Related papers: Redefining Revolutions
In many situations humans have to reason with inconsistent knowledge. These inconsistencies may occur due to not fully reliable sources of information. In order to reason with inconsistent knowledge, it is not possible to view a set of…
We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several equivalent statements} borrowed from algebra, real analysis, general topology, and non-standard analysis. We also…
Measures of irrationality are a numerical way of quantifying how far a given variety is from being rational (or rationally connected, uniruled, etc.). In the last two decades, there has been renewed interest in the study of these…
Consistent confirmations obtained independently of each other lend credibility to a scientific result. We refer to results satisfying this consistency as reproducible and assume that reproducibility is a desirable property of scientific…
Some thoughts are presented on the inter-relation between beauty and truth in science in general and theoretical physics in particular. Some conjectural procedures that can be used to create new ideas, concepts and results are illustrated…
The (extended) AGM postulates for belief revision seem to deal with the revision of a given theory K by an arbitrary formula, but not to constrain the revisions of two different theories by the same formula. A new postulate is proposed and…
Wigner's "unreasonable effectiveness of mathematics" in physics can be understood as a reflection of a deep and unexpected unity between the fundamental structures of mathematics and of physics. Some of the history of evidence for this is…
Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to…
This book invites readers to see mathematics not just as formulas and rules, but as the deepest expression of human thought. It begins by exploring the timeless idea of mathematics as a universal language, contrasting its precision with the…
This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. This is camera-ready copy prepared for publication as a book, but at the last minute I decided to publish it…
A few remarks on how mathematics quests for freedom.
Logic has its origins in basic questions about the nature of the real world and how we describe it. This article seeks to bring out the physical and epistemological relevance of some of the more recent technical work in logic and…
Established frameworks to understand problems with reproducibility in science begin with the relationship between our understanding of the prior probability of a claim and the statistical certainty that should be demanded of it, and explore…
Experimental mathematics is an experimental approach to mathematics in which programming and symbolic computation are used to investigate mathematical objects, identify properties and patterns, discover facts and formulas and even…
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of…
Interpretability is the study of explaining models in understandable terms to humans. At present, interpretability is divided into two paradigms: the intrinsic paradigm, which believes that only models designed to be explained can be…
In the paper by P. Quincey (2020) [1], the author claims that angles are not dimensionless quantities and that the radian should be a new independent base unit. However, the claim is not supported and the proposed redefinition will cause…
Mathematicians occasionally discover interesting truths even when they are playing with mathematical ideas with no thoughts about possible consequences of their actions. This paper describes two specific instances of this phenomenon. The…
Darwin's theory of evolution is considered to be one of the greatest scientific gems in modern science. It not only gives us a description of how living things evolve, but also shows how a population evolves through time and also, why only…
In this articles we consider very mathematical (if not very plausible) geometric marching. Our marchers will exhibit beautiful mathematics, some familiar and some less so. In a final summary section we discuss the point of all the fancy…