Abstract This article gives an account of a theoretical research subdivided into three parts. The first stage recalls the theorem of large numbers and the theorem a single number which demonstrate how the frequentist and the subjective models of probability can be used according to distinct assumptions. Based on these achievements, the second stage describes the temporal evolution of the random outcome (this article distinguishes the outcome from the random event or phenomenon that produces it). It is proved that the outcome of a single unpredictable event switches from the indeterministic to the deterministic state when the event winds up. The third stage means to show how the theoretical results apply to physical situations.
Abstract This article gives an account of a theoretical research subdivided into three parts. The first stage recalls the theorem of large numbers and the theorem a single number which demonstrate how the frequentist and the subjective models of probability can be used according to distinct assumptions. Based on these achievements, the second stage describes the temporal evolution of the random outcome (this article distinguishes the outcome from the random event or phenomenon that produces it). It is proved that the outcome of a single unpredictable event switches from the indeterministic to the deterministic state when the event winds up. The third stage means to show how the theoretical results apply to physical situations.
This article focuses on three themes concerning determinism and indeterminism. The first is observational equivalence between deterministic and indeterministic models. The article discusses several results about observational equivalence and presents an argument on how to choose between deterministic and indeterministic models involving indirect evidence. The second is whether Newtonian physics is indeterministic. The article argues that the answer depends on what one takes Newtonian mechanics to be and highlights how contemporary debates on this issue differ from those in the nineteenth century. The third theme is how the method of arbitrary functions can be used to make sense of deterministic probabilities. The article discusses various ways of interpreting the initial probability distributions and argues that they are best understood as physical, biological, and other quantities characterizing the particular situation. The fact that the method of arbitrary functions deserves more attention than it has received so far is also emphasized.
All our work in classical probability theory is to compute probabilities. The original idea in this research work is to add new dimensions to our random experiment, which will make the work deterministic. In fact, probability theory is a nondeterministic theory by nature; that means that the outcome of the events is due to chance and luck. By adding new imaginary dimensions to the event in the real set of probabilities R, we make the work deterministic and hence a random experiment will have a certain outcome in the complex set of probabilities and total universe G = C. It is of great importance that the stochastic system, like in the real-world problems, becomes totally predictable since we will be totally knowledgeable to foretell the outcome of chaotic and random events that occur in nature like for example in statistical mechanics or in all stochastic processes. Therefore, the work that should be done is to add to the real set of probabilities R, the contributions of M which is the imaginary set of probabilities which will make the event in G = C = R + M deterministic. If this is found to be fruitful, then a new theory in statistical sciences and in science in general is elaborated and this is to understand absolutely deterministically those phenomena that used to be random phenomena in R. This paradigm was initiated and developed in my previous 25 publications and research works. Moreover, this model will be related to my Theory of Metarelativity which takes into account faster-than-light matter and energy. This is what I called ‘The Metarelativistic Complex Probability Paradigm (MCPP)’ which will be elaborated in the present book.
On a very intuitive way of thinking, if it is already determined that some event will happen, then there is no non-trivial chance of it failing to happen, and if it is already determined that some event will not happen, then there is no non-trivial chance of it happening. A different sort of worry that arises for an actual frequentist analysis of deterministic chance is that such an analysis seems unsuited to theories like Boltzmannian statistical mechanics or Bohmian mechanics insofar as those theories are supposed to describe the evolution of the Universe as a whole. It should be clear, then, that there are options available for understanding deterministic chance that are in keeping with several of the leading metaphysical analyses of chance. It is widely accepted that whatever chances are, they are to be contrasted with individual credences . According to classical statistical mechanics, the fundamental laws are Newtonian and thus deterministic.
The outcomes of quantum measurements are generally considered to be random, but despite the fact that this randomness is an important element in quantum information theory, its nature is not well understood. In this thesis, we study several issues relating to the origin and certification of quantum randomness and unpredictability. One of the key results in forming our understanding of quantum mechanics as an intrinsically indeterministic theory is the Kochen-Specker theorem, which shows the impossibility to consistently assign simultaneous noncontextual definite values to all quantum mechanical observables prior to measurement. However, the theorem, under the assumption that any definite values must be noncontextual, only strictly shows that some observables must be value indefinite. We strengthen this result, proving a stronger variant of the Kochen-Specker theorem showing that, under the same assumption, if a system is prepared in an arbitrary state j i, then every observable A is value indefinite unless j i is an eigenstate of A. The indeterministic nature of quantum measurements does little to explain how the quality of quantum randomness differs from classical randomness. We show that, subject to certain physical assumptions, a sequence of bits generated by the measurement of value indefinite observables is guaranteed, in the infinite limit, to be strongly incomputable. We further discuss how this can be used to build a quantum random number generator certified by value indefiniteness. Next, we study the notion of unpredictability, which is central to the concept of (quantum) randomness. In doing so, we propose a formal model of prediction that can be used to asses the predictability of arbitrary physical experiments. We investigate how the quantum features of value indefiniteness and complementarity can be used to certify different levels of unpredictability, and show that the outcome of a single measurement of a value indefinite quantum observable is formally unpredictable. Finally, we study the relation between this notion of unpredictability and the computability-theoretic certification of quantum randomness.
Abstract The concept of objective probability — chance — has three distinctive features: chances are measured by statistics, but only with high chance, increasing with the number of trials involved; when the chances of the outcomes of a chance process are known, even in the context of a single trial, then one's subjective probabilities — credences — in those outcomes should be set equal to those chances; and finally, chance processes involve uncertainty as to the outcomes of those processes. The second of these features is established for ratios in the squared norms of branches by the Born-rule theorem, proved in Chapter 8. This chapter shows that the first and third hold for those same quantities. All three characteristics of chance thus apply to branching in the Everett interpretation. In consequence, branching processes should be identified with chance processes, and ratios in the squared norms with chances.