The numerical medium: on formerly undecidable propositions of mathematicians and related ilk.

To begin, the definitions of specialized terms used throughout this essay shall be presented before the main body of discussion. This is done for the sake of clarity in the description of ideas that would otherwise be rather cumbersome.

REALITY: the concept of the physical universe as perceived by general people.

AETHER: the concept of the mathematical universe as perceived by mathematicians.

EPHEMERALIZATION: the act of connecting Reality and Aether.

COSMOS: the resultant continuum where Reality and Aether are connected.

Ephemeralization is a term that was first coined by Robert Buckminster Fuller (1895-1983) to describe the movement towards doing "more with less." For instance, the invention of radio has ephemeralized communication. As Fuller would say himself, it is the transition "from tracked to trackless, from wired to wireless, from visible to invisible." (Kenner, 56). This idea serves well to illustrate the main goal of mathematicians. They are a people driven towards a specialized from of ephemeralization, whereby the two microcosms of Reality and Aether are joined together harmoniously. For one person in particular, Paul Erdõs, ephemeralization is the "uncovering of mathematical truth." (Hoffman, 26).

What is mathematical truth? The doctrine of Platonic Realism states that mathematical truth is an entity that exists independent of anything perceptible by the senses. Mathematical notions "are disembodied eternal Forms or Archetypes, which dwell in a distinctive realm accessible only to the intellect." (Nagel, 99). Accessible perhaps, only to mathematicians. Paul Erdõs refers to the Aether as the Book; the keeper of the Book is the Supreme Fascist, SF. "I’m always saying that the SF has this transfinite Book—transfinite being a concept in mathematics that is larger than infinite—that contains the best proofs of all mathematical theorems, proofs that are elegant and perfect." (Hoffman, 26). Apparently it is the SF who, in the beginning, divided Reality and Aether by the creation of a great chasm that would only be traversable by His following creation, the mathematician. Thus, Paul Erdõs is a mathematician.

It takes a special kind of person to be a mathematician. They must have the ability to move between the imperfect mechanical world of Reality and the ephemeral universe of the Aether. This is the discerning quality that separates Mathematician from Man. It takes tenacity to exist within the utterly malleable fabric of Platonic Realism, and to return unscathed. It is not easy to explore the Aether: "A land of rigorous abstraction, empty of all familiar landmarks, is certainly not easy to get around in.” (Nagel, 13). More than once, it has thrown many mathematicians into revolutionary convulsions when its utterly malleable fabric has seen fit (perhaps by command of the SF) to reveal yet another Platonic Oddity.

Observe Euclid and his fifth postulate, interchangeable with the fifth postulates of Lobachevsky, Bolyai, and the many axioms of Riemann. Indeed, space seemed to be able to take on as many shapes as the non-Euclidean geometers could think up. This sounded quite dangerous indeed, and Bolyai’s father warned him of this in a letter from 1820:

You should detest it just as much as lewd intercourse, it can deprive you of all your leisure, your health, your reset, and the whole happiness of your life. This abysmal darkness might perhaps devour a thousand towering Newtons, it will never be light on earth. (Struik, 166)

And only in this way can the majesty of the Aether manifest itself in the physical. Mathematicians are quite courageous indeed. Even then, their discoveries are sullied by the imperfections of the corporeal. For, "the triangular or circular shapes of physical bodies that can be perceived by the senses are not the proper objects of mathematics." (Nagel, 99). In this sense, mathematicians could be risking the crossing of the chasm, the "abysmal darkness", only for their own personal enjoyment. To Realists, this seems incredibly inane, if not blatantly masochistic. It may come as no wonder that mathematicians are often assumed to be vagrant minds with an ultra-tenuous grasp on Reality. But this is not entirely true; they simply see more of the Cosmos and budget their time in either microcosm accordingly. That is not to say, however, that many do not spend most of their time suspended somewhere between the two.

Whereas Paul Erdõs was a "mathematical monk" (Hoffman, 25), certainly one who rationed his time in the microcosms rather poorly, Buckminster Fuller was one who could comfortably perch himself at the intersections of the Appolonius lines of the Cosmos. When it was said that only Eulclid could have perceived the splendor the geometric universe, Buckminster Fuller was the exception. Never has anyone before him seen the vector lines within the polyhedra that govern the laws of force in the universe, and then continued to apply the insight toward the betterment of humanity. "Shelter for everyone," he says. We now have "a corrugated aluminum geodesic Zulu hut." (Kenner, 44). The discovery of what he calls the "Tensegrity Sphere" is like a tangible portal between the Aether and Reality. A collection of sticks and wire arranged in such a way that the tension and compression forces of either are in complete harmony. It appears as ephemeral as is possible to imagine, but it is as solid as physical Reality demands. When one man called it an insult to God, Fuller replied, "I cannot do anything nature does not permit." (Kenner, 93). Apparently, he has done absolutely everything that nature could ever hope to permit. Thus, Buckminster Fuller is as well a mathematician (not to mention an architect, engineer, and poet).

Mathematicians are indeed a special type of people. They were born to discover, apply, adapt, and understand within a Cosmos split in two. They were born to build the bridge that connects the two microcosms of Mathematics and Physicality. They seek to break the laws of Platonic Realism; they are dealers on the Universal Black Market of Knowledge, for quite often do their insights seem shady to the unwitting customer. They are adventurers of the high abstract planes, trailblazers of number lines and the unending perimeters of objects that defy dimension. Whether their goal is to have no goal at all, to remain afloat in the Aether; or to anticipate and improve upon humanity; or to calculate one more digit of pi; theirs are no different than the artist or the writer, the humanist, or one who simply wants to memorize one more digit of pi. For they are simply the mediums of another world, communicating and interpreting that, though invisible, has and always will exist.


Beckmann, Petr. A History of Pi. 1971, The Golem Press. United States of America.

Hoffman, Paul. The Man Who Loved Only Numbers. 1998, Paul Hoffman. United States of America.

Kenner, Hugh. Bucky. 1973, Hugh Kenner. United States of America.

Nagel, Ernest and Newman, James R. Gödel’s Proof. 1958, 1986, Ernest Nagel and James K. Newman. New York and London.

Struik, Dirk J. A Concise History of Mathematics. 1967, Dover Publishing. New York. Third Edition.