مجله فرهنگ و اندیشه ریاضی
سال چهل و دوم شماره 2 (پیاپی 73، پاییز و زمستان 1402)

Professor Taher Ghasemi Honary is one of the oldest graduates of the Institute of Mathematics, whose seriousness and emphasis on education is unmatched by his professor, Dr. Mosaheb. In this detailed interview, he recounts his valuable educational, research, and administrative experiences in the last seventy years for the readers.

In recent years, many efforts have been made to use mathematics to solve problems outside the field of mathematics. These fields go far beyond the topics traditionally studied in applied mathematics departments. Even parts of mathematics that until recently were too far-fetched to have concrete applications are now successfully used in various fields. The main question of this article is why mathematics is so useful. This is an old question about mathematics, and although it has been discussed a lot, it does not seem to have a concluding answer. Here we examine some features of mathematics that seem to play an important role in all its applications. We will also examine the effect of accepting some philosophical views on this issue. Among these philosophical views, realism and nominalism, Quine's naturalism, and Penelope Maddy's mathematical naturalism can be mentioned. The history of two important areas of mathematics, namely abstract algebra and formal logic, have been examined in detail as examples.

The purpose of the paper is to focus on the contribution of Greek mathematics in the development of mathematical ‎analysis‎‎. ‎The important elements in this study are analysis and synthesis‎, ‎number and magnitude‎, ‎and ratio and proportion‎, ‎all of which have roots in ancient Greek philosophical thought‎ and Euclid's Elements.

In this paper, the logical development of Euclidean geometry as presented in Euclid’s Elements, has been studied. Pointing out some gaps in the Euclid arguments, the need for modern axiomatic systems has been explored.

This is a translation of ‎last‎ part of Chapter 11 in I Want to be a Mathematician (1985)‎, ‎by Paul R‎. ‎Halmos.

In the so-called Aristotelian classification of sciences‎, ‎which was originally founded in the late Alexandrian age‎, ‎mathematics is one of the three main branches of theoretical philosophy and includes 4 branches‎: ‎arithmetic‎, ‎harmony (a part of theoretical music)‎, ‎geometry‎, ‎and astronomy‎. ‎In the 4th century BC and in the inner circle of the Pythagoreans‎, ‎the plural form of the word μαθήμα (μαθήματα)‎, ‎the root of European terms related to mathematics‎ - ‎which was originally used in the absolute sense of teaching/course‎ - ‎was first applied to‎ "‎4 disciplines altogether"‎, ‎and after that‎, ‎this word was mostly used in the meaning of‎ "‎mathematical teaching/course"‎. ‎Later Greek scientists‎, ‎at least from the 1st-6th centuries AD‎, ‎put forward some material to explain why these 4 disciplines came together under one name‎. ‎Later‎, ‎Arabic word‎ ‎"Taʿlīm"‎ ‎and especially‎ "‎ʿIlm al-Taʿālīm‎" ‎also met the same fate as Greek μαθήμα‎. ‎Later Greek scientists‎, ‎at least from the 1st-6th centuries AD‎, ‎put forward some material to explain why these 4 disciplines came together under one name‎. ‎Later‎, ‎Arabic word‎ ‎"Taʿlīm"‎ ‎and especially‎ "‎ʿIlm al-Taʿālīm‎" ‎also met the same fate as Greek μαθήμα‎. ‎The views of Alexandrian scholars about‎ "‎mathematics and its branches in the classification of sciences"‎, ‎were popularized among the scholars of Medieval Islam through Paul the Persian and Abū Alī Moskūyeh‎, ‎as well as Ibn Behrīz and al-Kindī‎. ‎In Medieval Islam‎, ‎Fārābi introduced the first original classification of the sciences in Iḥṣaʾ al-Ulūm‎, ‎but he mentioned the it’s philosophical base only in his moral works‎. ‎Abu Sahl al-Masīḥī‎, ‎Avicenna and a number of other philosophers‎, ‎in addition to determining the place of mathematics among the sciences‎, ‎listed branches for mathematics‎, ‎which over time were divided into two primary and secondary categories (Uṣūl and Furūʿ)‎, ‎and the number of its secondary categories such as ʿIlm al-Ḥiyal and ʿIlm al-Manāẓir (optics) also Increased.‎

This note was motivated by a student’s misconception that the average velocity over a time intervalof a particle in rectilinear motion is the arithmetic mean of its velocities at the ends of the interval. We presentan algebraic proof that this property holds only if the particle has constant acceleration, thereby recovering awell-known result. A characterization is also provided of differentiable functions on (−∞,∞) whose restrictionsto the intervals (-∞ 0]‎‎ and ‎‎[0,∞)‎‎ are quadratic polynomials. In addition, a calculus-free proof is presented toshow that a certain feature of the mean value theorem holds only for quadratic polynomials.

Years ago, Raymond Smullyan, a logician and master of puzzles, raised a problem that George Boolos called ``the hardest logic puzzle ever".  We present a generalization of the puzzle in the form of a problem.

In mathematics, chemistry , and nanoscience a graph model is considered for each molecule, so that the vertices are the atoms of the molecule and the edges are the bonds between the atoms. Graph topological indices are numerical parameters dependent on the graph, which are unique numbers and are fixed with respect to the isomorphism of the graphs. Many topological indices in Mathematical Chemistry are in terms of degrees of vertices. In this article, we study the relationship between some values of these topological indices and the golden ratio.

In this article, an important and widely used matrix decomposition is introduced. This decomposition, named the matrix polar decomposition, is actually a generalization of the polar representation of complex numbers to matrices. Several important applications of matrix polar decomposition in various fields are considered. Also, some iterative methods for computing the matrix polar decomposition and their order of convergence are mentioned.