Imagining Numbers (particularly the square root of minus fifteen)

I never saw the movie Titanic. Despite repeated urging by friends and colleagues who said I would enjoy it, I could never get over the fact that I knew how the story ends. I had a similar reaction to Barry Mazur's new book, and I suspect that many mathematicians will find themselves responding similarly. To which Mazur could reasonably reply, "But this book is not written for mathematicians."

If you are proficient at elementary algebra, know how to manipulate complex numbers, feel comfortable with the concept of an abstract number system, and have a good knowledge of the historical development of complex numbers, then, like me, you are probably going to find yourself continually frustrated by Mazur's slow, meandering approach, and by the frequent side excursions through literature and poetry. In response to which reaction I would line up with Mazur and echo his cry that this book is not written for you.

There is of course an innate interest that a book about the nature of mathematics written by one of the world's leading mathematicians holds for any mathematician. For instance, in "popular expositions" that have become minor classics, G. H. Hardy wrote about his own experiences as he came to the end of his research career and Jacques Hadamard reflected on the nature of mathematical thinking. Mazur's goal is to explain, to non-mathematicians, what it means to imagine a mathematical abstraction, specifically the imaginary numbers. This goal, I think, makes his book one that, unlike my other two examples, has little that will interest mathematicians.

The question is whether he will succeed in making a connection to his intended audience. He certainly pulls out many stops in his attempt. As a starting point, he takes the experience, which he reasonably assumes is one that his readers are sure to have had, of imagining in general — for example, imagining the color yellow. Then, with frequent allusions from literature and poetry, he weaves his way through the — to mathematicians — familiar pedagogic path that leads to complex numbers, part conceptual, part historical.

As someone who is very familiar with Mazur's topic, and indeed someone who has also put some effort into explaining it to non mathematicians, I am not at all suited to really judge how well he succeeds. My best guess, however, is that he might very well do so. The very diversions through literature and poetic imagination that I found, well, frankly diverting, might be exactly what will make his book work for his intended readers.

To give you a flavor of the fayre Mazur serves up, let me quote you two passages from the beginning of the book.

Consider the range of our imaginative experiences. Consider, for example, how immediate is the experience of imagining what we read. Elaine Scarry has remarked that there is no "felt experience" corresponding to this imaginary act. We experience, of course, the effect of what we are reading. Scarry claims that if we read a phrase like

 

the yellow of the tulip

we form, perhaps, the image of it in our mind's eye and experience whatever emotional effect that image produces within us. But, says Scarry, we have no felt experience of coming to form that image."

A short while later Mazur continues:

I want to think about the inner articulations of our imaginative life by "re"-experiencing a particular example ... It might be described as a moment of restless anticipation in the face of a slowly emerging act of imagining.

The example Mazur is referring to is the acceptance into the collective mathematical imagination of the square root of negative numbers. Insofar as Imagining Numbers has something to offer mathematicians, it is the approach he adopts, and the literary devices he employs, to try to convey something of the act of mathematical imagination to readers who have never knowingly experienced it. (I say knowingly here since any reader has surely learned to view counting numbers as concrete objects, even the negative ones, but that cognitive step is thrust upon us when we are very young and few will have paused in adult life to reflect on the nature of numbers.)

For me, once I had read along with Mazur for a while, and got a sense of his approach and style, I found myself simply skimming through the book, and had I not promised MAA Online editor Fernando Gouvêa to write a review, I would surely have given up at an early stage. My guess is that most mathematicians will do just that. Nevertheless, I am going to recommend that any mathematician who reads this review should buy a copy of the book. By all means take a look at it and, like me, see how Mazur goes about his task. But that's not why you should buy it. Rather, you should give the book to a friend, colleague, or family member who constantly tells you that they cannot imagine what it is like to do mathematics, and whom you suspect harbors the view that all you do is follow obscure rules for shuffling meaningless symbols on a page. I don't think there are any guarantees that Mazur's elegant and somewhat Quixotic prose will connect with them and cause a light bulb of comprehension to come on in their heads. But for some it surely will — particularly people with a literary streak. In which case, this little book will change the way the recipient views you. They will start to think of you as cool, artistic, and creative. Just as you yourself do. Not bad for just $22 in hard back.

Keith Devlin is a regular columnist for MAA Online ("Devlin's Angle") and the author of several books on mathematics for the general reader.