Dear FISers, In touch with Ludwig Wittgenstein's favourite example, let's play a chess game. Imagine that the chessboard is the information.
We have the whites, e.g., Jaynes, Logan, Kauffmann, Marijuan (more or less!), Loet, Chu-Hsi (Zhu Xi), Susskind's account of loss of information in black holes. (I also side with the whites, but I did not dare to put my name together with the great scientists I quoted!). And the blacks, e.g. Brillouin, Collier, Wheeler, Murray Gell-Mann, Lloyd, Layzer, Muller, Rizzo, Leydesdorff, Hawkins' account of absence of loss of information in black holes. They are all first-rank scientists. Whites do not believe very much in the foremost role of information in our world, blacks do. Who wins the game? Nobody wins. The two players are too strong and well-grounded to be defeated, and, weirdly, both logical and experimental results were not decisive in order to produce the winner. There is just a possibility to tackle the issue and see who wins: to change the rules of the chess game and the shape of the chessboard. The 2D chessboard must become… a 3D chessboard. Equipped with symmetries. The following text comes from our most important (according to me, of course) published (topological) paper. You can find the whole manuscript (with the mentioned references and the proper mathematical treatment) here: http://arturotozzi.webnode.it/products/a-topological-approach-unveils-system-invariancesand-broken-symmetries-in-the-brain/ Symmetry is a type of invariance occurring when a structured object does not change under a set of transformations (Weyl). Symmetries hold the key to understanding many of nature’s intimate secrets, because they are the most general feature of countless types of systems. Huge swathes of mathematics, physics and biology, including the brain, can be explained in terms of the underlying invariance of the structures under investigation. In physics, symmetries can be “broken”. Symmetry breaking consists of sudden change in the set of available states: the whole phase space is partitioned into non-overlapping regions (Roldàn, 2014), so that small fluctuations acting on a system cross a critical point and decide which branch of a bifurcation is taken. In particular, in spontaneous symmetry breaking (SSB), the underlying laws are invariant under a symmetry transformation, but the system as a whole changes. SSB is a process which allows a system cast in a symmetrical state to end up in an asymmetrical one. SSB describes systems where the equations of motion or the Lagrangian obey certain invariances, but the lowest-energy solutions do not exhibit them. “Hidden” is perhaps a better term than “broken”, because the symmetry is always there in such equations (Higgs). In case of finite systems with metastable states, the confinement is not strict: the system can “jump” from a region to another (Roldàn). Concerning the brain, that is the main issue of our FIS discussion, its activity is an example of an open system, partly stochastic due to intrinsic fluctuations, but containing islands at the edge of the chaos, which maintains homoeostasis or allostasis in the face of environmental fluctuations (Friston 2010). The brain retains the characteristics of a complex, non-linear system with non-equilibrium dynamics (Fraiman et. al., 2012), equipped with random walks (Afraimovich et.al., 2013); it operates at the edge of chaos (Tognoli et.al., 2014;) and lives near a metastable state of second-order phase transition, between micro- and macro-levels (Beggs et.al., 2012), characterized by infinite correlation length, countless dimensions, slight non-ergodicity, attractors (Deco et.al., 2012) and universal power laws, testified by the presence of spontaneous neuronal avalanches (De Arcangelis). In such a multifaceted framework, the Borsuk-Ulam theorem is useful. This theorem tells us that, if a sphere is mapped continuously into a plane set, there is at least one pair of antipodal points having the same image; that is, they are mapped in the same point of the plane (Beyer and Zardecki, 2004). Bain symmetries can be studied in a topological fashion, i.e. in terms of antipodal points on a hypersphere. If we enclose symmetries, equipped with antipodal self-similar points, into the abstract spaces of n-spheres, they can be evaluated in guise of projections on Sn-1, where they stand for the broken symmetry. This means that brain symmetries, hidden at a lower level, are detectable at a higher level of analysis, and vice versa. In other words, a symmetry break occurs when the symmetry is present at one level of observation, but “hidden” at another level. It must be emphasized that the symmetries are widespread at every level of organization and may be regarded as the most general feature of systems, perhaps more general than free-energy and entropy constraints too. Indeed, recent data suggest that thermodynamic requirements have close relationships with symmetries. The novel, interesting observation that entropy production is strictly correlated with symmetry breaking in quasi-static processes paves the way to use system invariances for the estimation of the brain metastable states’ free-energy and the energy requirements of neural computations and information processing (Roldán 2014). Thus, giving insights into symmetries provides a very general approach to every kind of brain function and dynamics. A shift in conceptualizations is evident in a brain theory of broken symmetries based on a BUT approach: the symmetries, in this framework, are hidden in a dimension and restored in a dimension higher, and vice versa. Coming back to our chessboard, this means that information can be studied in terms of systems’ symmetries and changes in dimensions, rather than in terms of entropies and energetic gradient descents. An object (on an event) embedded in an environment encompasses a certain amount of information, but such information increases when you add a further dimension to the environment (NOTE: non necessarily a spatial dimension, but also other possible ones, such as an increase of complexity). Indeed, a dimension more gives you a coordinate more, and therefore more information. To make the usual example, the 2D shadow of a cat encompasses less information than a 3D cat. Some authors start from a very low number of dimensions (e.g., the holographic Universe of t’Hooft and Susskind), others from an high number (claims dating back to Spinoza and Kant and going through our Universe made of eleven-dimensional Kaluza-Klein manifolds and subsequent decrease of dimensions, until our 3D plus time perceivable environment). It does not matter: when projecting among levels, information is always a function of the number of dimensions. In such a framework, the role of energy is different from the role of information: the energy is something "injected" from the external "environment" into the system, in order to produce the change of coordinates into the system; on the other side, the changes of information can be detected into the system, and depend on the energy just indirectly (second-hand). In other words: a) the system's change of dimensions dictates the change in information, whileb) the changes in energy dictate the projective mechanisms that allow the changes in system's dimensions. It is a subtle, but foremost difference, that can be highlighted just taking into account a very general topological framework. This is one of the examples of the importance of the topological meta-language in the study of science foundations. Arturo TozziAA Professor Physics, University North TexasPediatrician ASL Na2Nord, ItalyComput Intell Lab, University Manitobahttp://arturotozzi.webnode.it/
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