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A review of a book of mine called A Modest Collection of Impossibilities. It was written by Michio Sato and published in The Journal of Mathematics Education and Pedagogy.

A Modest Collection of Impossibilities

Karl Hede’s A Modest Collection of Impossibilities is a book written in the style of Euclid’s Elements, obviously inspired by Spinoza’s Ethica Ordinae Geometrico Demonstrata. Mathematical rigor is the main characteristic of the book, which gives it a rigid structure whose building blocks are formal definitions, theorems and proofs. But this doesn’t give A Modest Collection of Impossibilities the dull mood of a mathematics book at all. On the contrary, the witty style of Hede, combined with his highly original ideas, offers us an exiting reading experience.

The introduction, which is unexpectedly long compared to other introductions by the writer, begins with a quotation from Alice in Wonderland by Lewis Carroll. The Queen tells Alice that sometimes she believed in six impossible things before breakfast. Hede finds this claim utterly unrealistic, not because one cannot believe in impossible things, but because it is extremely difficult to find impossibilities. According to Hede, in a universe as rich as ours, a genuine impossibility is something quite rare and beautiful. Thus a collection of impossibilities is useful for both educational and artistic reasons. After supporting this point of view with examples from the history of science, our writer asks the following question:
“Is it possible to construct three dice, say A, B and C, such that, if you roll A and B, the possibility for A to win is higher; if you roll B and C, the possibility for B to win is higher; if you roll C and A, the possibility for C to win is higher. (Here, a die is a cube with natural numbers written on its faces. Different faces can have the same number. Higher numbers win.)

Then he proceeds as follows:
“Let’s paraphrase the problem: Is it possible to have three dice, A, B and C such that A is better than B, B is better than C and C is better than A? Since this would mean A is better than itself, which is absurd, we can conclude that it is impossible. Or can we?”

Hede plays with the reader and deceives him/her to illustrate his point. The issue is that there do exist such nontransitive dice –a simple example is 1, 4, 4, 4, 4, 4; 3, 3, 3, 3, 3, 6; 2, 2, 2, 5, 5, 5. So one cannot rely on intuition, which is a rather crude tool for delicate problems like this. Therefore formal methods are necessary. This is how Hede justifies the style of the book.

The rest of the book consists of actual impossibilities, which are ordered with respect to the complexities of their proofs. The ordering also allows Hede to use certain theorems as lemmas in later parts of the book. The first impossibility is the impossibility of kissing ones own nose on the mirror, derived as a corollary of the you-can-only-kiss-your-lips-on-a-mirror theorem. Although the statement seems obvious, the proof is not too short and it makes nontrivial use of optics and human anatomy.

In the first proofs each step is carried out carefully to demonstrate the difficulties of the formal method. Later, Hede does not deal with all the details and leaves some of them to the reader as an exercise. After a few more instructive examples, Hede moves to more technical topics. For instance, he deals with some of Murphy’s Laws and proves that no matter how hard you try, you cannot push a string. (The proof of this claim is slightly involved because a string can actually be pushed, but not by you.) Hede also proves a theorem justifying the name of MCI: It is impossible to have a comprehensive collection of impossibilities.

The unifying approach of the book allows Hede to reproduce some of the classical impossibility results from a diversity of disciplines. Mathematical logic, which is notorious for producing negative results, is the most notable of them. Hede mentions Goedel’s incompleteness theorem, Tarski’s theorem on the undefinability of truth and Turing’s theorem on the unsolvability of the Halting Problem. Bell’s theorem on nonexistence of a theory of ‘local hidden variables’ and Heisenberg’s uncertainty principle are the examples Hede took from physics. A somewhat surprising impossibility discussed by Hede is the economist Kenneth Arrow’s theorem which simply states the nonexistence of a perfect voting system. All these impossibilities are praised in the book due to their deep implications and the formality of methods used to reach them.

The last theorems of the book are dedicated to religious themes, mainly nonexistence of various kinds of gods. (Existence of god is one of the favorite issues of the writer. See, for example, Wine in The Book of Real and Imaginary Drugs.) In this part of the book, Hede first takes a look at the classical approaches used to prove the existence of god. After criticizing, or even ridiculing them, most notably the ontological argument of Anselm of Canterbury, he discusses Goedel’s proof. He thinks that Goedel’s derivation –a twelve steps proof in modal logic– is correct but his notion of ‘positive property’ is too vague. By restricting Goedel’s notion of positivity and adding a twist to his argument, Hede manages to prove the nonexistence of some ancient Greek gods, but he cannot go any further because of technical difficulties.

A Modest Collection of Impossibilities is an extraordinary textbook and, like most text books, it requires a respectable amount of effort. There are exercises which should be solved in order to understand certain arguments. But it is definitely worth it. Hede outlines a unique journey among impossibilities each of which is “as beautiful as a bottle made by Harry Eng.” If you are interested in formal methods used in unconventional ways as well as the famous negative results like the uncertainty principle, this book is going to be the ultimate source in your library.

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