## Mathematical metaphor

Last week, the book Fashionable Nonsense by Alan Sokal and Jean Bricmont emerged from its hiding place under the sofa. So I browsed a couple of chapters.

In the book, the authors take various sociologists, literary theorists, psychoanalysts, and others, mostly French, to task for using mathematical and physical ideas while rather obviously not understanding what they are talking about. Most of the targets march under the general banner of post-modernism, though the authors make clear that this is not a precise label, and some of the people criticised predate the advent of postmodernism (and so are modernists).

I read their account of the work of literary theorist Julia Kristeva, from the mid-1960s to the mid-1970s. She claimed to develop a theory of poetic language based on set theory, and invoking the Axiom of Choice and the Continuum Hypothesis. Certainly she does not draw the level of criticism directed at others like Jacques Lacan, since she makes it clear that her use of set theory is metaphorical. (Lacan, a psychoanalyst, uses the classification of surfaces – with or without boundary – in studying the unconscious of his patients, giving the impression that some neurotics actually have Möbius strips somewhere in their brains.)

Since I have no objection to mathematical metaphors, I pricked up my ears.

Kristeva’s argument is that poetry transcends (or “transgresses”, the fashionable word then) the 0 and 1 of Boolean logic. However, she proceeds to make two errors, which Sokal and Bricmont point out (though they admit that they are not sure thay have interpreted her correctly, and neither am I). First, she confuses the set {0,1} of Boolean values with the interval [0,1] of real numbers. Then she uses the interval [0,2] which transgresses the boundary 1 for poetic values, and seems to say that this interval has larger cardinality than [0,1].

At a second reading, I was struck by a couple of phrases which suggest a different interpretation, probably not one which Kristeva intended, but maybe a better metaphor for poetry.

In passages quoted by Sokal and Bricmont, she says,

Every sequence contains the message of the book,

and

[…] this literature […] puts forth its message in the smallest sequences; the meaning (φ) is contained in the mode of junction of words, of sentences …

This seems to be saying that the Kolmogorov complexity of poetry is very low: the entire poem can be generated from a small amount of information. I doubt this is what she meant, although certainly some effective poetry employs repetition.

But what I was reminded of was the countable structure which is the “universal permutation”, or generic pair of linear orders, which I have discussed here before. It has the property that each order is isomorphic to the rational numbers with the usual order (or, if you prefer, an open interval of the rationals), but together they have a remarkable property: any interval in one order, no matter how short, is densely distributed throughout the other order. I used this as a poetic metaphor in my previoius post: “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?” But perhaps it can serve as a metaphor for poetry?

We could imagine that one order describes the text of the poem, and the other describes its meaning or signification: each small part of one is densely scattered through the other. So, not the whole meaning, but a “dense subset” (suitably interpreted) of the meaning, is encoded in any short sequence of words. Probably, since both poems and brains are finite, some finitised version of this should really be used – but this is only a metaphor!

This feels, in some sense, congenial to me. I am older than the authors of this book, and was a student when Kristeva’s work was published. I knew nothing about it at the time, but those were “modernist” days, and many people were playing similar games. If you looked in my old diaries from those days, I think you would find arguments not a million miles from this one.

I count all the things that need to be counted.
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### 4 Responses to Mathematical metaphor

1. Fascinating! Metaphor is a curious thing. With it we can relate anything to anything else. It often employs transformation of, or partial elimination of, attributes or qualities. So in the case of a Möbius strip this could simply indicate ‘a closed loop with a twist’ – in terms of thinking simply going around in circles drawing wrong or false conclusions, and so on. Metaphorically {0,1} of Boolean values can be the same as the interval [0,1] of real numbers. Its like considering the absolute value of numbers – ignore the sign, or in this metaphoric case ignore the quality of (0,1), that is ignore (interval / Boolean), just consider 0,1. But of course such considerations lie outside the logic and formalism of mathematical language. Although not helpful for unambiguously accurate descriptions of ideas it can make great metaphorical poetry!

2. Walter Sinclair says:

I am reminded of a course I once took in Paradise Lost. John Milton apparently was attempting to impart this quality to his work, a quality he believed to exist in the Bible and which he wanted Paradise Lost to share. This was a belief he took from tradition, a belief common among exegetes of holy texts such as the Torah and the Quran. The connection made here suggests (to me any way) the possibility of some kind of advanced kabbala. But, as always, the Devil is in the details.

3. Jon Awbrey says:

And now for something entirely the same …

4. But we should not dismiss metaphor as some abberant undesirable phenomenon of human consciousness, an assault on logical truth. It provides a means of escaping the fixed boundary of logic and viewing from a different perspective. For instance: k/0=1 logically leads to the alternative (0 [] 1) =k viz. the multiplication of zero times infinity equals any number.
From a metaphoric perspective this contains a ‘logical’ truth. The zero carries a ‘quality of number’, but no magnitude. The infinity of that quality of number is the boundary or infinity of ‘all numbers of that quality’, so from this perspective there is a logical truth to the apparent contradiction. The metaphoric explanation lies in a dimensional sphere, the equation (0 [] 1) =k describes a state space of ‘all number’ of a particular quality, this is a dimension-like reality, and describes ‘the infinity of all number of a particular quality’. Mathematical logic operates inside the boundary, where magnitudes have unique distinction, and because the infinity contains the dimensional boundary of ‘quality, not ‘magnitude’, as described above, there is an inherent paradox. But we have only been able to approach this paradox metaphorically using an analysis based on dimension-like abstractions or ‘qualities’, and from outside the conventional mathematical logic, which operates ‘inside’ the infinity.

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