"The Library of Babel," which others call one of Jorge Luis Borges' most provocative works, is not a rational tale. The universe it conjures up, the Library itself, is an illogical place, riddled with inconsistency and self-contradiction. It is a place of life without food, birth without women, and Portuguese without the existence of Portugal. Yet there is an underlying internal logic to it, even if this logic bears no relation to that of Aristotle or Bertrand Russell. There are certain principles that govern its workings, even if they are no more concrete than the assertion that every possible book exists somewhere in its shelves (Borges 111). There is, to borrow a cliché, method in its madness. Upon closer examination, its madness begins to look very familiar indeed. The Library, as Borges describes it, corresponds to nothing less than the human mind itself. Ignacio Matte Blanco, a Chilean psychoanalyst, is responsible for a model of the mind that is based in part on Freud and in part on mathematical logic. Borges' Library is, in all important respects, identical to Matte Blanco's bi-logical model of consciousness.

Before tackling Borges, it is necessary to grasp the rudiments of this concept, which in turn requires some familiarity with a few basic terms from logic and set theory. As its name suggests, the heart of the bi-logical model is the presence of two different forms of logic under which the mind operates. The conscious mind uses that which we ordinarily think of as logic, often called Aristotelian logic or, in this context, asymmetrical logic. The term "asymmetrical" is applied because a statement is not always equivalent to its converse, that is, assuming "if x then y" is a true statement, "if y then x" need not be true also. When working asymmetrically, all our ordinary notions of reason apply. Individuals are viewed as distinct entities; a man's father is not, to his conscious, asymmetrical mind, identical to the set of fathers in general, or more generally the set of authority figures. A hand is a part of the body, but the entire body is not a part of the hand (Matte Blanco 31). Everything, in short, makes perfect sense.

The unconscious mind, however, does not always respect these laws. Psychoanalysts since the time of Freud have been able to identify several basic ways in which the unconscious operated differently, on a very fundamental level, than the conscious mind (36). Matte Blanco took this one step further, and deduced a set of axioms from which the logic of the unconscious seemed to follow. This he termed "symmetrical" logic, for one of its two basic precepts is the principle of symmetry: under symmetrical logic, a statement always implies its converse. This gains additional significance when considered in the light of Matte Blanco's second axiom, the principle of generalization. This states that the unconscious mind treats every object, person, or concept as though it was part of a more general class or set of things. Someone's father would belong to the set of fathers, which in turn would be a subset of the set of authority figures, and so on. Therefore, by combining these two principles, it follows that if John is an element, or subset, of the set of fathers by the principle of generalization, the principle of symmetry demands that the set of fathers thus be a subset of John. From this, Matte Blanco deduced that the unconscious mind can be treated as a collection of infinite sets, for this seeming contradiction, a set being placed in one-to-one correspondence with a proper subset of itself, is precisely the definition of an infinite set as put forth by the mathematician Dedekind (39).

It is clear that, operating under symmetrical logic alone, mental activity would become impossible. Carrying the principles of generalization and symmetry to their logical conclusion, every set would become identical to all its elements, and all of the sets that contain it. The unconscious would be unable to distinguish any two things from each other. Matte Blanco's model copes with this by specifying that the mind never operates using only symmetrical logic. Symmetrical logic coexists with asymmetrical logic in every situation, though as one delves "deeper" into the unconscious the symmetrical relations begin to play a larger and larger role (161). Hence, the mind is bi-logical, and the principles of symmetrical logic can be said to apply only under certain circumstances.

Returning at last to Borges, manifestations of bi-logic can be found throughout "The Library of Babel." Most obviously, the principle of generalization is apparent in the very structure of the Library. Borges describes it as being "composed of an indefinite, perhaps an infinite, number of hexagonal galleries...The distribution of the galleries is invariable" (Borges 108). The Library is described as a set, either infinite or incredibly large, of identical hexagons. Each hexagon, in turn, is basically a set of books: "Five shelves correspond to each one of the walls of each hexagon; each shelf contains thirty-two books of a uniform format; each book is made up of four hundred and ten pages; each page, of forty lines; each line, of some eighty black letters" (109). Books are more clearly seen as elements of the set of books a few pages on: "each book is unique, irreplaceable, but (inasmuch as the Library is total) there are always several hundreds of thousands of imperfect facsimiles-of works which differ only by one letter or one comma" (112). A hierarchy of sets of books is apparent here. Any single volume is an element of a set of books which differ from one another in only a single character, which is a subset of the set of books which differ from the original work in at most two characters, and so on until, at the ultimate level, all the sets of all the books are contained in the set of books in general. This conforms exactly to the principle of generalization, which states that the unconscious "treats an individual thing (person, object, concept) as if it were a member or element of a set or class which contains other members; it treats this class as a subclass of a more general class, and this more general class as a subclass or subset of a still more general class, and so on" (Matte Blanco 38).

The principle of symmetry, the complement of this, states that the unconscious "treats the converse of any relation as identical with the relation" (39). That is, in more tangible terms, if a book is a subset of the Library, then the Library is a subset of the book. Borges offers clear examples of this in the form of the catalogues. Amongst the contents of the Library are "the autobiographies of the angels, the faithful catalogue of the Library, thousands of false catalogues, a demonstration of the fallacy of these catalogues, a demonstration of the fallacy of the true catalogue...the interpolations of every book in all books" (Borges 111). There is also the concept of a "total book," which "is the cipher and perfect compendium of all the rest" (113). Every detail of the Library is contained in these catalogues and total books; thus, the Library is contained in its own proper subsets, which is a symmetrical relation.

A corollary of the principle of symmetry is that "all members of a set or class are treated as identical to one another...and are therefore interchangeable" (Matte Blanco 39). This is certainly the case with the hundreds of thousands of "imperfect facsimiles" of each book mentioned before, and also with the hexagons. "One of the free sides [of each hexagon] gives upon a narrow entrance way, which leads to another gallery, identical to the first and to all the others" (Borges 109). On this same page, the equivalence of the hexagons is made even clearer by the maxim that "the Library is a sphere whose consummate center is any hexagon, and whose circumference is inaccessible." Each hexagon, in addition to being physically identical to all the others, is positionally equivalent as well. All hexagons can view themselves as being at the center of the Library with equal validity.

Together, symmetry and generalization lead to the apparently irrational properties of the unconscious, and these properties too are echoed in the Library. The most obvious is the absence of time. Matte Blanco states that "the fact that the processes of the system Ucs. [the unconscious] are not altered by the passage of time seems to me a consequence of the fact that they are not ordered in time" (41). The principle of symmetry guarantees that the unconscious cannot recognize the concept of time, for it dictates that if event A happened after event B, then event B also happened after event A. The Library is also said to be timeless; it "exists ab aeterno. No reasonable mind can doubt this truth, whose immediate corollary is the future eternity of the world" (Borges 109).

Another consequence of symmetrical logic is the "lack of contradiction between two impulses which appear incompatible to Aristotelian logic" (Matte Blanco 44). In essence, this means that the unconscious is capable of accepting two ideas that appear to contradict one another, without negating either one. Psychoanalytically, someone can both love and hate their father simultaneously. In the Library, ideas and their negation coexist constantly. Hexagons include defecatory facilities, yet the identical nature of the hexagons dictates that food cannot be grown or raised, and thus not eaten (Borges 109). Nothing exists outside of the Library, as the Library is the universe (108). Yet the Portuguese, Yiddish, Guarani and Arabic languages all exist, which is clearly absurd since their existence is contingent on that of Portugal, Central Europe, Lithuania, and Arabia, which, not being parts of the Library, are all unreal (111).

"The impious assert that absurdities are the norm in the Library and that anything reasonable (even humble and pure coherence) is an almost miraculous exception" (113). This is a perfect characterization of the unconscious mind, from an asymmetrical point of view. It is equally valid, however, to say that it is symmetrical logic that is the norm, and the rational librarians themselves are the aberration. Using the intellectual tools developed by Matte Blanco, the illogical elements of the Library are seen to be the inevitable consequences of a bi-logical system, and the expressions of an order distinct from but nonetheless as real as that governing conscious thought. Though Borges wrote "The Library of Babel" some eighteen years before Matte Blanco began to publish his work, the correspondences are unmistakable (Matte Blanco 465). Those infinite hexagons and incomprehensible shelves of books make up not only the Library, but also a metaphor for the human mind itself.

Borges, Jorge Luis. "The Library of Babel." Trans. Anthony Kerrigan. Fiction: a Longman Pocket Anthology. 2nd ed. Ed. R. S. Gwynn. New York: Longman, 1998.

Matte Blanco, Ignacio. The Unconscious as Infinite Sets. Revised ed. London: Karnac, 1998.

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©1999 Luke Gutzwiller. I really mean it.