相关论文: How the continuum hypothesis could have been a fun…

The independence of the continuum hypothesis is a result of broad impact: it settles a basic question regarding the nature of N and R, two of the most familiar mathematical structures; it introduces the method of forcing that has become the…

This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…

Crucial experiments have a long history of contributions to progress in physics. Similarly, we claim that in the period roughly from 1955 to 1985 crucial calculations played a significant role in setting the agenda for elementary particle…

In the past century many fundamental results on unpredictability, undecidability and uncertainty have compelled scientists to grapple with the idea that some questions may never be resolved within our current theories. While this…

This talk presents foundations of mathematics as a historically variable set of principles appealing to various modes of human intuition and devoid of any prescriptive/prohibitive power. At each turn of history, foundations crystallize the…

This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and…

In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy's ideas of function, continuity, limit and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the…

Absolute continuity implies uniform continuity, but generally not vice versa. In this short note, we present one sufficient condition for a uniformly continuous function to be absolutely continuous, which is the following theorem: For a…

In the modern Bayesian view classical probability theory is simply an extension of conventional logic, i.e., a quantitative tool that allows for consistent reasoning in the presence of uncertainty. Classical theory presupposes, however,…

We study a well-known technique of using absoluteness for giving choice-free proofs to some statements which are known to be provable with the axiom of choice. The idea is to reduce the problem to an inner model where the axiom of choice…

Conceptual modeling is a strongly interdisciplinary field of research. Although numerous proposals for axiomatic foundations of the main ideas of the field exist, there is still a lack of understanding main concepts such as system, process,…

We discuss the repercussions of the development of infinitesimal calculus into modern analysis, beginning with viewpoints expressed in the nineteenth and twentieth centuries and relating them to the natural cognitive development of…

We show that the principle of entropy increase may be exactly founded on a few axioms valid not only for quantum and classical statistics, but also for a wide range of statistical processes.

Set-theoretical, physical, and intuitive notions of continuum are compared. It is shown that the independence of the continuum hypothesis determines status and properties of the set of intermediate cardinality. The intermediate set is a…

In this paper we study the interaction between logic and probability. In particular, we show that the convex hull of evaluations of a broad class of logics is always effectively axiomatizable. We define a Birkhoff-style calculus for…

A popular scientific contribution should not contradict any established facts and ought to be understandable. I complied with both these requirements and am offering a sufficiently full introduction to probability theory. Furthermore, I…

Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as…

Our collective views regarding the question "what is fundamental?" are continually evolving. These ontological shifts in what we regard as fundamental are largely driven by theoretical advances ("what can we calculate?"), and experimental…

Brief development of the idea of the very important notion of continuity is given. Continuity is often confused with contiguity, "drawing the graph in one go," "no gaps," etc. The author argues in support of using correct notions of…

The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop…