Mathematicians routinely pass judgements on mathematical proofs. A proof might be elegant, cumbersome, beautiful, or awkward. Perhaps the highest praise is that a proof is right, that is, that the proof fits the theorem in an optimal way. It is also common to judge that one proof fits better than another, or that a proof does not fit a theorem at all. This paper attempts to clarify the notion of mathematical fit. We suggest six criteria that distinguish proofs as being more or less fitting, and provide examples from several different mathematical fields.

Mathematicians routinely report that beauty is both a reward and a motivation for the work they do. However, how and to what extent children can appreciate mathematical beauty is an open question. ...

Many judgments are made about mathematical proofs, and among these we find the claim that a certain proof is beautiful or not. Beauty in mathematics is rarely denied, but it is also rarely explained. What is meant by the term, what criteria are needed for a proof to be beautiful, could there be objective qualities of a mathematical proof that map more or less onto the subjective experiences we might have when we read and understand it? While the concept of beauty itself is nebulous, the claim in this paper is that it might be related to a slightly more tractable quality of fit. We will discuss how fit arises in mathematics by looking at contrasting examples of proofs commonly held to be beautiful or not. We then take up the question about whether fit is a value, arguing that like justice and beauty, fit describes meaningful relationships and coherence, which make it a candidate for something to be valued not only in our mathematics classrooms, but also in the world at large.

Do we have an inbuilt maths drive, a bit like a sex drive, asks mathematician Manya Raman Sundstrom

This paper traces the history of compactness from the original motivating questions, through the development of the definition, to a generalization of sequential compactness in terms of nets and filters.

We discuss here the notion of mathematical fit, a concept that might relate to mathematical explanation and mathematical beauty. We specify two kinds of fit a proof can have, intrinsic and extrinsi ...

Beauty, which plays a central role in the practice of mathematics, is almost absent in discussions of school mathematics. This is problematic, because students will decide whether or not to continu ...

Summary This paper tells the story of what happened when students in the authors' sophomore-level introduction-to-proof classes were given a theorem to prove with no expectation about what proof method to use. The paper discusses the nine different student-produced proofs of the statement: If n is an integer such that n ≥ 3, then n3 > (n + 1)2.

We present two different proofs of Pick's theorem commonly held to be beautiful and attempt to identify features that give rise to the sensation of beauty. In particular, we discuss two concepts, generality and specificity, that appear to contribute to beauty in different ways. We also discuss possible implications of this work for the teaching of mathematics, especially in countries in which discussions of beauty and aesthetics are notably absent from curricular documents.

This paper poses two questions:Should mathematics textbooks try to convey the beauty of mathematics?If so, how?These questions are difficult, and subjective, and will not be settled in this paper, ...

This paper reports a study aimed at characterizing the conditions that lead to high achievement in mathematics in India. The study involved eight schools in the greater Mumbai region. The main result of the study is that the notion of high achievement itself is problematic, as reflected in the reports about mathematics achievement within and outside India as well as the complexities of a school system which, despite democratic ideals, has difficulty serving the needs of all students. This study provides snapshots of three mathematics classes that embody some of these complexities to provide a balanced account of conditions that either support or detract from the goal of high achievement in India.