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Review
. 2019 Oct;13(5):417-428.
doi: 10.1007/s11571-019-09539-8. Epub 2019 May 7.

Points and lines inside human brains

Arturo Tozzi  1 , James F Peters  2   3
Affiliations
Review

Points and lines inside human brains

Arturo Tozzi et al. Cogn Neurodyn. 2019 Oct.

Abstract

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 equations. 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.

Keywords: Continuum; Curvature; Infinity; Physical equations; Topology.

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Figures

Fig. 1
Fig. 1
Examples of sample vortices. Left: a Weierstrass-Zeta mathematical model for multiple surface spacetime vortices, in which each vortex is a funnel reaching down to a physical limit called a vertex. Right: black hole vortex (initial mass radius 0.44 km) formation. It results from steadily increasing strong gravitational force that pulls neighboring chunks of matter into its funnel winding down to the squish level. Therefore, massive chunks of matter are squeezed to the diameter of a vortex
Fig. 2
Fig. 2
Curvature changes in biological and physical systems. a given a physical system described by progressively increasing curves on a positive-curvature manifold, the occurrence of infinity (straight line) can be removed by taking into account progressively decreasing curves on a negative-curvature manifold. b by placing physical observables on a toroidal manifold, one achieves a correspondence between positive and negative curvatures, thus erasing the unwanted occurrence of infinity. c Time-reversal according to Lesovik et al. (2019). From the second (“developed state”) to the third (“time-reversed state”) panel, the Authors modified the phase of the wave-function in every tiny area, so that the lines with positive curvatures in the second panel become lines with negative curvatures in the third one
Fig. 3
Fig. 3
Conformable infinity at computable points near, but not at ideal points. Left side: Tractable point on spheres in spacetime that are sliced by an infinite plane. Ideal points on the infinite plane are replaced with non-ideal points that are computable. Right side: Those parts of a sheaf of infinite places that intersect with computable lines are computable up to but not including the ideal points
Fig. 4
Fig. 4
A point-free topology allows description of fractals. a the novel BUT variant, termed re-BUT, describes a pointless feature (blue shape) on a two-dimensional circle S1. When projected onto a sphere S2, two pointless features with matching description are achieved (blue shapes). b the pointless feature on S1 can be projected not just to S2, but also to the fractal structure S1.3. Therefore, we achieve two pointless features in S1.3 (blue shapes): they stand for two matching descriptions, because they depict the same fractal structure, self-similar at different magnifications and coarse-grained scales. (Color figure online)
See this image and copyright information in PMC

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