# METHODOLOGY, MATHEMATICS: PAPERS

**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

**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