مجله فرهنگ و اندیشه ریاضی
سال چهل و یکم شماره 2 (پیاپی 71، پاییز و زمستان 1401)

Algebra, in its basic form, that is the classification and solution of linear and quadratic equations, was founded around 820 AD with the composition of the book al-Jabr wa al-Maqabalah by Mohammad ibn Musa Khwarizmi. He defined algebraic entities for the first time and introduced mathematical operations on these entities as algebraic operations. He classified linear and quadratic equations into “6 standard forms” and presented the conditions for the existence of the root of these equations, and also provided geometrical proofs for the correctness of these solutions. He showed that all linear and quadratic equations can be converted into these “6 standard forms” by algebraic operations, and he presented the general method of solving all linear and quadratic equations. Some European historians of mathematics have tried to doubt the foundation of algebra by Khwarizmi. Some others have considered Diophantus Arithmetica as the source of inspiration for Khwarazmi’s Algebra, or at least both books as a continuation of the tradition rooted in Babylonian mathematics. But these views are not widely accepted today. In Medieval Islam, all the scholars agreed about the foundation of Algebra by Khwarizmi, and only a mathematician named Abu Barza considered his grandfather Abd al-Hamid ibn Wase‘ as the predecessor of Khwarizmi, although Abu Kamel Shuja‘ ibn Aslam strongly rejected this claim and others scholars of Medieval Islam have also emphasized the superiority of Khwarizmi.

Our aim in this paper is to introduce and study five common definitions of chaos in discrete dynamical systems and compare them to each other on compact intervals.These five different definitions that describe chaos from different points of view are Li-Yorke chaos, topological chaos, ω-chaos, Block-Coppel chaos, and Devaney chaos.

How important is the theory of categories? From the early days of its creation until today, this question has occupied the minds of mathematicians and different answers have been provided: For some categories are merely a tool. For others, they are a fundamental part of today’s mathematics. Generally, questions like these do not have definite and absolute answers. At least for this particular question, the mathematics community has not reached a common answer, and probably will never reach a resolution. Our goal in writing this article is to discuss the origins and developments of the theory of categories and also discuss its role in mathematics.

The Gauss quadratic reciprocity law is one of the most important theorems in number theory which Gauss proved it in nineteen years old. In this paper, we first prove some facts about the character of finite abelian groups. Then, using them, we provide a proof for the Gauss quadratic reciprocity law.

There exists a continuous family of disjoint smooth curves, partitioning the unit square, such that one can choose only one point from each curve to construct a measurable set whose area is equal to 1. This contradicts our intuition which resembles Cavalieri’s principle and Fubini’s theorem! So, what is the problem? In this note, we discuss the theorem (due to A. Katok) and answer this question.

Is arithmetic determinate and definitive? In other words, are there reasons for the correctness or falsity of each arithmetic sentence? For example, is there evidence that Goldbach’s conjecture is true or false, even if we don’t know about it? At first glance, the answer seems to be clearly yes. But how can you be sure? What does Gödel’s incompleteness theorem say about this? What is the relationship between independence of some axioms of set theory, such as the axiom of choice and the continuum hypothesis, with these questions? In this article, we will examine these questions. In addition, we will examine the impact of the presence of unconventional facilities such as computing machines that are able to perform an infinite number of instructions in a finite time, as well as proof systems equipped with infinite rules on the answers to the above questions.

Today, the application of game theory has been proven in many sciences. However, for many years, many sciences such as economics, sociology, and politics have been progressing in a less than favorable way; because according to the personal and social situation of the players, different results can be obtained from the same game. Meanwhile, knowing the behavior of each player and its mathematical inference requires prerequisites, which include knowing the beliefs and culture of the society in which he lives. Meanwhile, the value of the literary treasure of that society cannot be ignored. Therefore, the authors of this article decided to open the way by examining three anecdotes of Saadi’s Golestan in the form of a game.

In this article, the new concept of disk-recurrent operators is introduced. We prove that an operator is disk-recurrent if and only if it has a dense set of diskrecurrent vectors. We prove that these operators can be found in infinite dimensional and also finite dimensional Banach spaces. In addition, we show that the operator T is disk-recurrent if and only if T n is disk-recurrent. We also observe that if the diskrecurrence of the direct sum of two operators is disk-recurrent, then any of them is disk-recurrent and we express some results about this.

The author criticizes Bourbaki’s approach to fundamentals of mathematics, logic and set theory. He believes that despite being already aware of the impact of Gödel’s proof on Hilbert’s program of formalism, the group deliberately chose to ignore it, so far as they even refused to mention Gödel and address his work. He then points out the difference between arithmetic and geometric aspects of mathematics, and suggests that Zermelo’s set theory, which was acknowledged by Bourbaki, suffices only for the geometric part whereas the arithmetic side needs to rely on Zermelo-Fraenkel system of axioms.

Lakatos’s seminal work Proofs and Refutations introduced the methods of proofs and refutations by discussing the history and methodolog- ical development of Euler’s formula V − E + F = 2 for three dimensional polyhedra. Lakatos considered the history of polyhedra illustrating a good ex- ample for his philosophy and methodology of mathematics and geometry. In this study, we focus on the mathematical and topological properties which play a role in Lakatos’s methodological approach. For each example and counterexample given by Lakatos, we briefly outline its topological counter- part. We thus present the mathematical background and basis of Lakatos’s philosophy of mathematical methodology in the case of Euler’s formula, and thereby develop some intuitions about the function of his notions of positive and negative heuristics.

This article explores the history of the Eisenstein irreducibility criterion and explains how Theodor Schönemann discovered this criterion before Eisenstein. Both were inspired by Gauss’s Disquisitiones Arithmeticae, though they took very different routes to their discoveries. The article will discuss a variety of topics from 19th-century number theory, including Gauss’s lemma, finite fields, the lemniscate, elliptic integrals, abelian groups, the Gaussian integers, and Hensel’s lemma.

For an absolutely continuous (integer-valued) r.v. X of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order k holds. This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. X, expressions that seem to be known only in particular cases.

This is a translation of some parts of Chapter 10-11 in I Want to be a Mathematician (1985), by Paul R. Halmos.