Probability, Information, and Physics

The following sections are included:

• Dedication
• About the Author
• About the Book
• Contents
• Abbreviations and Acronyms
• Introduction

Frequently, gamblers are superstitious, and in the past they were familiar with talismanic gestures and signs to avoid bad luck. Their cultural background was often characterized by belief in supernatural influence, extracorporeal perceptions, and strange ideas about fate (Hald, 2003).

Over the centuries, gamblers made attempts to improve their chances of winning. They endeavored to obtain more serious forecasts even if historians have not found evidence of conscientious mathematical studies (Hacking, 2006). Let us cast a glance at two books and Pascal’s letters which provide the earliest bibliographical testimonies of calculations in gambling.

The term ‘probability’ (P) took its factual origin in judicial contexts as proved by the very etymology of the word ‘probable’ deriving from the Latin ‘probabilis’ that stands for something provable or believable before a court (Harper, 2000). Still today lawyers and magistrates accurately examine the probable elements in a lawsuit. The more a cue or evidence is probable, the better it can be accepted, and the better it serves as proof of an accusation or a defense.

The preceding chapter points the finger against the ‘direct theorization’ of probability and I imagine that the reader feels this criticism to be somewhat surprising and perhaps eccentric. The direct theorization method is universally shared by probabilists and, at this point, bibliographical quotations should conscientiously support the conclusions of the previous pages.

Chapter 2 shows how probability is a multifold concept and raises this inevitable query:

• What exactly does probability evaluate?

It seems that Charles Peirce claimed: “In no other branch of mathematics is it so easy for experts to blunder as in probability theory.”

The first part of this book shows how mathematicians underrated the project of Pascal, consequently they developed theories illustrating partial aspects and the dreamed ‘Geometry of Chance’, namely, the exhaustive probability theory, has not yet come to light. This chapter outlines a possible road to follow in order to get closer to the target.

Since long I have perceived the sad state of the present field and searched for a new framework by the end of the past century (Rocchi, 1999; Rocchi, 2001). Now and then I have developed some parts, so this book collects the scattered sections and provides the full illustration of my theoretical proposal that presents a distinguished trait: it begins with a precise primary notion that is the event and deduces all the results from it.

Obviously, the formulation of the event is to be redone from scratch.

Probability applies to material and abstract problems which require specific approaches. So far, these approaches have seemed impossible to bring into harmony, even conflicting. This chapter has the purpose of accomplishing just this.

Definition 8.3 says that the expected output is sometimes missing but does not say what happens when the elementary event fails. The triad E^(e) = (i, r; e) ignores the alternative outcomes to e.

This chapter addresses this issue and will provide the full account of random events. Dual explanations will be used to make the physical and cognitive perspectives explicit.

The present theory is grounded on the idea that an event is something that happens (Property 7.1) in the physical world or the human mind (Section 8.1).

David Hilbert (1902) presented a list of the problems that were unsolved in the early 1900s. The sixth problem was to formulate rigorously those branches of physics in which mathematics is prevalent, and probability theory ranked first because it lacked a coherent body of proofs.

This chapter will prove formulas (2.1), (2.2), and (2.3) which bettors and gamblers contrived, on the basis of properties 7.1, 7.2, and 9.1, and the subsequent achievements. This book follows the phenomenological logic, it analyzes the modes of being of events and how they unfold. This logic ignores the usual techniques of demonstration which focus on the outcomes and formalize them as subsets or propositions.

Testing is a well-established form of control which is typical of the scientific method and consists of a series of experiments done under exact conditions that also rely on feedback procedures (Figure 12.1).

A theory is not necessarily right and Popper (1963) emphasizes the importance of controlling any scientific claim. A truly scientific theory must offer the possibility of being falsified on the logical plane and the experimental plane alike. Predictions must be in agreement with observations and for Popper an empirical outcome opposed to the forecasts is sufficient to confute the entire theoretical construction. He concludes that the essential difference between science and pseudoscience is that the second seeks confirmations and the first seeks experimental falsifications.

The need for testing is central for the frequentists, while other schools of thought overlook this need. The present theory openly declares following the phenomenological approach (Section 6.2), therefore empirical data must corroborate or disprove the theoretical parameter P. The definitions and theorems illustrated in the previous pages must find a precise correspondence with facts in accordance with the scientific method.

The theorems of the previous chapter look to be of little significance or even trivial from the mathematical stance. The reader might be induced to underestimate their weight which instead provides precise answers to relevant and long-standing issues. Let us go through four conclusions derived from the theorems just mentioned.

Thinkers argue about opposed and sometime irreconcilable aspects of haphazardness that can emerge even in a single problem. The following pages are devoted to three distinct arguments and demonstrate how heterogeneous features cohabit without conflicts.

Conditioning is a noteworthy topic and this chapter reviews some significant questions in conformity with the phenomenological logic typical of this framework.

Soft disciplines – like sociology, managerial sciences, and political science – usually acquire qualitative data. When they use mathematical models, the experts tend to filter out the numerical results, they lean toward reinterpreting the numbers and often arrive at some compromise. The ambidexterity metaphor, recently emphasized as the ability to pursue opposite objectives (Junni et al., 2013), effectively illustrates the elastic thought of managers, sociologists, politicians, humanists, etc. who weigh the numbers using different scales. Naturally, the controversial meanings of probability come to be mitigated for them, and a certain relativism overcomes the frequentism/subjectivism dilemma…

The current chapter is arranged in four parts:

• Unit I illustrates the contents and aims of this study.
• Unit II applies the theorems of the structural theory to quantum mechanics.
• Unit III deals with the quantum experiments that corroborate the theoretical forecasts.
• Unit IV provides the conclusions.

Since my early studies on probability, I was inclined to believe that researchers’ quarrels stemmed substantially from narrow intellectual horizons.

The following sections are included:

• Appendices
• Index