The title of this book refers to circular reasoning. The snake that bites its own tail. This may play tricks on us with the logic of our arguments and it will almost always result in a paradox. In our binary Western culture we are used to things being either true or false. Circular arguments make us believe that our arguments result in a proposition being true, but that then implies that the same proposition must be false which implies that it must be true etc ad infinitum. A paradox. The simplest examples of such statements are "This sentence is false" or "I am always lying". These are examples that everybody knows and where the circularity if blatantly obvious. However, the same kind of circularity can be much more subtle and well hidden, which often leaves us perplexed by a confusing paradox.
Circularity can be both a curse and a blessing. It is a curse when it creates problems that are actually no problems or definitions that are non-definitions so that the wrong conclusions are drawn. On the other hand, when a paradox is a challenge that pokes the minds of bright philosophers or scientists to develop the proper framework to solve the paradox, it can be a blessing. In this book, Aharoni shows us both sides, the dark and the illuminating side of circularity.
Aharoni starts by giving several examples of paradoxes and points to the (sometimes hidden) circularity. What is tricking us is the self-reference. The solution of the paradox is often that we should not look at the proposition from the inside but we should place ourselves at the outside. It may then become clear that the proposition does not refer to itself but to something that is different from itself like in the above examples or a more explicit example: "the smallest number not defined by this sentence". More paradoxes can arise from logical errors, like if X is not true, then the implication "if X then Y" is always true, whether or not Y is true. It is wrong to argue that if Y is true, then X must also be true, just because the implication holds. This is an unpermitted inversion of the given implication. Zeno's paradox is based on the wrong assumption that an infinite sum of finite numbers must be infinite. The self-reference can be clarified by an analogy. Start with one grain of sand, which certainly is not a heap of sand. Keep adding one grain at a time. There is not a particular point where the set of grains magically turns into a heap of sand. So heaps of sand do not exist. Obviously a wrong conclusion. With every grain added, the definition of heap keeps shifting in our head. That's where the circularity is. Tricked by circularity, one can prove anything: the existence of the monster of Loch Ness or whatever. Some have tried to prove in this way the existence of God.
Next Aharoni introduces the reader to the problem of free will by starting with the Newcomb paradox. You have a 100 dollar bill and can drop it into a deep well or not. A never failing oracle says that it has predicted what you will do and acted accordingly in the past. A 1000 dollar was deposited on your bank account if you drop the 100 dollar. Nothing has been deposited if you keep it. What should you do? Can one change the past by doing something now? If not, you should keep the bill, if you believe the oracle, you should drop it. The origin of the dilemma was a problem in game theory. Formulated as above, it directly leads to the dilemma whether everything is predetermined and thus nobody is responsible for whatever he or she does (fatalism and the idle argument), or is there actually free will. Where is the circularity? Aharoni explains that past and future are linked through decisions that we make in the present. What causes the paradox is that we are trying to make a decision about the process of making a decision, which is self-referencing. Only Baron von Munchausen could pull himself up by his own hair. The deliberation is about a causal chain involving the deliberation itself.
The mind-body problem is similar. How one should link the non-physical sensation of pain and the physical event of the pricking of a needle? Here the circularity of all the philosophers discussing the mind-body gap is that they are talking about sensations in other people's minds, hence they refer to consequences or externalization of the sensation, which is quite different from the sensation itself. The sensation exists only in your own mind. It can only be experienced from the inside. You may observe a person having a sensation, which is different from observing the sensation into yourself. When the observed person and the observer are the same, then the observation of the sensation becomes the sensation itself.
So far for the dark side of paradoxes that play tricks on us. The second half of the book is about the bright aspects by bringing an account of stories about theories that resulted from successful attempts to resolve a paradox. These theories are better known to mathematicians and do not need an extensive clarification here. Aharoni however is addressing a general public. So he carefully explains countability and the diagonalization process to show that the rational numbers are countable and that the reals are not. Also the foundation of set theory was clarified thanks to the well known self-referencing Russell paradox of the set of all sets that are not a member of themselves. Zermelo and Fraenkel came to the rescue of the barber who cuts the hair of all the villagers who do not cut their own.
Pushing the limit somewhat further, after the introduction of Boolean algebra and the formality added by Frege, the question rose to mechanically derive all possible theorems. A genuine target it seems and Russell and Whitehead devotedly set to the task of writing their Principia Mathematica. It was however the incompleteness theorem of Gödel that caused a Copernican revolution in mathematics. Solving another case of self-reference: the (mathematical) reasoning about (mathematical) reasoning. The liar's paradox, equivalent with "this sentence is false" became "this sentence is not provable". Gödel's theorem is closely related to the mechanical (algorithmic) verification of the validity of a proposition, which links it to the Halting Problem. There the question is whether the the machine stop verifying the validity of its input after a finite number of steps or will it run forever? This was solved by Turing (with his Turing machine) and independently by Church (using Lambda calculus). The Turing machine is the theoretical machine that later became the general purpose computer. A most exciting serendipity if this may be considered the consequence of solving a paradox.
In an appendix some further explanation is given for those who feel a bit more mathematically inclined, explaining in more technical terms the diagonalization procedure and the proof that the reals are not countable. It is also clarified that there is no contradiction between Gödel's completeness theorem (of the Frege-Russell-Whitehead system) and his incompleteness theorem (of the set of Peano axioms).
The whole book is written in small chunks of only a few pages that introduce simple concepts, each one easily digestible, but at the subtle points one has to stay focussed not to loose track. No mathematical knowledge is assumed, not even for the appendices, but of course a straight mind to make sound logical deductions is needed. Much more than in the brief outline given above, Aharoni relates the material discussed to historical ideas and philosophers. It is pleasant reading, with a whiff of humor. The humor coagulates especially in a longer section where Aharoni discusses why this self-reference shows up in a particular type of jokes, and why we think of it as funny.
I did enjoy reading (and re-reading) this book very much. Reading it deserves a warm recommendation not only for mathematicians but for anybody (especially if you have the slightest interest in logic). Aren't we all are reasonable beings that are supposed to handle according to some logic most of the time. I can safely end with a self-referencing conclusion: this book makes you think about how and what you think you are thinking.