Strange Attractors

Barry Mazur wrote a book Imagining numbers:(particularly the square root of minus fifteen), which was intended to convey to non-mathematicians that the act of imagination in mathematics is quite comparable to that in poetry. Specifically, he wants to explain how the phrase “the square root of minus fifteen” can create just as vivid an image in the mind of a mathematician as the phrase “the yellow of the tulip” (an image from a poem by John Ashbery) can in a non-mathematician; and, also, to explain the long twisted road that brought mathematicians to the point where they have this very clear mental picture.

What is more yellow than a tulip? Yes, of course.

God being gone, love having left, our sole
remaining task is to define complex
numbers. We know they follow daffodil
but anticipate iris and dogwood.

That is a quote from “Orders of magnitude”, a poem by H. L. Hix, an extract from which is anthologised in a new book Strange Attractors: Poems of Love and Mathematics, edited by Sarah Glaz and JoAnne Growney. My regular readers will be aware of JoAnne’s contributions to my blog, including the tagline.

This is not a book review, but I need hardly say that if you love both mathematics and poetry (and especially if you practise both, as both the editors do), then you will certainly want this book. I just want to record the results of a little bit of random browsing in the anthology (and, as always, there will be far too much of me in it for a proper book review).

Here is a phrase to ponder, from “The Freezing Point of the Universe”, by Maureen Seaton:

the number of pebbles in Newton’s calculus

The number is ten, as we will shortly see, and ten has an important role in calculus, in the sense of calculation. The word “abacus” is thought to come from Hebrew (or possibly Arabic, I am not sure) “abaq”, sand or dust (sprinkled on a surface used as a writing board for calculation). The word “calculus” has a somewhat similar derivation: a stone, e.g. a bead on an abacus. But there are other stones in the universe. In his Dictionary, Doctor Johnson defined

calculus: The stone in the bladder.

Did Newton suffer from the affliction of ten stones in his bladder? (Ouch!) I do not know, but he had a reputation for crabbiness in his old age …

Now let’s pull back a little further in the poem:

… 10-4, Good Buddy. The difference
between the number of pebbles in Newton’s calculus
and this four-room house …

And there I get lost in the crackle of CB radio.

You can probably guess the commonest image of love in mathematics: either a curve and its tangent, or a curve and its asymptote. (These are of course equivalent in projective geometry since we can move any point to infinity.) This occurs frequently in the book. For example, Jonathan Holden’s Sex and Mathematics begins:

Making love we assume
may be defined by the equation
for the hyperbola y = 1/x

This seems a little sad to me. Love is either a point which we approach and never reach, or a point where we touch and then move away again.

Thinking about this, it struck me that my most recent theorem (described here) gives a metaphor for love that I like much better. Here is a very brief summary. Fraïssé’s Theorem gives necessary and sufficient conditions on a class C of finite relational structures for it to consist of all the structures embeddable in a countable homogeneous (i.e. highly symmetric) structure M. Moreover, if the conditions are satisfied, then M is unique up to isomorphism; then C is called a Fraïssé class, and M is its Fraïssé limit. If C consists of all finite structures of some type, then we might call M the generic countable structure of this type. For example, the random graph (about which I have written at length) is the countable generic graph.

Now the class of all finite ordered sets is a Fraïssé class; the generic countable ordered set is the set of rational numbers. According to a theorem of Cantor, this is characterised as the countable dense ordered set without endpoints.

Now consider the class of all biorders (finite sets carrying two unrelated orders). This is also a Fraïssé class; what does the generic biorder look like, apart from being countable and homogeneous? Each of the individual orders is isomorphic to the rationals, but they interpenetrate in a remarkable way. The analogue of Cantor’s theorem is that any interval, no matter how tiny, in one order is distributed densely throughout the whole of the other order.

If you have been in love, has it ever seemed to you that any feature of your beloved, no matter how tiny, is distributed throughout the whole of your being?

I count all the things that need to be counted.
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3 Responses to Strange Attractors

1. Jon Awbrey says:

He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.

— Robert Musil, The Man Without Qualities*

2. JoAnne says:

. . .
A mathematician
sees the world
as she defines it.
. . .

THANKS, Peter, for your attention to STRANGE ATTRACTORS.

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