Related papers: The Uses of Argument in Mathematics
There are various approaches to the problem of how one is supposed to conduct a statistical analysis. Different analyses can lead to contradictory conclusions in some problems so this is not a satisfactory state of affairs. It seems that…
Classes of linguistic paradoxes and linguistic tautologies are introduced with examples and explanations. They are part of the author's work on the Paradoxist Philosophy based on mathematical logic. The general cases exposed below are…
The specifics of data layout can be important for the efficiency of functional programs and interaction with external libraries. In this paper, we develop a type-theoretic approach to data layout that could be used as a typed intermediate…
We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code…
In recent years, large language models (LLMs) have made significant advancements in developing human-like and engaging dialogue systems. However, in tasks such as consensus-building and persuasion, LLMs often struggle to resolve conflicts…
We explore Leibniz's understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus of Newton and Leibniz have been a target…
Abraham Robinson's philosophical stance has been the subject of several recent studies. Erhardt following Gaifman claims that Robinson was a finitist, and that there is a tension between his philosophical position and his actual…
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such…
In a previous article we gave the general foundations of the theory of movement considered from a philosophical and mathematical point of view. Philosophical it meant to understand the opposition of the one and the multiple, mathematically…
The mathematical analysis was conceived in XVII century in Newton and Leibniz works. The problem of logical rigor in definitions was considered by Arnauld and Nicole in "Logique ou l'art de penser". They were the first, who distinguished…
Large language models (LLMs) demonstrate considerable potential in various natural language tasks but face significant challenges in mathematical reasoning, particularly in executing precise, multi-step logic. However, current evaluation…
This paper addresses the actual practice of justifying definitions in mathematics. First, I introduce the main account of this issue, namely Lakatos's proof-generated definitions. Based on a case study of definitions of randomness in…
Due to the remarkable language understanding and generation abilities of large language models (LLMs), their use in educational applications has been explored. However, little work has been done on investigating the pedagogical ability of…
This soliloquy outlines some naive philosophical arguments underlying the thesis that mathematics ought to be viewed simply as a universal set of languages, some of precise expression, and some of effective communication.
Translating expressions between different logics and theorem provers is notoriously and often prohibitively difficult, due to the large differences between the logical foundations, the implementations of the systems, and the structure of…
The use of argumentation in education has been shown to improve critical thinking skills for end-users such as students, and computational models for argumentation have been developed to assist in this process. Although these models are…
Mathematicians invented Mathematics to escape from words, but at last they depend on them just as much as everybody else. At the end, all basic definitions will be reliant on words, yet the mathematician believes that he's elevated from…
This preprint is a text for students and teachers on inequalities. Some standard topics are covered on application of calculus to inequality proving. Many examples are considered, stated, solved or partially solved. Some problems are…
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…
Aristotle considered particular quantified sentences in his study of syllogisms and in his famous square of opposition. Of course, the logical formulas in Aristotle work were not modern formulas of mathematical logic, but ordinary sentences…