Written: Dec. 11, 2007.
Prompted by Reuben Hersh's very positive review, published at the Dec. 2007 issue of the Notices of the American Mathematical Society, I bought, and read large parts of, William Byers' new book "How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics". This is really a love song to ambiguity, and how we should learn to love it, since it did so much good, as Byers amply illustrates with many captivating examples.
I was almost convinced that I should start to love ambiguity, but something felt wrong. The reason ambiguity was so fruitful was because we mathematicians hate it. More precisely, we love to hate it. And it is important that we continue to hate it and love to hate it. If we will start to love it, as Byers and Hersh suggest, it would lose all its potency.
Also, Ambiguity in mathematics is not all good. Lots of it sprang from human confusion. Of course, that's why Hersh loves this book so much, because it stresses the human aspect of mathematics, and this confirms his own pet philosophy of mathemtics, that it is human-made and fallible.
That's why we need computers. Computers abhor ambiguity, and trying to teach computers mathematics is also good for us humans, since it forces us to discover hidden ambiguities and resolve them.
Take for example, 1+1=2. Byers claims that it is ambiguous. The left side is
an operation while the right side is an integer. But if you think like a computer
does, then indeed 1+1=2 is neither true nor false. It is just an expression of
type "=". What is being meant is that if you go to Maple and type
you would get 2. So evalb(1+1=2); would evaluate to true.
Similarly, Byers mentions that students have trouble grasping that
.99999.....=1 because it is ambiguous. Byers calls it a category error. He is right, and so are the students. This is just shorthand for the statement that the sequence an:=9/10+9/102+ ...+9/10n is a convergent sequence, and its limit is 1. And this means that there exists an algorithm that given any ε > 0 outputs an N such that |1-a_n| ≤ ε for n ≥ N .
So a great remedy against ambiguity is to program it. I believe that all undergraduate mathematics major should learn how to program in Maple or Mathemtica (or the open-source Computer Algebra System SAGE), and it will do them lots of good.
So I really loved Byers' book, but I disagree with the bottom line. We should not unilaterally endorse ambiguity and start loving it as a great blessing. Once we start doing it, it will lose all its power. Let's continue to hate it. But this is too strong too. So let's compromise, and sometimes love it and sometimes hate it, sometimes love to hate it, and sometimes hate to love it, sometimes love to hate to love it and sometimes hate to love to hate it or even love to love to hate it, or hate to hate to love it and so on and so forth.