相关论文: Paradoxes of Randomness

Several examples are used to illustrate how we deal cavalierly with infinities and unphysical systems in physics. Upon examining these examples in the context of infinities from Cantor's theory of transfinite numbers, the only known…

We claim that human mathematics is only a limited part of the consequences of the chosen basic axioms. Properly human mathematics varies with time but appears to have universal features which we try to analyze. In particular the functioning…

This note will address the issue of the existence of God from a game theoretic perspective. We will show that, under certain assumptions, man cannot simultaneously be (i) rational and (ii) believe that an infinitely powerful God exists.…

One source of beauty in mathematics is totally unexpected connections between two fundamentally different objects. For instance, is it not surprising that the time period of a real simple pendulum is linked with a function arising out of…

For any particularly interesting theorem one proof is never enough. Instead, the first proof sets the challenge to find a more elegant method that illuminates subtle features of the math, is simpler to understand, or even avoids using…

There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real…

A number of well-known theorems, such as Cox's theorem and de Finetti's theorem. prove that any model of reasoning with uncertain information that satisfies specified conditions of "rationality" must satisfy the axioms of probability…

Mathematics cannot anymore be assimilated to a linguistic game, where formal proofs are strongly differentiated with conjectural thinking, without building any category of knowledge to understand the passage (Wittgenstein's gist). Nowadays,…

After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main epistemological consequences of the first incompleteness Theorem for the two fundamental…

We present a propositional logic to reason about the uncertainty of events, where the uncertainty is modeled by a set of probability measures assigning an interval of probability to each event. We give a sound and complete axiomatization…

We give a proof of the inconsistency of PM arithmetic, classical set theory and related systems, incidentally exposing an error in Goedel's own proof of Goedel's Theorems. The inconsistency proof, that formulae of the form R and ~R occur as…

Thirty original and collected problems, puzzles, and paradoxes in mathematics and physics are explained in this paper, taught by the author to the elementary and high school teachers at the University of New Mexico - Gallup in 1997-8 and…

We consider a randomised version of Kleene's realisability interpretation of intuitionistic arithmetic in which computability is replaced with randomised computability with positive probability. In particular, we show that (i) the set of…

This paper has several objectives. First, it separates randomness from lawlessness and shows why even genuine randomness does not imply lawlessness. Second, it separates the question -why should I call a phenomenon random? (and answers it…

We introduce a logic for reasoning about evidence, that essentially views evidence as a function from prior beliefs (before making an observation) to posterior beliefs (after making the observation). We provide a sound and complete…

Plausible reasoning concerns situations whose inherent lack of precision is not quantified; that is, there are no degrees or levels of precision, and hence no use of numbers like probabilities. A hopefully comprehensive set of principles…

The notions of potential infinity (understood as expressing a direction) and actual infinity (expressing a quantity) are investigated. It is shown that the notion of actual infinity is inconsistent, because the set of all (finite) natural…

We present a propositional logic %which can be used to reason about the uncertainty of events, where the uncertainty is modeled by a set of probability measures assigning an interval of probability to each event. We give a sound and…

Bertrand's paradox is a famous problem of probability theory, pointing to a possible inconsistency in Laplace's principle of insufficient reason. In this article we show that Bertrand's paradox contains two different problems: an "easy"…

A perplexing problem in understanding physical reality is why the universe seems comprehensible, and correspondingly why there should exist physical systems capable of comprehending it. In this essay I explore the possibility that rather…