The philosophy of mathematics stands at an interesting crossroads. To make sweeping generalizations: mainstream philosophy continues to pursue the legacy of twentieth century mathematical foundationalism while the mathematics educational community increasingly embraces a postmodern, humanist form of constructivism. The former tends to concern itself only with the content of mathematics, usually in some kind of formalist guise, and ignores the human creative process that is involved in knowing that content. The latter tends to focus on the acquisition of mathematical knowledge while dismissing (or confusing) important philosophical distinctions (e.g. ontology/epistemology). Meanwhile most practicing mathematicians are unconcerned with (and largely ignorant of) these diverging traditions.
This widening gap has created a landscape in recent years that is ripe for new approaches to the philosophy of mathematics — ones that acknowledge and account for both the objective and subjective nature of mathematics (i.e., both the content and the process). William Byers’ How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics is the latest such attempt, written for a popular audience. Unfortunately, it fails to deliver on many levels.
The first half of the book is devoted to an exploration of the nature of ambiguity, contradiction and paradox as they relate to mathematics. In each case Byers both broadens and obscures the meaning of the word. In particular, with ambiguity he stretches its domain of application so widely that it essentially loses any practical usefulness. These chapters are full of bold assertions that are not substantiated (for example that counting, as one of the primordial sources of mathematics, leads to algebra [p. 36]) and arguments that adopt the strategy of persuasion by obfuscation.
By the time I finished reading the chapter on ambiguity the only thing I was convinced of was that I no longer knew what it meant, and so felt more inclined to accept his exploitation of it later. The chapter on contradiction highlighted many illustrations of mathematical and logical contradiction, including proof by contradiction, but also embraced imagined contradictions as real (for example he describes zero as a one word oxymoron). Finally, the two chapters on paradox focused on the various manifestations of paradox surrounding the concept of infinity, from Zeno to Cantor. For the most part, these discussions accurately demonstrated the historical influence that paradoxes have had on mathematical development. However, there were also instances of over-reaching (for example, the principle of mathematical induction as inherently paradoxical).
In the second half of the book Byers uses ambiguity, contradiction and paradox to formulate the theory that mathematics is not so much about deduction, derivation, and computation (i.e., formalism) as it is about ideas. Now I should say that on its face I agree with this claim. In fact I also agree with many of the consequences that Byers draws from it: the inadequacy of formalism, the non-algorithmic nature of mathematics, the misguided objectives of radical artificial intelligence, and the contingency of mathematical truth (an honest formalist would affirm this last point in the sense that the truth of deductive arguments are contingent on the assumed truth of the axioms). However, despite my sympathy for many of his conclusions, I found the exposition and arguments to be weak and unconvincing. Byers repeatedly engages in a type of squishy logic and reasoning that is not uncommon amongst postmodern writers. One of the more egregious examples of this is where he conflates the technical concept of mathematical chaos in dynamical systems and its ordinary meaning of disorder: “In complexity theory there is a saying that interesting biological processes happen ‘at the edge of chaos.’ Now paradox is a form of chaos…” (p. 266). Ironically, the metaphorical analogy that he attempts to make becomes nonsensical in light of this ambiguity.
In another example, from the book’s introduction, Byers claims “There are eminent spokespeople for an opposing view, one that maintains that ‘The ultimate goal of mathematics is to eliminate all need for intelligent thought.’” (p. 4) This straw man is either sloppy scholarship or disingenuous. The citation is to Concrete Mathematics by Graham, Knuth and Patashnik. Although Graham et al. are indeed eminent, the quote itself is a (satirical) marginal note that the authors explain was submitted by a student in their course and is not a view that they necessarily endorse.
One of the more frustrating aspects of reading this book was its apparent lack of a copyeditor. Eliminating the redundant content and discussions could have easily reduced its length by 30–50%. There are numerous typographical errors, ranging from subscripts where exponents were intended, to sign errors in formulas, to duplicate numbering of paradoxes. More problematic is the utterly inconsistent mess that is made of the endnotes and bibliography. There are numerous sources either not referenced at all or referenced in the endnotes but absent from the bibliography. Finding bibliographic references is a further challenge, since the bibliography is only mostly in alphabetical order.
Despite all of its shortcomings, I expect that there will be many, particularly those in the field of mathematics education, who will herald this book as a significant contribution to the philosophy of mathematics. Nevertheless, I am confident that it will be completely ignored by mainstream philosophers.
David J. Stucki teaches computer science and mathematics at Otterbein College, in Westerville, Ohio. His most recent interests are in the history and philosophy of mathematics, and computer science education, although he also maintains an interest in artificial intelligence, theory of programming languages, and foundations/theory of computation.
ITRODUCTION: Turning on the Light 1
SECTION I: THE LIGHT OF AMBIGUITY 21
CHAPTER 1: Ambiguity in Mathematics 25
CHAPTER 2: The Contradictory in Mathematics 80
CHAPTER 3: Paradoxes and Mathematics: Infinity and the Real Numbers 110
CHAPTER 4: More Paradoxes of Infinity: Geometry, Cardinality, and Beyond 146
SECTION II: THE LIGHT AS IDEA 189
CHAPTER 5: The Idea as an Organizing Principle 193
CHAPTER 6: Ideas, Logic, and Paradox 253
CHAPTER 7: Great Ideas 284
SECTION III: THE LIGHT AND THE EYE OF THE BEHOLDER 323
CHAPTER 8: The Truth of Mathematics 327
CHAPTER 9: Conclusion: Is Mathematics Algorithmic or Creative? 368