Resonance: From Probability To Epistemology And Back

From Probability to Epistemology and Back

Krzysztof Burdzy

• Éditeur: ICP
• Année: 2016
• ISBN:
  9781783269228
• Sujet: Mathematics
• Sous-sujet: History & Philosophy of Mathematics

Resonance examines some building blocks of epistemology as a prelude to the careful analysis of the foundations of probability. The concept of resonance is introduced to shed light on the philosophical problems of induction, consciousness, intelligence and free will. The same concept is later applied to provide support for a new philosophical theory of probability.

Although based on existing ideas and theories, the epistemological concept of resonance is investigated for the first time in this book. The best-known philosophical theories of probability, frequency and subjective, are shown to be unrealistic and dissociated from the two main branches of statistics: frequency statistics and Bayesian statistics.

Written in an accessible style, this book can be enjoyed by philosophers, statisticians and mathematicians, and also by anyone looking to expand their understanding of the disciplines of epistemology and probability.

Contents:

• • Introduction
• Philosophy of Probability:
  • Main Philosophies of Probability
  • Skepticism
  • The Frequency Philosophy of Probability
  • The Subjective Philosophy of Probability
  • The Logical Philosophy of Probability
  • Common Issues
• Epistemology:
  • Epistemology
  • Religion
  • Science
• Science of Probability:
  • The Science of Probability
  • Decision Making
  • Frequency Statistics
  • Bayesian Statistics
• Miscellanea:
  • On Ideologies
  • Paradoxes, Wagers and Rules
  • Teaching Probability
  • Mathematical Methods of Probability and Statistics

Readership: Philosophers, statisticians and mathematicians, and readers who are interested in the fields of epistemology and probability.

Chapter 1

Introduction

My earlier book on philosophy of probability, [Burdzy (2009)], was focused on finding the best scientific formulation of the foundations of probability. This book presents much of the same program but it also tries to determine the main sources of the success of probability. This analysis will take me well beyond the narrow topic of philosophy of probability, to the frightening depths of epistemology.

1.1 Knowledge

“There is no truth” — this claim, in different forms, was made by a number of philosophers. What is surprising to me is that philosophers as different as Karl Popper and Thomas Kuhn, arguably the best known 20th century philosophers of science, shied away from the clear declaration that science constantly brings us closer to the truth about the objectively existing universe. I cannot prove that the objective universe exists or that we can find the truth about it. Nobody can. I am a 100% skeptic. But skepticism is a dead end in philosophy. The interesting direction in philosophy is to describe how we arrive at statements that we consider true. Once we understand the process, or rather many different processes, we can join other people in pursuing the truth in one of the established ways or we can seek our own alternative way. Scientists arrive at the truth in their own way. The prevalence of religion proves that there is not even a slightest chance, in the present society, for a consensus on how to find the truth. Despite the obvious lack of consensus on the truth and ways of attaining it, I believe that there is an important and universal element of knowledge acquisition (I will call it “resonance”) that received too little attention from philosophers. I will argue that resonance is the missing link in the known philosophical theories of probability. If resonance proves to be a viable concept outside philosophy of probability, I will consider this a welcome bonus.

1.2 Probability

Two and two makes four. Imagine a mathematical theory which says that it makes no sense to talk about the result of addition of two and two. Imagine another mathematical theory which says that the result of addition of two and two is whatever you think it is. Would you consider any of these theories a reasonable foundation of science? Would you think that they are relevant to ordinary life?

If you toss a coin, the probability of heads is 1/2. According to the frequency philosophy of probability, it makes no sense to talk about the probability of heads on a single toss of a coin. According to the subjective philosophy of probability, the probability of heads is whatever you think it is. Would you consider any of these theories to be a reasonable foundation of science? Would you think that they are relevant to ordinary life?

The frequency philosophy of probability is usually considered to be the basis of the “frequency” statistics and the subjective philosophy of probability is often regarded as the basis of the “Bayesian” statistics (readers unfamiliar with these terms should consult Chapter 18). According to the frequency philosophy of probability, the concept of probability is limited to long runs of identical experiments or observations, and the probability of an event is the relative frequency of the event in the long sequence. The subjective philosophy claims that there is no objective probability and so probabilities are subjective views; they are rational and useful only if they are “consistent,” that is, if they satisfy the usual mathematical probability formulas.

Von Mises, who created the frequency philosophy, claimed that ([von Mises (1957), p. 11]),

We can say nothing about the probability of death of an individual [within a year] even if we know his condition of life and health in detail.

De Finetti, who proposed the subjective philosophy, asserted that ([de Finetti (1974), p. x]),

Probability does not exist.

The standard education in probability and statistics is a process of indoctrination in which students are taught, explicitly or implicitly, that individual events have probabilities, and some methods of computing probabilities are scientific and rational. An alien visiting our planet from a different galaxy would have never guessed from our textbooks on probability and statistics that the two main branches of statistics are related to the philosophical claims cited above. I believe that the two cited philosophical claims are incomprehensible to all statisticians except for a handful of aficionados of philosophy. I will try to explain their meaning and context in this book. I will also argue that the quoted claims are not mere footnotes but they constitute the essence of the two failed philosophical theories.

1.3 Summary of the Main Claims

1.3.1 Resonance

The acquisition of information and creating knowledge (this includes both facts and theories) can be divided into two steps. I will call the first step “resonance” for reasons explained later in the book. This process is very fast in most cases, subconscious and very reliable in a great variety of situations. My guess is that resonance is not based on logic in any reasonable sense of the word “logic.” Resonance is a crude but reasonably reliable filter of information arriving at our senses.

Resonance is fallible in many situations recognized as significant to individual lives, society and science. Logic, probability, induction and all other named and unnamed ingredients of science provide the second filter, much more refined and reliable than resonance. Traditionally, philosophy was focused on the second filter because we have almost no access to resonance via our consciousness. This situation created various misconceptions concerning the sources of reliable truth. One of these is a tendency to ignore resonance despite the fact that resonance is at least as important to science and general knowledge as logic. Another common misconception is that resonance (under the name of “subjectivity”) is unreliable or not needed. Some other myths go in the opposite direction and invest intuition, subjective opinions and mystical experiences in powers that these sources of opinion and information do not have.

Resonance is a necessary first filter because it is impossible to process all information available to us in a logical way in a timely manner. This should not be interpreted as a claim that resonance was created deliberately by humans. Quite the opposite, resonance is the result of the blind evolution process selecting the fittest individuals.

I will illustrate the role of resonance in the knowledge acquisition process by analyzing four classical philosophical problems: induction, consciousness, intelligence and free will.

I will argue that the classical problem of induction is ill-posed. Our “knowledge” of facts is based on the same ontological and epistemological assumptions as our predictions of “unknown” events. The reliability of induction is a law of nature or rather the confluence of several laws of nature, including evolution. These laws of nature are specific to our universe so no general logical justification of induction can exist.

Consciousness is (among other things) an ability to observe, memorize and analyze one’s own information processing. Resonance is inaccessible to consciousness so it may appear to be irrational. This is a misleading impression. A process that does not follow the classical logic and is inaccessible to conscious analysis does not have to be arbitrary, subjective or unreliable.

Intelligence is the highest form of resonance. It does not have roots in observations of repetitive phenomena. Its essence is the ability to select facts or highly probable theories that are relevant to the current interests of the individual or society from the practically infinite amount of information and potential explanations of observations.

The analysis of free will cannot profit from relating free will to the deterministic or stochastic nature of our universe. Free will is an impression of one individual about another one due to the inability of a highly complex mind to create a model of another equally complex mind that would generate reliable predictions.

1.3.2 Critique of frequency and subjective philosophies of probability

In a nutshell, each of the two most popular philosophies of probability, frequency and subjective, failed in two distinct ways. First, both theories are very weak. The frequency philosophy of von Mises, developed in the first half of the 20th century, provides analysis of long sequences of independent and identical events only. The subjective philosophy of de Finetti (developed in parallel to that of von Mises, more or less) offers an argument in support of the mathematical rules of probability, with no hint on how the rules can be matched with the real world. Second, each of the two philosophical theories failed in a “technical” sense. The frequency theory is based on “collectives,” a notion that was completely abandoned by the scientific community long time ago. The subjective theory is based on an argument which fails to give any justification for the use of the Bayes theorem. Even one of the two types failures would be sufficient to disqualify these theories. The double failure makes each of the theories an embarrassment for the scientific community.

The philosophical contents of the theories of von Mises and de Finetti may be split into (i) positive philosophical ideas, (ii) negative philosophical ideas, and (iii) innovative technical ideas. There is nothing new about the positive philosophical ideas in either theory. The negative philosophical ideas are pure fantasy. The technical ideas proved to be completely useless. I will now discuss these elements of the two theories in more detail.

1.3.2.1 Positive philosophical ideas

The central idea in the frequentist view of the world is that probability and (relative) frequency can be identified, at least approximately, and at least in propitious circumstances. It is inevitable that, at least at the subconscious level, von Mises is credited with the discovery of the close relationship between probability and frequency. Nothing can be further from the truth. At the empirical level, one could claim that a relationship between probability and frequency is known even to animals, and was certainly known to ancient people. The mythical beginning of the modern probability theory was an exchange of ideas between Chevalier de Mere, a gambler, Pierre de Fermat and Blaise Pascal, two mathematicians, in 1654. It is clear from the context that Chevalier de Mere identified probabilities with frequencies and the two mathematicians developed algebraic formulas. On the theoretical side, the approximate equality of relative frequency and probability of an event is known as the Law of Large Numbers. An early version of this mathematical theorem was proved by Jacob Bernoulli in 1713.

The main philosophical and scientific ideas associated with subjectivism and Bayesian statistics are, obviously, the Bayes theorem and the claim that probability is a personal opinion. Once again, one can subconsciously give credit to de Finetti for discovering the Bayes theorem or for inventing the idea that probability is a subjective opinion. The Bayes theorem was proved by Thomas Bayes, of course, and published in 1763 (although it appears that the theorem was known before Bayes). De Finetti was not the first person to suggest that the Bayes theorem should be used in science and other avenues of life, such as the justice system. In fact, this approach was well known and quite popular in the 19th century.

Between Newton and Einstein, the unquestioned scientific view of the world was that of a clockwise mechanism. There was nothing random about the physical processes. Einstein himself was reluctant to accept the fact that quantum mechanics was inseparable from randomness. Hence, before the 20th century, probability was necessarily an expression of limited human knowledge of reality. Many details of de Finetti’s theory of subjective probability were definitely new but the general idea that probability was a personal opinion was anything but new.

1.3.2.2 Negative philosophical ideas

Both von Mises and de Finetti took, as a starting point, a very reasonable observation that not all everyday uses of the concept of probability deserve to be elevated to the status of science. A good example to have in mind is the concept of “work” which is very useful in everyday life but had to be considerably modified to be equally useful in physics.

One of the greatest challenges for a philosopher of probability is the question of how to measure the probability of a given event. Common sense suggests observing the frequency of the event in a sequence of similar experiments, or under similar circumstances. It is disappointing that quite often there is no obvious choice of “similar” observations, for example, if we want to find the probability that a given presidential candidate will win the elections. Even when we can easily generate a sequence of identical experiments, all we can get is the relative frequency which characterizes the whole sequence, not any particular event. The observed frequency is not necessarily equal to the true probability (if there is such a thing), according to the mathematical theory of probability. The observed frequency is highly probable to be close to the true probability, but applying this argument seems to be circular — we are using the concept of probability (“highly probable”) before we determined that the concept is meaningful.

Von Mises and de Finetti considered philosophical difficulties posed by the measurement of probability of an event and concluded that a single event does not have a probability. This intellectual decision was similar to that of a philosopher coming to the conclusion that God does not exist because the concept of God is mired in logical paradoxes. The atheist philosophical option has a number of intellectual advantages — one does not have to think about whether God can make a stone so heavy that He cannot lift it himself. More significantly, one does not have to resolve the apparent contradiction between God’s omnipotence and omnibenevolence on one hand, and all the evil in the world on the other. Likewise, von Mises and de Finetti do not have to explain how one can measure the probability of a single event.

While the philosophical position of von Mises and de Finetti is very convenient, it also makes their philosophies totally alienated from science and other branches of life. In practical life, all people have to assign probabilities to single events and they have to follow rules worked out by probabilists, statisticians and other scientists. Declaring that a single event does not have a probability has as much practical significance as declaring that complex numbers do not exist.

The claim that “God does not exist” is a reasonable philosophical option. The claim that “religion does not exist” is nonsensical. The greatest philosophical challenge in the area of probability is a probabilistic counterpart of the question “What does a particular religion say?” This challenge is deceptively simple — philosophers found it very hard to pinpoint what the basic rules for assigning probabilities are. This is exemplified by some outright silly proposals by the “logical” school of probability. While other philosophers tried to extend the list of basic rules of probability, von Mises and de Finetti removed some items from the list, most notably symmetry.

The fundamental philosophical claim of von Mises and de Finetti, that events do not have probabilities, was like a straitjacket that tied their hands and forced them to develop very distinct but equally bizarre theories. Their fundamental claim cannot be softened or circumvented. For a philosopher, it is impossible to be an atheist and believe in God just a little bit. Creating a philosophical theory of God that exists just a little bit is not any easier than creating a theory of God that fully exists. Similarly, creating a philosophy of probability which includes some events with a somewhat objective probability is as hard as inventing a philosophy claiming that all events have fully objective probability.

The two philosophies can be considered normative. Then their failure manifests itself in the fact that they are totally ignored. If the two theories are regarded as descriptive then they are complete failures because the two philosophers proved unable to make simple observations.

1.3.2.3 Innovative technical ideas

Von Mises came to the conclusion that the only scientific application of probability was in the context of long sequences of identical experiments or observations. Nowadays, such sequences are modeled mathematically by “i.i.d.” random variables (i.i.d. is an acronym for “independent identically distributed”). Since individual events do not have pr...

Table des matières

• Cover page
• Title page
• Copyright page
• Dedication
• Preface
• About the Author
• Contents
• 1. Introduction
• Philosophy of Probability
• Epistemology
• Science of Probability
• Miscellanea
• Bibliography
• Index