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The Best Writing on Mathematics 2020

Mircea Pitici, ed.
Princeton University Press
Publication Date: 
Number of Pages: 
The Best Writing on Mathematics
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Adhemar Bultheel
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As in the ten previous anthologies, this book collects the publications from roughly the previous year that were written about mathematics or mathematicians. In this case, the period from late 2018 through 2019 is covered by 20 contributions. The concept of the series has not changed and has been discussed already in reviews of the previous volumes 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, and 2019. What has changed is the cover design and the 2020 pandemic that disturbed fundamentally every routine. Nevertheless, Pitici once more did a marvellous editorial job as he has been doing since 2010. Unfortunately, in this exceptional year, he has not been able to be as exhaustive as he was before, but that isn't an appreciation of the quality of the papers nor of the selection made. What may have suffered most from the pandemic circumstances in which this collection had to be compiled, is the survey that Pitici gives of the avalanche of books, special journal issues, and papers that fit into the concept of the series, but that were not reproduced for space saving reasons. For electronic versions there is no problem when libraries are closed, but not all paper-only publications could be consulted as usual.
Recall that the papers eligible for these volumes are not the usual technical mathematics research papers. They are in a sense `meta'-mathematics papers, discussing general concepts, applications, historical, philosophical, computational, or recreational aspects. They are most often written by mathematicians, but the authors are addressing the general public and not their professional peers.
In the queer year 2020, it may not come as a surprise that the first paper selected is about the math of viral infections (in this case HIV). Next are some papers dealing with uncertainty, complexity and chaos. These enlighten why avalanches are explainable but unpredictable, why power and wealth accumulate spontaneously in an unregulated community, and why politicians move to the centre in a democratic regime. We learn also why politicians can make wrong decisions in their mobility policy if they promote cars with a low fuel consumption based on miles per gallon instead of gallons per mile, so that they are confusing arithmetic and harmonic means.
The recreational aspect is represented by a paper about the Rubik's Cube and one about the games of Sid Jackson. Everybody knows the beautiful fractals of the Mandelbrot and Julia sets, and exploring them has become partially recreational, but also the interaction of Möbius transforms give rise to similar fractals called Kleinian and quasi Fuchsian limit sets, as described by Chris King, and which can be enjoyed looking at his website.
Some papers dig a bit deeper into mathematics. Like the one where it is explained how algebraic geometry is a mathematical tool that allows us to move from classical Newtonian physics to modern quantum physics. Also, a discussion of the three-body problem, the explanation of a hyperbolic 3-manifold, and higher dimensional geometry used for model spaces are in this category of more mathematically flavoured contributions. And in the same vein, there is a computer science paper about the sensitivity of Boolean functions.
Questions from number theory are usually easy to formulate and often used in popular publications. In this case, it is represented by a paper about the historical evolution that resulted in a proof that there are only nine Heegner numbers. Also, a bit of integer arithmetic is needed to understand how the date of Easter can be computed and in particular how Gauss did it (and made a small mistake). And speaking of integer arithmetic, there is also a paper showing how to multiply large numbers in a fast way.
A bit of an ironic text is written about a paper that appeared in a medical journal in 1994 where a researcher discovered the trapezoidal rule to compute the area under a curve defined by a set of measurements. This is a serious warning that teaching an integral as an anti-derivative is dangerous and that today numerical tools should be learned as well.
Towards the end, more papers deal partially with education. One paper emphasizes that mathematics can prove a certain theorem and we can learn how to compile that proof, but it is equally important that we can put ourselves outside the proof and explain why something is true. Another paper about statistical inference is a plea to use uncertainty intervals rather than hypothesis testing.
Finally, I mention a philosophical paper by Paul Thagard, known for his many books about the philosophy of science. Here he gives an excerpt from one of his books. It deals with mathematics, which takes a particular position within science, and its relation to the real world differs from other sciences. His considerations relate to the old question of whether mathematics is invented or discovered, and he discusses also Tegmark's claim that we live in a physical reality that is just a mathematical structure.
I cannot imagine that, if you are only remotely interested in mathematics and its role in our society, then your curiosity will not be triggered by some if not most of the topics that I just mentioned. The great advantage is that you do not have to spend hours in the library looking for all these papers. There is no specialized journal that collects most of this kind of papers. One has to search for them sometimes in the most remote corners of the vast literature. The enormous advantage of these yearbooks is that Pitici has done it all for you, and the papers are presented to you in a nice uniform format. It is always a pleasure to discover what the new collection brings and to browse through the list of his additional suggestions to discover what else there is to be explored.
Adhemar Bultheel is emeritus professor at the Department of Computer Science of the KU Leuven (Belgium).  He has been teaching mainly undergraduate courses in analysis, algebra, and numerical mathematics.  More information can be found on his home page.