# Pragmatic Hypotheses in the Evolution of Science

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## Abstract

**:**

## 1. Introduction

## 2. Gallais’ Hexagonal Spirals and the Evolution of Science

- ${A}_{1}$- Extant thesis: This vertex represents a standing paradigm, an accepted theory using well-known formalisms and familiar concepts, relying on accredited experimental means and methods, etc. In fact, the concepts of a current paradigm may become so familiar and look so natural that they become part of a reified ontology. That is, there is a perceived correspondence between concepts of the theory and “dinge-an-sich” (things-in-themselves) as seen in nature [29,30].
- ${U}_{1}$- Analysis: This vertex represents the moment when some hypotheses of the standing theory are put in question. At this moment, possible alternatives to the standing hypotheses may still be only vaguely defined.
- ${E}_{1}$- Antithesis: This vertex represents the moment when some laws of the standing theory have to be rejected. Such a rejection of old laws may put in question the entire world-view of the current paradigm, opening the way for revolutionary ideas, as described in the next vertex.
- ${O}_{2}$- Apothesis/ Prosthesis: This vertex is the locus of revolutionary freedom. Alternative models are considered, and specific (precise) forms investigated. There is intellectual freedom to set aside and dispose of (apothesis) old preconceptions, prejudices and stereotypes, and also to explore and investigate new paths, to put together (prosthesis) and try out new concepts and ideas.
- ${Y}_{2}$- Synthesis: It is at this vertex that new laws are formulated; this is the point of Eureka moment(s). A selection of old and and new concepts seem to click into place, fitting together in the form of new laws, laws that are able to explain new phenomena and incorporate objects of an expanded reality.
- ${I}_{2}$- Enthesis: At this vertex new laws, concepts and methods must enter and be integrated into a consistent and coherent system. At this stage many tasks are performed in order to combine novel and traditional pieces or to accommodate original and conventional components into an well-integrated framework. Finally, new experimental means and methods are developed and perfected, allowing the new laws to be corroborated.
- ${A}_{2}$- New Thesis: At this vertex, the new theory is accepted as the standard paradigm that succeeds the preceding one (${A}_{1}$). Acceptance occurs after careful determination of fundamental constants and calibration factors (including their known precision), metrological and instrumentational error bounds, etc. At later stages of maturity, equivalent theoretical frameworks may be developed using alternative formalisms and ontologies. For example, analytical mechanics offers variational alternatives that are (almost) equivalent to the classical formulation of Newtonian mechanics [31]. Usually, these alternative worldviews reinforce the trust and confidence on the underlying laws. Nevertheless, the existence of such alternative perspectives may also foster exploratory efforts and investigative works in the next cycle in evolution.

## 3. Pragmatic Hypotheses

**Definition**

**1.**

**Example**

**1**

**.**Let $\mathbf{Z}=({Z}_{1}\dots ,{Z}_{d})\sim N(\theta ,{\Sigma}_{0})$ be a random vector with a multivariate Gaussian distribution:

**Definition**

**2**

**.**Let $\widehat{\mathbf{Z}}:\Theta \to \mathcal{Z}$ be such that $\widehat{\mathbf{Z}}\left({\theta}_{0}\right)$ is the best prediction for $\mathbf{Z}$ given that $\theta ={\theta}_{0}$. For example, one can take

**Example**

**2**

**.**Let $\mathcal{Z}={\mathbb{R}}^{d}$, ${\mu}_{\mathbf{Z},\theta}=\mathbf{E}\left[\mathbf{Z}\right|\theta ]$, ${\Sigma}_{\mathbf{Z},\theta}=\mathbb{V}\left[\mathbf{Z}\right|\theta ]$ and $\mathbf{S}$ be a positive definite matrix. Define the quadratic form induced by $\mathbf{S}$ to be ${\parallel \mathbf{z}\parallel}_{\mathbf{S}}^{2}={\mathbf{z}}^{T}\mathbf{S}\mathbf{z}$ and

**Example**

**3**

**.**Consider Example 1 and let ${\delta}_{\mathbf{Z},{\theta}_{0}}\left(\mathbf{z}\right)$ be as in Example 2. It follows from Equation (4) that when $\mathbf{S}$ is the identity matrix,

**Definition**

**3**

**.**Let ${\widehat{\theta}}_{{\theta}_{0},{\theta}^{*}}:\mathcal{Z}\to \Theta $ be such that

**Example**

**4**

**.**Consider Examples 1 and 3, when $d=1$, ${\Sigma}_{0}={\sigma}_{0}^{2}$, obtain

#### 3.1. Singleton Hypotheses

**Definition**

**4**

**.**Let ${H}_{0}:\theta ={\theta}_{0}$, ${d}_{\mathbf{Z}}$ be a predictive dissimilarity function and $\u03f5>0$. The pragmatic hypothesis for ${H}_{0}$, $Pg(\left\{{\theta}_{0}\right\},{d}_{\mathbf{Z}},\u03f5)$, is

**Example**

**5**

**.**Consider Examples 1 and 3 when $d=1$, ${\Sigma}_{0}={\sigma}_{0}^{2}$ and $g\left(x\right)=\sqrt{x}$. It follows from Equations (1), (7) and (8) that

#### 3.2. Composite Hypotheses

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7**

**.**The agnostic test based on the region estimator R for testing ${H}_{0}$, ${\varphi}_{{H}_{0}}^{R}$, such that

**Definition**

**8.**

- If ${\varphi}_{Pg\left(B\right)}\left(x\right)=0$ for some $B\subseteq A$, then ${\varphi}_{Pg\left(A\right)}\left(x\right)=0$.
- If ${\varphi}_{Pg\left({A}_{i}\right)}\left(x\right)=1$ for every $i\in I$ and $A\subseteq {\cup}_{i\in I}{A}_{i}$, then ${\varphi}_{Pg\left(A\right)}\left(x\right)=1$.

**Theorem**

**1.**

**Lemma**

**1.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**2.**

**Corollary**

**1.**

**Theorem**

**3.**

- (i)
- ${\left(Pg({\Theta}_{0},K{L}_{m},\u03f5)\right)}_{m\ge 1}$ and ${\left(Pg({\Theta}_{0},C{D}_{m},\u03f5)\right)}_{m\ge 1}$ are non-increasing sequences of sets
- (ii)
- $Pg({\Theta}_{0},K{L}_{m},\u03f5)\stackrel{m\to \infty}{\to}{\Theta}_{0}$ and $Pg({\Theta}_{0},C{D}_{m},\u03f5)\stackrel{m\to \infty}{\to}{\Theta}_{0}$.

## 4. Applications

**Example**

**6**

**.**Consider the setting from Example 5, but with ${\sigma}^{2}$ unknown and $0<{\sigma}^{2}\le {M}^{2}$. In this case, the parameter is $\theta =(\mu ,{\sigma}^{2})$. Consider the composite hypothesis ${H}_{0}:\left\{{\mu}_{0}\right\}\times (0,{M}^{2}]$, which is often written as ${H}_{0}:\mu ={\mu}_{0}$. In this case, let ${\theta}_{0}=({\mu}_{0},{\sigma}_{0}^{2})$ and ${\Theta}_{0}=\left\{{\mu}_{0}\right\}\times (0,{M}^{2}]$. Proceeding as in Example 5, it follows that

**Example**

**7**

**.**Let $\mathbf{Z}\sim Multinomial(m,\mathit{\theta})$, where $\mathit{\theta}=({\theta}_{1},{\theta}_{2},{\theta}_{3})$, ${\theta}_{i}\ge 0$, and ${\sum}_{i=1}^{3}{\theta}_{i}=1$. The Hardy–Weinberg (HW) hypothesis [43], ${H}_{0}$, which is depicted in the red curve in Figure 5 satisfies

**Example**

**8**

**.**Assume that $\mathbf{Z}=(X,Y)\sim N(({\mu}_{1},{\mu}_{2}),{\sigma}^{2}{\mathbb{I}}_{2})$, with σ known. We derive the pragmatic hypothesis for ${H}_{0}:{\mu}_{1}={\mu}_{2}$, that is, for $\{({\mu}_{1},{\mu}_{2})\in {\mathbb{R}}^{2}:{\mu}_{1}={\mu}_{2}\}$. Such a test might be used in a bioequivalence study, where X and Y are the concentrations of an active ingredient in a generic (test) drug medication and in the brand name (reference) medication [45], respectively. As ${H}_{0}$ is composite, it helps to derive the pragmatic hypothesis of its constituents.

## 5. Final Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**3.**

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**Figure 3.**$\varphi \left(x\right)$ is an agnostic test based on the region estimator $R\left(x\right)$ for testing ${H}_{0}$.

**Figure 4.**Pragmatic hypotheses in Example 6 for ${H}_{0}:\mu =0$ with KL (upper), CD (lower), $\u03f5=0.1$, and ${M}^{2}=2$. ${H}_{0}$ is represented by a red line in all figures.

**Figure 5.**Pragmatic hypotheses obtained for the HW equilibrium, depicted in red, using $m=20$, $\u03f5=0.1$ for BP and CD and $\u03f5=0.01$ for KL. The blue regions indicate the pragmatic hypothesis for HW and $p=\frac{1}{3}$ (top) and for HW (bottom). The lower, middle, and right panels were obtained, respectively, with BP, KL, and CD. The green regions in the right panels represents 80% HPD regions for the genotype distribution of each of the eight groups collected by Brentani et al. [44] and two simulated datasets.

Vertex | Orbital Astronomy | Chemical Affinity |
---|---|---|

${I}_{1}$- Enthesis/ ${A}_{1}$- Thesis | Ptolemaic/Copernican cycles and epicycles | Geoffroy affinity table and highest rank substitution |

${U}_{1}$- Analysis | Circular or oval orbits? | Ordinal or numeric affinity? |

${E}_{1}$- Antithesis | Non-circular orbits | Non-ordinal affinity |

${O}_{2}$- Apothesis /Prosthesis | Elliptic planetary orbits, focal centering of sun | Integer affinity values, for arithmetic recombination |

${Y}_{2}$- Synthesis | Kepler laws! | Morveau rules and tables! |

${I}_{2}$- Enthesis ${A}_{2}$- Thesis | Vortex physics theories, Keplerian astronomy | Affinity + stoichiometry substitution reactions |

${U}_{2}$- Analysis | Tangential or radial forces? | Total or partial reaction? |

${E}_{2}$- Antithesis | Non-tangential forces | Non-total substitutions |

${O}_{3}$- Apothesis/Prosthesis | Radial attraction forces, inverse square of distance | Reversible reactions, equilibrium conditions |

${Y}_{3}$- Synthesis | Newton laws! | Mass-Action kinetics! |

${I}_{3}$- Enthesis/ ${A}_{3}$- Thesis | Newtonian mechanics & variational equivalents | Thermodynamic theories for reaction networks |

AA | AD | DD | Decision | |
---|---|---|---|---|

1 | 4 | 18 | 94 | Agnostic |

2 | 6 | 53 | 74 | Accept |

3 | 57 | 118 | 100 | Agnostic |

4 | 58 | 97 | 48 | Agnostic |

5 | 120 | 361 | 194 | Agnostic |

6 | 206 | 309 | 142 | Accept |

7 | 110 | 148 | 44 | Accept |

8 | 34 | 22 | 12 | Agnostic |

9 | 198 | 282 | 520 | Reject |

10 | 641 | 314 | 45 | Accept |

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Esteves, L.G.; Izbicki, R.; Stern, J.M.; Stern, R.B. Pragmatic Hypotheses in the Evolution of Science. *Entropy* **2019**, *21*, 883.
https://doi.org/10.3390/e21090883

**AMA Style**

Esteves LG, Izbicki R, Stern JM, Stern RB. Pragmatic Hypotheses in the Evolution of Science. *Entropy*. 2019; 21(9):883.
https://doi.org/10.3390/e21090883

**Chicago/Turabian Style**

Esteves, Luis Gustavo, Rafael Izbicki, Julio Michael Stern, and Rafael Bassi Stern. 2019. "Pragmatic Hypotheses in the Evolution of Science" *Entropy* 21, no. 9: 883.
https://doi.org/10.3390/e21090883