Related papers: Adventures in Mathematical Reasoning
New understandings of the functioning of human brains engaged in mathematics raise interesting questions for mathematics educators. Novel lines of research are suggested by neuroscientific findings, and new light is shed on some…
Mathematical reasoning is a fundamental aspect of human intelligence and is applicable in various fields, including science, engineering, finance, and everyday life. The development of artificial intelligence (AI) systems capable of solving…
The book "A Course in Constructive Algebra" (1988) shows the way of understanding classical basic algebra in a constructive style similar to Bishop's Constructive Mathematics. Classical theorems are revisited, with a new flavour, and become…
We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a…
Large Language Models (LLMs) have demonstrated impressive reasoning abilities through test-time computation (TTC) techniques such as chain-of-thought prompting and tree-based reasoning. However, we argue that current reasoning LLMs (RLLMs)…
Can mathematics help us find our way through all the wonders and mysteries of the universe? When physicists describe the laws governing the physical world, mathematics is always involved. Is this due to the fact that the universe is, at…
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…
Astronomy is an observationally-led subject where chance discoveries play an important role. A whole range of such discoveries is continually made, from the trivial to the highly significant. What is generally needed is for luck to strike…
I am most honoured to have the privilege to present the Foreword to this fascinating and wonderfully varied collection of contributions, concerning the nature of computation and of its deep connection with the operation of those basic laws,…
This article illustrates pedagogy through training in the handling of abstractions. Mental arithmetic is not limited to numerical calculation; one can mentally calculate primitives and simplify analytical expressions. Even if there is…
I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete…
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…
An expository hitchhikers guide to some theorems in mathematics.
The classical platonist / formalist dilemma in philosophy of mathematics can be expressed in lay terms as a deceptively naive question: \emph{Is new mathematics discovered or invented? Using examples from my own mathematical work during the…
Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to…
We state the defining characteristic of mathematics as a type of symmetry where one can change the connotation of a mathematical statement in a certain way when the statement's truth value remains the same. This view of mathematics as…
Logic really is just algebra, given one uses the right kind of algebra, and the right kind of logic. The right kind of algebra is abstraction algebra, and the right kind of logic is abstraction logic.
This book covers the history of probability up to Kolmogorov with essential additional coverage of statistics up to Fisher. Based on my work of ca. 50 years, it is the only suchlike book. Gorrochurn (2016) is similar but his study of events…
Magic: the Gathering is a popular and famously complicated card game about magical combat. Recently, several authors including Chatterjee and Ibsen-Jensen (2016) and Churchill, Biderman, and Herrick (2019) have investigated the…
Mathematics is changing. Computers are verifying proofs, checking calculations, and exploring complex structures that would overwhelm human effort. Yet curiosity-driven research is where tomorrow's breakthroughs are quietly prepared. In…