Book Review: Arieh Ben-Naim, Entropy Demystified

 The author explains entropy essentially with a dice-rolling experiment: Given N dice, e.g. all with the six on top, choose any dice (chance no. 1) and roll (chance no. 2). If the experiment is carried out often enough, the sum of the numbers on the dice approaches a value quite stably. It is very likely that the value 6N will not be reached again if N is sufficiently large.

It is clear from the experiment what is meant by (generally not directly measurable) specfic events (the numbers on the individual dice) and by (directly measurable) "dim" events (the sum of the numbers on the dice). I would have been interested to know whether the relationship between specific and measured events must always be linear.

The fact that 6N will most likely not be reached again is an illustration of the "arrow of time".

There are many more specific events that lead to a sum 3N or 4N than to 6N. This seems to me to be the main point of the book.

The author equates "entropy" with search cost, i.e. as a measure of missing information. I would have liked to see a proof of equivalence to the usual definition of entropy (sum of p log p over all p).

These are definitely important insights for me. I'm not sure if I needed to read this at this length to understand it. Why does this need to be discussed for 2, 4, 10, 100 and 10000 dices? And then again imagine that it's not numbers of dice but colours, smells, tastes or sounds?

I don't find the book entertainingly written. The examples from physics (Bose-Einstein configurations, Fermi-Dirac configurations) did not help me as a non-physicist. While reading, I had the idea that one could write a book "Entropy Mystified", where the many applications of this ingenious concept are presented.

Book Review: David Foster Wallace, Everything and More. A Compact History of Infinity.

 I came across David Foster Wallace through his famous speech "This is Water". I then read some of his essays, about lobsters, about cruises, about severe depression and about how few good books there are on mathematics that can be understood by lay people.

The last essay in particular, "Rhetoric And The Math Melodrama", made me curious about how Wallace himself would write such a book on mathematics. And indeed, Everything and More is a unique non-fiction book.

I like the personal references: Wallace's niece is mentioned, the high school teacher gets a place of honour. I like how Wallace sketches the human side of the mathematicians (Kronecker, Cantor, Weierstrass, Dedekind et al) with one paragraph, I had an immediate image, and contrary to some biographies, I think these images are plausible. 

I also like how he takes elements of textbooks on mathematics and plays with them. Abbreviations suddenly appear that have to be remembered, proofs, "interpolations". It may give a layman a sense of how mathematics is often written then and now.

The many footnotes and "IYI" ("if you're interested") insertions make the revision processes visible to me. Sometimes there is direct reference to notes from the editor, sometimes a footnote nullifies itself, "but in an interesting way". This brings Wallace closer to me, I am not only concerned with the text and its content, but also with Wallace, with the thoughts that (might) have led to the text.

From a mathematical-philosophical point of view, I find §1c particularly interesting, where two types of abstraction are presented: one where the concrete is inferred to the abstract ("horse", "forehead", "horn") and one where different abstractions are linked together ("unicorn"). I also find interesting, even if I don't yet know exactly what I will do with it, the criticism of the "Theory of Types" in §7f, according to which it is be a philosophically bad idea to derive definitions from paradoxes. 

Mathematical induction, the epsilon delta technique and the diagonal proof are presented as techniques for dealing with the infinite. From my point of view, the compactification is missing, although the idea was touched upon. I would also have liked more on the axiom of choice. But I can understand that in a book where Cantor is the focus, it is only mentioned. (For those interested in mathematics: Eric Schechter in Handbook of Analysis and its Foundations really goes into this topic intensively from a practical point of view).

In short: a beautiful book that invites you to pay attention not only to the content but also to the form.