- 1 All translations are mine, unless otherwise indicated in the appropriate place.
1One way of approaching Henri Poincaré’s thought and work is by looking at their interactions with other thinkers and researchers, both ones that inspired Poincaré and those who were influenced by him. This essay aims at discussing one such interaction. I will trace various links between some of Poincaré’s ideas (both philosophical and mathematical) and the philosophy of a Jewish-Polish thinker of the early 20th century, Joachim Metallmann (1889-1942). Metallmann was a very original philosopher who, despite his claim that “philosophy is a function of science,”1 was critical of neopositivism and logicism then dominant in Polish philosophy of science (especially the logicism of łukasiewicz and Leśniewski) [see Miskiewicz 2018]. Metallmann’s thought, concerned with the very foundations of science (addressing problems such as determinism, the relation between representation and reality, the nature of scientific concepts and conventions, or evolution in nature and paradigm shifts and science), has developed from a predominantly methodological approach to a particular metaphysics of nature. I will show how some crucial ideas and concepts in Metallmann’s later thought (emergentism, structuralism, holism) may have been influenced by Poincaré’s thinking. Grasping this intellectual link challenges the opinion that Poincaré’s impact on the philosophy of science is mainly due to him being an early conventionalist. More generally, this link is an example of an interesting interdisciplinary interaction between mathematics and philosophy.
- 2 For more information on Metallmann’s life, see [Woleński 2000], [Mączka 2002a].
- 3 On the beginnings of experimental psychology and philosophy in Poland, see [Miskiewicz 2015].
2Joachim Metallmann was born on June 6, 1889 in Cracow, the son of Menasse and Regina Metallmann (de domo Friedman2). He studied physics and philosophy at the Jagiellonian University under Witold Rubczyński, Maurycy Straszewski and, most importantly, Władysław Heinrich. The latter, a student of Richard Avenarius, and a pioneer (along with Kazimierz Twardowski) of experimental psychology in Poland,3 influenced Metallmann’s early thought, probably suggesting to him the topic of his PhD thesis. In 1912, having obtained his PhD in philosophy, Metallmann began studying natural sciences, focusing on mineralogy, zoology and botany. This early interest in science accompanied Metallmann until the end of his career. As a philosopher he addressed problems relating to the very foundations of science, such as the question of determinism and the problem of structure. During his whole life he kept up to date with the recent scientific knowledge of his time, despite its rapid progress.
3In 1918, Metallmann passed the state examination allowing him to take up the position of a school teacher and began teaching in various secondary schools. At this time, soon after Poland had regained its independence, many Polish philosophers worked as high school teachers at the early stages of their careers, including Zygmunt Zawirski, Stanisław Leśniewski, and Tadeusz Kotarbiński, to name just a few.
4In 1933, Metallmann defended his habilitation on Whitehead’s philosophy of nature and epistemology. After the defence of the thesis he was granted a one-year scholarship in France where he met Emil Meyerson, the well-known Polish-Jewish philosopher [Cieślar 2014]. After his return to Poland he started lecturing at the Jagiellonian University as an associate professor. In 1939 he was about to be promoted to become a titular professor, but the outbreak of war thwarted these plans. One of the first decisions of the German general governor of the occupied Polish territories, Hans Frank, was to arrest the teaching staff of the Jagiellonian University on November 6, 1939 [Dąbrowski 1946, 10]. The professors, Metallmann among them, were put in a detention center in Wroclaw and were subsequently moved to concentration camps. Joachim Metallmann was eventually murdered by the Germans in Buchenwald on September 21, 1942 (the Jewish religious community’s website points to a different date: August, 21, 1940).
- 4 A complete bibliography of Metallmann’s published works prepared by Janusz Maczka can be found in (...)
5The work of Metallmann displays impressive coherence regarding its main focus. From the very beginning until his last writings, the author of Determinizm w biologii [2002] was predominantly concerned with the philosophical problems underlying the natural sciences.4 He published three books (the fourth one was edited and published posthumously in 2002 by Janusz Maczka), 19 articles (in Polish and German) and numerous reviews and papers. In his work he displayed a lively interest and expertise in the most contemporary intellectual achievements of the time, both in philosophy and science. He commented on the then recent philosophical theories by Mach, Avenarius and Whitehead, he was acquainted with the quantum theory, with both special and general relativity theories, as well as with the recent advances in biology, especially in genetics.
6Metallmann’s philosophical approach towards science was original and atypical for a philosopher of his time. The dominant positions represented two general approaches: either attempting to subordinate philosophy to science (typical of neo-positivism and logicism), or trying to establish the autonomy of philosophy by downplaying science (for example in the philosophy of life, phenomenology and hermeneutics). Metallmann was aware of this polarity but he took a different attitude which gradually evolved during the course of his work. Initially he was a philosopher of science, predominantly focused on methodology, who sought the confirmation of his philosophical insights within the field of science itself. Later, largely due to switching his interest from physics to biology, Metallmann developed a more general and autonomous philosophy that veered towards a form of metaphysics of nature and was based on the idea of an emergent order. Even then, however, he did not give up the realist conviction that science and philosophy were concerned with the same world and that they both approached it in a rational way and therefore needed each other.
7Metallmann’s PhD thesis and his first book, Zasada ekonomii myślenia, jej historia i krytyka [On the Principle of Economy of Thought, its History and Criticism] [1914], contains important polemical references to Poincaré. These are connected to ideas that later became the most important and distinctive of Metallmann’s thought.
8The principle of economy of thought (PET) was introduced by Ernst Mach in the 1860s and it rapidly became popular among both philosophers and scientists (Avenarius, Duhem, Poincaré, Cornelius) which led to some terminological confusions, as Metallmann notes. He approaches the topic by pointing out the hidden contradictions within the discussed theories and disclosing their metaphysical assumptions. These analyses lead him to understanding laws of nature as “even parts of reality and not just tools to explore it” [Metallmann 1914, 74] and consequently to highlighting the difference between laws and theory. Laws belong to the metaphysical order and allow us to know the world and make scientific predictions (we assume they must exist because science works, although we do not necessarily know their content). Theories, on the other hand, belong to the cognitive or epistemological order as the human ways of (more or less accurately) grasping these laws. Theories may contain discovered and articulated laws but they mix them with hypotheses.
9Another conclusion drawn by Metallmann was that “the scientific thinking originates in the common, vulgar thinking that responds to the everyday, relatively primitive needs of man” [Metallmann 1914, 77]. PET proves that there is a continuity between science and pre-scientific thinking as they both employ the principle of economy. Both common thinking and science have a tendency to maximise the cognitive gain and to minimise the amount of effort required. From this point of view, the value of thinking, scientific or not, lies predominantly in explaining the experience of the world and in helping us to cope with it.
10This leads Metallmann to a critique of Poincaré’s and Duhem’s conventionalism. Metallmann challenges Poincaré’s interpretation of non-Euclidean geometries expressed by the author of Science and Hypothesis in the claim that “one geometry cannot be more true than another; it can only be more convenient” [Poincaré 1905, 50]. To this Metallmann answers that “creating other geometries was just an act of creating other worlds, other experiences serving as bases for these geometries. The possibility of conceiving different spatial worlds does not invalidate the claim that in application to our experience the axioms of our geometry are the only possible ones.” Therefore, “Euclidean geometry is not ‘more true’ than other geometries in the sense that it is more logically consistent and strict; but it is more ‘true’, so to speak, in the ‘experiential’ sense” [Metallmann 1914, 89]. Here Metallmann opts for an empirical and realistic understanding of geometry positioning himself against Poincaré’s position, which he considers too abstract. This attitude is consistent with his general view of science and knowledge as extensions of practical needs.
11According to Metallmann, the difference between Euclidean and Non-Euclidean geometries is not conventional but essential. Euclidean geometry is not just the simplest and most convenient way of describing reality, as Poincaré claimed in Science and Hypothesis [1905], but one that fits the very laws of this reality and the ways of experiencing it. Metallmann claims that Poincaré is wrong when he limits the laws of nature to the cognitive operations of the mind, because “a scientific law can be conceived only as a real relation between the phenomena” [Metallmann 1914, 97]. A theory, however, consists not only of laws but also of some assumptions and hypotheses convenient for the researcher. Accordingly, two different theories may express the same relations. According to Metallmann, Poincaré, in his concept of a “scientific principle” (from The Value of Science [Poincaré 1958, 93n]), confuses the conventional and arbitrary elements present in every theory with the real and objective relations between phenomena expressed by the laws of nature. It seems, however, that Metallmann did not take full account of the detailed analyses of the concept of a phenomenon (or a fact) provided by Poincaré in The Value of Science. There, Poincaré points out various difficulties with this fundamental concept. He points out that different scientific disciplines have their own ways of extracting phenomena from reality—astronomy, for example, does it very differently from particle physics.
12Metallmann’s criticizes Poincaré both on the methodological and metaphysical level. In the latter field he accused Poincaré of idealism:
For Poincaré the world ‘exists as a dependent’ of the knowing ‘mind’ and only as such. This is why, according to him, the belief in ‘outside’ objects or in the circular movement of the Earth, is only convenient. But moreover, accepting a given claim as convenient, he also accepts it as true, exact and exposing the real relations. [Metallmann 1914, 103, emphasis added – M.R.]
13The theory of the French thinker is presented as an example of incorrectly expanding the psychological principle of the economy of thought (represented in his vocabulary by the idea of convenience) beyond its proper range which is secondary and subjugated to the more important principle of fidelity to experience. Poincaré is being juxtaposed to Husserl as well as to Neo-positivistic logicists who, on the other hand, completely downplay the role of the principle of economy of thought in the process of cognition and in that respect are also wrong.
14Metallmann’s dialogue with Poincaré, outlined above, seen from the perspective of his later works, seems to be crucial for his philosophical development. The critique of conventionalism apparently inspired and strengthened the metaphysical elements in Metallmann’s philosophy of science. It also may have had an impact on the structuralist features of his thinking that became fully visible in his next book.
15Published in 1934, Determinizm nauk przyrodniczych [Determinism in the natural sciences] is Joachim Metallmann’s longest book and is considered his major work. Despite the general title, this ambitious and impressive work is predominantly focused on methodology, statistics and theoretical physics, since it was meant as the first volume of a larger whole. The main problem addressed in the book is both still up to date and fundamental for the philosophy of science. Metallmann rightly observes that the spectacular progress of natural sciences has not been accompanied by philosophical reflections on the ideas of determinism and causality, despite the crucial status of these ideas for any science. The sciences are getting better and more accurate in predicting events, but they still cannot fully explain the phenomenon of causality itself, which is the necessary condition of their own validity. This problem became especially urgent in the light of the then recent quantum mechanics and both theories of relativity which undermined the classical mechanical interpretation of determinism. Unlike Heisenberg, however, Metallmann did not want to completely abolish determinism.
16In a manner typical of his earlier work, Metallmann carefully analysed the very idea of determinism in order to avoid any conceptual misunderstandings. This led him to distinguish two aspects of determinism:
- A pattern of prediction: it is assumed that there is a constant correlation between experience and the principles of causation and statistics; this correlation allows for predicting future facts and laws.
- An ontological component: two extra-experiential assumptions that need to be made: (a) the principle of induction; (b) the principle of “partial identity” [Metallmann 1934, 1–7].
17Regarding philosophical interpretations of determinism, he rejects both poles which he considers extreme. On the one hand, empiricism (in its radical forms) does not explain the relations between experience and its inherent order. On the other hand, transcendentalism completely detaches the principle of causality from experience (and as such is impossible to maintain after Einstein) [Metallmann 1934, 37–42]. Metallmann proposes instead that the content of the principle of causality should be considered as changing over time and these changes are dependent both on scientific progress and on philosophical analyses. He also suggests that Whitehead’s theory of “extended contemporaneity” within an event can help explain causal relations.
18The principle of causality is where the connection between philosophy and science can be seen most clearly, according to Metallmann. Physics reaches this principle starting with experience. Philosophy finds it in the process of analysing the metaphysical assumptions of science. Therefore, science should never be opposed to philosophy. The former contains ontological (e.g., philosophical) assumptions, and the latter (conceived as metaphysics in constant development) remains in accordance with the principles and ideals of science. Such a complementary relation between science and philosophy reveals that they both address the same basic structures of reality.
19Metallmann identifies two basic ontological assumptions of science. The first one, dubbed by him “the principle of induction,” can be recognized solely within the structure of experience and not within its content:
The principle of induction is suggested to us both by experience and by reason. Induction is both dependent on and independent of experience, and, in a similar way, it is dependent on and independent of the mind. It is dependent on experience insofar as we become aware of it through the analysis of experience. More precisely: it is through this analysis that we become aware of the material for the analysis of induction, for we find that relations such as repeatability, identity of elements in time and space, are not less accessible to us than these elements themselves. [Metallmann 1934, 378–379]
Induction is taken as “an ontological axiom”:
It is not a question of the fact that the principle of induction can be explained by psychology. What is essential, is that the principle of induction manifests itself as the first principle from the logical point of view and as ontological with regard to its object. [Metallmann 1934, 376]
20The other basic principle, “the principle of partial identity,” states that not only facts may be identified with each other but also relations between them. According to Metallmann, “elements are repeatable within relations or patterns and not individually” [Metallmann 1934, 406] and “if one element repeats itself, then there is another one that does the same” [Metallmann 1934, 409].
21These two principles are Metallmann’s answer to the problem of the philosophical interpretation of determinism. Unlike the transcendental idea of a universal mind that unifies all experience, they remain faithful to the variety and everchanging structure of nature. The concept of determinism, then, is a dynamic and historical one and it may change in the future along with scientific and philosophical progress. Crucial for this whole reasoning was the concept of structure which was also the topic of Metallmann’s habilitation lecture given in [1933]. As I will try to prove in what follows, this very concept might have been inspired by Poincaré.
22Most scholars commenting on Metallmann’s thought point out that his philosophy displays or anticipates features typical of structuralism, see [Woleński 2000], [Mączka 2002b], [Dombrowski 2011], [Schaff 1983]. Adam Schaff identifies three such characteristic qualities: (1) Taking the subject of inquiry as a whole that dominates over its elements; (2) This whole needs to have its structure which is real and independent from the knowing mind; (3) The inquiry focuses on the laws of coexistence (e.g., structural or morphological laws) [Schaff 1983, 15–23]. All these features are indeed clearly present in Metallmann’s philosophy. He is a holist (1), namely he sees the universe as a coherent unity governed by regular patterns. These patterns are real and independent from the knowing mind (2), as his early polemics with Poincaré proves. These patterns (or laws of nature) are also synchronical (3)—they determine the sufficient or necessary conditions of some events happening simultaneously with others. This is exactly what Metallmann expressed in his book on determinism by introducing the principle of partial identity described above.
23In his habilitation lecture from 1933, Metallmann’s proto-structuralist intuitions were further developed. The philosopher rightly observes that the theoretical reflection on science as such has been predominantly modelled on physics and other sciences dealing with inorganic matter. He realizes that adopting a different approach and focusing on sciences oriented towards living organisms, in the light of their recent developments, may inspire a new approach to thinking about science and reality as such. It is biology (especially genetics and the studies on evolution) that further proves the structural and holistic approach by encouraging a new way of conceiving the relation between the whole and its parts. An organism is more than just a sum of its parts because here “(a) the whole ultimately determines its parts and (b) the whole possesses a specific quality which is impossible to draw from its parts” [Metallmann 1933, 337].
24Subsequently, Metallmann proceeds to the field of psychology and makes similar points: sensual data are recognizable only in the context of a whole. Acts and experiences of a subject are “meaningful compounds.” As such, they demand the existence of a realm of intersubjective and trans-individual values and meanings in which the individual self is embedded [Metallmann 1933, 341]. If natural sciences need to make a minimal assumption that there exists a certain reality to be known and that it has some repeatable patterns to be identified, then psychology needs to make a similar assumption regarding culture, e.g. the realm of values. Culture is a “tool allowing individual personalities to multiply,” as well as a necessary intermediary in communication between individuals: what we communicate to each other are not individual feelings but objective meanings. Also, history needs to be conceived as based on structures that are related to values: “historical process is the fulfillment of values” [Metallmann 1933, 341] as opposed to it being just a sum of facts.
25Metallmann describes his understanding of structure as “holocentric.” By this he means that structure is not just a strict assignment of the elements from one set to the elements of another (such an understanding would be “mereologic”). On the contrary, in a holocentric structure (or holostructure) the whole not only determines the meaning of the elements and makes them intelligible, but it is equally real and concrete as its parts: “the whole and its parts are equally accessible in an empirical way; the whole is neither a sum of certain parts, nor their cause, nor vice versa” [Metallmann 1933, 348]. He concludes that the concept of structure, thus understood, is “predestined to take up a central position and play a crucial role in contemporary scientific and philosophical thought” [Metallmann 1933, 353], challenging the still dominant principle of causality. It is, according to Metallmann, universal enough to bridge the gap between natural sciences and the humanities, and it may even be applied to science itself.
26These ideas, essential to Metallmann’s philosophical position, may have been inspired by Poincaré or at least are parallel to his position. It is clear that the Cracow thinker was well versed in the work of Poincaré, as numerous references in his writings show. Both thinkers were struggling with the very same problem: how objective knowledge and its continuous scientific progress were possible in spite of seemingly disruptive changes, such as the one brought on by quantum mechanics and the theory of relativity. For Poincaré the solution to this problem was combining “the ideas of convention and conventionality with the ideas of invariance, groups, transformations, etc.”, as Jerzy Giedymin writes in Science and Convention. Essays on Poincaré’s Philosophy of Science [Giedymin 1982, ix]. Giedymin claimed that Poincaré’s epistemology emphasising the relational nature of knowledge is not tantamount to instrumentalism (with which it was often associated) and may be instead called structural realism or conventionalist structuralism [Giedymin 1982, xi]. According to Giedymin, these “structural” tendencies in Poincaré can be linked to some of his mathematical ideas, especially to the theory of groups and invariants. Poincaré also claimed that some qualities of arithmetic groups reflect the nature of thinking. Similarly to Metallmann, he also assumed that the structural or formal aspects were as important a part of a given theory as its numerical predictions.
27Having finished Determinizm nauk przyrodniczych, Metallmann immediately started working on the second volume of the work that would cover the issue of determinism in biology. However, due to the tragic fate of the author, the book remained unfinished and was edited and published only in 2002 by Janusz Maczka as Determinizm w biologii [Determinism in Biology] [Mączka 2002a]. Metallmann began with noting that two rival positions were dominant in the biological sciences regarding the nature of life as such. According to the first one, dubbed by him “physicalism,” life can be fully explained by physics and chemistry. According to the other, “vitalism,” life completely transcends the laws governing inorganic matter and is explainable only on its own terms. Metallmann finds both these approaches unsatisfactory and reformulates the question. How is it possible, he asks, to explain the creativity and novelty inherent to life while still maintaining some form of determinism which is essential for any scientific explanation and prediction?
28The answer to this is his hypothesis of emergence. It states that living nature acts in accordance with laws, but the number of these laws is ever increasing. “In the process of emergence,” states Metallmann, “new laws of a higher order emerge on the basis of which a given multitude of things is unified and is elevated to a higher level of complexity” [Metallmann 2002, 188]. To further explain this idea, Metallmann refers to the notion of “discontinuous time”—it is as if the time of old laws ceased to flow and gave way to time in which new laws emerge and are added to the previous ones.
29The hypothesis of emergence explains the possibility of novelty and creation within the living world and yet it does not require “suspending” the regular laws of physics. As Janusz Maczka observes, it only requires the presence of non-linear laws. Actually the very idea of such laws can be traced back to the concept of “non-linear dynamic structures” introduced by Poincaré [Mączka 2002b, 178]. These in turn are based on non-linear equations which allow for describing chaotic processes as well as systems in which the whole exceeds the mere sum of its parts. The emergent processes described by Metallmann can also be linked to non-linear thermodynamic models developed by Euler-Poincaré.
30Poincaré’s influence on the philosophy of science is often limited to viewing him simply as an early conventionalist. Among others, Kazimierz Ajdukiewicz (a conventionalist himself) adhered to this interpretation [see Ajdukiewicz 1985, 175, 304]. The example of Metallmann proves, however, that the impact of the French philosopher should be considered more broadly. In this case not only the methodological ideas of Poincaré but also the purely mathematical ones (the theory of groups, non-linear dynamics, chaos) have inspired a highly original and novel metaphysics of nature developed by the author of Determinizm nauk przyrodniczych.