The idea that there is a parallel between mathematics and art is not new and many have tried to express the experience of beauty in both, but it is not easy to make it explicit and to convey to nonbelievers. For visual arts, beauty is related to order, patterns, and symmetry, but not too much either, so that it doesn't become dull and boring. Also the link between music and mathematics has been explored a lot ever since mathematics was around. The link with poetry is less explored, and this is what this book is about.
Clearly, if this book should be readable by mathematicians and poets, the mathematics can't be too complicated. What we find here are often the usual topics that one may find in books on popular, hence easily understandable, mathematics such as games, history, and paradoxes. There is however also some probing in more difficult topics like topology, transcendental numbers, Gödel's incompleteness theorems, etc., and there are even proofs. Brilliantly elusive ones! Of course, each time a parallel is drawn between some mathematical concept or tool and the corresponding poetical one.
There are three parts in the book: (1) Order, (2) How mathematicians and poets think, (3) Two levels of perception.
The first part about order is clearly an attempt to define what arises our sense of pleasure or beauty in mathematics and in poetry. One element is that sometimes a seemingly complicated problem all of a sudden can become an stunningly simple solution by looking at it from a totally different angle. But the opposite is also true: sometimes a very simple conjecture may require a proof that is totally out of proportion. Historical examples of the latter are the proofs for trancendence of some real numbers, but also sphere packing, or as yet unsolved ones like the Collatz conjecture, and twin prime conjecture.
In this part one may also find a statement that not everybody may agree with. It is a classical subject of discussion: is mathematics discovered or invented? Aharoni says that mathematics is discovered and poetry is invented. His argument is that if a mathematician misses a theorem or a proof, then sooner or later another one will discover it. However, if a poem is not written, then it will be lost forever. Hence Aharoni is a supporter of formalism like Einstein, Hilbert, Cantor, etc., but there are equally illustrious mathematicians like Hardy, Penrose, and Gödel who thought otherwise. Personally, I am more a believer that mathematics is a result of both components like also Mario Livio does in Is God A Mathematician? (2009).
In the second and most extensive part, Aharoni explores the minds of both mathematicians and poets. Both their minds use images, sometimes use an oblique approach, compress statements, and they play a ping-pong game between the concrete and the abstract until some leap of insight erupts. Some mathematical proofs rely on a law of conservation like for example the number of transpositions in a sequence being even does not change no matter how much exchanges you make. The `conservation of truth' in poetry is related to fate like the inevitable fatum in Greek tragedies. The metaphor is a form of cross fertilization mixing concepts from different senses. It is quite often used in poetry, but this kind of insights can also be obtained in mathematics for example by solving a problem in discrete mathematics by applying a theorem from topology. Fantasy, imagination, and analogies are main ingredients of poetry as well as mathematics. However, care must be taken in mathematics, not to generalize too easily based on intuition and analogy. Tautologies and symmetries are are familiar to both the poet and the mathematician. With the hyperbole, one approaches the impossible, the infinitely small or infinitely large, so that paradoxes and oxymorons may be imminent, like the ones dealt with by Cantor and Gödel.
In the last short part, the tension between what we know and what we do not completely understand is explored. A poem may be confusing, extracting beauty from the strange and the unknown. Progress in mathematics is only possible by leaving the familiar and the habitual. When the known framework leads to results causing alianation and estragement, a leap of faith into the unknown becomes inevitable. After a while however, it will turn out to be a step that brings us one rung higher on the ladder of understanding.
After all these parallels between mathematicians and poets have been explored, Aharoni places ithis a bit in perspective because one obvious difference between both is that mathematicians most often collaborate and learn from each other, while creating a poem is a lonely and individual activity.
The book is the English translation of the original Hebrew edition from 2008. Of course there are many poems or quotes from poems included, most often in an English translation. Several of them are analyzed in view of the point that the author wants to illustrate. Although he has written several books with a philosophical inclination, he is a mathematician and I believe this book is in the first place written by a mathematician for mathematicians. In any case the mathematical aspects treated in the book are more elaborated than the poetical aspects. It is not a deep theoretical philosophical work, but basically a collection of exemplary illustrations of the point the author wants to make. As I mentioned above, many of the examples are also found in other popular math books, but it is of course most inspiring to see them placed against a background of poetical analogs and vice versa. Apart from the fact that the details and the history of the mathematical stories are most informative, and without them, the book would be skinny and probably not very enjoyable, I do not always see their direct relevance to make the link with poetry. There are some appendices explaining some of the mathematical terminology and poetical terms and figures of speech, but since there are so many different facts and persons appearing, it is difficult to trace them if you pick up the book at a later stage. It would have been nice to have a subject index to facilitate that. But I very much enjoyed reading the book the first time though.