Related papers: Mathematical Proof Between Generations
Present day mathematics is a human construct, where computers are used more and more but do not play a creative role. This situation may change however: computers may become creative, and since they function very differently from the human…
This is a draft of a chapter on mathematical logic and foundations for an upcoming handbook of computational proof assistants.
In proof theory the notion of canonical proof is rather basic, and it is usually taken for granted that a canonical proof of a sentence must be unique up to certain minor syntactical details (such as, e.g., change of bound variables). When…
The ability to read, write, and speak mathematics is critical to students becoming comfortable with statistical models and skills. Faster development of those skills may act as encouragement to further engage with the discipline. Vocabulary…
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
Computation, the use of a computer to solve, simulate, or visualize a physical problem, has revolutionized how physics research is done. Computation is used widely to model systems, to simulate experiments, and to analyze data. Yet, in most…
The information technology explosion has dramatically increased the application of new mathematical ideas and has led to an increasing use of mathematics across a wide range of fields that have been traditionally labeled "pure" or…
Informal logic is a method of argument analysis which is complementary to that of formal logic, providing for the pragmatic treatment of features of argumentation which cannot be reduced to logical form. The central claim of this paper is…
In mathematics information is a number that measures uncertainty (entropy) based on a probabilistic distribution, often of an obscure origin. In real life language information is a datum, a statement, more precisely, a formula. But such a…
We describe the first results of a project of analyzing in which theories formal proofs can be ex- pressed. We use this analysis as the basis of interoperability between proof systems.
The traditional foundation of science lies on the cornerstones of theory and experiment. Theory is used to explain experiment, which in turn guides the development of theory. Since the advent of computers and the development of…
In this essay we examine some aspects of the classical theory of definition as codified in Aristotle's \emph{Topics} and Porphyry's \emph{Eisagog\^e} in the light of the way definition is carried out in modern mathematical practice. Our…
A sketch of some of the fundamental notions related to the nature of knowledge is offered, with special focus on the role of mathematics and my own opinions. No single idea exposed here is entirely original; indeed, this topic has been…
It is introduced the concept of Superiority Degree one competitive decision over another. On the basis of this concept the mathematics theoretic structure is developed, which is part of pairs comparisons branch in modern decision making…
In an emerging computing paradigm, computational capabilities, from processing power to storage capacities, are offered to users over communication networks as a cloud-based service. There, demanding computations are outsourced in order to…
When mathematicians present proofs they usually adapt their explanations to their didactic goals and to the (assumed) knowledge of their addressees. Modern automated theorem provers, in contrast, present proofs usually at a fixed level of…
The understanding of probability can be difficult for a few young scientists. Consequently, this new mathematical symbol, related to binomial coefficients and simplicial polytopic numbers, could be helpful to science education. Moreover,…
The question of the definition of what is an algorithm is recurrent. It is found in teaching, at different levels and particularly in secondary education because of the recent evolutions in high school, with immediate consequences in higher…
Proof competence, i.e. the ability to write and check (mathematical) proofs, is an important skill in Computer Science, but for many students it represents a difficult challenge. The main issues are the correct use of formal language and…
A primary goal of physics is to create mathematical models that allow both predictions and explanations of physical phenomena. We weave maths extensively into our physics instruction beginning in high school, and the level and complexity of…