The Best Writing on Mathematics 2018
Here is number nine in this series in which Pitici collects previously published papers about mathematics and how it relates to society. See also the reviews of the previous volumes on this EMS site. As usual the papers are harvested from magazines and journals that are not in the mainstream of specialised mathematical research journals. Although they are about mathematics, they do not really involve new mathematical results as such. They are rather about mathematicians and the way they do mathematics. The topics are familiar for those who know (some of the) previous volumes. The papers deal with for example the teaching and the history of mathematics, or things that inspire(d) mathematicians. Others are about paradoxes, games, or recreational aspects. Eighteen papers were reprinted in a uniform way for this volume, all published originally in 2017. What follows is a quick survey.
The first paper is a contribution by Francis Edward Su, past president of the MAA and strongly involved in methods to transfer mathematics to students. His paper is a plea arguing that practising mathematics cultivates virtues that bring humans to completion (he uses the term "human flourishing") because mathematics answers to five basic human desires: play, beauty, truth, justice, and love.
Margaret Wertheim argues that people often think of mathematics as being formulas and abstractions, i.e., fabrications of human intelligence. However, much of our mathematical inspiration comes from nature. Nature does not have the advanced human intelligence, but it is just "living the mathematics". It does expose all the juicy mathematical charateristics such as patterns, fractals, hyperbolic geometry, etc.
On a more philosophical level is a contribution by Robert Thomas who explains that the aesthetics of mathematics is not only beauty. Just being interesting is also an important aesthetic category. If your paper is not interesting, it will not get published.
Computers certainly have influenced the way we do mathematics. Satisfiability (or SAT) is a concept of computer science. A Boolean formula is satisfiable if it is possible to find a model or interpretation of the variables that makes the formula true. Here computers can come to the rescue giving automated proofs that recently solved open problems where traditional proofs are infeasible because too many cases have to be considered separately. Marijn Heule and Oliver Kullmann give an introduction to SAT and make a point of the fact that this may give insight in the complexity of a problem and it may open some perspectives for Ramsey theory. So, what is needed is that the proofs themselves should become objects of investigation. Also in the realm of computer science is the paper by Peter Denning who gives his definition of the recent paradigm of computational thinking and discusses its advantages and limitations.
Mathematics can be inspired by physics, as Robbert Dijkgraaf illustrates with quantum physics. The concept of mirror symmetry such as particle and wave interpretations of the same (quantum) mechanical phenomenon may have consequences for mathematics. These different interpretations require different mathematical tools and methods. So perhaps mirror symmetry also connects different mathematical disciplines as two sides of the same reality.
But also the playfulness of mathematicians can lead to interesting problems. Three examples can be found here. Erik Demaine et al. discuss planar configurations that can be produced in Tangle toy by joining quarter circle pieces together to form a closed loop. Like in other papers in which father and son Demaine are involved, there are some interesting mathematical questions that can be asked about something that starts from playful amusement. And these are just the planar problems, predicting much more involved questions since the loops can also be constructed in 3D. James Grimm describes the design of dice to play a game where the winning strategy is nontransitive, meaning that one state winning over another may lead to circular arrangements like in rock, paper, scissors. And Arthur Benjamin et al. analyse the statistics of a kind of Bingo game to explain a paradoxical outcome. In the same vein but less straightforward is the analysis of the Sleeping Beauty problem by Peter Winkler. The problem was formulated around mid 1980's and has caused a lot of controversy since. The Sleeping Beauty has to undergo a sleeping experiment, with a final result that depends on a coin toss. She has to estimate the probability of the outcome of a coin toss when she wakes up at the end of the experiment (the details can be looked up on the web). Winkler does not favour one solution but he gives arguments to support the different possible probabilities that have been proposed in the past.
There are also papers of a more historical nature. José Ferreiros discusses the paper by E. Wigner in which he introduces his often quoted "unreasonable effectiveness of mathematics in natural sciences".
Daniel Mansfield and N. Wildberger show that the Babylonians knew Pythagoras' theorem long before Pythagoras (which is probably not a surprise), but also that their study of (rectangular) triangles was based on ratios of the sides of the triangle. Since they used a number system in base sixty, they were able to produce some rather accurate tables that we now call goniometric tables, but the surprise is that they did it independently of the angles.
Isidore of Seville (560-636 CE) was a Spanish bishop and scholar. He authored the encyclopedic Etymology. In his time mathematics was part of the priesthood education. It is via sources like this that classical Greek mathematics was passed on to the Middle Ages. Isabel Serrano et al. discuss what sources he may have used for his description of the classical Quadrivium.
Michael Barany shows how after World War II mathematical awareness became an issue (and it still is today) and that Mina Rees has been instrumental as the first president of the American Association for the Advancement of Science (AAAS) to promote the idea.
In the vein of education and teaching, there are four papers: Methods, concepts, and techniques of interdisciplinary teaching is discussed by Chris Arney. Nancy Emmerson Kres goes through a list of essential questions to ask when solving a problem. Benjamin Braun et al. explain what active learning means when teaching mathematics. And finally Caroline Yoon wants to remove the dichotomy between mathematics and languages (sometimes almost considered to be opposites of each other) by illustrating that there is a clear parallel between them: writing a text can be seen as a task of modelling (getting insight), of problem solving (reorganizing the text), and even of giving a proof (convincing your reader).
Pitici gives also a list of around sixty interesting books, and a long list of "notable writings" (journal papers, or special issues, book reviews, teaching tips, interviews, biographies, obituaries). The selection from this vast amount of the eighteen papers included here is obviously a personal choice made by Pitici. If you are interested in just one of the broad set of topics that were touched upon in this volume and you want to know more, then you will certainly find a lot of inspiration to read. There is certainly enough material to keep you reading till the next volume of "The Best Writing in Mathematics" to which I am already looking forward to now.