— Numerologies —
[Continued from here.]
However, we were not basically looking for the meaning of numerator per se, we were looking for recurrence, and our quick side-glance towards the blind blank and blacken’d speculum-punctum of n, ’N.N.’ et consortes provides an example of ’recursive discourse’, which is the course set out for our curriculum in this excursion.
Metaphor — Mirror of Meaning
What n and N.N. are in the context of finding a formula for recursion could fall under the general headline of metaphor. Etymologically metaphor comes from words giving the idea of ferrying across (transfer/phore, across/meta). A ferry is a bridge over troubled water, the metaphor carries meaning from one shore to another, contains the meaning as a vessel contains a passenger. It is a Scylla and Charybdis-challenge to guide such a ship over the open ocean of understandings and ununderstandings.
To succeed, the captain of metaphor must navigate successfully between presenting too opaque a metaphor which might crash the vessel of meaning against an impenetrable mass of words, but just as much the navigator needs to abstain from getting carried away on overly wordy waves of inspiration, no matter how clear the waters. Navigare necesse est, there is no escaping that. Too long time upon the sea and the crew will finish all apples and oranges before the final port, get scurvy, die off and leave the captain to navigate all on his own even should he be lucky to survive, meaning time is of essence or the recipient the meaning is meant for will lose their patience, no patient Penelope awaiting Odysseus when he returns home, the reader having already let their glance glide off the page, the listener already lost the red thread that should have guided them through the Labyrinth. A metaphor must carry meaning, must not let it sink like a rock into a whirlpool. If it does, the captain is a no-good nobody, the captain being the meaning that must steer firmly the ship. The medium is the message, the metaphor the meaning.
Metaphor clarifies something unknown by what it is not, but which we know. It basically says that the unknown, X, which we want to explain, this unknown X is pretty much like Y. By equating the unknown with something known, we intimate the unknown. It’s like asking Numerius to present us to their attractive younger sibling Nomen, whom we have not been introduced to but would very much like to get to know.
X or Y or…?
The metaphor is incomplete, though, as they all are, by definition, as every metaphor explains one thing by another, meaning these two are unidentical. It is like building a bridge from nowhere to nowhere else. You have a ferry going from Scylla to Charybdis? No ferryman would get passengers for such a ferryboat.
This mirror X-sample of metaphor is an extreme case of incomplete metaphor, as we explain one unknown, ’X’, by another unknown, ’Y’. Even the confused souls who might want to metaphorize the seas with such a ferryman could find no way to get onboard.
You just cannot ask Numerius to introduce you to attractive Nomen if you do not even know or recognize Numerius. You also do not know if Nomen would reject or accept no men or women, which though un-universal is a consideration we know is quite often brought into the equation in these pretty please present me to your pretty friend/sibling-schemes. A bridge from Scylla to Charybdis cannot be built.
And attempting to explain X by Y, we discover X may be XX as well as XY.
Universal and Un-Universal Unknown Unknowns
In our previous mirror metaphor, we found out that we, ourselves, have a line of sight intersecting with the line of sight of ’viewer’, and that neither we (w), nor Viewer (v), can see the point of intersecting lines. Which means that we do not see Numerius, we do not see Nomen, and we do not even see ourselves or our imago, being as hidden away as they are. Any and all siblings of our Nomen Nescio and our Numerius Negidius are now accompanied by an A.N. Other, yet another unknown N-name. Now we only need to find Mr. X, and we can set up a barbershop quartet.
As noted previously, any Mr. X may turn out to in reality be Ms. X, and for that matter in a situation where we do not know even who we are ourselves, we know nothing about any participant. We are all X in this together in this particularly peculiar barbershop quartet, meaning some X may be Y.
However, besides changing what we call the numerator and metaphorizing about mirrors, there does not seem to be much we can do about n, so let us leave it aside for now.
Let us move on and look at what is below the vinculum, in denominator land.
Below the Vinculum
Under the vinculum bar first item is t divided by t. Originally one, in the first form given above, for the purpose of having one term only, we removed the numerical one and replaced it with t divided by t, which expresses the same but in a non-numerical manner. The point of doing so is that we now have a thoroughly self-similar and self-referencing expression. As this particular t describes division, we shall call it d. As we might as well have chosen to translate 1 into t times t it could also be represented by, say, m (multiplication).
However, we must choose, it can be represented by either, but logically not both at the same time. To divide is not to multiply and to multiply is not to divide. The first term thus is an illustration of the logical expression EITHER/OR. This, in turn, means that the specific numerical value of 1 is an expression of a particular logical operation. In everyday terms, EITHER/OR means choice. Choice is inevitable where one is.
However, as 1 could express either d or m it also means that it is an exception. A particular and strange case of its own rule, it — 1 that is — is the value which does allow equally for either variation of the identity to come forth. All numbers multiplied with one become themselves. All numbers self-divided by themselves become one too. The value 1 thus contains not only EITHER/OR but also the logical idea of AND. If one was a verb, it would mean you could both multiply and divide at the same time; ”multivide”.
Consequently, announcing 1 as fundamental among integers, every numerical value being both a multiplication of one with itself and a division by one, every integer differing from every other by one, by definition if nothing else, we conclude that every integer may be seen as its own reciprocal as well.
The numerological series continues here.