Wednesday, September 13, 2023

“The Infinite and the Infinitesimal” • by Christopher Degni

In June of 1986, upon the death of the great Argentine mathematician Luis Davila,
the sole trustee of his estate discovered a small enchiridion among his personal effects. A prominent inscription on the title page consisted of a single decimal number of 106 digits, close to ⅓, but not matching any known mathematical constants: .33263638—, along with a request for his library of unpublished works to be donated to the Universidad Nacional de Mar del Plata. The University, in conjunction with the governing board of the International Congress of Mathematicians, appointed a panel of five esteemed algebraists, geometers, and logicians to review and analyze Davila’s private collection and report on any significant findings.

Early in his career, Davila had written several seminal works in set theory, comprising results that had imposed order on the foundations of mathematics, but he’d abruptly stopped publishing and begun to explore chaos theory upon receiving tenure. Professors of his stature would normally allow their graduate students to publish small findings in their stead, but Davila had famously never taken a protégé. The panel discovered that, in fact, the majority of his results hadn’t escaped the thousands of impeccably kept notebooks arranged in a series of interlocking hexagonal shelves in his modest home office.

Beyond troves of new theorems in combinatorics, entropy, and fractal geometry, some of which opened up entire new subfields of scholarship, they found a peculiar result related to literature, contained in one set of notebooks, seventeen in number and bound together under the label Lo infinito y lo infinitesimal (“The infinite and the infinitesimal”).

The first notebook defined a basic scheme that informed all that followed, and the next three traced its natural consequences. The Spanish alphabet can be encoded by assigning each letter a two-digit number, A = 11, B = 12, and so on, with the remaining two-digit numbers being assigned to other orthographic symbols, such as spaces, punctuation marks, and mathematical symbols. Davila used this simple system rather than a universal Gödel-based numbering system for reasons known only to him, though he did note that a similar scheme would work for every known language. Any written text can then be represented by a number formed by replacing the symbols with their corresponding values; the process is reversible by translating any number two digits at a time back into orthographic symbols. No two texts correspond to the same number, and no two numbers correspond to the same text, though there are numbers that correspond to no earthly texts.

Davila’s simple stroke of genius was in placing these derived numbers behind a decimal point. Now instead of yielding unboundedly large values, every text produced a value greater than zero, but less than one. For instance, using the scheme above, the text consisting of the single word ‘BAD’ would correspond to 121114, or, with decimal, .121114.

In this manner, Davila’s scheme placed every story ever written—or ever to be written—into the compact space of a line segment of length one, the so-called “unit interval.” By the same process, this unit interval includes all non-fiction writings as well: biographies of everyone who ever lived, monographs describing solutions to all the world’s problems, even the text of this very record. Davila himself noted that the Library of Alexandria, as vast and varied as it were, would occupy a vanishingly small measure—what the mathematicians call a “set of measure zero”—in his system.

Developing an ordered method of discovering these works presented Davila his next challenge. Fully twelve of the notebooks were dedicated to this inquiry. Did the common constants—the Zuckerman-Collatz threshold, the inverse Feigenbaum number, Chaitin’s infinite construction—decode to any of the great works of antiquity? Or, in the other direction, were any famous texts describable by primitive mathematical or physical constants?

Davila encountered false start after false start, although he occasionally discovered the tiniest bits of meaning, if nothing close to the transcendental works of literature. One special value of an L-function decoded to elegante esperanza before descending into gibberish. Davila’s scrawls, meanwhile, exhibited evidence of an increasing tendency towards desperation.

Here, knowing that many of his choices were arbitrary, Davila worked in multiple languages and encoding schemes, applying common and esoteric transformations, but his notebooks showed no significant literary theorems. Davila opined that such a result would prove the existence of a higher Meaning, an oasis of Order in a universe of Chaos, and, as much as one can intuit mental state from a sheaf of mathematical notebooks, at this stage, Davila seemed defeated, despite his great theoretical contributions.

But the final notebook, written some seven years after the first sixteen, showed evidence of a completely different strategy. Davila, as was his style, began with a simple, even trivial, observation: that every decimal number can be divided by two. The original number would be recoverable, by doubling. Thus, in dividing everything by two, all the literary works fit not only between zero and one, but in fact between zero and one-half. There was no obstruction to recursion: repeating the process placed everything between zero and one-quarter, one-eighth, one-sixteenth…

Davila’s final step was to introduce an infinitesimal system of mathematics. The great literature of the past and the future, the research monographs and the autobiographies, the works of recondite sects of monks, and even the instructions to derive the theorems he had been struggling with now all managed to fit into no space at all, an infinitude into a void, a feat both monumental in itself, as well as made inevitable by the very mathematics Davila used to delineate it.

After reviewing the seventeen notebooks, the panel were able to decode Davila’s original inscription: voy a la nada, y me regocijo, porque todo reside allí.

“I go forth into the nothingness, and rejoice, for everything resides there.”


Christopher Degni is a 2019 graduate of the Odyssey Writing Workshop. He writes about the magic and the horror that lurk just under the surface of everyday life. He lives south of Boston with his wife (and his demons, though we don't talk about those). You can find more of his work in, Sherlock Holmes and the Occult Detectives, 99 Tiny Terrors, the upcoming 99 Fleeting Fantasies, and of course, here on Stupefying Stories.


Find out what you’ve been missing!
Buy the three latest issues on Kindle with just one click!



Karin Terebessy said...

Wow! So this is the first time in thirty years I have been grateful for taking that early morning elective Number Theory class in high school. It was worth it to understand this story! Brilliant!

Made in DNA said...


Richie said...

Worthy of Jorge Luis Borges - muy bueno!

Richard J. Dowling