and Borges:
from order
to disorder

Michele Diego

A porcelain vase held in our nervous hands. A frictional force is established between our fingertips and the surface of the vase, which increases the tighter we hold it. The muscles in our shoulders and arms counteract the force of gravity by which the Earth’s center attracts the vase to itself. But the vase slips out of our grasp. Its speed increases as it falls at a constant rate. Hitting the ground, there is a sudden deceleration; the kinetic energy of the vase is partly absorbed by the ground as heat and partly converted into vibrations of the vase’s atoms, which causes it to break.
But why can’t we observe the opposite phenomenon, as we would in a reverse video? Why can’t the pot shards absorb heat from the floor, cooling it, and then convert this heat into motion so they reassemble themselves into the original pot and jump into our hands?

According to the laws of kinematics and dynamics, which describe the motion of objects and their causes, there is no real reason why this phenomenon cannot occur. Every law that describes the motion of a body is temporally invertible without compromising its validity. But then why are there phenomena, such as the vase breaking, that do not happen in reverse? Where does the temporal symmetry of the vase breaking, which indicates the existence of a unique direction of time’s flow, come from?
The answer lies in the second principle of thermodynamics, which establishes the constant increase of entropy in the universe. If in any physical process energy is constant, entropy is – if anything – always increasing. In mathematical terms this means:


However, the second principle of thermodynamics, as we have seen, is not an infallible law. It is valid in probabilistic, practical, visible and factual terms. It is almost impossible to violate it, but does our universe really distinguish between the impossible and the infinitely improbable?

But what is entropy? Intuitively it defines the degree of disorder of a system. The higher the disorder, the higher the entropy. The second principle of thermodynamics then tells us that in the universe every transformation tends to increase the disorder. If we call Sᶤ the initial entropy of a system and Sᶠ its final entropy after it has undergone some transformation, we have Sᶠ-Sᶤ =ΔS≥0. The phenomena for which ΔS is exactly equal to zero are the reversible ones, so we are not surprised to see them in reverse as well (e.g. a rubber ball bouncing on the ground and back into our hand). For all other phenomena, ΔS is greater than zero and they occur in only one time direction.
But why does the universe “prefer” disorder to order? The answer is in mathematics, in numbers, and probability. An ordered system has less possible configurations than a disordered one, and therefore disorder will always tend to establish itself.
Physicist Brian Greene explains this concept with a clear example. Supposing we have 100 coins on a table, all turned on their “heads” side. We hit the table so that the coins start to quake, jump and eventually turn over showing “tails”.
How many possible configurations of the system exist that show 100 heads? Only 1, the one in which all the coins are turned on the same side. How many configurations exist with 99 heads and 1 tail? There are 100 different ones. With 98 heads and 2 tails? 4950 different configurations. And 50 heads and 50 tails? 1008913445564193334812497256 different configurations.
Therefore, it is clear that even starting with all the coins ordered from the “heads” side, while banging the table there is a very high probability that the system will evolve towards disorder, i.e. towards a state where there is an equal number of heads and tails. In principle there is no real impossibility that the hundred coins will all turn to “heads”, decreasing the entropy of the system, but it is incredibly unlikely.
Moreover, phenomena of nature are much more complex than the example of coins and involve a number of particles tremendously higher than one hundred, making the possibility of the entropy’s decrease even more unlikely.
However, the second principle of thermodynamics, as we have seen, is not an infallible law. It is not, as for the conservation of energy, narrowly an inviolable principle. It is valid in probabilistic, practical, visible and factual terms. It is almost impossible to violate it, but does our universe really distinguish between the impossible and the infinitely improbable? Is it possible to think that, somewhere in the universe, a throw of one hundred coins gives one hundred “heads”?
Thanks to his formidable mathematical intuition, Jorge Luis Borges asked himself this same question. In his short-story “The Library of Babel”, the Argentinean writer imagines a library-universe, in which there are all possible books of 410 pages (each with 40 lines of 40 letters) written through the permutations of 25 characters (22 alphabetical letters, a full stop, a commas and a space). Therefore, somewhere in Borges’s universe there is a book that contains only the letters “aaaaaa…”, others in which there are only letters “a” except for one letter “b” (“baaaa…”, “abaaa…”, “aabaa…”, etc.), and so on, with all the possible combinations of characters that exist. In this labyrinthine library man is busy looking for a trace of order in the chaos that overwhelms him: books in which complete sentences can be found. Such books exist, mathematics requires their existence, but so far very few sentences have been found in books. Yet somewhere, among the shelves filled with books, there has to be a book that fully covers whatever subject comes to mind: “the autobiographies of the archangels, the faithful catalogue of the library, thousands and thousands of fake catalogues, the proof of the fakeness of those fake catalogues, a proof of the fakeness of the true catalogue, The Gnostic Gospel of Basilides, the commentary upon that gospel, the commentary on the commentary on that gospel”. Including books that talk about us, about our lives and even books that are about every possible variation of any topic. Therefore, there must also be an unmitigated book that contains a supreme, absolute truth. But can we really say that this book exists if no one will ever be able to find it, buried in a chaotic universe of random letters?
Equally, in the infinite complexity of our universe, the principle of increasing entropy could fail. The chances of this happening are infinitesimal, but not null. Borges wishes someone could read this unmitigated book. I wish that someone’s broken vase will return to their hands intact.

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