The poetry of the universe, the periodic table, and the scientific progress: a review of new studies on the periodic table of the elements

In 1869, two distinguished scientists, Dimitri Mendeleev and Lothar Meyer, discovered a certain periodicity among the chemical characteristics of the then known elements. Both developed first versions of the periodic table, independently. In the wake of the 150th anniversary, UNESCO proclaimed 2019 the “International Year of the Periodic Table of Chemical Elements”. Two lucid and detailed studies on the periodic table—accompanied by smaller studies on the occasion of the anniversary—have been published, recently, one of them analysing the scientific history, contributing to the (philosophical) theory of science (Eric Scerri), the other analysing the structures, patterns, and irregularities of the table (Geoff Rayner-Canham). Both studies are profound and vivid examples how scientific progress works. They illustrate that even in hard sciences—mirroring Merton’s concept of middle range theory—the required degree of exactness can remain on an intermediate level, as imperfection allows interpretations which could not (yet) be reached by pure mathematics and logic. Both of these brilliant studies provide valuable material, especially for a social science, to better understand how scientific ideas develop, how the power of visualization helps shape ideas, and how contingency is absorbed by the scientific process.

publications that gave Mendeleev his breakthrough in the gradually globalizing scientific community were published in German, which was still one of the leading scientific languages at the time, mainly because of the university landscape in chemistry and physics. Later he switched to French, Russian, and especially English as languages of science. 1 The parallel discoverer was his contemporary Lothar Meyer (*1830 †1895), 2 a professor of chemistry in Karlsruhe and Tübingen (both Germany). For his part, he designed the first periodic tables in order to arrange the elements known at that time according to a pattern (Boek 2021).
The scientific breakthrough for modern chemistry, which was still comparatively young at that time, was that the periodic table linked certain chemical properties with a periodicity and, thus, established a scientific order within the rapidly growing number of known elements, which made it possible to make initial predictions as to how an element would behave approximately. In particular, Mendeleev's pioneering work was largely based on an arrangement comparing atomic masses. The electron configurations actually relevant for the chemical "behavior" were not yet known to him. The electron was not discovered and described until 1897 (see for the progress Scerri 2020, 203; also Marks/Marks 2021). Systematic integration into chemical models did not begin until the early twentieth century (Scerri 2020, 227).
Nevertheless, the periodic table has survived as a scientific concept of order to this day and is essentially unchallenged. Compared to the upheavals and fundamental shifts in the natural sciences during the long twentieth century, this stability of a scientific idea is remarkable. While Newtonian mechanics has been overtaken by modern physics since the beginning of the twentieth century, the periodic table has not yet experienced any serious shock in chemistry that would have called its validity into question (Scerri 2020, 26). On the contrary, quantum chemistry, which emerged from physical quantum mechanics as a hybrid discipline (Gavroglu/Simões 2012), moves with its explanations within the grid of the periodic table. The fundamental upheavals caused by the theory of relativity have rather increased the explanatory potential of the periodic table, because relativistic effects can also be integrated (Scerri 2020, 364).

Periodic table as an object of the philosophy of science with historiographic means
Eric Scerri, a distinguished theorist of the periodic table, presents-now in the second edition-a material-rich and comprehensive (XXVII & 472 pages) study that uses the history of the periodic table as a source of fundamental questions in the philosophy of 1 D. Mendelejeff, Ueber die Beziehungen der Eigenschaften zu den Atomgewichten der Elemente, Zeitschrift für Chemie 12 (1869), 405; Dmitri Iwanowitsch Mendelejew, Die periodische Gesetzmäßigkeit der chemischen Elemente, Annalen der Chemie und Pharmazie, VIII. Supplementband (1871), 133. The paper reviews the periodic table presented in 1869 before the newly founded Russian Society of Chemistry. It was published in parallel in Russian. A French and an English version followed 10 years later: Dmitri Iwanowitsch Mendeleïev, La loi périodique des éléments chimiques, Le Moniteur scientifique 21 (1879), 691; Dimitri Mendeleev, The periodic law of the chemical elements, Chemical News 40 (1879) science. Scerri is a chemist and philosopher of science who has been seminal in initiating the emerging philosophy of chemistry as a branch within the multifarious philosophy of science (compare for the subject Schummer 2006;Vančik 2021). Until now, philosophy of chemistry remains a rather peripheral scientific discipline, especially in comparison with the traditionally leading discipline of philosophy of physics. Nonetheless, philosophy of chemistry, as an independent scientific subject, has been growing since the 1990s, and Scerri is co-editor of the archival journal Foundations of Chemistry (in existence since 1999). If you haven't noticed yet, you're reading it right now!

The chemical thought
Scerri traces the problem of whether the history of the periodic table can be used to identify a proprium of chemical thought. Or as the British science theorist (and chemist) David William Theobald formulated the question in disciplinary-comparatist terms as early as the mid-1970s (Theobald 1976): Are there certain styles of thinking and procedures that distinguish chemistry as a science from its sister disciplines physics and biology? While the closeness of theoretical physics and philosophy is obvious and can refer to a long tradition of connecting debates, chemistry is focused on less fundamental questions and is therefore often (and-as Scerri impressively shows-wrongly) considered intellectually not sufficiently attractive and neglected in philosophical discourse. It is also remarkable that chemistry-which is not addressed by Scerri with his clear focus on inorganics-regains its attractiveness for philosophical fundamental debates where it becomes much more concrete and is supposed to answer questions of molecular macro-organization in the life sciences. A fictitious philosophy of biochemistry, however, is ultimately, according to its questions, one of biology and no longer of chemistry. In accordance with this approach, Scerri's investigation focuses on fundamental questions concerning the epistemology and identity of chemistry as a scientific subject, in the center of which the periodic table of elements still stands today-both in terms of subject function and iconic symbolism.

Is the fundamental really indivisible?
Scerri uses the periodic table of the elements as a visual object to demonstrate the fundamental relationship and the potentials of differential reducibility with regard to a chemical problem (Scerri 2020, XIX). He shows in rich detail and material how the periodic table has always served as a focus for fundamental questions-not least in atomic physics. Experimental atomic physics has always relied on powerful chemistry, which must ultimately isolate and identify relevant atoms (Herrmann 1995). Apparently, theoretical chemistry, in turn, is inconceivable without physics, both theoretical and experimental. For example, without quantum mechanics there would be no molecular orbital theory. Beyond this practical linkage, however, there is a gray area that offers a variety of connections to both scientific and philosophical questions (Scerri 2016): Can chemistry ultimately be reduced to a special field of physics? Or does a unique proposition of chemistry remain, whose reductionist dissolution into mere "shell physics" would lead to a loss of scientific knowledge? Scerri is probably inclined to the latter. Not to be separated from this is the scientific-theoretical question of what degree of exactness and fundamentality a discipline requires in order to do justice to its scientific task.
Repeatedly, the philosophical question is raised whether elements are really indivisible basic building blocks of the physical world or only simple substances which cannot be 1 3 chemically decomposed any further for the time being. With the synthetic creation of new elements-by bombardment of heavy elements with atomic nuclei, e.g. α-particles (for the methods Moody 2013)-also completely new philosophical questions arise (Scerri 2020, 347). The elementary suddenly becomes a malleable mass, which breaks with all philosophical ideas of the atomic since antiquity (Scerri 2020, 3). Scerri carefully traces this in the context of the history of science for all synthetic elements-most recently 118 Og. He does not refrain from making the half-ironic remark that, with synthetic elements, the refuted and obscure transmutation teachings of medieval alchemists (Scerri 2020, XVIII) suddenly make sense again under completely different circumstances (Scerri 2020, 350).

Not just physics of the electron shell
The fact that chemistry is not yet reducible to a special physics of the electron shell is repeatedly demonstrated by the limited predictability of chemical properties of not yet studied elements on the basis of their periodicity. Since Mendeleev, the prediction of certain (undiscovered) elements and their properties was the acid test on which the periodic table could prove itself and allowed to negotiate fundamental questions about the nature of the elements (Scerri 2020, 124, 137).
In addition, the periodic table was repeatedly adjusted when predictions were falsified or new findings permitted more precise observations. Behind this are also necessary shifts in perspective in the context of time. At the beginning of modern chemistry, problems arose in extracting the individual elements from compounds, separating them and identifying them. Since the middle of the nineteenth century, elements had already been scientifically classified on the basis of their chemical behavior or atomic masses. With the discovery of the electron and the development of the first atomic models in the first quarter of the twentieth century (Scerri 2020, 208), it became possible to detach chemical properties from atomic mass. Already Mendeleev had to find out that mass does not always allow reliable statements. First of all, this was noticed by the anomaly that 52 Te has a (slightly) higher mass than 53 I in spite of the lower atomic number (Rayner-Canham 2020, 12). To explain this puzzling observation, of course, the discovery of the neutron was required, which could not have been known to Mendeleev. First (and rather imperfect) descriptions of the neutron emerged only in the 1930s (Chadwick 1932). Apparently, there remained gaps in explanation. Finally, the electron configuration came into view as the anchor of chemical properties.

Not always exciting: "Bohrium is boring"
Ideally, properties of new atoms turn out to be predictable. For example, about the artificial element 106 Sg 3 it was reported almost disappointedly that it behaved chemically as expected like the other two group 6 (VI B) elements 74 W and 42 Mo (Lougheed 1997). Laconically, the unspectacular, quite precisely predicted chemical behavior of 107 Bh was commented on by the statement: "Bohrium is boring " (Service 2020). Of course, this is not always the case. Often there are significant deviations, which can often be explained by relativistic influences, especially for elements with high mass numbers, which in turn make 1 3 predictions of the rather volatile chemical properties from the lanthanides ( 57 La upwards) difficult (Lougheed 1997, 21). In addition, sometimes calculated relativistic effects turn out experimentally different than expected.

Naming at the vanity fair
Woven into the lively presentation by Scerri are numerous vignettes of the history of science, such as the cheerful naming of new elements, which in the politicized competition spirit of the Cold War always had a touch of the political. The International Union of Pure and Applied Chemistry (IUPAC), organized as a private professional society, traditionally grants the right to propose names to the discovering institution. This is not a legally vested right, but it is a powerful line of tradition. For decades, this was a race between Russian and US scientists, later joined by other countries like Germany and Japan (Scerri 2020, 355). Most recently, the names of 113 Nh, 115 Mc, 117 Ts, and 118 Og were adopted in November 2016. 4 The fact that elements 107-112 were named according to German proposals (which was then reflected in the personalized honors given by 109 Mt, 111 Rg, and 112 Cn) is due to discoveries made by the Gesellschaft für Schwerionenforschung (Society for Heavy Ion Research, now GSI Helmholtzzentrum für Schwerionenforschung) in Darmstadt (Hesse, Germany), 5 which in turn explains the geographical names of 108 Hs and 110 Ds (Scerri 2020, 359). Naming elements forces chemical research into social interactions that cannot be broken down to scientifically resolvable technical issues. Scerri perceptively notes that the substitution of technical arguments for consensus (in the relevant committee of IUPAC) is an anomaly, but one that can have repercussions for the organization of the scientific process (Scerri 2020, 349). The sociology of science, as the secret antagonist of the philosophy of science, could certainly gain insightful examples here that demonstrate the social patterns according to which scientific competition works.

Scientific Self-image and epistemic iconography
One central aspect of Mendeleev's periodic table was that "it is represented graphically" (Robinson 2021, 25). Every science needs its inner iconography, a visualization that does not exclusively follow strict scientific derivations, but contains symbols and illustrations of disciplinary identity or a matrix for the specific structure of thought. The influence of the visual on the formation of scientific thought structures has recently received increased attention (Hentschel 2019, 362;Matlin 2022, 279;Ramharter 2019). Many groundbreaking discoveries began with attempts to visualize the invisible in order to create projections for one's imagination to work with. "Intuition and imagination play an important part in the scientific method" (Pauling 1988, 15). The discovery of the structural formula of the benzene ring would be a striking example from organic chemistry (Rocke 2010). The social can never be excluded-despite all epistemological rationalization. Thus, the systematization of the elements in a periodic table-as Scerri demonstrates again and again very vividly-is based on a gradually contingent developmental path of representing scientific knowledge in a certain way, i. e. in a specific graphical order. The underlying physical parameters (such as significantly the nuclear charge number) are empirically measurable and representable within the framework of a theory. In contrast, the (visual) "mapping" in a system is an artificial representation (Mazurs 1975) to make correlations visible and to predict properties. In this respect, the periodic table is a visual instrument. It can be replaced by other representations, without revealing the physical-theoretical fundamentals, in order to make better predictions or to show other interrelationships.
For any science, but also for the way legal institutions deal with science, it is crucial to distinguish between the contingent and cultural as well as socially influenced context of discovery on the one hand, and the context of justification serving rationalized evidence on the other (Feigl 1964, 472;Popper 2002, 7;Reichenbach 1938, 6;Siegel 1980). Of course, visualization and aesthetics cannot replace the rational-scientific approach to the world, the scientific justification of a hypothesis. Nor should they. But they have always catalyzed the progress of knowledge, because the human mind needs bridges to make the invisible tangible. It was rightly emphasized that the periodic law had not yet been established and corroborated by experiments when Mendeleev first wrote about it in 1869, but he had the right hunch "that the periodic system could assist in future research" (Robinson 2021, 21). As an alternative to the established periodic table of elements, there would be other tangible visualization-without abandoning the rational context of justificationwhich are discussed in detail (with graphical illustrations) in the study by Scerri (2020, 373). Recently, for example, another attempt has been made to draw a map of the elements, which should allow predictions about material properties and is based on multiple parameters such as atomic radius, Pauling electronegativity, polarizability, and valence (Allahyari et al. 2020). Which representation prevails then mostly follows simple pragmatics, which, however, always remains a mirror of background styles of thinking. It is therefore no coincidence that the periodic table is a symbol of chemical science and its disciplinary selfimage. These socio-cultural projections can provide valuable insights into how scientific thinking works in a discipline and how this affects the styles and methods through which observations are handled.

Periodic table as a puzzle
Geoff Rayner-Canham, a Canadian-based professor (now: emeritus), inorganic chemist and historian of science, 6 presents an extremely condensed (XIV & 296 pages 7 ) study. A longstanding and fascinating research interest has been the patterns in the behavior of atoms behind the periodic table. Consequently, his new study focuses on the patterns and anomalies of the periodic table.

The patterns of the seemingly random
Rayner-Canham traces common and separating features within the elements under quite different aspects: Electronegativity, isotope distribution, ratio between protons and neutrons in the nucleus, "magic" numbers of the stability of atomic nuclei etc. The analysis is furthermore fanned out for individual groups, especially in detail for the main group elements (Rayner-Canham 2020, 121) and transition metals (Rayner-Canham 2020, 151; previously Rayner-Canham 2018). All this is not analyzed in depth or even mathematized. Numerous tables, graphs and comparisons of individual chemical properties enrich a rather spartan, yet easily readable and immediately catchy text. The book remains as a condensed set of presentations, which confront the reader in staccato rhythm with very different phenomena of the periodic table, in order to demonstrate, above all, one thing again and again: The professional focus of chemists on the electron structure of the elements neglects the core structure, which holds many surprises, but also allows meaningful predictions. Almost sadly he must state: "Chemists so often overlook the fascinating world of nuclear structure" (Rayner-Canham 2020, 21). Rayner-Canham wants to raise awareness that the nuclear structure should also be a ponderous chemists' interest because numerous properties of chemical elements (indirectly) depend on the structure of the atomic nucleus (Rayner-Canham 26-49) Thus, holistic view of the elements (beyond the electron shell) serves epistemic interests not only of physics but also of chemistry.

Let's play chess: the knight's move phenomenon
Rayner-Canham is particularly fascinated by the Knight's Move phenomenon, which was only discovered in the 1990s (Rayner-Canham 2020, 210) 8 and has so far been difficult to explain (Rayner-Canham 2020, 195;Rayner-Canham/Oldford 2007;also Scerri 2020, 414). Atoms of some heavy elements behave in a strikingly similar way to other elements that are not in the same group but in the next higher period two groups to the right-like a chess move with the knight piece. At first glance, the analogy seems rather like reading anthropomorphic figures in constellations of the stars, but it is a serious finding that requires a rational explanation. The figurative representation provides attention to the problem at least.

The shine of gold: relativistic effects of heavy nuclei
With ascending atomic mass numbers, relativistic effects appear repeatedly, which are paradigmatic for the interaction between the atomic nucleus and the electron shell underlined by Rayner-Canham, which (indirectly) determines the chemical reactivity, as a result of changes in energy levels of atomic orbitals (Rayner-Canham 2020, 46-47; see also Giuliani, Matheson, Nazarewicz, Olsen, Reinhard, Sadhukhan, Schuetrumpf, Schunck, and Schwerdtfeger 2019, 10). The best-known relativistic effects are the atypical color of gold 79 Au (Scerri 2020, 366) and the liquid state of mercury 80 Hg (Jansen 2005;McKelvey 1983;Norrby 1991;Pyykkö 1988;Schwerdtfeger 2002;Thayer 2005). The relativistic explanation for such anomalies is that as the nuclear charge increases, the velocity of the inner shell electrons increases and approaches the speed of light (Bardají and Laguna 1999, 202;McKelvey 1983, 115;Kratz and Lieser 2013, 12). The accompanying mass effects and contraction of the inner s orbitals lead to better shielding of the atomic nucleus and destabilization of diffuse orbitals of outer shells, associated with a shift in energy levels. This has consequences: Without this effect, 79 Au would probably look similar to 47 Ag (Pyykkö 1988, 583), which is one period above it in the 11 th group, i.e.: silvery-white. Alternatively, p-electron pairs can become "inert". Thus, noble gas properties of 112 Cn are discussed. In contrast, it is still difficult to quantify the concrete fraction of relativistic effects in relation to "normal" orbital characteristics. For example, for the well known and experimentally proven lanthanide contraction there are so far only mathematized estimates. The question of a share of relativistic effects was raised almost 35 years ago by Pekka Pyykkö, probably the most important pioneer in this field of research, but was left in abeyance (Pyykkö 1988). Quantum chemical calculation models suggesting exactness have always been met with reservation (Jansen 2005(Jansen , 1473: "Of course, quantum chemical calculations intended to be close to physical reality have to consider the relativistic variation of the electron mass with its velocity. However, neither theoretical nor experimental measures seems to exist, that would allow to impartially differentiate between the contributions of conventional orbital effects and of relativistic effects to the 6 s orbital contraction." With the gradual exploration of the chemistry of superheavy elements, there is also an increasing interest in such relativistic influences, which are usually not relevant for the lighter elements that are usually in focus (Thayer 2005(Thayer , 1725. The big problem for chemistry, of course, is that superheavy atomic nuclei are so unstable that the half-lives of radioactive decay are usually in the range of milliseconds. For example, 293 Lv as the most stable Lv isotope has a half-life of 53 ms, 294 Ts of 78 ms, 294 Og of less than 1 ms. Already the artificial generation is difficult and usually only a few atoms can be generated (Le Naour et al. 2013). However, a chemistry of single atoms (instead of manageable amounts of substances) under conditions of extremely fast decay is still in its infancy. Measurement results allow only indirect constructions of chemical properties, which can never be observed in a really experiential way. This is not only a practical but also an epistemological problem (Scerri 2020, 347). What does it actually mean to speak of the existence of an element (or of a certain nuclide of the element) in terms of scientific theory? Chemistry is confronted here with similar epistemic problems as astrophysics, which seeks to explore spaces that can never be practically experienced. The problem of the unobservable is not new for chemistry (Chang 2016;Gavroglu and Simões 2012, 181), but the limits of the experimental and empirical become clearer with the increase of knowledge here. This has the consequence that theoretical approaches are needed, which often touch fundamental philosophical problems of epistemology.

The romance of the unpredictable
In his overarching, multifaceted, and sometimes surprising account of galloping through the periodic table, Rayner-Canham repeatedly puts the importance of abstract patterns into perspective. He emphasizes the volatile peculiarities of the individual elements and the limited predictability of chemical behavior on the basis of the periodic table. A constant of inorganic chemistry is the "individuality" of the elements, specific properties that are associated with an element that can sometimes only be put into an abstract order to a limited extent. This feature allows a search for possible patterns, but does not support the expectation of systemic coherence (Rayner-Canham 2020, 200). Chemistry is too complex to allow linear predictions. Therefore, the scientific goal is usually only to estimate chemical behavior, but this does not obviate the need for experimental confirmation and specification.
In a sober tone, Rayner-Canham ultimately engages in a "puzzling" of chemistry-in some way: a re-enchantment bringing a certain kind of modern magic of the imperfect back into a disenchanted world. The book plays with the subjectivity of the observer, testifies to passion, fascination, and a magical attraction of the unfathomable, the enigma. It is a declaration of love for the dazzling diversity of inorganic chemistry, which is usually overshadowed by the molecular multiplicity of organic chemistry. Thus, the fact-oriented style conceals a small cabinet piece of neo-romanticism inherent in dedicated research. It is linked to another side of scientific knowledge, a driving force behind science, which is deliberately underestimated. It is a passionate search for ravishing beauty of the unknown, for a silken poetry hidden under the fabric of the universe, the unraveling of conundrums. The theorist of science Ernst Peter Fischer has painted a colorful picture of how much impetus scientific progress owes to these rather opaque forces of passion and curiosity (Fischer 2003;. These are experiences that a social science (like jurisprudence) will never have.

Some remarks from a lawyer's perspective: the beauty of imperfection and the vulnerability of the scientific process
Both authors have presented very different studies in terms of style, concept, and level of detail, but they are both immensely worth reading. They illustrate how scientific knowledge processes evolve across language and system boundaries from a long-term perspective on the basis of an ordering model that is equally central to the natural sciences, in both practical and theoretical terms. What can be learned from this? Perhaps one or the other reader may have wondered: Why should a trained lawyer like me worry about?

Law and science: the periodic table as a model for scientific progress
Why is it also worthwhile for lawyers to be interested in the history and theory of the periodic table? Scientific knowledge plays a central role in many areas of law. For example, in environmental law, health law, or pharmaceutical law, dealing with scientific issues (like chemistry, physics, and biology) is daily routine. In constitutional law, for example, we need to assess whether statutory legislation enacted to protect against chemical risks can be based on sufficiently plausible assumptions. Science in its social context, in turn, is fragile and rests in the hands of a political order. Law influences the scientific process. Legal rules may, for example, require verification that standards of good scientific practice are being met. If the (legally secured) freedom of science is to protect the scientific process, we need a picture of how science works. Lawyers, of course, do not have to understand knowledge at a scientific level. That is neither possible nor necessary. Such knowledge is provided by external expertise. What is necessary, however, is a fundamental understanding of how knowledge arises in the sciences, how epistemic processes take place, and how knowledge is epistemically structured, how it changes, and how it is theorized.
Different scientific disciplines can best communicate with each other by looking at their foundations. Lawyers are typically laymen for everything. Therefore, they cannot delve deeply into disciplinary discourse beyond the law. What is needed, however, is visual material on how other sciences work and are structured. Differences in the epistemological approach have to be addressed and understood. For example, scientific hypotheses are recognized as theories according to different rules than evidence is presented in legal proceedings. Interdisciplinary approaches through which legal scholarship can address another scientific subject include epistemology, which looks at a subject's methods; sociology of science, which can contextualize and culturally situate discourse; and history of science, which can show where ideas and explanations come from and how they have changed over time. History positions scientific ideas in the context of the social, of which the law is also a part. History and theory of natural sciences can illustrate contingency and dependence of representations, but also show how complex and how fragile science is, and which epistemic degrees of hardness different disciplines and methods can acquire. Scientific thinking is still poorly understood in the legal sciences. Its value is consistently underestimated (Gärditz 2022).
The periodic table of the elements provides suitable illustrative material for several reasons: The periodic table has a central key position in chemistry. Its development and updating are closely connected with the history of modern chemistry as an independent part of sciences. The periodic table is a representation that systematizes empirically based knowledge and makes correlations tangible for rational discussion. I would like to illustrate this in a little more detail below.

Sciences as process in the wake of image and imagination
Scientific progress lives from its claim to unfinished knowledge, as the German Federal Constitutional Court has put it rather poetically: "In order that research and teaching can be oriented unhindered to the endeavor for truth as 'something not yet completely found and never completely to be found' (Wilhelm von Humboldt), science has been declared to be an area of personal and autonomous responsibility of the individual scientist, free from external state regulation. This also means that a constitutional right to freedom of science […] 9 does not seek to protect a particular conception of science or a particular theory of science. Rather, its guarantee of freedom extends to every scientific activity, i.e. to everything which, according to content and form, is to be regarded as a serious, planned attempt to determine the truth. This follows directly from the principle of the incompleteness of all scientific knowledge." 10 Scientific progress is not damaged by instructive dead ends, but by overestimating oneself or by a lack of error culture. The history of the periodic table, which has accompanied the modern natural sciences for over 150 years and is still of decisive importance for understanding the world today, provides rich illustrative material for this. Of course, misinterpretations also occurred again and again in the development and updating of the periodic table-for example, in the supposed identification of the first transuranic elements (Scerri 2020, 350) or in the assumption, refuted as recently as 2003, that bismuth is stable (Rayner-Canham 2020, 18). In fact, radioactive decay of bismuth only has an extremely long half-life (Hollemann, Wilberg, and Wilberg 2017, 942: the naturally occurring 209 Bi has a half-life of 19 × 10 18 years), for the detection of which there was simply no sufficiently sensitive technology available before. Above all Scerri shows again and again, very vividly, that dead ends and erroneous paths are not operational accidents of science, but necessary components of an always incomplete process of cognition that learns from mistakes. If you really want to understand the successes of science, you have to deal with its well-founded aberrations. Or as the physicist, philosopher and science theorist Ernst Mach put it: "As a corrective, the clearly recognized error is just as conducive to knowledge as positive knowledge" (Mach 1917, 110).
In this respect, the periodic table offers overwhelming illustrative material of how an evolutionary and responsive scientific discourse handles errors, integrates new findings and rationally processes them further within a visualized system of interpretation. Law, which I represent as a subject, can also learn from this. Science in a constitutional sense comprises "what is to be regarded as a serious and planned attempt to ascertain truth according to content and form". 11 This includes the process of scientific cognition 12 and also-without compromising the claim of scientific search for truth-the methodically disciplined errors, which unavoidably occur. The history of science shows that a free society and its law must protect not only the successes of scientific research, but its crooked paths, its sometimes more or less eccentric inspirations, and its dead ends. Scientific statements are not based on certainties, but always on probabilistic judgments (Lepsius 2020), which then change continuously. This can be demonstrated, in particular, by hard science, which is often made of very soft wood and follows crooked ways.

Cliché of exactness and the gnarled charm of muddling through
Science has very different degrees of exactness and is sometimes-contrary to the typical urge for specification and differentiation-dependent on precisely reducing complexity. Relations of the elements of the periodic table to each other are-as both studies showmuch more complex than the initially groundbreaking and roughly sustainable periodicity suggests (Scerri 2020, 44). Whether complexity and the resulting need for differentiated disciplines can, at some point, be resolved reductionistically and mathematized is one of the major future questions of a philosophy of science of chemistry. Potentially system-relevant "borderline questions" are already being raised today, for example, about an absolutely limiting horizon of the periodic table by nuclear stability, about the possible resolution of the orbital structure of electrons (Jerabek et al. 2018;Scerri 2020, 367), which is constitutive for chemistry, or about possible islands of stability among still undiscovered superheavy nuclides (Scerri 2020, 367).
Significantly, the predictions with which atomic number reaches the limits of nucleus stability have changed continuously throughout the history of the periodic table (Karol 2018, 10, 14-16). Up to now, chemical research has mainly reacted pragmatically to these inherent trouble spots of uncertainty. This is unlikely to change. Even fundamental new findings in atomic physics would not eliminate the practical significance of the approximate assumptions of modern chemistry. Just as Newtonian mechanics has been superseded by the theory of relativity but is still suitable for everyday use, the boundary issues of theoretical chemistry are unlikely to affect the vast majority of research fields. What is striking about chemistry-in comparison to physics-is, for example, the still rather low degree of mathematization (Theobald 1976, 213). However, this is also a difference that pervades the social sciences and their explanatory horizons. The periodic table was neither derived from an abstract theory nor born in a moment of genius. Rather, it arose from conclusions drawn from almost 20 years of iterative observation of already known as well as new elements that were constantly being discovered at the time, independently of scientific research (Bensaude-Vincent 1986, 17).

Middle range theories and the basis of normativization
A medium level of abstraction and a renunciation of hyper-theoretical reduction at the expense of practical operability can be precisely an epistemological gain. Obviously, even a theorized discipline cannot do without interpretative horizons of the descriptive. Recently, chemistry was even certified as a "basic subject" precisely because it is comparatively closely connected with concrete experiments and practices, i.e. because it corresponds to a "realistic" ideal type of scientific discovery (Müürsepp et al. 2020). Significantly, science theory attributes an intermediate level of physical complexity to chemistry (Theobald 1976, 204). Theoretical models of chemistry-e.g. of molecular orbitals-have to cope with extreme complexity of subatomic interaction and therefore have to operate with a limited degree of precision (Chang 2016, 240), which obviously has not harmed the progress of knowledge. The specifically molecular perspective of chemistry (Theobald 1976, 210) allows to leave out other-for their part highly complex-questions (for instance of the particle physics of the atomic nucleus).
Models of chemistry, therefore, are typically less fundamental than practically useful in explaining certain processes (Chang 2016, 250). This withdrawal of the claim to abstraction, based on the division of labor between different scientific disciplines, finds its counterpart in the Mertonian concept of middle range theory of social science provenance (Merton 1986, 39), which, incidentally, has also been demanded for the legal sciences (Lepsius 2014). What is then needed is a scientific ethics of second-best solutions. With the gradual withdrawal of the claim to abstraction, the degree of differentiation of scientific disciplines increases correspondingly. The necessity of precisely addressing the concrete functional conditions and argumentation patterns of the specific subjects has long since prevailed in the philosophy of science over holistically abstracted attempts to interpret the world. Science law, too, must pay increased attention to this, for example, when professional standards of good scientific practice are transposed into legally binding rules.

Epistemic aesthetics and the fragility of science
Both studies discussed here are works about the visualization of knowledge as an aesthetic part of the cognitive process. The periodic table has shaped scientific thinking precisely in its visualization. Scientific beauty lies in the imperfect, which has its own aesthetics that even modern rationalism could never really banish from science. The inherent aesthetics of science is an ideal playing field for interdisciplinary approaches. Even distant disciplinessuch as e.g. art history-can contribute to shedding light on the context in which scientific cognition arises (Bredekamp 2005;Voss 2007). Representation is part of an epistemic culture and contributes to making knowledge tangible, but also to making it possible to criticize (Krohn 2006).
The difference in scientific theory between the context of discovery and the context of justification is also a legally indispensable functional condition for the juridico-political institutions of a democratic constitutional state (Gärditz 2021). Good science needs the effort to reduce the unavoidable contingency of the context, from which knowledge originates (the social conditionality of topics, interests and methods), in the context of justification as far as possible by rationalization. In addition, however, there remains a context of presentation which, as a third level, focuses on the social mediation of scientifically founded knowledge and is thus particularly socially contextualized, i.e. unavoidably contingent. This can have a powerful impact on development paths, styles of thought (Fleck 2015, 109), and modes of problem perceptions. For this very reason, it is crucial that freedom of science also protects science communication in its very different forms of expression. Law and social sciences are typically prone to clichés of exactness and linearity attributed to the natural sciences, which often distorts perceptions and obscures the view of the real practices and challenges that a society must address when seeking an appropriate approach to science. This is mostly due to a simple ignorance of how scientific progress works in sophisticated disciplines such as chemistry, physics or biochemistry. However, clichés are always a source of misperception and misunderstanding. Disciplinarily open and context-sensitive works like the magnificent ones discussed here are therefore of central importance to enlighten and promote mutual understanding through interdisciplinarity.
Funding Open Access funding enabled and organized by Projekt DEAL.
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