Classical Probability

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There are two parts. The notion of an "event" is reinterpreted from being an epistemological state of indefiniteness to being an objective state of indefiniteness. And the mathematical framework of finite probability theory is recast as the quantum probability calculus for QM/sets. The point is not to (...) clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features in a classical setting. (shrink)

Although probabilistic statements are ubiquitous, probability is still poorly understood. This shows itself, for example, in the mere stipulation of policies like expected utility maximisation and in disagreements about the correct interpretation of probability. In this work, we provide an account of probabilistic predictions that explains when, how, and why they can be useful for decision-making. We demonstrate that a calibration criterion on finite sets of predictions allows one to anticipate the distribution of utilities that a given policy will yield. (...) Based on this, we specify assumptions under which expected utility maximisation is a sensible decision criterion. We also introduce the notion of prediction methods, highlighting the dependence on schemes of abstraction, and argue that all probabilities are outputs of such prediction methods. This helps explain how the calibration criterion can be satisfied and shows that also supposedly objective probabilities are model-dependent. We compare our account of probability with conventional interpretations and show that it recovers key intuitions behind the latter. We, thus, provide a general account of what probabilities are and how they can enable successful decision-making under uncertainty. (shrink)

We prove that quantum mechanics is the unique theory compatible with normalization- induced curvature on complex Hilbert space. Two rigidity theorems establish: (1) Lin- earity: continuous, reversible, scale-invariant flows necessarily have linear generators, and (2) Born rule: the assignment P (ϕ|ψ) = |⟨ϕ, ψ⟩|2 is the unique probability mea- sure compatible with cone geometry, unitary invariance, and orthogonal additivity. These are not interpretations but uniqueness results—no other mathematical struc- tures are possible given the geometric assumptions. We derive the Schr¨odinger equa- (...) tion, interference, Berry phase, and uncertainty relations as consequences of the master equation CurvC(X, Y ) = [X, Y ] − ⟨∇N,[X,Y ]⟩ ⟨∇N,x⟩ x applied to the quantum cone. The re- construction inverts standard quantum foundations: rather than postulating Hilbert space structure, we show it emerges from reversible dynamics on normalized states. (shrink)

According to contemporary consensus, physical probabilities may be “non-uniform”: they need not correspond to a uniform measure over the space of physically possible worlds. Against consensus, I argue that only uniform probabilities connect robustly to long-run frequencies. Suppose, for example, that a uniform measure assigns a probability of x to a coin landing Heads on any given toss (independently of the outcomes on other tosses). Then, in almost all physically possible worlds containing many tosses, the frequency of Heads outcomes approximates (...) x. Thus, if the physical probability of Heads is not x, the frequency deviates from this probability in almost all physically possible worlds. But this would undermine the robust connection between long-run frequencies and physical probabilities. Conclusion: the uniform measure assigns the correct physical probability to Heads outcomes! This paper makes the argument rigorous and general, and evaluates its significance. (shrink)

Probabilidad Imposible Vol 2 reúne los fundamentos de una investigación desarrollada entre 2001 y 2018, cuyo propósito es replantear las bases del conocimiento racional y su relación con la probabilidad empírica y teórica. La obra introduce conceptos como el Impacto del Defecto y el Segundo Método, proponiendo un marco alternativo frente a la epistemología de Karl Popper y orientado hacia la búsqueda del conocimiento puro. Este proyecto, originado en el cruce entre matemáticas, inteligencia artificial y psicología, constituye una semilla teórica (...) para el desarrollo de la Inteligencia Artificial Global y para la construcción de una sociedad tecnológica basada en un nuevo paradigma del saber. -/- Files. (shrink)

The work presented here, Introduction to Impossible Probability, is essential for understanding the original meaning of the Global Artificial Intelligence. Its origins can be traced back to my writings in 2002 on Astro-ecology, the science that understands the cosmos as a single environment that must be studied from a unified, single science. In essence, this is what the Global Artificial Intelligence seeks to achieve. At the same time, in 2002 I developed the formulations of Impossible Probability, beginning in the early (...) hours of September 11, 2001 with the Impact of the Defect, which was later expanded into the Second Method on October 16, 2002. (shrink)

Brentano never published on Bertrand’s paradox but claimed to have a solution. Adrian Maître has recovered from the Franz Brentano Archive Brentano’s remarks on his solution. They do not give us a worked demonstration of his solution but only an incomplete and in places obscure justification of it. Here I attempt to identify his solution, to explain what seem to me the clearly discernible parts of his justification and to discuss the extent to which the justification succeeds in the light (...) of current work on Bertrand’s paradox. (shrink)

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand’s chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand’s Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the (...) philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author’s proposed solution—a solution in the spirit of Bertrand’s, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference. (shrink)

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an agent values (...) accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings. (shrink)

Certain mathematical problems prove very hard to solve because some of their intuitive features have not been assimilated or cannot be assimilated by the available mathematical resources. This state of affairs triggers an interesting dynamic whereby the introduction of novel conceptual resources converts the intuitive features into further mathematical determinations in light of which a solution to the original problem is made accessible. I illustrate this phenomenon through a study of Bertrand’s paradox.

Review of Sylla's translation of Jacob Bernoulli's Art of Conjecturing, emphasising Bernoulli's success in understanding multiple quantifiers to formulate and prove a law of large numbers.

Bangu (2010) claims that Bertrand’s paradox rests on a hitherto unrecog nized assumption, which assumption is sufficiently dubious to throw the burden of proof back onto ‘objectors to [the principle of indifference]’ (2010: 31). We show that Bangu’s objection to the assumption is ill-founded and that the assumption is provably true.

Different inferences in probabilistic logics of conditionals 'preserve' the probabilities of their premisses to different degrees. Some preserve certainty, some high probability, some positive probability, and some minimum probability. In the first case conclusions must have probability I when premisses have probability 1, though they might have probability 0 when their premisses have any lower probability. In the second case, roughly speaking, if premisses are highly probable though not certain then conclusions must also be highly probable. In the third case (...) conclusions must have positive probability when premisses do, and in the last case conclusions must be at least as probable as their least probable premisses. Precise definitions and well known examples are given for each of these properties, characteristic principles are shown to be valid and complete for deriving conclusions of each of these kinds, and simple trivalent truthtable 'tests' are described for determining which properties are possessed by any given inference. Brief comments are made on the application of these results to certain modal inferences such as "Jones may own a car, and if he does he will have a driver's license. Therefore, he may have a driver's license.". (shrink)

It is shown that quantum mechanics cannot be formulated as a stochastic theory involving a probability distribution function of position and momentum. This is done by showing that the most general distribution function which yields the proper quantum mechanical marginal distributions cannot consistently be used to predict the expectations of observables if phase space integration is used. Implications relating to the possibility of establishing a "hidden" variable theory of quantum mechanics are discussed.

In the manner of the Philosophiæ Naturalis Principia Mathematica, we present here a rigorous investigation into the true nature of acceleration. Since Newton, acceleration has been understood merely as “the rate of change of velocity”—a definition adequate for calculation but insufficient for understanding. We demonstrate that acceleration is not fundamentally a temporal derivative, but rather a geometric invariant: the measure of deviation from geodesic flow in the underly- ing manifold. This perspective unifies classical mechanics, general relativity, electromag- netism, and quantum (...) mechanics under a single geometric framework. Our method is axiomatic and deductive. We begin with minimal assumptions about the structure of space and the nature of motion, and derive from these the various man- ifestations of acceleration across physics. We prove that what appears as “radiation” in unbounded spaces and “oscillation” in bounded spaces are dual expressions of a single geometric principle: the Projection-Curvature Law. (shrink)

Extending the ideas from (Hofer-Szabó and Rédei [2006]), we introduce the notion of causal up-to-n-closedness of probability spaces. A probability space is said to be causally up-to-n-closed with respect to a relation of independence R_ind iff for any pair of correlated events belonging to R_ind the space provides a common cause or a common cause system of size at most n. We prove that a finite classical probability space is causally up-to-3-closed w.r.t. the relation of logical independence iff its probability (...) measure is constant on the set of atoms of non-0 probability. (The latter condition is a weakening of the notion of measure uniformity.) Other independence relations are also considered. (shrink)