Related papers: De Divino Errore
I'll discuss how Goedel's paradox "This statement is false/unprovable" yields his famous result on the limits of axiomatic reasoning. I'll contrast that with my work, which is based on the paradox of "The first uninteresting positive whole…
In 1931 de Finetti proved what is known as his Dutch Book Theorem. This result implies that the finite additivity {\it axiom} for the probability of the disjunction of two incompatible events becomes a {\it consequence} of de Finetti's…
Many early-modern mathematical books incorporated at least a part of Diophantus' Arithmetica, from Jacques de Billy's Diophanti Redivi Pars prior et posterior to John Kersey's Third and Fourth Books of the Elements of algebra or Jacques…
We give a popular account of the Banach-Tarski paradox and its connections with the axiom of choice.
In this short note, we point out a mistake in G.Cybenko's proof of his version of the universal approximation theorem which has been widely cited. This mistake might not be easily fixable along the idea of his proof and it also leads to an…
Raymond Smullyan came up with a puzzle that George Boolos called The Hardest Logic Puzzle Ever.[1] The puzzle has truthful, lying, and random gods who answer yes or no questions with words that we don't know the meaning of. The challenge is…
Credit allocation in the mainstream bibliometrics is fundamentally flawed and the popular indicators have been misleading science for decades. Originally a simple technical mistake has become an integral part of our culture and is very…
The apparently trifling unexpected hanging paradox has generated an enormous philosophical literature. We introduce the mathematician to this literature, paying special attention to aspects that involve nontrivial mathematics. This xxx…
Goedel's results have had a great impact in diverse fields such as philosophy, computer sciences and fundamentals of mathematics. The fact that the rule of mathematical induction is contradictory with the rest of clauses used by Goedel to…
The first seeds of mathematical intuitionism germinated in Europe over a century ago in the constructive tendencies of Borel, Baire, Lebesque, Poincar\'e, Kronecker and others. The flowering was the work of one man, Luitzen Egbertus Jan…
Dutch book arguments have been applied to beliefs about the outcomes of measurements of quantum systems, but not to beliefs about quantum objects prior to measurement. In this paper, we prove a quantum version of the probabilists' Dutch…
Let $k\geq 1$ be an integer. Let $\delta_k(n)$ denote the maximum divisor of $n$ which is co-prime to $k$. We study the error term of the general $m$-th Riesz mean of the arithmetical function $\delta_k(n)$ for any positive integer $m \ge…
We find it necessary to advise the interested and active instructor of Physics on the wrongness of some computations in the aforementioned article. Surprisingly, the Journal refuses to even publish an erratum on the paper, which naturally…
This article discusses epistemological problems in the philosophy of mathematics and issues concerning the reliability of the mathematical literature.
We describe various errors in the mathematical literature, and consider how some of them might have been avoided, or at least detected at an earlier stage, using tools such as Maple or Sage. Our examples are drawn from three broad…
Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? The author and Pieter Moree, building on work of Zhang, Cobeli, and Zaharescu, gave heuristics for…
Three versions of Luca Pacioli's 'De Dvina Proportione' remain: a manuscript held in Milan, another in Geneva and a printed version edited in Venice. A recent book, 'Antologia della Divina Proporzione', has all three in one volume, allowing…
The main hypothesis about Thomas Bayes's intentions to write his famous Essay on probability is that he wanted to refute the arguments of David Hume against the reliability of the occurrence of miracles, published in 1748. In this paper we…
Considerable thought has been devoted to an adequate definition of the class of infinite, random binary sequences (the sort of sequence that almost certainly arises from flipping a fair coin indefinitely). The first mathematical exploration…
The aim of this paper is to show that partial probability can be justified from the standpoint of subjective probability in much the same way as classical probability does. The seminal works of Ramsey and De Finetti have furnished a method…