# Does the Differential Structure of Space-Time Follow from Physical Principles?

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Why Is Mathematics Considered an Invention?

#### 2.1. What Cantor Did

**c**to that of $\mathbb{R}$. (Note: Fraenkel [3] used ℵ for the cardinality of $\mathbb{R}$, reserving $\mathbf{c}$ for the generic cardinal. Hausdorff in his classic text [10] also used ℵ for the cardinality of $\mathbb{R}$.) The cardinality of a finite set is, of course, the number of elements in the set.

#### 2.2. Russell’s Paradox and Its Resolution

#### 2.3. Physics and the Power Set Construction

#### 2.4. The Mathematics Needed by Physics

## 3. From the Discrete to the Continuous

**Definition**

**1.**

**Example**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Example**

**2.**

**Example**

**3.**

## 4. Putting the Pieces Together

#### 4.1. Is the Continuum a Discovery or an Invention?

#### 4.2. Is the Differentiable Structure of ${\mathbb{R}}^{N}$ a Discovery or an Invention?

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Wigner, E.P. The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math.
**1960**, 13, 1–14. [Google Scholar] [CrossRef] - Cantor, G. Über unendliche, lineare Punktmannigfaltigkeiten, Part V. Math. Annalen
**1883**, 21, 545–596. [Google Scholar] [CrossRef] - Fraenkel, A.A. Abstract Set Theory; North-Holland: Amsterdam, The Netherlands, 1953. [Google Scholar]
- Hilbert, D. Über das Unendliche. Math. Annalen
**1926**, 95, 161–190. Available online: https://zbmath.org/?format=complete&q=an:51.0044.02 (accessed on 14 January 2024). [CrossRef] - Alexandrov, A.D. Filosofskoe soderzhanie i znachenie teorii otnositel’nosti. Voprosy Filosofii
**1959**, 1, 67–84. [Google Scholar] - Zeeman, E.C. Causality implies the Lorentz group. J. Math. Phys.
**1964**, 5, 490–493. [Google Scholar] [CrossRef] - Donaldson, S.K. An application of gauge theory in 4-dimensional topology. J. Differ. Geom.
**1983**, 18, 279–313. [Google Scholar] [CrossRef] - Gompf, R. Three exotic ℝ
^{4}s and other anomalies. J. Differ. Geom.**1983**, 18, 317–328. [Google Scholar] [CrossRef] - Borchers, H.-J.; Sen, R.N. Mathematical Implications of Einstein-Weyl Causality; Lecture Notes in Physics, No. 709; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 2006. [Google Scholar]
- Hausdorff, F. Set Theory; Chelsea Publishing Co.: New York, NY, USA, 1957. [Google Scholar]
- Wikipedia. Russell’s Paradox. 2023. Available online: https://en.wikipedia.org/wiki/Russell’s_paradox (accessed on 14 January 2024).
- Fraenkel, A.A.; Bar-Hillel, Y.; Levy, A. Foundations of Set Theory, 2nd ed.; Elsevier: Amsterdam, The Netherlands, 1973. [Google Scholar]
- Weil, A. Sur Les Espaces a Structure Uniforme et sur la Topologie Générale; Hermann: Paris, France, 1937. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sen, R.N.
Does the Differential Structure of Space-Time Follow from Physical Principles? *Entropy* **2024**, *26*, 179.
https://doi.org/10.3390/e26030179

**AMA Style**

Sen RN.
Does the Differential Structure of Space-Time Follow from Physical Principles? *Entropy*. 2024; 26(3):179.
https://doi.org/10.3390/e26030179

**Chicago/Turabian Style**

Sen, Rathindra Nath.
2024. "Does the Differential Structure of Space-Time Follow from Physical Principles?" *Entropy* 26, no. 3: 179.
https://doi.org/10.3390/e26030179