Physics as the science of the possible: Discovery in the age of Gödel Dragutin T. Mihailović, Darko Kapor, Siniša Crvenkovic, Anja Mihailovic This book represents a continuation, an elaboration, and possibly a clear explanation of the ideas which were expounded in the previous book Time and Methods in Environmental Interfaces Modeling published by the Elsevier. In that book as well as in whole of our published scientific work we were either implicitly or explicitly driven by a need to understand how the space between the human mind and observed physical reality is bridged. Here we use synonymously the terms physical reality and reality since the reality is all of physical existence, and concepts related to it as opposed to those products of our mind which remain on the level of mind. Relying on that book we add our new experiences in research in which physics plays a dominant role. To these experiences we attached some epistemological features as well as a view of physics through the optics of Gödel Incompleteness Theorems (Gödel 1931). In the Prolegomena (Chapter 1) we consider some aspects of generality of physics (1.1 Generality of physics). We work on the book with a provisional title “Physics of complex systems as the science of the possible: Discovery in the age of Gödel”. We decided to upload the Chapter 1 on arXiv to have feedback from scientific community Chapter 1 Prolegomena 1.1 Generality of physics (arXiv:2104.08515 [physics.hist-ph]) 1.2 Physics: a crisis that has been lasting over a century! Is that really so? 1.3 Physics and complex systems: how to define this connection? 1.4 Physics and mathematics walking together on a narrow path 1.5 Computational physics as an opportunity for closer relation to other sciences Subchapter 1.2 will appear very soon while 1.3 (and other subchapters) will come later. In this text we deal with term emergent in complex system. The text is a compilation of comments and thoughts related to this term used in 1.3. Comment one In this book, we will deal with complex systems in physics. Therefore, we have to elucidate that term by releasing it from ambiguous interpretations. How are complex systems defined in physics? Does that definition correspond to the definition of complex systems in modern science? Most complicated systems in physics are not complex. Certainly, for example, we can say that particle physics is more or less complicated. On the other hand it shows no signs of intricacy. In the standard theory of elementary particles interactions between particles always happen in the same way (Holovatch et al 2017). However, it is true that most complex systems are complicated (Gell-Mann and Lloyd 1996). The extensional definition of complex systems, and also in physics, introduces us into the problem of having to decide which systems are in the set of complex systems and which not. In answering this question, for example, a physicist and a biologist often have quite different perspectives. This difference is determined by subjective disciplinary bias. Comment two In my response on the RG from June this year I commented definition of emergence given by Jeffrey Goldstein (Goldstein, 1999) since the emergence (weak or strong) plays a central role in theories of integrative levels and of complex systems. Emergent properties are properties that manifest themselves as the result of various system components working together, not as a property of any individual component. Fromm (Fromm 2005) says: „Since emergence is an ambiguous word, it is important to start with a clear definition. In the following, emergent and emergence are defined like this: a property of a system is emergent, if it is not a property of any fundamental element, and emergence is the appearance of emergent properties and structures on a higher level of organization or complexity (if “more is different” (Anderson 1972)). This is the common definition that can be found in many introductory text books on complex systems (Flake 2000; Bar-Yam 1997)“, monographs (Mihailovic et al 2016) and encyclopedias (O’Connor 2020). We think that in physical community Anderson's explanation of emergent property in complex systems is the most comprehensive and indicates his deep understanding of physics and its limits. Comment three Condensation phenomena in particle systems typically occur as one of two distinct types: either as a spontaneous symmetry breaking in a homogeneous system, or as an explicit symmetry breaking in a system with background disorder. Figure 1.2. Explanatory diagram showing how is symmetry breaking works. Diagram in figure 1.2. simply shows how symmetry breaking works. At a high enough energy level, a ball settled in the center (lowest point), and the result has symmetry. At lower energy levels, the center becomes unstable, the ball rolls to a lower pint – but in doing so, it settles on an (arbitrary) position and the result is that symmetry is broken – the resulting position is not symmetrical. This diagram can help us to clearly address two questions. The first one is: “Why does image A turn into image B?” When asking this question, we should always keep it in mind that we stay all the time stay in the domain of events as physics sees them. It can be set as (i) epistemological, (ii) practical and (iii) due to restrictiveness of physical low. In this comment we discuss only (i). (i) The epistemologically stated question means that we should determine the criteria for knowledge so that we can know what can or cannot be known. Transferred to our example of symmetry breaking it means that we are trying to find out what caused image B to come after image A. Is the initiation of emergence in the domain of our knowledge or is it, by analogy with the fact that there are truths which cannot be proved within known physical formalism. The direction of the black arrow shows that the process of emergence definitely goes towards the symmetry breaking. According to some macroscopic manifestations of the system, the process of emergence can be explained with a consistent theory. However, what interest us is the point from which black arrow begins unexpectedly to go in the direction of symmetry changing. The occurrence of that point arises from Bohm's following thinking: “We see, then, that it is appropriate to speak about objectively valid laws of chance, which tell us about a side of nature that is not treated completely by the causal laws alone. Indeed, the laws of chance are just as necessary as the causal laws themselves (footnote: Thus, necessity is not to be identified with causality, but is instead a wider category). For example, the random character of chance fluctuations is, in a wide variety of situations, made inevitable by the extremely complex and manifold character of the external contingencies on which the fluctuations depend. Moreover, this random character of the fluctuations is quite often an inherent and indispensable part of the normal functioning of many kinds of things, and of their modes of being.” (Bohm, 2005) (emphasis added by the authors). Comment four Anderson (1972) concisely elaborated why “More is different?” Recently Bar-Yam (2017) elaborated his opinion about one of the hardest things in complex system science, i.e. to explain is why complex systems are actually different from simple systems (Why complexity is different?). In that paper he commented the separation of scales and why complex systems need a new mathematics. In physics of complex everything is, basically, a question of a scale and corresponding mathematics. This is literally described in “The Oxford Murders” by Argentinean mathematician and writer Guillermo Martinez. “that exactly the same kind of phenomenon occurred in mathematics, and that everything was, basically, a question of a scale. The indeterminable propositions that Gödel had found must correspond to a subatomic world of infinitesimal magnitudes for normal mathematics. The rest consisted in determining right notation of scale. What I proved, basically, is that if a mathematical question can be formulated within the same “scale” as the axioms it must belong mathematicians’ usual world and possible be prove or refute. But if writing on different scale, then it risks belonging to the world-submerged, infinitesimal, but latent in everything-of what can neither be proved nor refuted.” In paper by Mihailović et al. (2021) in Radiation Physics and Chemistry is found “breaking point” after which system complexity declines rapidly. Fig. 11. The running complexity of the neutron and gamma time series with 3 and 7 PAHN plates. The “breaking points” are rounded by ellipses. The position of sample on the x – axis is normalized by its the length of a time series. BP is notation for “breaking point”. KC complexity denotes running complexity of the system while indicates scaling level (Mihailović et al., 2021; Radiation Physics and Chemistry). Fig. 2. The running mean KC complexity of time series for a fixed window (of sizes 200, 300, 400, …, 2800). Calculations were performed for six-time series, each of length 3000, covering an interval of r from 3.948 to 3.953 with an increment of 0.001 (blue circles and squares). Black circles and squares corresponds to the values obtained for = 3.950. Red circle indicates position of the “breaking point”. References Anderson P W 1972 More is different Science l77 393-396 Bar-Yam Y 1997 Dynamics of Complex Systems (Boston: Addison-Wesley) Bar-Yam Y2017 Why complexity is different, New England Complex Systems Institute (March 16, 2017). Bohm D, 2005: Causality and Chance in Modern Physics (London: Taylor & Francis e-Library) Flake GW 2000 The Computational Beauty of Nature (Cambridge: The MIT Press) Fromm J 2005 Types and Forms of Emergence arXiv:nlin/0506028 Gell-Mann M and Lloyd S 1996: Information measures, effective complexity, and total information Complexity 2 44-52 Goldstein J 1999: Emergence as a Construct: History and Issues Emergence 1 49–72 Holovatch Y Kenna R and Thurner S 2017: Complex systems: physics beyond physics Eur. J. Phys. 38 Mihailović DT Balaž I and Kapor D 2016: Time and Methods in Environmental Interface Modeling: Personal Insights (Amsterdam: Elsevier) O’Connor T 2020 Emergent Properties ed E N Zalta The Stanford Encyclopedia of Philosophy (Stanford: The Metaphysics Research Lab, Center for the Study of Language and Information Stanford University) Comment five For further reading Guillermo Martinez The Oxford Murders MacAdam/Cage, San Francisco, USA pp. 36-37 “I met Sarah, Beth’s mother, at that time. She had just started studying physics and she was already engaged to Johnny, the Eagletons’ only son. The three of us would go bowling or swimming together. Sarah told me about the uncertainty principle in quantum physics. You know what I’m referring to, of course: that the clear, tidy formulas governing physical phenomena on a large scale, such as the motion of celestial bodies, or the collision of skittles, are no longer valid in the subatomic world of the infinitesimal, where everything is far more complex and where, once again, logical paradoxes even arise. It made me change direction completely. The day she told me about the Heisenberg Principle was strange, in many ways. I think it’s the only day of my life that I can recall hour by hour. As I listened, I had a sudden intuition, the knight’s move, so to speak,” he said, smiling, “that exactly the same kind of phenomenon occurred in mathematics, and that everything was, basically, a question of scale. The indeterminable propositions that Gödel had found must correspond to a subatomic world, of infinitesimal magnitudes, invisible to normal mathematics. The rest consisted in defining the right notion of scale. What I proved, basically, is that if a mathematical question can be formulated within the same ‘scale’ as the axioms, it must belong to mathematicians’ usual world and be possible to prove or refute. But if writing it out requires a different scale, then it risks belonging to the world—submerged, infinitesimal, but latent in everything—of what can neither be proved nor refuted. As you can imagine, the most difficult part of the work, and what has taken up thirty years of my life, has been sh wing that all the questions and conjectures that mathematicians from Euclid to the present day have formulated can be rewritten at scales of the same order as the systems of axioms being considered. What I proved definitively is that normal mathematics, the maths that our valiant colleagues do every day, belongs to the ‘visible’ order of the macroscopic.” “But that’s no coincidence, I think,” I interrupted. I was trying to link the results that I had presented at the seminar with what I was now hearing and find where they fitted in the large figure that Seldom was now drawing for me. “No, of course not. My hypothesis is that it is profoundly linked to the aesthetic that has been promulgated down the ages and has been, essentially, unchanging. There is no Kantian forcing, but an aesthetic of simplicity and elegance which also guides the formulation of conjectures; mathematicians believe that the beauty of a theorem requires certain divine proportions between the simplicity of the axioms at the starting point, and the simplicity of the thesis at the point of arrival. The awkward, tricky part has always been the path between the two—the proof. And as long as that aesthetic is maintained there is no reason for indeterminable propositions to appear ‘naturally’.”
