WILLIAM BYERS. How mathematicians think: Using ambiguity, contradiction, and paradox to create mathematics

Without wishing to suggest that professional philosophers would regard the book as philosophy, I can report that this book is definitely philosophical. Most of the book pertains to mathematical invention, but not just the psychology thereof, with many examples of the way in which mathematical advances move from two different and incompatible ways of viewing something to a higher viewpoint on it that makes better sense and better mathematics. A simple example of this is the invention of zero, where the two incompatible viewpoints are that numbers are for counting and that there is nothing to count. The number one exemplified almost the same degree of blockage for the ancient Greeks, for whom the least number was two. It is perhaps unfortunate that the word that the author chose to represent the presence of such resolvable cognitive difficulties is ‘ambiguity’. As ambiguity is severely shunned by mathematicians and as there is none of it—as the word is normally used—in such situations as are described either before, when there are the two viewpoints, or later when there is a higher one, the use of ‘ambiguity’ would be misleading if it were not so adequately explained not to mean ambiguity. The excuse for using the word is claimed to be the genuine ambiguity of one of the simplest examples discussed, 3 + 4, with indifferently the meanings ‘add four to three’ and 7. While at first I thought that the author was right that it is useful to be able symbolically to denote both the addends and the sum the same way and that to do so is genuinely ambiguous, further reflection has led me to the conclusion that ‘add four to three’ is a meaning that one grows out of when one learns algebra. As soon as one is solving x + 1 = 0 one knows that those symbols represent the sum and are not an instruction to add, since there one cannot add. One cannot add 1/2 + 1/4 + …, nor is one instructed to. As this is an empirical question, I hope I'm right. And anyway what he is almost always talking about is much more complicated and interesting than ambiguity. He makes a good case that a lot of mathematical advances at levels from ancient arithmetic to present-day research do involve such resolutions as to count debts as well as assets by extension of the system of integers beyond zero rather than by using positive numbers and different coloured inks. The increasing number of persons interested in basing philosophy of mathematics on mathematical practice cannot afford to ignore this serious reflection on cognitive processes.

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