# Symmetry, Invariance and Ontology in Physics and Statistics

## Abstract

**:**

Objectivity means invariance with respect to the group of automorphisms.Hermann Weyl. Symmetry, (1989, p.132).

Let us turn now to the relation of symmetry or invariance principles to the laws of nature... It is good to emphasize at this point the fact that the laws of nature, that is, the correlations between events, are the entities to which the symmetry laws apply, not the events themselves.Eugene Wigner. Symmetry and Conservation Laws, (1967, p.16).

We cannot understand what is happening until we learn to think of probability distributions in terms of their demonstrable information content.Edwin Jaynes. Probability Theory: The Logic of Science, (2003, p.198).

## 1. Introduction

## 2. Minimum Action Principle and Physical Invariants

#### 2.1. Variational Principles

#### 2.2. Euler–Lagrange Equation

#### 2.3. Hamilton’s Principle and Classical Mechanics

## 3. Positivism vs. Objective Ontologies

#### 3.1. Positivism, Subjectivism and Solipsism

Positivism assumes that the only primary statements which are immediately evident are those describing direct sensual impressions. All other statements are indirect, theoretical constructions to describe in short terms the connections and relations of the primary experiences. Only these have the character of reality. The secondary statements do not correspond to anything real, and have nothing to do with an existing external world; they are conventions invented artificially to arrange and simplify “economically” the flood of sensual impressions.[The] strictly positivistic standpoint: The only reality is the sense impressions. All the rest are “constructs” of the mind. We are able to predict, with the help of the mathematical apparatus …what the experimentalist will observe under definite experimental conditions …But it is meaningless to ask what there is behind the phenomena, waves or particles or what else.We have before us a standpoint of extreme subjectivism, which may rightly be called “physical solipsism”. It is well-known that obstinately held solipsism cannot be refuted by logical argument. This much, however, can be said, that solipsism such as this does not solve but evades the problem. Logical coherence is a purely negative criterion; no system can be accepted without it, but no system is acceptable just because it is logically tenable. The only positive argument in support of this abstract type of ultra-subjectivism is [that] the belief in the existence of an external world is irrelevant and indeed detrimental to the progress of science.

#### 3.2. Born’s Criticism of Positivism

Many physicists have adopted [the positivist] standpoint. I dislike it thoroughly …The actual situation is very different. All great discoveries in experimental physics have been due to the intuition of men who made free use of models, which were for them not products of the imagination, but representatives of real things. How could an experimentalist work and communicate with his collaborators and his contemporaries without using models composed of particles, electrons, nucleons, photons, neutrinos, fields and waves, the concepts of which are condemned as irrelevant and futile?However, there is of course some reason for this extreme standpoint. We have learned that a certain caution is necessary in using these concepts …Modern theories demand a reformulation. This new formulation is slowly evolving, but has probably not reached a final expression.

#### 3.3. Ontology by Invariance and Autonomy

I think the idea of invariant is the clue to a rational concept of reality, not only in physics but in every aspect of the world. The theory of transformation groups and their invariants is a well established part of mathematics. Already in 1872 the great mathematician Felix Klein discussed in his famous ‘Erlanger Program’ the classification of geometry according to this point of view; the theory of relativity can be regarded as an extension of this program to the four-dimensional geometry of space-time. The question of reality in regard to gross matter has from this standpoint a clear and simple answer.Thus we apply analysis to construct what is permanent in the flux of phenomena, the invariants. Invariants are the concepts of which science speaks in the same way as ordinary language speaks of “things”, and which it provides with names as if they were ordinary things.The words denoting things are applied to permanent features of observation or observational invariants.The invariants of [a] theory have the right to be considered as representations of objects in the real world. The only difference between them and the objects of everyday life is that the latter are constructed by the unconscious mind, whereas the objects of science are constructed by conscious thinking.

## 4. Bayesian Statistics

#### 4.1. Fisher’s Metric and Jeffreys’ Prior

#### 4.2. Posterior Convergence

**Theorem**

**4.1.**

**Theorem**

**4.2.**

#### 4.3. Subjective Coherence vs. Objective Priors

We found that objectivity means invariance with respect to the group of automorphisms …This is what Felix Klein called a ‘geometry’ in the abstract sense. A geometry, Klein said, is defined by a group of transformations, and investigates everything that is invariant under the transformations of this given group.

## 5. Exchangeability and De Finetti Theorems

#### 5.1. De Finetti’s Subjectivism

Every time that probability is given by a mixture of hypotheses, with independence holding for each of them respectively, it is possible to characterize the inductive relevance that results on the basis of [the mixture] equation in terms of exchangeability …[This] shows how the concept of exchangeability is necessary in order to express in a meaningful way what is usually referred to as “independence with constant but unknown probability”. This expression is not correct because in fact (a) there is no independence; (b) the probability is not constant because the composition of the urn being unknown, after learning the outcome of the draws, the probability of the various compositions is subject to variation.

Some of you might have expected me, as a confirmed Bayesian, to restrict the meaning of the word ‘probability’ to subjective (personal) probability. That I have not done so is because I tend to believe that physical [objective] probability exists and is in any case a useful concept... The philosophical impact of de Finetti’s theorem is that it supports the view that solipsism cannot be logically disproved. Perhaps it is the mathematical theorem with most potential philosophical impact.

## 6. Simple Symmetric Probability Distributions

#### 6.1. Cauchy’s Functional Equations

#### 6.2. Radial Symmetry and the Gaussian Distribution

#### 6.3. Homogeneous Discrete Markov Processes

**Theorem**

**6.1.**

## 7. Minimum Information or MaxEnt Principle

#### 7.1. Maximum Entropy under Constraints

**Bregman’s Algorithm:**

- -
- The Binomial distribution is characterized as the distribution of maximum entropy for $i\in \{1\dots n\}$, given the expected value of the mean, relative to the combinatorial prior $C(n,i)$.
- -
- The Normal distribution is characterized as the distribution of maximum entropy on ${R}^{n}$, given the expected values of its first and second moments, i.e., mean vector and covariance matrix.
- -
- The Wishart distribution:$$f\left(S\phantom{\rule{0.166667em}{0ex}}\right|\phantom{\rule{0.166667em}{0ex}}\nu ,V)\equiv c(\nu ,V)exp\left(\frac{\nu -d-1}{2}log(det\left(S\right))-{\sum}_{i,j}{V}_{i,j}{S}_{i,j}\right)$$$$\phantom{\rule{0.166667em}{0ex}}E\left({S}_{i,j}\right)={V}_{i,j}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}E(log(det\left(S\right)))={\sum}_{k=1}^{d}{\Gamma}^{\prime}\left(\frac{\nu -k+1}{2}\right)\phantom{\rule{4pt}{0ex}}$$
- -
- The Dirichlet distribution$$f\left(x\phantom{\rule{0.166667em}{0ex}}\right|\phantom{\rule{0.166667em}{0ex}}\theta )=c\left(\theta \right)exp\left({\sum}_{k=1}^{m}({\theta}_{k}-1)log\left({x}_{k}\right)\right)$$

## 8. Cognitive Constructivism

#### 8.1. Objects as Eigen-Solutions

#### 8.2. Semantics by Construction and by Correspondence

## 9. Conclusions and Further Research

## Acknowledgements

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**Figure 1.**Variational problem, $q\left(x\right)$, $\eta \left(x\right)$, $q\left(x\right)+\u03f5\eta \left(x\right)$.

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Stern, J.M.
Symmetry, Invariance and Ontology in Physics and Statistics. *Symmetry* **2011**, *3*, 611-635.
https://doi.org/10.3390/sym3030611

**AMA Style**

Stern JM.
Symmetry, Invariance and Ontology in Physics and Statistics. *Symmetry*. 2011; 3(3):611-635.
https://doi.org/10.3390/sym3030611

**Chicago/Turabian Style**

Stern, Julio Michael.
2011. "Symmetry, Invariance and Ontology in Physics and Statistics" *Symmetry* 3, no. 3: 611-635.
https://doi.org/10.3390/sym3030611