The Inconceivable Middle Page
At the end of The Library of Babel by Jorge Luis Borges, there is a footnote in which he describes a book containing, “an infinite number of infinitely thin leaves,” and then, almost as an afterthought, adds, “the inconceivable middle page would have no reverse.” I paused, fatefully, on those words. I tried to understand, but could not. Why does the middle page have no reverse? There followed sleepless nights, scribbled diagrams, fretful and broody walks, unhappiness. And then finally it was over.
It may be that the two explanations proposed below are invalid, even absurd. That hardly matters. They succeeded in appeasing the ghost of Borges who, soon after, loosened his stranglehold on me. In The Fearful Sphere of Pascal, Borges speaks of an infinite sphere “whose centre is everywhere and whose circumference is nowhere.” It was at first tempting, therefore, to conclude that a book with an infinite number of infinitely thin pages does not have a middle page, or that every page is the middle page. But then I remembered that the book is not infinite. Only the number and the thinness of its pages are infinite and a middle page could occupy the infinitely small medial point between the first and last page. For this reason, and since Borges speaks of a middle page, I decided that I should proceed on the assumption that there was one. I The first problem is to locate the middle page. To turn to it, starting from the front or back of the book, is impossible: Because the pages are infinite in number, and because ∞ ÷ 2 = ∞, the number of pages preceding and succeeding the middle page are also infinite. However many of these infinitely thin pages are turned, the increase in the thickness of all the pages turned (and the decrease in the thickness of all pages yet to be turned) would be zero. The middle page is forever approached and never reached. This is expressible by the following illustrations, where ∞ is the number of pages yet to be turned, and x is the number of pages already turned, which may be any finite real number, however large: If you let the book fall open at random x number of times, you will encounter either situation A or B. The probability of a third outcome, in which the book falls open at the middle page, is expressible as 1:∞; that is, approximately analogous to finding a mustard seed on a desert planet of infinite size. This probability 1:∞ is, I think, equal to zero, but the result of the miraculous event could be represented as follows,
Situation A allows for an increase in the number of pages on the verso while the recto remains numerically unalterable; situation B for an increase in the number of pages on the recto while the verso remains numerically unalterable. In situation C, however, because ∞–x=∞ and ∞+x=∞, the pages on either side of the middle page are equal, infinite, and immutable.
The middle page is either impossible to find or impossible to escape. Turn the middle page. Neither its subtraction from the recto nor its addition to the verso has any numerical effect. It is absorbed into, and obliterated by, the infinite pages of the verso, but, at the same time, “returned” to the recto. The middle page has no reverse. The following diagram shows the evolution of the middle page through a single page turn, Bertrand Russell has said that the rear of a moving object is being annihilated at the same speed as the front is being formed. That image might help in trying to visualise the middle page: The page before the middle page becomes the middle page when the middle page becomes the page after the middle page. It is as if the obverse of this middle page, like a particle in quantum physics, has no definite position but is “smeared out” between infinite pages, while its reverse, each time it is laid down on the verso, repeatedly ceases to exist.
There is one final attribute of our middle page that is worth pointing out. Since real books consist of folded and bound leaves, which result in an equal number of pages, there is no middle leaf. A book with an infinite number of pages, however, functions as though it had both an even and odd number of pages. When it is open, and all its pages are horizontal, it is as though it had an even number of pages and no middle page. When you turn a page of the book, the page you are turning becomes the middle page (is flanked by an equal number of pages) and the book functions as though it had an odd number of pages. The middle page exists only while it is being turned. II So far, nothing has been said of the fact that, as well as being infinite in number, the pages of our hypothetical book are infinitely thin. However, approaching the riddle from this angle allows for a second solution. Although we tend to think of a piece of paper as a plane having zero thickness, it is, properly regarded, a three dimensional object with six surfaces, much like a flattened rectangular solid: Even a razor blade, which has a cutting edge of approximately 0.05 mm, can be considered a flattened parallelepiped. An infinitely thin piece of paper, on the other hand, has a thickness which is computable at zero. This means its edge could be magnified an infinite number of times without yielding surface area,
Imagine a microscopic insect crawling from one side of an infinitely thin page to the other: It will arrive at the other side without having passed an angle, edge or surface perpendicular to the plane of the page. Therefore, the edges of an infinitely thin page are functionally convex, and the page itself forms a single, continuous surface like a sphere.
But this precept applies not only to the edges of any given page. It applies equally (inconceivably) to edges formed by any number of adjacent pages. In other words, because –∞ + –∞ = –∞, there is no difference in thickness between one page and x number of pages, where x is any finite real number, however large.
The entire book forms a single surface with no reverse. To picture this, imagine the book has been stood up with its front and back covers bent back until they touch, and the pages are spread out. A torus is a solid object which has a single surface and two sides. The cylindrical body before you is a single surface with an infinite number of radially projecting sides. Before I even finished these solutions, I saw there was a problem with everything I said with respect to “x number of pages” because a single page is an infinite number of other pages, and it is no less impossible to find a single page than it is to find the middle page. Take the first page, for example. It falls apart into innumerable other pages; you seize the first of these; it falls apart into innumerable pages; you seize the first of these...
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