Future, Present, & Past:

~~ Giving itself latitude and leisure to take any premise or inquiry to its furthest associative conclusion.
Critical~~ Ready to apply, to itself and its object, the canons of reason, evidence, style, and ethics, up to their limits.
Traditional~~ At home and at large in the ecosystem of practice and memory that radically nourishes the whole person.

Oυδεὶς άμουσος εἰσίτω

Sunday, April 20, 2014

ɸ among the integers

The integers form an ordered series, with properties which may strike one as trivial. For instance, any given element [n] of the series occurs once and only once, and in a specifiable location, i.e., the nth place. Thus, e.g., 5 occurs only in the 5th place and nowhere else in the series of integers. These characteristics practically constitute a definition of the series of integers. Listed off in a series, each of the integers has two immediate neighbors, N+1 and N-1. But what if we imagine each integer N as having precisely N “neighbors”?

To determine this, we’ll sketch the first few stages of a process by which the integers are imagined as “generated” out of each other. This thought-experiment is based on John Conway’s in On Numbers and Games and on Donald Knuth’s exposition of Conway’s notions, in which numbers are imagined as being born discretely, one at a time. Conway’s constructions take us very quickly to the infinitieth step and well beyond. We, however, will stay safely within the finite. We are after a different phenomenon.

The integers will generate each other by a specific mechanism. Each element [N] of the series is entitled to have precisely N “links” to other elements of the series. We will see that this produces an interesting pattern.

In the graphics below, the colors here are not indicative of any absolute differences. The element [N] whose "turn" it is will be shown in gold. Each new element will in turn link to other elements, generating new elements as necessary until it has the number of connections specified for it by the rule that it have precisely N such connections.

When an element first is generated, it will be shown linked in red to the element that has generated it. Pre-existing links between an element [N] whose turn it is and other elements -- that is links which come into being before a given turn -- will be shown in green. (This means that green links will always be to elements less than N.) New links to pre-existing elements will be shown in blue. New links to new elements, i.e., "generating" links, will be in red, as mentioned. Blue and red links will therefore be to elements greater than N. (Any links to an element whose turn is not current will be in black.)

We begin with element [1].

This first element [1] needs to be connected to precisely one other element. We will establish these connections in order; so the first element [1] now “generates” a second element [2]. This generation (shown in red) also suffices as the “link” between [1] and [2].

This second element [2], in turn, needs to be connected to two other elements of the series (two, precisely because it is the second element). One connection already exists (shown in green) – its connection with [1]. It therefore generates a further element of the series, element [3], and is thus linked to both [1] and [3].

“The Tao produced One; one produced Two; Two produced Three. And three produced the Ten Thousand Things,” says the Tao Te Ching (ch. 42). And if we read “the ten thousand things” as “more than one,” this is what happens.

[3], as the third element of the series, requires connections to three other elements. It is already linked to [2] (shown in green), but it needs two more links. It cannot link to [1] (because [1] already has the single connection which exhausts its quota). So [3] generates, and is thereby linked with, [4] and [5].

[4] in turn requires four connections. One exists already, to [3]. It cannot link to [1], nor can it link to [2], because [2] already has two connections, but it can link to [5], which is does forthwith. (In the graphic, a new link to a pre-existing element is shown in blue.) That makes two links. To get it to its requisite four, [4] then generates [6] and [7].

By now you are probably starting to see how things work. We cannot skip on to [6] or [7] yet – it is not their turn. Before we get to them, we must see to [5].

[5] already has two (green) links – to [3] and to [4]. It needs three more. [6] and [7] are both available, so [5] links (in blue this time, because these elements are already there) to [6] and to [7], and then, to finish off its fifth link to which it is entitled, it generates (in red) a new element, [8].

Now we may proceed with [6]. [6] has pre-existing links to [4] and [5]. It forges two new links, to [7] and to [8] respectively, and then generates elements [9] and [10].

[7] has links to [4], [5], and [6]. It makes links to [8], [9] and [10] and then generates a new link to [11].

[8] has pre-existing links to [5], [6], and [7]. It establishes links to [9], [10], and [11] and then makes new links to [12] and to [13].

You have almost certainly glimpsed something in the graphics by now. Whenever an element generates a new element (the ones in red here) (as opposed to linking to an element that has already been generated), it generates either one or two of them. In our graphics, these are always put to the right of the “parent” element whose turn it is. These pile up, until the turn moves back down to the bottom of the next stack, and the piling-up starts over. This process always leaves a top element in any given pile, and those numbers probably look familiar. [1], [2], [3], [5], [8], [13]. . . that’s right, you in front waving your hand! It is indeed the Fibonacci series. Very good.

The Fibonacci series, as you know, generates each next term by summing the two previous terms. So (3+2)=5, (5+3)=8, (8+5)=13, and so on. The ratio between any two adjacent terms converges, as the series goes on, upon the golden ratio, that splendid number also known by the letter phi, ɸ, the ratio defined by (a+b)/a = a/b, whose decimal expression is 1.618033988. . ., and which is also expressible as (1+√5)/2, the value of the lovely continued fraction 1 + (1/(1+(1/(1+(1/1+…)))), and any number of other nifty tricks.

I have in fact traced our linked-integer series a few steps further, just for fun, but as you can imagine, the graph gets pretty tangled and hard to follow after this stage. However, the Fibonacci pattern remains for as far as I followed, and I can’t see why it wouldn’t. I presume it is a trivial implication of the mathematics involved, and that this way of deriving the series is well-known in relevant circles. I don’t move or read in those circles; all I can report is that in those popular-mathematics accounts with which I am familiar, this way of deriving the golden section is not described. (I'd love to hear of any references.) I find it of interest because it shows how the series emerges from the overlay of two ways of construing the integers – as cardinals (one, two, three…) and as ordinals (first, second, third…). It is, moreover, interesting to see the Fibonacci numbers emerge from a set of well- (and minimally-) defined relations (a.k.a. “links”, above).

As mentioned, new elements are generated either singly or in pairs. Fans of recreational mathematics may find it interesting to trace the pattern that emerges in the way these variations continue to play out.


  1. Nice! Now, not to be disrespectful: it occurs to me that, if Christian theologians had understood the procession of the persons of the Trinity as formally similar to the generation of which you write, they would have been led to posit a Fibonacci sequence of persons within the Godhead, though that would have considerably outrun the givens of scripture. I guess I’ll find myself pondering the similarities and the differences of the orthodox Eastern, the orthodox Western, and the Fibonaccian understandings of procession/generation.

  2. I'd be lying if I said that something along the lines of a Trinitarian analogy had not occurred to me. (It didn't seem especially germane to the post, as I was in a rare mood of feeling like being succinct). But it wasn't indefinite generation of divine persons. It's just that after three, the expansion begins to be multiple. (Hence the "Ten Thousand Things" remark). And of course the fact that the generation here is occasioned precisely by the predicate of being-related-to-, has a theological undercurrent to it.

  3. Understood. But am I wrong in thinking that the predicate, being-related-to-, is not enough to give rise to three and only three relata? After all, if there were only one existent or being, (x)(Ixx) would still hold true. And, on the other hand, if "S" abbreviates "is superior to," then:

    (x)(y)(z)(w)((Sxy & Syz & Szw) --> Sxw).

  4. The Church Fathers, of course, warned against too-readily making arithmetical analogies for the three-ness of the Trinitarian relations (there is no "superiority" in the Trinity in one sense). But that aside, not only does being-related-to- not restrict things to three, it does the opposite (in this post, anyway). Certainly within the parameters of the exercise detailed here, it generates the infinitude of the positive whole numbers.

  5. If you are interested in some new ideas on religious pluralism and the Trinity, please check out my website at www.religiouspluralism.ca, and give me your thoughts on improving content and presentation.

    My thesis is that an abstract version of the Trinity could be Christianity’s answer to the world need for a framework of pluralistic theology.

    In a constructive worldview: east, west, and far-east religions present a threefold understanding of One God manifest primarily in Muslim and Hebrew intuition of the Deity Absolute, Christian and Krishnan Hindu conception of the Universe Absolute Supreme Being; and Shaivite Hindu, Buddhist, Taoist apprehension of the Destroyer (meaning also Consummator), Unconditioned Absolute, or Spirit of All That Is and is not. Together with their variations and combinations in other major religions, these religious ideas reflect and express our collective understanding of God, in an expanded concept of the Holy Trinity.

    The Trinity Absolute is portrayed in the logic of world religions, as follows:

    1. Muslims and Jews may be said to worship only the first person of the Trinity, i.e. the existential Deity Absolute Creator, known as Allah or Yhwh, Abba or Father (as Jesus called him), Brahma, and other names; represented by Gabriel (Executive Archangel), Muhammad and Moses (mighty messenger prophets), and others.

    2. Christians and Krishnan Hindus may be said to worship the first person through a second person, i.e. the experiential Universe or "Universal” Absolute Supreme Being (Allsoul or Supersoul), called Son/Christ or Vishnu/Krishna; represented by Michael (Supreme Archangel), Jesus (teacher and savior of souls), and others. The Allsoul is that gestalt of personal human consciousness, which we expect will be the "body of Christ" (Mahdi, Messiah, Kalki or Maitreya) in the second coming – personified in history by Muhammad, Jesus Christ, Buddha (9th incarnation of Vishnu), and others.

    3. Shaivite Hindus, Buddhists, and Confucian-Taoists seem to venerate the synthesis of the first and second persons in a third person or appearance, ie. the Destiny Consummator of ultimate reality – unqualified Nirvana consciousness – associative Tao of All That Is – the absonite* Unconditioned Absolute Spirit “Synthesis of Source and Synthesis,”** who/which is logically expected to be Allah/Abba/Brahma glorified in and by union with the Supreme Being – represented in religions by Gabriel, Michael, and other Archangels, Mahadevas, Spiritpersons, etc., who may be included within the mysterious Holy Ghost.

    Other strains of religion seem to be psychological variations on the third person, or possibly combinations and permutations of the members of the Trinity – all just different personality perspectives on the Same God. Taken together, the world’s major religions give us at least two insights into the first person of this thrice-personal One God, two perceptions of the second person, and at least three glimpses of the third.

    * The ever-mysterious Holy Ghost or Unconditioned Spirit is neither absolutely infinite, nor absolutely finite, but absonite; meaning neither existential nor experiential, but their ultimate consummation; neither fully ideal nor totally real, but a middle path and grand synthesis of the superconscious and the conscious, in consciousness of the unconscious.

    ** This conception is so strong because somewhat as the Absonite Spirit is a synthesis of the spirit of the Absolute and the spirit of the Supreme, so it would seem that the evolving Supreme Being may himself also be a synthesis or “gestalt” of humanity with itself, in an Almighty Universe Allperson or Supersoul. Thus ultimately, the Absonite is their Unconditioned Absolute Coordinate Identity – the Spirit Synthesis of Source and Synthesis – the metaphysical Destiny Consummator of All That Is.

    For more details, please see: www.religiouspluralism.ca

    Samuel Stuart Maynes

  6. Now get every number that is one level above the initial number line to absorb all that are above it. (Addition by absorption is allowed in dealing with infinities, why can't we extend the notion to 'regular' integers.) Yet, allow the resulting number to retain a memory of what it absorbed (I bet Kronus remembered his children) and you get a model of what I call ML. But say no more, and most certainly don't ask about the memory mechanism and how it may be stored in a simple integer. That kind of insanity may only be communicated in person.