历史与综述
Programming is deeply embedded in contemporary mathematical practice, yet its epistemic status in university mathematics teaching remains contested. Little is known about how mathematicians themselves understand the legitimacy of…
We review the memoir \emph{heorie der Parallellinien} by Johann Heinrich Lambert, written in 1766. Lambert, a victim of the prejudices of his time, conceived this memoir as an attempt to prove the so-called parallel postulate of Euclid's…
Artificial intelligence (AI) is the name popularly given to a broad spectrum of computer tools designed to perform increasingly complex cognitive tasks, including many that used to solely be the province of humans. As these tools become…
Mathematics curriculums at most universities tend to perpetuate a belief that higher mathematics is historically and culturally European. First Nations and minority students may not see their identities and cultures reflected in the…
This text is a reworked version of a recorded interview with Bernard Teissier conducted in his house in Paris, on 28 and 29 September 2024.
In this expository note we present an elementary direct rigorous definition and the simplest properties of the winding number. This definition is simpler than the one given in some textbooks. We show how to compute the winding number…
We show that the field of complex numbers $\mathbb C$ contains non-zero infinitesimals by observing that $\mathbb C$ contains non-Archimedean subfields. Our observation is based on an old theorem in algebra due to E. Steinitz, discussed in…
This paper investigates the concept of Nothing from both philosophical and mathematical perspectives, distinguishing between absolute non-being (nihil) and relational negation as a principle of difference. It explores how mathematics…
This paper re-evaluates Jozsef Sutak (1865-1954), a Hungarian scholar-priest and professor, as a grey eminence rather than a genius, offering a counter-narrative to the history of Hungarian university mathematics. By examining his career -…
All men are created equal, proclaimed Jefferson in 1776 -- but some are more equal than others, added Orwell in Animal Farm in 1945. So what's the probability that two skaters are exactly equal, to the third decimal places, after four…
Mathematical proofs are often said to justify their conclusions by indicating the existence of a corresponding formal derivation. We argue that this widespread view relies on an under-examined notion of correspondence, or what it means for…
The main purpose of this paper is using a very simple constructive method to study an old number theory problem related to the Legendre symbol modulo p, and completely solved it. The proving method of the result is purely elementary and has…
The constant $\pi$ has fascinated scholars throughout the centuries, inspiring numerous formulas for its evaluation, such as infinite sums and continued fractions. Despite their individual significance, many of the underlying connections…
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…
This short essay celebrates the mathematical meaning of Pi Day through Euler's formula \[ e^{ix}=\cos x+i\sin x, \] from which Euler's identity \[ e^{i\pi}+1=0 \] follows immediately. We briefly note the historical background of the…
G\"unter Hellwig was the author of influential textbooks on PDEs and differential operators of mathematical physics, an enthusiastic and inspiring teacher to generations of engineers, organiser of PDE conferences at Oberwolfach and a…
The aim of this book is to introduce the reader to the beauty of Algebra, through a journey from the natural numbers to prime fields and finite fields, with some detours. Many books are devoted to the construction of these fields from the…
This paper was motivated by the worldwide May 12 initiative that aims to celebrate, encourage, and inspire women in mathematics. It presents in short how the May 12 initiative has arisen, what are some of the events in the first years, in…
These Course Notes provide an introduction to mathematical proofs for undergraduate students transitioning from computational calculus to abstract mathematics. Topics include propositional logic, proof techniques, mathematical induction,…
This paper shares some experience in advanced mathematical education. We show how a high school student can be naturally and gradually introduced to basic steps of scientific research: developing intuition by finding and correcting mistakes…