*Coming To Our Senses* is an important book that delivers its promised message on the importance of “perceiving complexity to avoid catastrophes” quite well. The book is very well written: the author weaves a complex web of skillfully interconnected tales in which the recurrent theme is that failure to perceive complexity leads to disasters on a multitude of levels and circumstances. The book serves as a warning for practitioners, scientists, and lay people on the use and misuse of knowledge. Having said that, in my view, the exposition neglects many issues in the development of the sciences, an editorial choice that may be perceived as bias.

There are numerous reviews of the book that address it from the point of view of its intended audience, namely, non-scientists. Since this review is intended for mathematicians and other scientists, I will do a disservice to the book and present a review which scrutinizes the book as read by a mathematician. I choose to do so also for my own comfort, as it allows me to bring the book to my playing field. I apologize in advance to the author, who can certainly be forgiven for her somewhat controversial choice of mathematical point-of-view, but perhaps less so for her rather simplified, and perhaps biased, presentation of scientific lore.

Starting on page 115, the author compares fractal geometry with Euclidean geometry, stating that

Mandelbrot’s fractal geometry is to conventional Euclidean geometry what the direct perception of the structural configurations that specify real-world items is to our limited pictorial representation of those items.

The discussion starts off with the comment that

Benoit Mandelbrot’s fractal geometry emerged directly from his perceptions of the natural world, not from a set of axioms, as is more typical in mathematics.

I must first address this comment at some length. The emergence of a new theory in mathematics is never “from a set of axioms.” Instead, a new theory is the specification of *new* axioms. Hilbert’s strict view of mathematics as a purely formal game may only begin once the players agree on the axioms, but mathematical creativity and genius lie not just in finding clever proofs from the axioms but also in laying down good axioms. There is an immense difference between investigating a theory, which might indeed be a completely formal (yet creative) process, and creating new theories. It is never the case that a mathematician wakes up in the morning, chooses some axioms at random, and start investigating this “new theory.” Out of a vast infinite ocean of axioms only a select few are chosen as the basis of meaningful theories worthwhile of investigation.

Mathematics is about modeling in a very broad sense. Axioms are chosen to model something. The resulting theory is deemed worthy if its predictions match sufficiently well with what we are trying to model. So, for instance, three dimensional Euclidean space is a model of the familiar three dimensional space we live in. Mathematical theorems about what is possible or impossible with the mathematical model translate to predictions about the world. It is famously said that all models are wrong (otherwise they would not be models), and that is quite fine. For instance, the Banach-Tarski paradox is a curiosity of the mathematical model, not a fatal blow to its applicability.

Which brings me to the comparison between fractal and Euclidean geometry. Let's do it by addressing the following paragraph from page 118, with my comments in square brackets:

For example, consider the problem of measuring a coastline to make decisions about defending the country’s shores or shipping its goods. Traditional measurements of physical phenomena typically use whole numbers [I do not know what the author meant to say, as this is clearly incorrect], which reduce the complexities of those things to fit the limitations of our numbering system [again, I do not understand what the author intended to say, as any measurement will have an outcome in some system of measurement, and thus any measurement reduces a physical phenomena to something much simpler]. Fractals, as their name implies, use fractions that lie between the whole numbers 1 and 2 to measure uneven structures such as trees, mountains, or coastlines [this is unfortunately phrased. Fractals don’t “use numbers.” Rather, fractal dimension need not be an integer (and it certainly need not lie between 1 and 2)]. Fractal measurements (called fractal dimensions) can record, for example, the many ins and outs of a particular coastline while whole numbers cannot [one should not confuse fractal measurements with fractal dimension!]. When fractal dimension is used, we get a much more accurate measurement of the coastline in question [the use of the term “more accurate” suggests that accuracy is an objective feature. It is not. The accuracy of a model depends on what the model intends to model, on which features of the original phenomena are suppressed, and on the problem one aims to solve using the model. Higher precision is not always a virtue. In fact, good models always neglect a lot of information]. Thus the coastline of England has a fractal dimension of 1.26, while the much more serrated coast of Norway is considerably longer and has the higher fractal dimension of 1.52 [the author seems to suggest that the length of the coast line has something to do with its fractal dimension, but this is not the case], giving it many more affordances for mooring boats or running contraband. [but the fractal dimension of a coastline is a completely useless fact for the sake of mooring, shipping goods, or defense!]

The author paints a picture according to which Euclidean geometry is a cheap version of something much better, and mathematicians keep Euclidean geometry alive for no good reason, failing to accept its fractal successor. But conventional Euclidean geometry is a model, and an extremely successful one. It neglects numerous intricacies of the real world but it retains enough information for the European Space Agency to be able to send the Philae and Rosetta spacecraft across an unfathomable distance to land on on the Churyunov-Gerasimenko comet. The fractal dimension of the comet, of earth, of the fabric of space through which the spacecraft traveled, would not have helped in the design of the journey. In fact, that extra information would have only clouded the scientists’ view. Fractal geometry is also a model. It neglects other numerous intricacies of the real world. The coast of England is not really a fractal. It can be approximated by Euclidean geometry, or by fractal geometry, and each has its merits.

The book rightfully alerts the reader of the potential dangers of misusing a model, but it is very easy to see those dangers clearly after they have happened. The book would have been more accurate, and more useful, had it included an analysis of the difference in circumstances between the many cases where models were used disastrously, and equally numerous (probably much more numerous) cases where models have been, and still are, used very successfully. Standing on the shoulders of giants, we can easily see the pitfalls they stumbled into. But we are still on their shoulders.

A minor unfortunate slip of the keyboard that I must point out is the claim on page 159 that the world is not only complex, but it is also continuous. At the very least, one can say that the nature of space-time at the smallest scale is unsettled, and that the claim of the continuity of the world is unfounded.

Another curious example of an expository decision appears on page 166, discussing the beginning of the theory of continental drift. I find again that from the comfortable seat on top of the shoulders of giants it is all too easy to ridicule the position of the scientific community towards continental drift, as today every child knows that the continents drift. But extraordinary claims require extraordinary proof, which is precisely what was missing from Wegener’s theory of continental drift. The matching of the contours of various continents separated by thousand of miles suggests continental drift, but it would be irresponsible for the scientific community to have accepted such a substantial theoretical shift based on this alone. When the mechanism for continental drift was found, things changed. Scientists very often have a completely wrong understanding of a particular phenomenon, and may reject what is later found to be the correct explanation. But the rejection can still be well-founded. One must have evidence.

One final exemple of what I perceive as bias, or at least severe neglect (and without getting into a detailed criticism), is the last paragraph on page 180, which scrutinizes the atomistic and separatist approach to problem solving, while forgetting to mention the numerous examples where such an approach is extremely successful.

To conclude, I find the book entertaining and very well-written. But, especially from the point-of-view of a mathematician, I find the exposition to be geared solely to getting its point across, at the cost of reducing the quality of its important message and of the occasional mathematical inaccuracy.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.