# William Byers: How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics.

William Byers

How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics.

Princeton, NJ: Princeton University Press 2007.

Pp. 415.

US\$35.00 (cloth ISBN-13: 978-0-691-12738-5).

In this book Byers focuses on the hermeneutical side of the philosophy of mathematics. How can we understand what mathematicians do when they do mathematics? What has to happen in order that a new interesting result sees the light of day? Byers tries to argue that mathematics has been and still is widely misunderstood. Far from being algorithmic and fixed on proofs, the real core of mathematics is creativity. And creativity is tied to 'great' or (simple) 'mathematical ideas'. These ideas are not only placed at the centre of mathematical understanding, which Byers calls 'turning on the light', but they also propel mathematical progress.

Byers presents a couple of examples in which a crucial step forward in the development of mathematics depended on the presence of two at first sight unrelated or even barely compatible perspectives on some mathematical structure. He starts with the discovery of the irrational numbers, like v2, where v2 is clearly present as a geometric object--the length of the hypotenuse of the right angled triangle with unit length sides--but is not allowed for by (early Greek) arithmetic. The real numbers 'provide a context' (38) in which the two perspectives are unified. Another famous example is the Fundamental Theorem of calculus, which says 'that there is in fact one process in calculus that is integration when it is looked at in one way and differentiation when it is looked at in another' (50). The core of mathematics, according to Byers, is finding such situations and being able to understand them by providing a more comprehensive view. This process is creative and not algorithmic. Proofs only sum up the discovery and preserve the result in text books. Mechanical proofs Byers sees as 'trivial' (373), whereas 'deep' proofs are framed in expressing some (great) 'idea'. Re-ordering the terms in an infinite series of additions and subtractions, for instance, makes it obvious to see a sum formula. Good mathematicians are, therefore, those who hit on 'ideas', like Cantor hitting on diagonalization and the continuum hypotheses. Even more revolutionary are 'great ideas'. An example of a great idea is formalism. Formalism provided a unifying perspective on the whole of mathematics. When Hilbert started with formalizing Euclid's geometry, 'formalism was born and, in the process, the whole notion of truth was radically transformed' (291). A great idea is then inflated (as in Hilbert's claims on behalf of formalism) and then again delimited in a wider perspective (as when Godel's theorems hit formalism). Because ideas are outbursts of creativity, 'the answer to the question of whether a computer could ever do mathematics is clearly "No!"' (369). Byers finally relates his view to the question of how mathematics is to be taught, namely by getting students to understand the ideas so as to 'turn on the light'.

In the introduction Byers remarks that, unluckily, mathematicians are not the best source to account for how mathematics works or what mathematicians are doing. This cliche about the working scientist unfortunately applies to Byers as well, who is a mathematician himself. On the one hand his central concept of 'ambiguity' remains far from being clearly developed, and on the other hand the reader is constantly provided with a subtext airing some post-modern world view, sometimes bordering on post-modern mumbo jumbo.

Ambiguity can consist in one expression having several meanings, 'bank' being a paradigm example. Some of the ideas Byers mentions are of this type. Godel's discovery that some number theoretic statements are about the number theoretic proof system itself depends on the ambiguity of reading these statements as at one time being about numbers and at one time being about the proof system itself. Reading these statements in the latter way employs the Godel numbering semantically coded in the meta-language. The majority of the cases Byers uses are, however, not of this type. The core aspect here seems to be that one and the same mathematical object or structure can be seen from two perspectives (these perspectives often being theories of different mathematical fields). The two perspectives unified shed further light on the structure in question, but they (still) refer to the same structure. This does not seem to fit well our folk concept of ambiguity. Nor need it do so; but a more worked out concept of, say, 'multiperspectivity' or 'perspective integration' would strengthen Byers' analysis of the examples he presents. (In fact several examples do not even refer in detail to his supposed methodological tool set of ambiguity, contradiction, etc.)

To such criticism Byers certainly could answer, as he in fact on occasion does, that 'the definition of ambiguity is itself ambiguous' (31). This highlights his more post-modern inclination to pseudo-deep remarks. One more example (introducing the idea of contradictions): 'The contradictory is an irreducible element of human life as we all experience it.... In this way and others we are all walking contradictions' (80). Whatever this means--and even if it is true--it relates only very vaguely to the concept of contradiction used in the formal sciences. Starting with his confession that he found the unity of Zen and mathematics, up to his equation of formula and other 'metaphors' found in literature, Byers throws fog on otherwise interesting analyses.

Byers ends up with an anti-realistic understanding of mathematics, one that stresses constructive ideas: 'Knowing and truth are not two; they are different perspectives on the same reality. There is no truth without knowing and no knowing without truth' (343). This even outdoes intuitionism.

Byers stresses that his is also a qualified constructivism. But this will not do: calling both anti-realist and realist theories 'perspectives', far from being a compromise in which both theories equally cede ground, in fact simply caves in to the anti-realist.

A reader who ignores Byers' post-modern musings--or skips a few chapters--will find a few challenging ideas on the practice of mathematics in this book. The examples are often lucidly presented. They await a more thorough going analysis along the line of Byers' main ideas of mathematical creativity in the integration of perspectives.

Manuel Bremer

Universitat Dusseldorf