresponds to Poincaré’s philosophical views: the basis of mathematics is intuition, and not everything in mathematics lends itself to formalization and analytical exposition. When explaining the theory of surfaces, Puzyna appeals to the reader’s geometric intuition and uses rich illustrative material. Note that a systematic presentation of the theory of surfaces based on set-theoretic topology would hardly be possible at that time, as it needed systematic development of the necessary topological apparatus, primarily homeomorphisms and deformations (homotopies).
11It can be speculated that Puzyna’s book envisaged two approaches to the creation of mathematics: a more formalized one and one in which less attention was paid to formalization and intuition came to the fore. With a certain convention, the first approach can be attributed to Hilbert, and the second to Poincaré.
12We note here that the set theory was further developed in Lwów by WacŃaw Sierpiński, who began teaching at the Lwów University in 1908. Zygmunt Janiszewski later began to develop the geometric sections of topology, in particular the continuum theory.
13The leaders of the Lwów Mathematical School were Stefan Banach and Hugo Steinhaus.
14Stefan Banach (1892-1945) is considered one of the creators of functional analysis. Some publications are devoted to philosophical aspects of Stefan Banach’s contributions to the field of functional analysis, in particular, to normed spaces and linear operators [see, e.g., Jaëck 2017, 2020]. The concept of Banach space is one of the central concepts of modern mathematics. The Banach Contraction Principle became the basis for future results in the theory of fractals. The Banach-Tarski paradox about the sphere doubling, discovered in 1924, became widely known. In total, several dozen mathematical objects have been named after Banach [see, e.g., Kaluża 1996].
15Hugo Steinhaus (1887-1972) worked in functional analysis, measure theory, geometry, mathematical logic, and applied mathematics. He made important contributions to the foundations of probability theory, as well as game theory (see Steinhaus’ book [2016]).
16Stanisław Mazur (1905-1981), one of Banach’s students, worked in linear and nonlinear functional analysis. Together with Banach, he gave the first example of an infinite positional game of perfect information [see, e.g., Telgárski 1987, on the history of this game].
17Stanisław Ulam (1909–1984) is primarily known as the creator of the hydrogen bomb. His mathematical interests during the Lwów period included topology, topological algebra, set theory, and measure theory. Among his widely known achievements is the antipode theorem, which he stated as a hypothesis and which was proved by Karol Borsuk [1933]. Memories and reflections about Lwów School of Mathematics can be found in the autobiographical book by Ulam [1976].
18Although Kazimierz Kuratowski (1896–1980) is considered a representative of the Warsaw School of Mathematics, he spent 6 years in interwar Lwów and his activities had a tangible impact on the Lwów School of Mathematics [Kuratowski 1980].
19Juliusz Paweł Schauder (1899–1943) was one of the creators of nonlinear functional analysis. He also worked in the field of differential equations and mathematical physics. The Leray-Schauder principle [see Leray & Schauder 1934] provides a method for solving partial differential equations.
20Among the well-known representatives of the Lwów School of Mathematics, we also mention the names of Feliks Barański (1915-2006), Władysław Orlicz (1903–1990), Stanisław Saks (1897–1942), Józef Schreier (1909-1943), Mark Kac (1914–1984), Antoni Łomnicki (1881–1941), Stefan Kaczmarz (1895–1939), Herman Auerbach (1901–1942), Włodzimierz Stożek (1883–1941), Stanisław Ruziewicz (1889–1941), Eustachy Żyliński (1889–1954). Duda’s book [2014] is an important source of information about the biographies and results of these mathematicians.
21Speaking of the Lwów School of Mathematics, the name of Miron Zarycki (1889-1961) is mentioned only in passing. However, the subject of his research follows some of Kuratowski’s results and therefore fits perfectly into the activities of the school. A part of Zarycki’s results is devoted to the axiomatization of the concept of the boundary of a set in a topological space [see Zarycki 1927]. Zarycki showed, in particular, that his system of axioms for the boundary is equivalent to Kuratowski’s known system of axioms for the concept of closure of a set in a topological space, that is, it can be used to define a topological space. Note that the axioms for the boundary operator have been repeatedly rediscovered [Albuquerque 1941], [Gravett & Scott 1956]. Zarycki’s results are also of philosophical interests, they were cited in modern papers on formal ontology [Varzi 1997], [see Zarichnyi & Ptashnyk 2017, for further references].
22A big role in the development of science is played by setting open problems that motivate and direct the efforts of researchers. This is especially important for mathematics when compared to the natural sciences, in which topics for future research are shaped by real problems that arise when studying natural objects. One of the most famous collections of open problems in mathematics was Hilbert’s 23 problems, formulated in 1900 at the International Congress of Mathematicians in Paris. In 1912, Edmund Landau proposed four major problems in prime number theory that are still open. There are also collections of open problems that become crucial for certain area of mathematics. Each solution to a problem that has a certain history behind it usually means a breakthrough in the relevant field of knowledge.
23An important aspect of the activity of the Lwów School of Mathematics was the formation of a collection of open mathematical problems under the name The Scottish Book. The book is named after the “Scottish Café”, a place where mathematicians used to gather after meetings of the Lwów branch of the Polish Mathematical Society. Starting from 1935, 193 problems were formulated, and rewards were offered for solving some of them.
24Later, thanks to Stanisław Ulam, who converted the manuscript into a printed text, The Scottish Book was distributed throughout the world (see the monograph [Mauldin 2015], which not only contains a list of problems together with information on whether a given problem is solved or not, but also contains lectures by several authors describing the atmosphere of informal meetings of mathematicians in Lwów). Some problems from The Scottish Book proved to be very difficult, some (about a one third to date) remain unsolved. Before inclusion in the collection, the problems were discussed at sessions, they reflected the scientific interests of the school participants and, in a certain sense, established a value scale that provided guidelines for further research. The importance of The Scottish Book is especially evidenced by the fact that some of them initiated new areas of mathematics. In particular, Borsuk’s problems from it led to the development of the topology of infinite-dimensional manifolds based on the model of the Hilbert cube. The concept of a random graph first appeared implicitly in a problem formulated by Stanisław Ulam. Let us also note Mazur’s problem about an infinite game in topological space, which was solved by Stefan Banach.
25Most of the problems refer to functional analysis and then to set-theoretic topology. Some branches of mathematics, for example, number theory, are practically not represented.
26Today, The Scottish Book is one of many similar collections of open problems, but its historical significance remains important. This is a valuable document for studying the history of mathematics in the 20th century. After WWII, The Scottish Book continued in Wrocław in the form of the New Scottish Book, a manuscript containing about a thousand open problems. It is symbolic that modern mathematicians of Lviv have published their collections of open problems in the field of topology [see Pearl 2007]. Another project of Lviv mathematicians is the Lviv Scottish Book, an online collection of open problems from various sections of mathematics (http://www.math.lviv.ua/szkocka/). It should be noted that one problem from the Lviv Scottish Book was ingeniously solved by Fields medalist Terrence Tao.
27In 1920, in accordance with Janiszewski’s program for the development of Polish mathematics mentioned above, the journal Fundamenta mathematicae was published in Warsaw. It was the world’s first specialized mathematical journal, its topics covered set theory, set-theoretic topology, mathematical logic, and the foundations of mathematics. The idea arose to create a journal specializing in functional analysis. This is how Studia mathematica appeared, which began to be published in Lwów in 1929.
28This journal was one of the few Polish mathematical journals published in world languages; this made it possible to spread it all over the world and thereby widely popularize the scientific results of mathematicians in Poland and, in particular, Lwów. In turn, the university library received about 150 foreign publications in exchange for Studia mathematica. The first 9 volumes of the journal were published in Lwów. After the war, the journal began to be published in Warsaw. Volume 10 contains publications of the results obtained during the war, in particular, posthumous articles by Stefan Banach.
29Studia mathematica is now considered one of the most influential publications in the field of functional analysis.
30The peculiarities of the activity of the Lwów School of Mathematics attracted the attention of historians and philosophers of mathematics. First of all, we should mention the book by Roman Duda [2014], which is entirely dedicated to the school. It contains a brief overview of the history of mathematics in Lwów at the beginning of the 20th century, the birth of the school, the main results obtained in it, as well as a description of the dramatic period of decline. A lot of attention is paid to outstanding personalities of Lwów mathematics of that period. Of course, it is difficult to achieve completeness in such a book, and the review [Maligranda 2016] provides an important additional information concerning the school. We also mention the recent edition [Domoradzki, Stawiska-Friedland et al. 2021].
31Philosophical questions of mathematics (and logic) related to the activities of Polish mathematicians (and logicians) were considered in the publications of R. Murawski [2013, 2011, 2014]; his article [Murawski 2013] is devoted to the topic of the philosophy of the Lwów School of Mathematics. The text below presents a slightly different view on the same question: the philosophy of a mathematical school can be judged not only from the rather stingy, well-known statements of the members of the school, but also from the forms of creating mathematics, from the essence of the results obtained.
32First, representatives of the school rarely spoke systematically on the topic of philosophy, although some materials of this type have reached us. In addition, often the scientific activity of a scientist can allow for an ambiguous philosophical interpretation and not fully correspond to the declared philosophical views.
33It is hypothesized that, along with the Lwów-Warsaw Philosophical School, the Lwów School of Mathematics served as a model body of thought for the famous microbiologist and philosopher of science Ludwig Fleck [see, for example, Cohen 1986]. These schools were the center of intellectual life in Lwów between the two world wars, and Fleck maintained contacts with their representatives. As for the mathematical school, Fleck collaborated with Steinhaus in the statistical evaluation of his microbiological observations, and was also probably acquainted with the logician Leon Chwistek, who is also considered a representative of the Lwów School of Mathematics.
34The volume of our publication does not allow us to dwell on all aspects of the scientific activity of the Lwów School of Mathematics, here we will focus only on the question of the constructiveness of the methods used in the school.
35Existence theorems are widespread in mathematics, where the proofs say nothing about the methods by which the desired object can be constructed. Actually, this is the unconstructive nature of such theorems. (For example, note that the proof of the well-known Banach theorem on a fixed point is quite constructive, it contains a method of approximating the desired point.)
36Applied mathematicians figuratively say that they are statements like “there are fish in the ocean” and therefore have no utility. In fact, existence theorems of this type also have an applied value, because from them it is possible to derive statements about the consistency of the applied models.
37One of the dominant non-constructive methods of proof at the Lwów School of Mathematics was the method based on the Baire Category Theorem. This theorem, proved in 1999 by the French mathematician René Baire (for Euclidean spaces), made it possible, informally speaking, to distinguish among all metric spaces the so-called spaces of the second category, that is, in some sense, “large” spaces. It is asserted that the spaces of the second category cannot be represented as countable unions of their “small” subspaces.
38A very schematic application of the Baire Category Theorem can be described as follows. To prove the existence of an object with certain properties among a given set of objects, we first endow this set with a certain metric. In the formed metric space, we select a set that corresponds to objects with the required properties. If such a set turns out to be of the second category (that is, it is a countable intersection of open dense sets), and the mentioned metric space is complete, that is, it contains the limits of its potentially convergent sequences (called Cauchy sequences), then the Baire theorem asserts that this set of objects is nonempty, i.e., guarantees the existence of an object with the desired properties.
39We can see that this method of proof does not provide any example of an object with the given properties. The advantage of the method is that it demonstrates that the set of objects with the given properties is “large”, that is, we know that there are many such objects, but we cannot find an explicit example.
40As an illustration, one can compare Banach’s proof of the existence of a continuous nowhere differentiable function on a segment [see Banach 1931] with the well-known Weierstrass’ constructive example of such function.
41Note that the Baire Category Theorem is also used to prove the principles of functional analysis mentioned above: the open mapping theorem, the closed graph theorem, the Banach-Steinhaus theorem, etc.
42Unlike Banach’s fixed point theorem also mentioned above (Banach’s Contraction Principle), most fixed point theorems have non-constructive proofs. This was the case with Brouwer’s fixed point theorem, as well as the KKM (Knaster-Kuratowski-Mazurkiewicz) lemma obtained within the framework of the Lwów School of Mathematics, which is equivalent to Brouwer’s theorem.
43Here we should also mention Schauder’s fixed point theorem for compact convex subsets in infinite-dimensional vector spaces. The initial version of the proof given by Schauder contained a gap, it worked smoothly only in the locally convex case. The proof of the general case given by Robert Cauty turned out to be wrong. Cauty [2005] claimed that his theory of algebraic absolute neighborhood retracts led to a correct proof.
44The source of non-constructivity is also the Axiom of Choice in set theory. It asserts that for an arbitrary family of nonempty sets, a set can be created by choosing one element from each set of the given family. Non-constructivity is manifested here in the absence of a selection algorithm.
45The Hahn-Banach theorem, one of the principles of functional analysis, about the extension of linear functionals from a subspace to the entire space, is based on non-constructive methods, and the Kuratowsky-Zorn lemma is used for its proof. The latter, as is known, is equivalent to the Axiom of Choice.
46The Banach-Tarsky paradox is one of the most famous achievements of the Lwów School of Mathematics. It consists in the fact that a sphere of unit radius in three-dimensional space can be divided into five parts, from which two spheres of unit radius can be assembled, if only isometric transformations of these parts are allowed, that is, shifts and rotations. The opposite of this statement to the properties of the surrounding world is obvious, since it is impossible to create matter from nothing. At the same time, the Axiom of Choice, which is key in proving this paradox, does not cause reservations at first glance.
47Note that the Banach-Tarski paradox can be viewed as a non-existence theorem: there is no invariant measure defined for all sets in three-dimensional space. At the same time, Banach showed that a motion-invariant measure exists for all subsets of a real line and plane.
48From the list of results, which are considered basic for the Lwów School of Mathematics, it can be seen that mathematical objects, for which we only have proven existence, have the same status as constructed objects.
49The non-constructive methods are clearly more sensitive to formalization than the constructive proofs, which tend, in particular, to intuitive reasoning.
50Since the constructive methods of proof are related to the philosophy of constructivism, it can be said that the philosophy of the Lwów School of Mathematics differs from constructivism, as well as from intuitionism.
51In connection with the issues of the relationship between formalism and intuitionism, we mention the name of the mathematician łucjan Böttcher (1872-1937), who is considered one of the creators of holomorphic dynamics. Some mathematical objects are named after him: Böttcher equation, Böttcher function (coordinate, mapping) [Milnor 2006].
52Note that Böttcher is also the author of the first example of a chaotic mapping (a rational mapping of a sphere, the chaotic set of which is the entire sphere). Twenty years later, such a mapping was rediscovered by Samuel Lattès.
53Many years of Böttcher’s mathematical creativity passed in Lwów, but his name is not mentioned when talking about the Lwów School of Mathematics. Despite the undoubted value of his results, Böttcher could not get a professorship at the Lwów University. He made such attempts twice and met with serious criticism of his publications.
54In our opinion, one of the main reasons for rejecting Böttcher was his style of presentation of mathematical results. Böttcher’s philosophy was closer to the views of Poincaré, it was much more based on an intuitive understanding of concepts and constructions than on a set-theoretic basis. Only the lack of proper formalization prevented Böttcher’s contemporaries from seeing the essence and importance of his mathematical achievements. Below is an excerpt from the committee’s decision on Böttcher’s application [see Domoradzki & Stawiska 2014]:
The method used by the Candidate in his works cannot be considered scientific. The author works with undefined, or ill-defined, notions (e.g., the notion of an iteration with an arbitrary exponent)...
The shortcoming, or rather lack of rigor of the definition of iteration with an arbitrary exponent introduced by the candidate met with justified and clearly written criticism by Dr. Stanisław Ruziewicz in Wektor, Warsaw 1912, 5 [On a problem concerning commuting functions]. [Domoradzki & Stawiska 2014]
55Here we allow ourselves some speculative considerations related to the above-mentioned Puzyna’s monograph. Recall that in the presentation of the material, the author combined a set-theoretical, formal approach with an intuitive-visual approach. The Lwów School of Mathematics developed mainly according to the first of them. However, despite the importance of formal methods, they do not exhaust all mathematics. It is interesting to look at the situation from the point of view of later discussions about the role in mathematics of intuitive reasoning of incomplete proofs, hypotheses, speculations, etc. (see [Jaffe & Quinn 1993] and also [Atiyah, Borel et al. 1994]).