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Starting from Greg Moore's description about Physical Mathematics, a framework is proposed in order to understand it, based on Gilles Châtelet's philosophy. It will be argued that Châtelet's ideas of inverting, splitting, augmenting and virtuality are crucial in the discussion about the nature of Physical Mathematics. Along this line, it will be proposed that mirror symmetry is a natural study case to test Châtelet's ideas in this context. This should be considered as a first step in a long term project aiming to study the relations among mathematics, physics and philosophy in the construction of a global understanding of the structure of the universe, as it was envisioned by Grothendieck in the late 80's of the last century and it was started to be developed independently by Châtelet in the beginning of the 90's. The main suggestion of the essay is that it is in the relations between mathematics, physics and philosophy that new knowledge arises.

In 1911, Alfred North Whitehead published a short book "Introduction to Mathematics" (IM) intended for students wanting an explanation of the fundamental ideas of mathematics. Whitehead's IM has enduring value because it was written not long after he and Bertrand Russell published their monumental three-volume work "Principia Mathematica" (PM) -- a publication of immense historical significance for mathematics. IM sheds light on Whitehead's view of mathematics at that time. Whitehead's book places proofs in predicate logic as the mythical starting point of mathematics, although Whitehead himself was slow to understand the significance of symbolic predicate logic.

ChatGPT, an Artificial Intelligence model, has the potential to revolutionize education. However, its effectiveness in solving non-English questions remains uncertain. This study evaluates ChatGPT's robustness using 586 Korean mathematics questions. ChatGPT achieves 66.72% accuracy, correctly answering 391 out of 586 questions. We also assess its ability to rate mathematics questions based on eleven criteria and perform a topic analysis. Our findings show that ChatGPT's ratings align with educational theory and test-taker perspectives. While ChatGPT performs well in question classification, it struggles with non-English contexts, highlighting areas for improvement. Future research should address linguistic biases and enhance accuracy across diverse languages. Domain-specific optimizations and multilingual training could improve ChatGPT's role in personalized education.

In this paper, we situate the educational movement of "Ethics in Mathematics," as outlined by the Cambridge University Ethics in Mathematics Project, in the wider area of mathematics ethics education. By focusing on the core message coming out of Ethics in Mathematics, its target group, and educational philosophy, we set it into relation with "Mathematics for Social Justice" and Paul Ernest's recent work on ethics of mathematics. We conclude that, although both Ethics in Mathematics and Mathematics for Social Justice appear antagonistic at first glance, they can be understood as complementary rather than competing educational strategies.

Origami is the art of paper folding, and it borrows its name from two Japanese words \emph{ori} and \emph{kami}. In Japanese, {ori} means folding, and the paper is called {kami}. While origami is just a hobby to most, there is a lot more to it. If you fold a square sheet of paper into any of the traditional origami model (for example the flapping bird) and unfold it, you can see crease patterns. These crease patterns tell us that there is a lot of geometry hidden behind the folds. In this article, we investigate the symbiotic relationship between mathematics and origami. The first part of this article explores the utility of origami in education. We will see how origami could become an effective way of teaching methods of geometry, mainly because of its experiential nature. Complex origami patterns cannot be created out of thin air. They usually involve understanding deep mathematical theories and the ability to apply them to paper folding. In the second part of the article, we attempt to provide a glimpse of this beautiful connection between origami and mathematics.

This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what is, or isn't, a foundation; and get clues as to how a foundation can be optimized for effective human use. For this we turn to history and professional practice of the subject. We have no asperations to Philosophy. The first section gives a short abstract discussion, focusing on the significance of consistency. The next briefly describes foundations, explicit and implicit, at a few key periods in mathematical history. We see, for example, that at the primitive level human intuitions are essential, but can be problematic. We also see that traditional axiomatic set theories, Zermillo-Fraenkel-Choice (ZFC) in particular, are not quite consistent with mainstream practice. The final section sketches the proposed new foundation and gives the basic argument that it is uniquely qualified to be considered {the} foundation of mainstream deductive mathematics. The ``coherent limit axiom'' characterizes the new theory among ZFC-like theories. This axiom plays

Constructivists (and intuitionists in general) asked what kind of mental construction is needed to convince ourselves (and others) that some mathematical statement is true. This question has a much more practical (and even cynical) counterpart: a student of a mathematics class wants to know what will the teacher accept as a correct solution of a homework problem. Here the logical structure of the claim is also very important, and we discuss several types of problems and their use in teaching mathematics.

The old lie of mathematical inadequacy of Indigenous communities has been curiously persistent despite increasing evidence shows that many Indigenous communities practiced mathematics. Attempts to study and teach Indigenous mathematical knowledge have always been questioned and even denied validity. The Aboriginal and Torres Strait Islander Histories and Cultures cross-curriculum priority in the F-10 Australian schools curriculum, from 2022 onwards, includes content elaborations related to Indigenous mathematics, which have been developed and refined by expert Indigenous advisers. We celebrate this initiative, but experience also tells us to expect some resistance from sectors of the education communities who hold to an exclusively Anglo-European provenance of mathematics. Through this review article we seek to constructively forestall potential pushback and address concerns regarding the legitimacy and pedagogical value of Indigenous mathematics, by countering with evidence some published claims of mathematical inadequacies of Australian First Nations cultures.

Recent developments in computer programming and in mathematics suggest that there is a strong case for a new way of introducing programming to enhance the learning of school mathematics. The article describes a collaboration of mathematics and computer science teachers to solve the Josephus problem. We demonstrate how a programming approach based on both types and functions can make a vastly improved contribution to learning mathematics than the less successful use of conventional computer programming in Scratch.

A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the physical universe? We review the well-known fact that the symmetries of the laws of physics are their defining properties. We show that there are similar symmetries of mathematical facts and that these symmetries are the defining properties of mathematics. By examining the symmetries of physics and mathematics, we show that the effectiveness is actually quite reasonable. In essence, we show that the regularities of physics are a subset of the regularities of mathematics.

"Phase-locking" is a fundamental phenomenon in which coupled or periodically forced oscillators synchronise. The Arnold family of circle maps, which describes a forced oscillator, is the simplest mathematical model of phase-locking and has been studied intensively since its introduction in the 1960s. The family exhibits regions of parameter space where phase-locking phenomena can be observed. A long-standing question asked whether "hyperbolic" parameters~-- those whose behaviour is dominated by periodic attractors, and which are therefore stable under perturbation~-- are dense within the family. A positive answer was given in 2015 by van Strien and the author, which implies that, no matter how chaotic a map within the family may behave, there are always systems with stable behaviour nearby. This research was a focal point of a pioneering collaboration with composer Emily Howard, commencing with Howard's residency in Liverpool's mathematics department in 2015. The collaboration generated impacts on creativity, culture and society, including several musical works by Howard, and lasting influence on artistic practice through a first-of-its-kind centre for science and music. We describ

A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics usually constructions and algorithms are more simple with respect to their traditional analogs. We especially examine algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other "good" operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent and semirings are examined. In this case, basic algorithms are polynomial as well as their interval versions. This situation is very different from the traditional linear algebra, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.

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 Coronavirus lockdown, I argue that there is also a third way: new mathematics can also be inherited. And entering into possession, making it your own, could be great fun.

In the Soviet Union a reform movement in mathematics education was triggered by Andrey Kolmogorov in the 1970s, and followed by a counter-reform. This movement was rooted in the very different socioeconomic conditions of that time and place, and followed a strategy with very significant contrasts to similar programs in the USA, England, or France. This provides an interesting case study which may illuminate the way such movements arise and succeed or fail, and, at the social level, certain fundamental commonalities of constraints as well as significant differences according to local conditions. We shall show that the principal reasons of the failure of the Kolmogorov reform were political: (1) The reform ignored the reality of the socio-economic conditions of the country; (2) The human factor was ignored, and very little attention was given to professional development and retraining of, and methodological help to, the whole army of teachers; (3) An attempt to transfer mathematical content and methods from the highly successful advanced extension stream for mathematically strong and highly engaged children to mainstream education was an especially grievous error.

Many have wondered how mathematics, which appears to be the result of both human creativity and human discovery, can possibly exhibit the degree of success and seemingly-universal applicability to quantifying the physical world as exemplified by the laws of physics. In this essay, I claim that much of the utility of mathematics arises from our choice of description of the physical world coupled with our desire to quantify it. This will be demonstrated in a practical sense by considering one of the most fundamental concepts of mathematics: additivity. This example will be used to show how many physical laws can be derived as constraint equations enforcing relevant symmetries in a sense that is far more fundamental than commonly appreciated.

The Department of Applied Mathematics at the University of Nottingham Malaysia Campus has a responsibility for delivering mathematics modules for engineering students. Due to the significantly large number of students, two methods of teaching delivery--parallel teaching and block teaching--have been implemented. This article discusses some pros and cons between these two methods, particularly for the Foundation programme and the first year of the Undergraduate programme in Engineering. Whether parallel teaching or block teaching is implemented, feedback comments from the students indicate that some areas need to be paid attention to.

The learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they imple

We introduce Voevodsky's univalent foundations and univalent mathematics, and explain how to develop them with the computer system Agda, which is based on Martin-Löf type theory. Agda allows us to write mathematical definitions, constructions, theorems and proofs, for example in number theory, analysis, group theory, topology, category theory or programming language theory, checking them for logical and mathematical correctness. Agda is a constructive mathematical system by default, which amounts to saying that it can also be considered as a programming language for manipulating mathematical objects. But we can assume the axiom of choice or the principle of excluded middle for pieces of mathematics that require them, at the cost of losing the implicit programming-language character of the system. For a fully constructive development of univalent mathematics in Agda, we would need to use its new cubical flavour, and we hope these notes provide a base for researchers interested in learning cubical type theory and cubical Agda as the next step. Compared to most expositions of the subject, we work with explicit universe levels.