# Weird maths. At the edge of infinity and beyond

This book is a joint venture of an experienced science writer (Darling) and an exceptionally bright young math student (Banerjee). The result is this book popularising mathematics by presenting a set of curious/interesting/surprising (I would not call them weird) mathematical facts in such a way that they are easily accessible for the layman. The topics they cover are close to the topics that are also discussed in other books written with the same objective. There is of course always a new approach and there is always something to learn from a surprising fact or an unexpected link that is made. The authors have adopted the rule that also Hawking used: avoiding mathematical formulas in a popular science book. Each formula would presumably halve the number of readers. Besides symbols like ℵ and ω (when discussing infinities) and notations like 3↑↑3↑↑3 (when discussing large numbers) there are only very few equations or formulas. The authors state in the preface: If we can not explain it in plain language, then we don't properly understand it. This does not mean that hard topics are avoided since there is quantum theory, cosmology, and physics, as well as the foundations of mathematics with for example Gödel's theorems. According to the preface, Darling is responsible for the philosophical and anecdotal aspects, the relation to music, and he polished everything into the final text. Banerjee was more involved with the technical aspects including large numbers, computation, and prime numbers. The result is a pleasant read that anyone with only a remote interested in mathematics will enjoy.

The breadth of the topics covered is too wide to enumerate them all, but to give a rough idea, what follows is a fistful of topics that are discussed.

Historically mathematics is of course inspired by the necessity to count and by our surrounding physical world and the stars up above. But how would a four-dimensional being see our three-dimensional world? For us hard to imagine, but mathematics has no problem to function in higher dimensions.

With probability one can simply explain the birthday paradox, but it is also essential in quantum theory which is hard to understand, certainly when it eventually leads to vibrating strings in an attempt to construct a theory of everything. In chaotic systems such as the weather, the smallest perturbations, in spite of all the laws of probability, may prevent any valid prediction. On the other hand probabilistic systems can obey simple rules like in Brownian motion, or it may generate complex structures such as fractals. Think of automata like the Turing machine or Conway's Game of Life. If we can model a system, this does not mean that it is practically computable because one can hit the boundaries of complexity like problems of class NP.

Music and prime numbers are classical topics for books like this one. With the title "music of the spheres", there is a reference to Kepler of course but the story of that chapter also meanders by mentioning the music disk sent into space in the SEFI project as well as singing whales. Obviously there is an obligatory extensive discussion of the mathematics of music. The next chapter is discussing the unavoidable prime numbers. We meet for example the 17 year cycle of cicadas, the Ulam spiral, the Riemann hypothesis, and the twin prime gap.

Two more classical recreational topics are game theory and logical paradoxes. Game theory is discussed in connection with computers playing chess against humans and later also the more complex game go, but game theory can of course be applied to other games as well. One may for example investigate whether winning strategies exist when the human player can start? Game theory may have been developed to help people win an entertaining game, but when it was applied to very real economics, politics and other modelling and optimization problems it became a serious mathematical subject and John Nash won even the Nobel Prize in economics and the Abel Prize with his results. Paradoxes, the foundations of mathematics, and logic get their separate chapter. They may be applied to entertaining logical puzzles, but when digging a bit deeper, one bumps into much harder problems and the chapter ends by mentioning surprising mathematical results such as for example the Banach-Tarski theorem (a solid ball can be cut up into 5 pieces such that these can be reassembled into two solid balls of the same size as the original).

With large numbers and transfinite numbers we are back in the realm of numbers and mathematics. Infinity and the orders of infinity are discussed including the continuity hypothesis and the existence or not of the absolute infinity Ω. Big numbers (the really really big ones) like googol, googolplex, power towers (as introduced by Knuth), Graham's number, TREE(3), and several other numbers are featuring in a chapter that is less common in popular math books. Yet there is a subculture of googologists challenging each other in competitions to define ever larger (finite) numbers.

The last two chapters dive somewhat deeper into the less elementary mathematical topics. Of a more geometric nature is a chapter on topology with objects like the Moebius band, the Klein bottle, and different kinds of geometry and how this is applied to our universe. Finally we arrive at fundamentals discussing the completeness of the mathematical system, Gödel's theorems, proof theory, the axiom of choice, the Peano calculus, ZFC axioms, etc. These are clearly among the more advanced topics discussed in this last chapter.

As can be seen from this (largely incomplete) enumeration of topics, the discussion is rather broad and sometimes also touching upon deeper theoretical problems. The text remains however very readable, even when more advanced problems are carefully dissected. This is a very nice addition to the popular math literature deserving a warm recommendation.

**Submitted by Adhemar Bultheel |

**5 / Feb / 2019