— Pseudo-Mathematics —
When you are a Bear of Very Little Brain, and you Think of Things, you find sometimes that a Thing which seemed very Thingish inside you is quite different when it gets out into the open and has other people looking at it.
— A.A. Milne, Winnie-the-Pooh
Perhaps recursion is a fact of life. Let us assume there exists an ultimate recursivity formula. Let us assume that this formula gives self-similar results regardless of input. Let us assume that this formula ends up reproducing the conditions by which it can come into existence; that is, that it is self-recursively recursive.
Now let us attempt to prove it.
If the formula itself is self-recursive, the variables should be of no particular importance as long as they are relatively formula-compliant in context. If a formula, it should be possible to mathematize and then analyze mathematically our object of investigation.
The mathematical archetype for recurrence is exemplified in the Fibonacci numbers, produced by the following formula:
F(n) = F(n-1) + F(n-2)
Interpretation: A Fibonacci number, F(n), is made up of the one preceding it, F(n-1), and the one preceding the preceding one, F(n-2).
The formula is the corollary of the following expression (below), requiring no equation, meaning it expresses only itself:
t / (1-t-t^2)
I shall allow myself to rephrase this slightly.
t / (t/t — t — t^2)
This certainly looks to be self-referencing and to provide us with the opportunity of recursion, so it may be that we have here the formula we were looking for, already, in the very beginning our journey.
However, it is opaque, so let us take it apart and look at each part individually. We need to understand it if we are to prove it anyhow.
The expression is really a fraction, essentially of the same kind as, say, 1/2 or 4/5. The first part is the numerator, in these cases 1 and 4. In our two examples 2 and 5 are denominators. The bar separating t and the rest of the expression — (t/t — t — t^2) — is called a vinculum. This Latin word means bond, meaning that what is above it and what is below it ties together to form a particular unit.
Vinculum As Speculum
Perhaps we could think of this vinculum, this bond, as a speculum. We see the mirror image viewed from the side, showing us, a third-party observer, how a reflection looks different to not the viewer, nor that which is viewed, but someone seeing both viewer and reflection from a position perfectly perpendiculum to the reflection. What is above the vinculum would be the observing viewer, the spectator. Below it the image of the viewer, the spectaculum. Seeing both and without any other clues about which is which, we would have no way of knowing if it was the numerator or the denominator who was the viewer. We could think of them both as images or both as real.
We are Flatlanders
In reality, being positioned with a line of sight exactly coinciding with the plane of the mirror, whatever image it showed would be hidden from us. Having excluded all other lines of sight, we would see neither viewer, nor image. We would be Flatlanders not seeing what is orthogonal to our optical field, which would be compressed and confined to a line; the vinculum and the relationship it embodies.
Above the Vinculum
The numerator t is for term, meaning it is one in a series or sequence. Being numerator, expressing number, we might as well call it n. Perhaps we will.
Keeping in mind the mirror metaphor, n could be shorthand for nomen nescio, generally abbreviated N.N., as not only we are unable to see the viewer, the viewer themselves are also unable to see the viewer. What is true for us about not seeing them is as true for them. Looking at a mirror, no one sees themselves. Looking at a mirror is to look at an image, a reflection.
From knowing this, we can choose to think of the mirror as not only a plane and a line coinciding, but as two lines of sight coinciding in a single point — much like the intersection of X-axis and Y-axis in a Cartesian coordinate system. And what our analogy tells us is that Descartes was right all along; X and Y intersecting at 0,0 we now see means that this is a spot which cannot be seen; X, Y is a blind spot in the field of vision. We cannot see a point perpendicular to our line of sight, and 0,0 is perpendicular to all orthogonal lines of sight. The blind spot is like a black hole, subsuming into itself all light, reflecting none of it.
OK, so Descartes did not say exactly that, but anyhow.
N.N. was in Roman times in jurisprudence sometimes unabbreviated, or perhaps we should say reabbreviated, into a metaphorical name; Numerius Negidius, a word-play echoing more recent 20th-century Can’t Pay, Won’t Pay, or vice versa.
’Numerius’ is formed on numerare, meaning to pay but also to enumerate, and the related noun number is numerus. So, yes, we would find in Numerius a name, a prænomen, representing number. Negidius we recognize as a variation on negat, ’deny’, in modern English found in words like negation. So N.N. not only means no name, it can also be read as no number.
Not that we needed to do the Latin detour to show that N.N might be read No Name or No Number as it works to n-abbreviate and N.N.-unabbreviate in English anyhow, ab ovo, and there is no need for un-English languages, English now essentially being a lingua franca, not an insular island language at all.
Is continued here.
- Zebra / CC