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Operational Approach to the Topological Structure of the Physical Space

Abstract

Abstract definitions and explanations frequently come together and permeate almost all fields of knowledge. This does not exclude mathematics, even when these definitions hold clear links and close connections with our physical world. Here we propose a rather different perspective. Making operational physical assumptions, we show how it is possible to rigorously reconstruct some features of both geometry and topology. Broadly speaking, assuming this operational and more concrete philosophy we not only are capable of defining primitive concepts like points, straight lines, planes, angles and spaces, but we also go all the way up to more complex constructions. We show that within our method it is also possible to come up with what we call a compass-based topology as well as a normed and a metric space. We hope this operationalist point-of-view can assist other researchers and students to have a clearer understanding of these quite abstract concepts.

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Notes

  1. 1.

    This postulate is also called the parallel postulate because of Playfair’s reformulation of it, stating that, “given a line and a point outside it, it is possible to draw a line parallel to the first containing the initial point.

  2. 2.

    If your compass is such that it cannot differentiate a given marked point from another, by all means you should regard these two different points as being equal. Physically, if your measurement apparatus cannot differentiate between two objects, one should consider them as being the same.

  3. 3.

    To the reader who is not familiar with the concept of equivalence classes and relations, we recommend Halmos (1960), which is sufficiently formal. For a more intuitive interpretation, see Júnior et al. (2018).

  4. 4.

    We used a piece of dental floss as an in-extensible thread to represent the line segment, tied to two plastic weights.

  5. 5.

    It is important to notice that the notion of a compass can vary depending on the geometry of interest.

  6. 6.

    We find the same structure in relativity theory. Minkowski’s space-time \(\mathbb {M}\) has the structure of a four-dimensional affine space (O’Neill 1983).

  7. 7.

    Technically, the exigence of excluding reflections means asking that the mapping \(R_P\) be continuous and that it can be deformed continually to the identity operator. We will discuss the operational meaning of continuity in a subsequent work.

  8. 8.

    Rotations are described by theory of representations of the group SO(3). The details can be seen in Rocha et al. (2013).

  9. 9.

    As a matter of fact we should have considered not only \(\tau\) but an extension of it containing both the empty set and \({\mathcal {E}}_{SR}\) itself.

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Acknowledgements

This work is supported by Programa Institucional de Bolsas de Iniciação o Científica - XXXI BIC/UFJF-2018/2019, project number ID45249 as well as by grant number FQXi-RFP-IPW-1905 from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation. CD has been supported by a fellowship from the Grand Challenges Initiative at Chapman University. BFR would like to express his gratitude for the warm hospitality of the Institute for Quantum Studies and the Grand Challenges Initiative at Chapman University, where this work was concluded. CD thanks J. Ahern-Serond for helpful discussions.

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Correspondence to B. F. Rizzuti.

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Rizzuti, B.F., Gaio, L.M. & Duarte, C. Operational Approach to the Topological Structure of the Physical Space. Found Sci (2020). https://doi.org/10.1007/s10699-020-09650-8

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Keywords

  • Operationalism
  • Hausdorff spaces
  • Normed spaces
  • Operational constructions in mathematics