# METHODOLOGY and/or MATHEMATICS

**Tozzi A. 2020. Random walks are not so random, after all…**

Physical and biological phenomena are often portrayed in terms of random walks, white noise, Markov paths, stochastic

trajectories with subsequent symmetry breaks. Here we show that this approach from dynamical systems theory is not

profitable when random walks occur in phase spaces of dimensions higher than two. The more the dimensions, the

more the (seemingly) stochastic paths are constrained, because their trajectories cannot resume to the starting point.

This means that high-dimensional tracks, ubiquitous in real world physical/biological phenomena, cannot be

operationally treated in terms of closed paths, symplectic manifolds, Betti numbers, Jordan theorem, topological

vortexes. This also means that memoryless events disconnected from the past such as Markov chains cannot exist in

high dimensions. Once expunged the operational role of random walks in the assessment of experimental phenomena,

we take aim to somewhat “redeem” stochasticity. We suggest two methodological accounts alternative to random

walks that partially rescue the operational role of white noise and Markov chains. The first option is to assess

multidimensional systems in lower dimensions, the second option is to establish a different role for random walks. We

diffusely describe the two alternatives and provide heterogeneous examples from boosting chemistry, tunneling

nanotubes, backward entropy, chaotic attractors. **PDF**

**Tozzi, A., Peters, J.F. Information-devoid routes for scale-free neurodynamics. Synthese (2020). https://doi.org/10.1007/s11229-020-02895-7. **

Neuroscientists are able to detect physical changes in information entropy in the available neurodata. However, the

information paradigm is inadequate to describe fully nervous dynamics and mental activities such as perception. This paper suggests explanations to neural dynamics that provide an alternative to thermodynamic and information accounts. We recall the Banach–Tarski paradox (BTP), which informally states that when pieces of a ball are moved and rotated without changing their shape, a synergy between two balls of the same volume is achieved instead of the original one. We show how and why BTP might display this physical and biological synergy meaningfully, making it possible to model nervous activities. The anatomical and functional structure of the central nervous system’s nodes and edges makes it possible to perform a sequence of moves inside the connectome that doubles the amount of available cortical oscillations. In particular, a BTP-based mechanism permits scale-invariant nervous oscillations to amplify and propagate towards widely separated brain areas. Paraphrasing the BTP’s definition, we could state that: when a few components of a self-similar nervous oscillation are moved and rotated throughout the cortical connectome, two self-similar oscillations are achieved instead of the original one. Furthermore, based on topological structures, we illustrate how, counterintuitively, the amplification of scale-free oscillations does not require information transfer.

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**Tozzi A. 2020. Are Borders Inside or Outside? Found Sci. https://doi.org/10.1007/s10699-020-09708-7.**

When a boat disappears over the horizon, does a distant observer detect the last moment in which the boat is visible, or the first moment in which the boat is not visible? This apparently ludicrous way of reasoning, heritage of long-lasting medieval debates on decision limit problems, paves the way to sophisticated contemporary debates concerning the methodological core of mathematics, physics and biology. These ancient, logically-framed conundrums throw us into the realm of bounded objects with fuzzy edges, where our mind fails to provide responses to plain questions such as: given a closed curve with a boundary (say, a cellular membrane) how do you recognize what is internal and what is external? We show how the choice of an alternative instead of another is not arbitrary, rather points towards entirely different ontological, philosophical and physical commitments. This paves the way to novel interpretations and operational approaches to challenging issues such as black hole singularities, continuous time in quantum dynamics, chaotic nonlinear paths, logarithmic plots, demarcation of living beings. In the sceptical reign where judgements seem to be suspended forever, the contemporary scientist stands for a sort of God equipped with infinite power who is utterly free to dictate the rules of the experimental settings.

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**Tozzi A, Peters JF. 2020. A Topological Approach to Infinity in Physics and Biophysics. Found Sci.**

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**Points & lines, skepticism & infinity in the brain: a physical answer to ancient questions**

Starting from the tenets of human imagination, i.e., the concepts of lines, points and inﬁnity, we provide a biological demonstration that the skeptical claim ‘‘human beings cannot attain knowledge of the world’’ holds true. We show that the Euclidean account of the point as ‘‘that of which there is no part’’ is just a conceptual device produced by our brain, untenable in our physical/biological realm: currently used terms like __‘‘lines, surfaces and volumes’’ label non-existent, arbitrary properties__. We elucidate the psychological and neuroscientiﬁc features hardwired in our brain that lead us humans to think to points and lines as truly occurring in our environment. Therefore, our current scientiﬁc descriptions of objects’ shapes, graphs and biological trajectories in phase spaces need to be revisited, leading to a proper portrayal of the real world’s events: miniscule bounded physical surface regions stand for the basic objects in a traversal of spacetime, instead of the usual Euclidean points. Our account makes it possible to erase of a painstaking problem that causes many theories to break down and/or being incapable of describing extreme events: __the unwanted occurrence of inﬁnite values in equation__s. We propose __a novel approach, based on point-free geometrical standpoints__, __that banishes inﬁnitesimals__, leads to a tenable physical/biological geometry compatible with human reasoning and provides a region-based topological account of the power laws endowed in nervous activities. We conclude that __points, lines, volumes and inﬁnity do not describe the world, rather they are ﬁctions introduced by ancient surveyors of land surfaces__. PDF

**“The same”: the principle of identity reloaded**

https://doi.org/10.1016/j.pbiomolbio.2017.10.005.

__A unifying principle underlies the organization of physical and biological systems__. It relates to a well-known topological theorem which succinctly states that an activity on a planar circumference projects to two activities with “matching description” into a sphere. Here we ask: __what does “matching description” mean__? Has it something to do with “identity”? Going through different formulations of the principle of identity, we describe diverse possible meanings of the term “matching description”. We demonstrate that __the concepts of “sameness”, “equality”, “belonging together” stand for intertwined levels with mutual interactions__. By showing that “matching” description is a very general and malleable concept, we provide a novel testable approach to “identity” that yields helpful insights into physical and biological matters. Indeed, we illustrate how a novel mathematical approach derived from the Borsuk-Ulam theorem, termed __bio-BUT, might explain the astonishing biological “multiplicity from identity” of evolving living beings__ as well as their biochemical arrangements. PDF

**How to solve decision limit problems... with holes**

__<span lang="EN-US" style="font-size:10.0pt; font-family:" times="" new="" roman","serif";times="" roman";="" "="">Tozzi A, Peters JF. 2020 Removing uncertainty in neural networks. Cognitive Neurodynamics. https://doi.org/10.1007/s11571-020-09574-w.__

Starting from unidentified objects moving inside a two-dimensional Euclidean manifold, we propose __a method to detect the topological changes that occur during their reciprocal interactions and shape morphing__. This method, which allows the detection of topological holes development and disappearance, makes it possible __to solve the uncertainty due to disconnectedness, lack of information and absence of objects’ sharp boundaries__, i.e., the three troubling issues which prevent scientists to select the required proper sets/subsets during their experimental assessment of natural and artificial dynamical phenomena, such as fire propagation, wireless sensor networks, migration flows, neural networks’ and cosmic bodies’ analysis. PDF

**Projections vs cause/effect**

Causal relationships lie at the very core of scientific description of biophysical phenomena. Nevertheless, observable facts involving changes in system shape, dimension and symmetry may elude simple cause and effect inductive explanations. Here we argue that numerous physical and biological phenomena such as chaotic dynamics, symmetry breaking, long-range collisionless neural interactions, zero-valued energy singularities, and particle/wave duality can be accounted for in terms of purely topological mechanisms devoid of causality. We illustrate how simple topological claims, seemingly far away from scientific inquiry (e.g., “given at least some wind on Earth, there must at all times be a cyclone or anticyclone somewhere”; “if one stirs to dissolve a lump of sugar in a cup of coffee, it appears there is always a point without motion”; “at any moment, there is always a pair of antipodal points on the Earth’s surface with equal temperatures and barometric pressures”) reflect the action of non-causal topological rules. To do so, we introduce some fundamental topological tools and illustrate how phenomena such as double slit experiments, cellular mechanisms and some aspects of brain function can be explained in terms of geometric projections and mappings, rather than local physical effects. We conclude that unavoidable, passive, spontaneous topological modifications may lead to novel functional biophysical features, independent of exerted physical forces, thermodynamic constraints, temporal correlations and probabilistic a priori knowledge of previous cases. PDF

**Deformation is not a topological invariant: a critique to topology**

It is well-known that topology deals with the properties of space preserved under continuous deformations, such as stretching, twisting, bending and so on. This means that two shapes of genus zero (or one, or two, and so on) are topologically invariant under homeomorphisms, i.e., they share matching topological description. Here we ask: is this tenet true?Take a positive-curvature active surface, such as a spherical soap bubble. Due to the Borsuk-Ulam theorem, __the bubble’s surface displays at least two antipodal points with the same description __(e.g., two antipodal points with the same value of surface tension, the latter standing for a continuous function on the 2D surface of the 3D bubble). When a spontaneous or a mechanical stress (e.g., an internal or external force, or a torque) is applied within and onto the surface, the subsequent instability leads to the production of a deformed bubble. The formation of this bubble’s nontrivial surface shape leads to the loss of the above-mentioned antipodal points with matching description. Therefore, __once a spherical manifold’s curvature is modified, an algebraic topological feature gets lost__, i.e., the two antipodal points with matching description.

**Pairwise comparison and the infinity problem**

In this study, we provide mathematical and practice-driven __justification for using [0, 1] normalization__ of inconsistency indicators in pairwise comparisons. The need for normalization, as well as problems with the lack of normalization, is presented. A new type of paradox of infinity is described. PDF16 Kolkodz - pairwise comparison.pdf (303,7 kB)

**Debunking Poppers’s falsifiability**

It has been stated that "a founding principle in science is the ability to falsify your theory". This logical, Popperian tenet, dating back to the first half of the 20th Century, has been fully discarded, in particular by Lakatos, and then by Sokal, Bartley III, and so on. __A scientific theory does not need to be falsifiable, rather simply requires experimentally testable, quantifiable previsions__ that must be treated with statistic methods to evaluate their probability. To give an example related to the scientific (not philosophical!) theory of the multidimensional brain, __the "geometric codes that map information domains" can be tested by looking at the required hidden symmetries__, possibly endowed in the real neurodata provided by currently-available techniques, such as EEG, fMRI.

**Novel versions of the Borsuk-Ulam theorem**

**Borsuk-Ulam theorem on concave manifolds: **Tozzi A. 2016. __Borsuk-Ulam Theorem Extended to Hyperbolic Spaces__. In Computational Proximity. Excursions in the Topology of Digital Images, edited by J F Peters, 169–171. doi:10.1007/978-3-319-30262-1. PDF

**Re-BUT: **Peters JF, Tozzi A. 2016. Region-Based Borsuk-Ulam Theorem. arXiv.1605.02987.

This paper introduces a region-based extension of the Borsuk-Ulam Theorem (denoted by reBUT). A region is a subset of a surface on a finite-dimensional n-sphere. In topology, an n-sphere is a generalization of the circle. For a continuous function on an n-sphere into n-dimensional Euclidean space, __there exists a pair of antipodal n-sphere regions with matching descriptions__ that map into Euclidean space Rn. The main results include a number of different region-based forms of the classical Borsuk-Ulam Theorem as well as the Straecker digital Borsuk-Ulam Theorem and the Burak-Karaca digital Borsuk-Ulam Theorem. Applications of reBUT are given in the evaluation of brain activity and quantum entanglement.

**String-BUT: **Peters JF, Tozzi A. 2016. String-Based Borsuk-Ulam Theorem. arXiv:1606.04031.

This paper introduces a string-based extension of the Borsuk-Ulam Theorem (denoted by strBUT). A string is a region with zero width and either bounded or unbounded length on the surface of an n-sphere or a region of a normed linear space. In this work, an n-sphere surface is covered by a collection of strings. For a strongly proximal continuous function on an n-sphere into n-dimensional Euclidean space, __there exists a pair of antipodal n-sphere strings with matching descriptions__ that map into Euclidean space Rn. Each region M of a string-covered n-sphere is a worldsheet (denoted by wshM). For a strongly proximal continuous mapping from a worldsheet covered n-sphere to Rn, strongly near antipodal worldsheets map into the same region in Rn. An application of strBUT is given in terms of the evaluation of Electroencephalography (EEG) patterns.

### KILLING THE VERB to BE: A PRAGMATIC LANGUAGE FOR SCIENTIFIC PURPOSES