Roi Wagner is an Israeli philosopher from Tel Aviv University (PhD in 2007). He is strongly influenced by the French post-structuralism (Derrida, Deleuze, Lacan, Baudrillard). This is a philosophical movement of the 1960's. Structuralism of the 1950's analysed the dynamics underlying texts or, in this case, mathematics, via logical and scientific methods. Post-structuralism starts from the premise that understanding these dynamics is only possible by understanding a broader system of knowledge embedded in a constrained social, cultural, historical and practical framework.
In this book, rather than trying to define what mathematics is, or finding out whether it is real, or whether it is true, he explains what mathematicians do when they do mathematics, i.e., mathematics in motion, a dynamic rather than a static vision. There is an echo in this of the title of Reuben Hersch's book Experiencing Mathematics: What do we do, when we do mathematics? (AMS, 2014). However Hersch is a mathematician and deals with the philosophy of mathematics like many mathematicians would. Therefore his book is considered philosophical drivel by most professional philosophers. Back to Wagner. Think for example about what mathematicians did when a new concept such as infinitesimals was introduced. Some new concepts, like negative numbers, may have been used in computations, even before they emerged as new concepts. Or think about how mathematical signs did catch a certain meaning and how that meaning did change over time or by the introduction of the new concepts. Hence Wagner is placing mathematics in a broader historical and social context, and constrained by practical considerations.
Much of what he develops in the book is illustrated with an example that he explains as a vignette in the introduction: the Black-Scholes formula for option pricing. Although it does not work in all circumstances, it immediately was accepted by many practitioners. It became a common reference and was the reason for awarding the 1997 Nobel prize in economics. It just felt right. It had all the qualities that mathematicians could agree upon.
To place his vision against more traditional philosophies of mathematics, a first chapter introduces some of the tensions that are intrinsic to these classical approaches and that are relevant for further discussion. The analytic philosophy of Quine (1908-2000) with a tension between natural order and conceptual freedom. Kant (1724-1804) placed mathematics in between the empiricist and the rationalist philosophical views. The tension caused by mathematical 'monsters' that were originally banned, but gradually were integrated in the system like the irrationals, or the negative and complex numbers. Most of these were the result of the introduction of algebra replacing or completing geometry, while algebra and geometry had been considered distinct fields before. But other examples of these monsters exist. And finally the problem of authority: are mathematicians the only standard to decide on what mathematics is about?
In a second chapter, some of these controversies are illustrated in a case study: the introduction of abbacus and the algebra during the Renaissance. This is mostly about the 'monsters' now known as negative numbers, that were alien to mathematics in those days. However, they worked fine in computations and could be interpreted as a debt rather than a possession. However, this didn't make them 'numbers' yet. The progress came by combining both interpretations of well known ordinary subtraction and the alien concept of negative numbers. But this tension about accepting or banning a monster is not the only tension that is illustrated, also the other kind of tensions from the previous chapter are discussed here.
Chapter three is basic and describes the core of Wagner's approach. It sketches a constraint based approach to the practice of mathematics. Mathematics is a field of knowledge where meaning, reality, interpretation, truth, are all accepted or not based on negotiation over several constraints (natural, social, practical, cognitive,...) which may give rise to different mathematical cultures. In fact the choices that have to be made in the historical approaches of chapter one, are just the choices that one has to make between different constraints. Although mathematicians may have different interpretations of what is important, there is usually a strong consensus about accepting a mathematical statement or not, which creates a sense of a deeper truth represented by mathematics. This consensus is based upon a formalism in a broad sense that was agreed upon. Formalism in a structural sense may still give different meanings to a 'sign' that is used in a mathematical formula as a consequence of abstraction.
This semiosis is the subject of two case studies in the next chapter. For example the x in an infinite sum may represent a variable, a constant, a place holder, etc. just as plus or minus signs and other operators can be overlayed with different meanings. Another example is the combinatorial problem, known as the stable marriage problem. It can be explained in many different contexts, and one of them is the problem of pairing elements in a graph, an abstract formulation with many possible interpretations depending on a social an cultural context.
A more in-depth analysis of cognitive aspects is the subject of the next chapter. Here Wagner summarizes the views of Dehaene and Walsh about the mental representation of numbers. We are indeed born with an innate 'number sense' for small quantities. Next Lakoff and Nuñez theory on mathematical metaphors: a mechanism that allows to use the structure in one conceptual domain (e.g. geometry) to reason in another one (e.g. algebra). There are nevertheless some deficiencies like paradoxes, or boundaries of conceptual domains or the direction in which the transfer takes place, etc. The conclusion is a discussion and an interpretation of Deleuze's 'haptic vision'.
This cognitive vision is again illustrated in the next chapter with some case studies. This chapter illustrate the limitations of the theory when applied to mathematical metaphors. A number of historical examples relate to the algebraization of geometry which show that the metaphor cannot be isolated from the context. Also Lakoff's basic metaphor of infinity (BMI) seeing a continuous process as a limit of an iterated finite process is shown to fail both in the context of limits and in covering all notions of infinity.
In a final chapter, Wagner follows three post-Kantian thinkers (Fichte, Schelling, Cohen) to illustrate that mathematics also changes the environment in which it evolves and hence also the constraints which shape it. This brings him to his own explanation for Wigner's 'unreasonable applicability of mathematics on natural sciences'. Since mathematics is derived from what we experience, and our experience of nature depends on our mathematical tools, models and reasoning, it is not so surprising that some model will be applicable in quite different situations. More on Wigner's dictum can be found in another recent book The Pythagorean World (Jane McDonnell, Springer, 2016).
With this book, Wagner had three types of readers in mind: mathematicians, philosophers, and those who are users of mathematics, without being mathematicians. It is not an easy read for mathematicians, mainly because of some of the mon-mathematical vocabulary, but Wagner does explain clearly what he means when some concept or some philosophical theory is used. Sometimes, this explanation is only selective, just picking those elements that are relevant for the current narrative. I can imagine that many philosophers may not concur with what Wagner proposes in this book. There have always been philosophical schools and different interpretations. Wagner has clearly chosen for a post-structural kind of approach. For the non-mathematician, some mathematical examples could be a bit too advanced, but again, things are well explained and it should not be a serious problem. All in all, a not so easy readable but very interesting philosophical view on the activity and the essence of mathematics. The effort of reading and assimilating the book is more than worth it.