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

Human kind

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, *H*^{1}(*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 *E*_{6}, *E*_{7}, and *E*_{8} 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.

Isn’t this the inverse of ambiguity? An ambiguous statement is one that has several logically distinct readings – this may or may not say something about our ability to make the logical distinction at a more intuitive level. A multifaceted mathematical theorem, on the other hand, has several apparently different statements, but they turn out to represent ‘the same thing’ at some deeper level. In other words, one has a superficial distinction and underlying unity, rather than superficial unity with an underlying ‘subtle’ distinction.

On the other hand, mathematicians often use ambiguous words when talking about aims and achievements. ‘Classify’ and ‘describe’ come to mind.

Without trying to be facetious, I think that the two things you describe (one underlying structure that can be seen from several aspects, and several different structures that share an aspect) are not so dissimilar; you just have to decide which is up and which is down (or inside/outside if you prefer that metaphor). Maybe the word “ambiguity” is ambiguous, having at least these two meanings (as well as the one you hint at in the second paragraph, which is just lack of clear definition, and the one that arises when the same word is used with different meanings (e.g. “A graph with a regular automorphism group is regular”)).

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Ambiguity is a very broad subject in mathematics, in the sciences (especially quantum mechanics) and in everyday language. Is there an underlying phenomenon that is universal? Ambiguity can be understood, not simply as a ‘vagueness’, but a reality that exists in two or more states, which are, or appear to be, inherently incompatible. we do not have to look into the esoteric realms of complex mathematics to uncover it, a point well made in William Byers book How Mathematicians Think:

Using Ambiguity, Contradiction, and Paradox to Create Mathematics.

Square roots having both positive and negative forms are inherently ambiguous, zero and infinity are inherently ambiguous, paradoxical and so on.

What I find so interesting is that ambiguities and paradoxes occur when realities are dimensionally partitioned and co-exist in two or more partitioned realms. Inside / outside is particularly relevant to certain forms of infinity. We can, for instance, consider the infinity of natural numbers (integers) to be unbounded in magnitude, yet bounded by the quality of, or class of ‘integer number’. Whence this infinity can be understood as the ‘realm of all integer number’, limited by quality of numberness but unlimited in magnitude.

From this point of view we view from the outside the infinite realm of ‘all integer number’ – a unified class or realm of (infinite) numberness, from the inside an infinite set of (particulate) integer numbers. The ambiguity continues in that each particulate integer number can be considered as a unique set of distinctions or a realm of specific numberness.

(five distinctions / a realm of fiveness)

From this it also follows that as particulate numbers approach infinity in magnitude they become ‘more attribute like’, or ‘dimension-like’, and as approaching the upper limit condition suggests transformation, we may also expect differing properties of numbers as we approach the lower limit condition zero, 1 as the universal factor, clustering of primes, root 2 as irrational, pi for instance…