JoAnne Growney has recently referred again to our small disagreement about the role of ambiguity in mathematics and in poetry. The fact is that I really agree with her, but it is very hard to make a good argument for the case that ambiguity is important in mathematics, and fatally easy to argue the reverse.
So let me do the easy part first.
T. S. Eliot, in “Burnt Norton”, the first of the Four Quartets, says
Cannot bear very much reality.
He is taken to task for this by Ursula K. Le Guin in The Lathe of Heaven:
There is a bird in a poem by T. S. Eliot who says that mankind cannot bear very much reality; but the bird is mistaken. A man can endure the entire weight of the universe for eighty years. It is unreality that he cannot bear.
Of course, I think they are both right in some sense (though my sympathy is with Eliot, I think). But this reminds me of a story (perhaps apocryphal) about a famous mathematician who once gave a conference talk entitled, “Some new applications of Cauchy’s inequality”. He started by writing on the board: “Cauchy’s Inequality: 〈stuff〉 ≤ 〈other stuff〉”. A hand was raised in the audience, and someone said, “Shouldn’t that be ≥?” The lecturer stared at the board for a moment and then said, “Talk cancelled”.
It is a bit of a stretch to imagine Le Guin raising her hand during a reading of Four Quartets by Eliot and saying “Shouldn’t that be unreality?”, and getting the response “Reading cancelled”.
So here is another try at presenting an example of ambiguity in mathematics. This concerns “Hilbert’s Theorem 90”: see here for the Wikipedia page. The theorem is due to Kummer, but was Theorem 90 in the report on number theory that Hilbert wrote for the DMV. The theorem is rather technical; it is an ingredient in the proof that a finite Galois extension with cyclic Galois group is a radical extension, itself used in the proof that a polynomial with soluble Galois group can be solved by radicals.
The Theorem is often regarded as the foundation of cohomology theory. Long before algebraic topology or homological algebra, it says that a 1-cocycle in the multiplicative group of L, where L is a cyclic extension of K, is a 1-coboundary; that is, H1(G,L×) = 0, where L/K is a finite extension with cyclic Galois group G. In this form it was generalised to arbitrary Galois extensions by Emmy Noether.
But the theorem is ambiguous because it has other interpretations. The Wikipedia article points out that, applied to the simple case of the complex numbers over the real numbers (a Galois extension with Galois group of order 2) it reduces to the classification of Pythagorean triples.
Another example, which I keep promising to discuss but not getting around to, is the famous ADE-classification given by the Coxeter–Dynkin diagrams. Do these describe 3-dimensional rotation groups and singularities (so that E6, E7, and E8 represent the tetrahedron, octahedron and icosahedron), or root systems in higher dimensions (so that these three diagrams represent beautiful objects in Euclidean spaces of dimension 6, 7 and 8)? This ambiguity seems to be at the heart of the ubiquity of these diagrams in mathematics.