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.
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References
Aleksandrov AD (1969) Non-Euclidean geometry. In: Alexsandrov AD, Kolmogorov AN, Lavrent’ev MA (eds) Mathematics: its content methods and meaning. The MIT Press, Cambridge. ISBN 0-486-40916-3
Armenta Salas M, Bashford L, Kellis S, Jafari M, Jo H et al (2018) Proprioceptive and cutaneous sensations in humans elicited by intracortical microstimulation. Elife 7:e32904. https://doi.org/10.7554/elife.32904
Autrecourt, Nicholas of. About 1340. The universal treatise. Marquette University Press, Milwaukee, Wisconsin, 1971
Barranca VJ, Huang H, Li S (2018) The impact of spike-frequency adaptation on balanced network dynamics. Cogn Neurodyn 13(1):105–120
Bergmann PG (1989) Quantum gravity at spatial infinity. Gen Relativ Gravit 21(3):271–278
Bollinger T (2018) Fundamental as fewer bits. FQXi essay contest 2017, Jan 2018. https://fqxi.org/data/essay-contest-files/Bollinger_FQXi_Essay_2017_.pdf
Bonzon P (2017) Towards neuro-inspired symbolic models of cognition: linking neural dynamics to behaviors through asynchronous communications. Cogn Neurodyn 11(4):327–353
Borsuk K (1958) Concerning the classification of topological spaces from the standpoint of the theory of retracts. FundamentaMathematicae XLVI:177–190
Borsuk K (1969) Fundamental retracts and extensions of fundamental sequences. FundamentaMathematicae 1:55–85
Bradwardine T (1330) about 1330, De continuo (On the Continuum). In: Murdoch JE (ed) geometry and the continuum in the fourteenth century: a philosophical analysis of Thomas Bradwardine’s Tractatus de continuo. Ph.D. thesis, University of Wisconsin, 1957
Chamberlain BP, Clough J, Deisenroth MP (2017) Neural embeddings of graphs in hyperbolic space. arXiv:1705.10359
Chan B, Kawashima Y, Katouda M, Nakajima T, Hirao K (2016) From C60 to infinity: large-scale quantum chemistry calculations of the heats of formation of higher fullerenes. J Am Chem Soc 138(4):1420–1429
de Arcangelis L, Herrmann HJ (2010) Learning as a phenomenon occurring in a critical state. Proc Natl Acad Sci 107:3977–3981
de Arcangelis L et al (2006) Self-organized criticality model for brain plasticity. Phys Rev Lett 96:028107
de Cusa N (1997) 1440. De doctaignorantia. English translation. In: Bond HL (ed) Nicholas of Cusa: selected spiritual writings, classics of western spirituality. Paulist Press, New York
de Haro S, Dieks D, ‘t Hooft G, Verlinde E (2013) Forty years of string theory reflecting on the foundations. Found Phys 43(1):1–7
Di Concilio A (2013) Point-free geometries: proximities and quasi-metrics. Math Comput Sci 7(1):31–42. https://doi.org/10.1007/s11786-013-0140-2
Di Concilio A, Guadagni C, Peters JF, Ramanna S (2018) Descriptive proximities. Properties and interplay between classical proximities and overlap. Math Comput Sci 12:91–106. https://doi.org/10.1007/s11786-017-0328-y
Duff M (1996) M-theory (the theory formerly known as strings). Int J Mod Phys A 11(32):5623–5641
Duffy KR, Hubel DH (2007) Receptive field properties of neurons in the primary visual cortex under photopic and scotopic lighting conditions. Vis Res 47(19):2569–2574
Ehresmann C (1950) Les connexions infinit´esimales dans un espace fibrée differentiable. Colloque de Topologie. Bruxelles, pp 29–55
Fournier J, Müller CM, Schneider I, Laurent G (2018) Spatial information in a non-retinotopic visual cortex. Neuron 97(1):164–180. https://doi.org/10.1016/j.neuron.2017.11.017
Fox KC, Spreng RN, Ellamil M, Andrews-Hanna JR, Christoff K (2015) The wandering brain: meta-analysis of functional neuroimaging studies of mind-wandering and related spontaneous thought processes. Neuroimage 1(111):611–621. https://doi.org/10.1016/j.neuroimage.2015.02.039
Frankel T (2011) The geometry of physics: an introduction, IIIrd edn. Cambridge University Press, Cambridge. ISBN 978-1-107-60260-1
Frauendiener, 2000. Conformal infinity. Living Reviews in Relativity, http://www.springer.com/us/livingreviews/articles/volume3/2000-4frauendiener
Friston K (2010) The free-energy principle: a unified brain theory? Nat Rev Neurosci 11(2):127–138
Geroch R, Kronheimer EH, Penrose R (1972) Ideal points in space-time. Proc R Soc London A 327:545–567
Helmholz H (1858) Uber integrale derhydrodynamicsGleichungenwelchederWirbelbewegungentsprechen. J fur die reine und angewandteMathematik, 55, 25–55, [On the integrals of the hydronamical equations, which express vortex motion], trans. by Tait PG. Phil Mag 33: 485–512
Iyer SV, Petters AO (2007) Light’s bending angle due to black holes: from the photon sphere to infinity. Gen. Relativ. Grav. 39:1563–1582
Jirsa VK et al (2014) On the nature of seizure dynamics. Brain J Neurol 137:2210–2230
Johnstone PT (1983) The point of pointless topology. Bull Am Math Soc 8(1):41–53
Kelvin L (Thomson W) (1867) On vortex atoms. Proc R Soc Edinb 6:94–105
Kim S-Y, Lim W (2017) Dynamical responses to external stimuli for both cases of excitatory and inhibitory synchronization in a complex neuronal network. Cogn Neurodyn 11(5):395–413
Langevin P (1908) Sur la théorie du mouvementbrownien. C R Acad Sci Paris 146:530–533
Lesovik GB, Sadovskyy IA, Suslov MV, Lebedev AV, Vinokur VM (2019) Arrow of time and its reversal on the IBM quantum computer. Sci Rep 9:4396
Levi-Civita T (1917) Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura Riemanniana. Rend Circ Mat Palermo 42:73–205. https://doi.org/10.1007/bf03014898
Li H, Fang Q, Ge Y, Li Z, Meng J, Zhu J, Yu H (2018) Relationship between the dynamics of orientation tuning and spatiotemporal receptive field structures of Cat LGN Neurons. Neuroscience 377:26–39. https://doi.org/10.1016/j.neuroscience.2018.02.024
Lübeck S (2004) Universal scaling behavior of non-equilibrium phase transitions. Int J Mod Phys B 18:3977–4118
Mazade R, Alonso JM (2017) Thalamocortical processing in vision. Vis Neurosci 34:007. https://doi.org/10.1017/s0952523817000049
Milstein J et al (2009) Neuronal shot noise and Brownian 1/f2 behavior in the local field potential. PLoS ONE 4:e4338
Mizraji E, Lin J (2017) The feeling of understanding: an exploration with neural models. Cogn Neurodyn 11(2):135–146
Muxin H (2011) Cosmological constant in loop quantum gravity vertex amplitude. Phys Rev D 84(6):064010
Nagel T (1974) What is it like to be a bat? Philos Rev 83(4):435–450. https://doi.org/10.2307/2183914
Naimpally SA, Warrack BD (1970) Proximity spaces. Cambridge University Press, Cambridge
Papo D (2014) Functional significance of complex fluctuations in brain activity: from resting state to cognitive neuroscience. Front Syst Neurosci 8:112
Pavese F, Charki A (2016) Some important features of the proposed new definition of the International System of Units (SI): realization and hierarchical problems that the users should know about. Int J Metrol Qual Eng 7:403. https://doi.org/10.1051/ijmqe/2016023
Peters JF (2014) Topology of digital images: visual pattern discovery in proximity spaces. Springer, Berlin. https://doi.org/10.1007/978-3-642-57845-2
Peters JF (2016a) Computational proximity. Excursions in the topology of digital images. Springer International Publishing, New York City. https://doi.org/10.1007/978-3-319-30262-1
Peters JF (2016b) Two forms of proximal, physical geometry.Axioms, sewing regions together, classes of regions, duality and parallel fibre bundles. Adv Math Sci J 5(2):241–268
Peters JF, Tozzi A, Ramanna S, İnan E (2017) The human brain from above: an increase in complexity from environmental stimuli to abstractions. Cogn Neurodyn 11(4):391–394
Popkin RH, Maia Neto JR (eds) (2007) Skepticism: an anthology. Prometheus Books, Amherst. ISBN 1591024749, ISBN 13: 9781591024743
Pritchard WS (1992) The brain in fractal time: 1/f-like power spectrum scaling of the human electroencephalogram. Int J Neurosci 66:119–129
Rovelli C, Smolin L (1988) Knot theory and quantum gravity. Phys Rev Lett 61(10):1155–1958
Seiden A (2005) Particle physics. A comprehensive introduction. Addison-Wesley, Boston. ISBN 0-8053-8736-6
Sengupta B, Tozzi A, Cooray GK, Douglas PK, Friston KJ (2016) Towards a neuronal gauge theory. PLoS Biol 14(3):e1002400. https://doi.org/10.1371/journal.pbio.1002400
Shapiro S, Hellman G (2017) Frege meets aristotle: points as abstracts. Philos Math 25(1):73–90. https://doi.org/10.1093/philmat/nkv021
Simon M (1901) Euclid und die sechsplanimetrischen Bucher, vol 8. Teubner, Leipzig
Smirnov JM (1952) On proximity spaces. Math. Sb. (N.S.) 31(73):543–574 (Engish translation, Am Math Soc Trans Ser 2(38): 5–35 (1964)) (in Russian)
Sommers P (1978) The geometry of the gravitational field at space-like infinity. J Math Phys 19:549–554
‘t Hooft G (1971) Renormalizable Lagrangians for massive Yang-Mills fields. Nucl Phys B 35:167–188
Tait PG (1877) On knots. Trans R Soc Edinb 28:273–317
Tozzi A (2019) The multidimensional brain. Phys Life Rev. https://doi.org/10.1016/j.plrev.2018.12.004
Tozzi A, Peters JF (2016) A topological approach unveils system invariances and broken symmetries in the brain. J Neurosci Res 94(5):351–365. https://doi.org/10.1002/jnr.23720
Tozzi A, Peters JF (2017a) What does it mean “the same”? Prog Biophys Mol Biol. https://doi.org/10.1016/j.pbiomolbio.2017.10.005
Tozzi A, Peters JF (2017b) From abstract topology to real thermodynamic brain activity. Cogn Neurodyn 11(3):283–292
Tozzi A, Sengupta B, Peters JF, Friston KJ (2017) Gauge fields in the central nervous system. In: Opris J, Casanova MF (eds) The physics of the mind and brain disorders: integrated neural circuits supporting the emergence of mind. Series in cognitive and neural systems. Springer, New York. ISBN 978-3-319-29674-6
Tozzi A, Peters JF, Jaušovec N (2018) EEG dynamics on hyperbolic manifolds. Neurosci Lett 683:138–143. https://doi.org/10.1016/j.neulet.2018.07.035
Van Hooser SD, Heimel JA, Nelson SB (2005) Functional cell classes and functional architecture in the early visual system of a highly visual rodent. Prog Brain Res 149:127–145
Viswanathan P, Nieder A (2017) Visual receptive field heterogeneity and functional connectivity of adjacent neurons in primate frontoparietal association cortices. J Neurosci 37(37):8919–8928. https://doi.org/10.1523/jneurosci.0829-17.2017
Whitehead AN (1929) Process and reality. An essay in cosmology. The Free Press, New York
Wigner EP (1960) The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959. Commun Pure Appl Math 13:1–14
Witten TA, Li H (1993) Asymptotic shape of a fullerene ball. Europhys Lett 23:51–55
Zaka O, Peters JF (2019) Isomorphic-dilations of the skew-fields constructed over parallel lines in the Desargues affine plane. arXiv, 1904, no. 01496, 1–15 DOI: arXiv:1904.01469v1
Zenginoglu (2007) A conformal approach to numerical calculations of asymptotically flat spacetimes. Dissertation, Mathematisch-NaturwissenschaftlichenFacultat der Universitat Potsdam and Max Planck Institut fur Gravitationphysic Albert Einstein Institute and arXiv 0711.0873v2, 2007
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Tozzi, A., Peters, J.F. Points and lines inside human brains. Cogn Neurodyn 13, 417–428 (2019). https://doi.org/10.1007/s11571-019-09539-8
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DOI: https://doi.org/10.1007/s11571-019-09539-8