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Barzakh, Bibliothetic Order & Borges's Library in Babel

Vivek Iyer

The books inBorges's Library of Babel are a random distribution of unique character strings of equal

length. One way to generate the Library would be to start with the book that consists entirely of theletter a, then the one which is entirely a except that its last letter is b and so forth. This would generatethe lexical ordering of the Library in a time factorial to that of producing one book. The actual order

of books is random and we don't know where the Library starts and finishes. Still, we can justarbitrarily choose an anchor book and say, this is in the right place and every other book must be

moved to correspond to the lexical ordering. More generally, for any bibliothetic ordering, twovariables arise for any given book- first, its vector distance from the anchor book, and second itsvector distance from its assigned place- both of which have polynomial form.

Thus catalogs of the Library have existential Second Order Complexity which,byFagin's theorem,isnon deterministically computable. But, since we know the books have a lexical ordering such that

there is a least fixed point, it follows that algorithmic orderings are deterministically computable inexponential time. Moreover, if the catalog of the Library must also be a book in the Library then,

since we know that there is no'sparse language' in the one case which is not in the other we havenogrounds for hoping that some non deterministic computation would be shorter than exponential time.This is one approach, a pessimistic one, to answering the question-'Could there be a specified bibliothetic order for Borges's library that you or I could create anddescribe in, say, a 100-page document? Not a complete cataloguing, obviously, but an algorithm thatsensibly placed the books in an order explainable to a bright and interested human? I don't think so

at least I wasn't able to dream anything up!but nor did I see any path to a proof that no suchordering could be possible. And maybe I'm being redundant, but it wouldn't need to specify hexagonand shelf for each possible book, but rather a compelling way that set the books so that were you tofind yourself in a particular hexagon, you'd have a good sense of what might be in nearby hexagons. Igot nuthin'.'

What happens if we abandon the idea of a universal descriptive language and replace it by somethingarising out of the theory of co-evolution? Do we thereby escape the tyranny ofTime hierarchytheorems?One reason to hope so is that co-evolution is stochastic, not deterministic, and generates'advicestrings'.-i.e. an extra input becomes available- in a particularly useful manner. The Red Queen, inLewis Caroll's 'Through the looking glass', manages to run just as fast as the scenery- which is in 'statespace explosion'- thus staying in the same place.How might this work in practice?Suppose 'interesting books' in the Library of Babel are traded and used as a store of value and supposepreferences are epigenetically canalized to not-too-much, not-too-little diversity (GraciellaChichilnisky- an Argentine Mathematical Economist- did path-breaking work on this in the Seventies)then over infinite time we are likely to see all sorts of bibliothetic orderings corresponding to allpossible Wealth distributions which would exhaust polynomial complexity. Now, if a steady stateobtains at any point, then we know from a result by Axtell, that this is also a Walrasian equilibrium-which is an exponential time algorithm for fixed points. Under certain conditions- e.g. if we assumethat the race of Librarians have homothetic preferences and thus the same satiation point- this

corresponds to the notion that a bibliothetic ordering has been implemented.

My feeling, however, is that there is more to this. Essentially, the Economic evolution of biblotheticordering itself adds meaning and, what's more, the fitness landscape (assume guys more successful inthe book trade live longer or have more progeny) for hermeneutics now has a sort of built in 'RedQueen' predator-prey type driver for complexity. I think this means that though any given bilateraltrade is still only of complexity class P, the co-evolved complexity of the system is much greater.

Indeed, Borges-the-writer-who-was-also-a-librarian's own sequential bibliothetic orderings of thebooks that simultaneously createdhis precursors - which corresponds to a non computable real- could

http://www.amazon.co.uk/Unimaginable-Mathematics-Borges-Library-Babel/dp/0195334574http://www.amazon.co.uk/Unimaginable-Mathematics-Borges-Library-Babel/dp/0195334574http://en.wikipedia.org/wiki/Descriptive_complexity_theoryhttp://en.wikipedia.org/wiki/Descriptive_complexity_theoryhttp://en.wikipedia.org/wiki/Descriptive_complexity_theoryhttp://en.wikipedia.org/wiki/Sparse_languagehttp://en.wikipedia.org/wiki/Sparse_languagehttp://en.wikipedia.org/wiki/Sparse_languagehttp://dl.acm.org/citation.cfm?id=808769http://dl.acm.org/citation.cfm?id=808769http://dl.acm.org/citation.cfm?id=808769http://dl.acm.org/citation.cfm?id=808769http://en.wikipedia.org/wiki/Time_hierarchy_theoremhttp://en.wikipedia.org/wiki/Time_hierarchy_theoremhttp://en.wikipedia.org/wiki/Time_hierarchy_theoremhttp://en.wikipedia.org/wiki/Time_hierarchy_theoremhttp://en.wikipedia.org/wiki/Advice_%28complexity%29http://en.wikipedia.org/wiki/Advice_%28complexity%29http://en.wikipedia.org/wiki/Advice_%28complexity%29http://en.wikipedia.org/wiki/Advice_%28complexity%29http://en.wikipedia.org/wiki/Advice_%28complexity%29http://en.wikipedia.org/wiki/Advice_%28complexity%29http://en.wikipedia.org/wiki/Time_hierarchy_theoremhttp://en.wikipedia.org/wiki/Time_hierarchy_theoremhttp://dl.acm.org/citation.cfm?id=808769http://dl.acm.org/citation.cfm?id=808769http://en.wikipedia.org/wiki/Sparse_languagehttp://en.wikipedia.org/wiki/Descriptive_complexity_theoryhttp://www.amazon.co.uk/Unimaginable-Mathematics-Borges-Library-Babel/dp/0195334574

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have a representation. (This is because a few typos don't make much difference, so there can be a lotof book sequences corresponding to how specific works stood in relation to each other on Borges'smental book-case at different points of time)Elsewhere in the library, variorum editions of Schelling focal 'interesting books' might themselves endup with higher prices because of baked in hysteresis effects. Parallel to this is that 'important' bookdealers have epigenetic canalisation- i.e. they behave like each other even though they have differentorigins and preferences. I think this means that a lot of the time we are going to see fractal patternswith provincial 'collections' mimicking (mutatis mutandis) those of the 'metropolitan' MerchantPrinces. My guess, or particular brand of Economic Romanticism, is that this still wouldn't give us asubstantive 'catalog' except as a degenerate state.

Still the co-evolved descriptive complexity I am attributing to the librarians could, of course, be aproperty of a discrete math simulation. We just need to tinker with the rules of the game till we get asufficiently rich steady state fractal. But would the result be something we would find humanlycognizable or intuitive?The problem is that, whereas librarians can have a canalised tropism to value 'interesting' books- aguy running a simulation doesn't have that ability to pick out 'interesting books' in the same way. I

suppose one could breed genetic algorithms to do something which looks like Librarian Economicsbut the only way of knowing if what they produce is something interesting is to look inside the thing.

But that's clearly illegitimate. I mean why not just put in a predator which goes through the lexicalorder and either quarantines or eats anything that looks meaningless when compared to the GoogleBooks database?

Another approach, of whose legitimacy I'm not sure, is to appeal to Intutitionistic mathematics.

Suppose there is an ideal ordering such that the Expected value of the sum of randomly teleportedvisitor bewilderment is minimized (i.e. some visitors, by reason of their lucky location, can quicklywork out the ideal order while some, who are stuck in low structure stacks, take a very long but finitetime to do so). Suppose further that the visitors are of a specific mathematical race such that Totalbewilderment is a function of the spread of choice sequences they construct in comparing adjacent

books.Muth rationalityis the notion that one's choices or expectations conform to the prediction ofthe correct theory. Since the best thing for the visitor race as a whole is that the library be ordered to

minimize total bewilderment, it makes sense if their choice functions reflect this (i.e. they choose tobe bewildered by any empirical deviation from the ideal order). So we know there is an ideal ordering(albeit only for ideal visitors).We also know there are better than random orderings- e.g. the lexical.I am tempted to say that by the Brouwer's continuity principle (an illegitimate use in this context?)

this means there is a canonical bibliothetic ordering for all visitors who are the product of naturalselection. In other words, there is some way to extract phenomenological information about us- on theassumption that we have evolved in a Darwinian manner- and make it available to Mathematics in thesame way that the Anthropic principle might yield information for Physics.I also think, because of the nature of choice sequences, one can keep changing the genotype of the

visitors so that there is a trade-off between Total bewilderment minimization and other things wevalue- e.g. the chance for a lucky few to quickly get to a 'good' book-stack.

But, can Brouwer's principle be used in this way?Neither books nor information about ordering poses a difficulty. Everything is well defined andtherefore continuous. But is it legitimate to stipulate that choice sequences be constructively Muthrational?- doesn't that make them impredicative?On one horn of the dilemma- the Kantian horn- we can have a priori Muth rationality but no intuitiveidea of how the choice sequence is constructed, on the other horn- the Darwinian horn- we know thatchoices are epigenetically canalised and thus something like ex poste Muth rationality obtains only wecan never be sure it is optimal, and therefore really 'rational', in any sense.One way to resolve the dilemma is to embrace a peculiar sort of monadology in which whatever isinterstital is constantly populated by virtual particles which themselves spontaneously create choicesequence bridges. This, I suppose, is the Ibn Arabi's concept of the barzakh- the incompossible,phantasmal Limbo which unites what it separates- the world of the 'command', i.e. what is possible,

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and the world of 'creation'- i.e. what exists- which is also the pure virtuality represented by the'superficies of the Mirror' in the Library of Babel of which Khwaja Mir Dard was surely speakingwhen he wrote-

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