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Circularity: A Common Secret to Paradoxes, Scientific Revolutions and Humor

Ron Aharoni
World Scientific
Publication Date: 
Number of Pages: 
[Reviewed by
Darren Glass
, on

As I write this review, it is graduation season, and many of my students are looking for their first “real” job. And they are discovering the harsh fact that many employers only want to hire someone who already has experience. But how can you get experience if you need experience to get a job that will give you experience? Ron Aharoni describes this phenomenon as “circularity”, and he has dedicated his new book to the topic. Circularity: A Common Secret to Paradoxes, Scientific Revolutions and Humor takes a wide range of topics, primarily in philosophy and mathematics, and frames them in terms of this notion.

I do mean a wide range of topics. The first part of the book, entitled “Magic”, introduces the idea of circularity and discusses it in terms of paradoxes, giving an overview of a number of examples. I suspect that most of these examples will be familiar to readers of MAA Reviews, but are probably less familiar to the general audiences that this book is designed for. Aharoni discusses Zeno and Cicero, tortoises and heaps of sand, Winnie the Pooh and the Loch Ness Monster. At several points in these chapters, the author invokes Martin Gardner and Raymond Smullyan, both of whom wrote at various times about the same kinds of issues that Aharoni is interested in. Unfortunately, these invocations only serve to remind this reader that Gardner and Smullyan are better writers who engaged the topics with more detail and wit than Aharoni does.

The next two parts of the book examine two questions that philosophers have been looking at for many years, discussing them through the lens of circularity. First he tackles the question of whether humans have free will, mostly focusing on Newcomb’s Paradox. He then goes on to discuss The Mind-Body Problem, which he phrases as the question of how physical concepts can be constructed by our mind, even though our mental constructs are created by physical things, leading to a different type of circularity. His discussions of these topics are interesting and touch on many different approaches to reconciling these seeming paradoxes. They are brief and sometimes his connections to the idea of circularity seemed a bit forced, but did I learn something new, perhaps because I have not been trained as a philosopher.

I have, however, been trained as a mathematician, and the next three parts of the book deal with ideas of circularity in mathematics, so again did not contain much information that I (and likely you if you are reading MAA Reviews) found new. The first of these parts deals with the nature of infinity, and in particular the notion of different sizes of infinity. Aharoni chooses not to get too mathematically rigorous, but he does describe the nature of bijections of sets and cardinality. This all builds to his own version of Cantor’s Diagonalization Argument that the power set of any set is larger than the set itself, which he proves with an argument using Mr. Potatohead toys. His argument is cute, but I am not sure makes things easier to understand rather than harder. He discusses the more standard version of the proof, as well as some of the mathematical details of his various arguments, in a final part of the book (essentially an appendix) entitled “For The Experienced Hiker”.

After Cantor and Infinity, the next part of Aharoni’s book is about Gödel’s Incompleteness Theorem, a topic which seems to be endlessly fascinating to the general public despite the fact that it is notoriously difficult to explain well. Given that he dedicates less than twenty pages to the topic and wants to keep technical details to a minimum, Aharoni does a pretty good job of getting the basic ideas across. At the same time, I suspect that most readers will be frustrated either with the details he does include or with the details he chooses to leave out, and the target audience that will want this level of detail seems to me to be small.

The sixth part of the book is called “Turing Invents The Computer”, and indeed it does start with a discussion of the Halting Problem and its connection with the work of Gödel and Cantor. He quickly moves on, however, to a very brief description of artifical intelligence, and tries to bring all of these ideas back to the notion of circularity, which he had moved away from somewhere along the way. The last chapters of this part take a very different turn, though, and discuss the relationship of circularity to humor. Or at least that is what the author claims he is doing, but mostly this chapter seems to serve for him to list some of his favorite jokes. Admittedly, humor is subjective, but clearly Aharoni’s sense of humor is not the same as mine; to me this chapter read like a late-night conversation with an Uncle who likes to alternate between corny puns and somewhat risqué anecdotes. In some ways, the whole book reminded me of this type of conversation: there were parts I found interesting, but just as often as I wanted to learn more I often lost sight of the big picture.

You will notice that I keep referring to “parts” of Aharoni’s book rather than “chapters”. That is because each of these parts is divided into 8–10 chapters that are each only two or three pages long. At times this feels like a nice feature of the book, since you can dip in and just read a short bite before moving on. But at other times I think it forces the author to be overly brief, and so not dig as deeply into a given topic as it deserves. It also meant that I often was left unclear on how certain topics fit into the big story that Aharoni clearly wants to tell. He clearly has some interesting ideas and at times I found his writing to be quite engaging. Perhaps in a different format this book would have worked for me, but as it is I feel uncertain who would benefit from reading the book.

Darren Glass is a Professor of Mathematics at Gettysburg College. He can be reached at In the spirit of this book he wants to note that it is worth reading other reviews of it, such as this one.

  • The Protagonist of the Story
  • A Nation Enamored with Paradoxes
  • From Proof to Paradox
  • From Paradox to Proof
  • The Paradoxes of the Direction of Time
  • The Paradox of All Paradoxes — the Mind-Body Problem