Mathematical definition of Folgezettel

System

This discussion was created from comments split from: Antinet Zettelkasten.

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Edit @ctietze 2021-10-16: Shortened the title (I came up with when moving the posts) by request of @ZettelDistraction 👍

ZettelDistraction

20210924172458 Luhmann adds backlinks to notes

This note supercedes my previous notes on this topic.

Often the context in which we are working suggests a multiplicity of links to other notes. This is especially the case when the card index is already voluminous. In such cases it is important to capture the connections radially, as it were, but at the same time also by right away recording back links in the slips that are being linked to. In this working procedure, the content that we take note of is usually also enriched.

-- Niklas Luhmann. Communicating with Slip Boxes: An Empirical Account

The proposed checklist should still distinguish between adding a note and revising a note. However, the addition of a note is followed by revisions of the notes it links to, if we adhere to Luhmann's procedure. The links to the pre-existing notes that a new note links to are the radial connections that Luhmann mentions.

Now the solution to the digital Folgezettel problem is to observe that at various stages in the development of a digital ZK, notes can be added or revised. Addition is now in two steps:

1. When a new note is added, the decision is made about "where" the note relates to the prior notes. The adverb 'where' refers to a collection of immutable IDs of notes that provide the context for the new note. The note itself should contain the links it relates to, for context.
2. The notes that the new note links to are revised by adding backlinks to the new note.

To accommodate this additional structure I need to modify the tripartite format of my notes so far. The header will need to include the links to the pre-existing notes that we have decided on in Step 1. This addition will go after the H1 header with the immutable ID and the title. Here is the structure, consisting of a header, body and footer.

Part 1: Header

The optional YAML header

---
reference-section-title: Bibliography
---

occurs at the beginning of the note in case there are Pandoc style bibliographic references (I'm using Zettlr+Pandoc+MikTeX+Zotero+BetterBibTex). This YAML header places the Bibliography after the keywords in the footer of a note. Next we have the H1 header, the immutable id, and the title, followed by the context links, indicated by FROM (where the note comes FROM).

# ImmutableID Title
FROM [[immutable IDa]] [[immutable IDb]] ...

Part 2: Body

The format of the body of a zettel is unchanged.

Part 3: Footer

The footer is organized then by mandatory keywords; optional links back to notes for which the current note provides (some of) the context [for lack of a better name I'm calling this TO (where this note leads TO), the immutable ID of any notes arising in step 1 will go here, in step 2]; and, an optional Bibliography (provided the YAML header above for Zettlr is present and Pandoc citation style references are also present).

#keyword1 #keyword2 ...
TO [[immutable ID1]] [[immutable ID2]] ...

Bibliography

Well, I can't think of anything simpler at the moment that corresponds digitally to Luhman's physical setup, at least for the digital implementation I settled on. I'm incorporating the generalization of @prometheanhindsight and @Will. If you don't like step 1(2) omit it along with the line beginning with the keyword FROM(TO) and the links that follow. If you omit both, you have what I started with, which would approximate Luhmann's system by accident, but not by design, if the body of each note contained the necessary links. Not likely.

PS. Please forgive the edits--it's a struggle.

PPS. Mathematically, the Zettels of a digital Zettelkasten correspond to triples

(x, From, To)

where x is an element of a linearly ordered set ID of "immutable identifiers" and From and To are disjoint subsets of ID\{x}, which is the set of IDs excluding x. The triple satisfies the relation

From < x < To

which means that x is greater than every element of From and less than every element of To. So now I am assuming some structure on the set of immutable identifiers, with apologies to @bradfordfournier, since I previously stated that in my implementation, zettel IDs were immutable but had no other structure. Now they have some additional structure.

PPPS. The condition that

From < x < To

precludes the insertion of a new note between two existing notes. However, the condition might be relaxed to just

From < x

to allow the kind of extension "arbitrarily to the right," corresponding to the addition of notes in Luhmann's slip box of the form 1/2, 1/3, 1/3a, 1/3b and so on. We want just enough structure to implement what Luhmann was up to, but no more than that, in case we break something.

PPPPS. I probably should write down a Set coalgebra for the structure that corresponds to the digital set up and Luhmann's slip box. That should put to rest any disagreement about what might be missing from digital Zettelkasten.

ZettelDistraction

Corrections: relaxing the constraint allows looping and parallel arrow configurations. There should be a rule, which is that FROM and TO links are only added when a new note is added to the ZK. Any other link to a lower numbered note should be in the body of the note. That TO can be a set gives you the expansion from 1/2, 1/3 to 1/2, 1/2a, 1/3, in which a note is inserted after 1/2 when 1/3 is present. It's only necessary to relax the inequality constraint if you want 1/2a to be part of the FROM context for 1/3 after the fact. This update would be precluded by the rule that TO and FROM links are only updated when a new note is added. Their purpose is to indicate context when the note was created, but not after. Any additional interpretation or context would require linking in the bodies of the affected notes. I should keep this waffling between me and my Zettelkasten...

ctietze

(I split this off of the Antinet discussion to not derail it when this is being discussed)

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@ZettelDistraction said:
PPPPS. I probably should write down a Set coalgebra for the structure that corresponds to the digital set up and Luhmann's slip box. That should put to rest any disagreement about what might be missing from digital Zettelkasten.

I think that'd be fun

Why you introduced and how you use x in your triples (x, From, To) where all three are subsets of ID, the set of all identifiers, that didn't "click" for me.

How the < operator defined? Just size/element count?

ZettelDistraction

@ctietze said:
(I split this off of the Antinet discussion to not derail it when this is being discussed)

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@ZettelDistraction said:

Why you introduced and how you use x in your triples (x, From, To) where all three are subsets of ID, the set of all identifiers, that didn't "click" for me.

Only From and To are subsets of ID; x is an element of ID. x is the ID of the (new) Zettel.

How the < operator defined? Just size/element count?

I should have said that the linear order on the set ID is <. Or that the pair
(ID, <) is linearly ordered, where < is the order on ID. This is the case with timestamps--the '<' operator is the "usual" ordering of timestamps by time.

The point was to keep track of the order of addition of notes to the ZK. Just enough to simulate adding a note in the context of others, instead of another disconnected note. The Zettels in Luhmann's slip box don't change their IDs, though the slip box can grow "sideways" at least notationally. When deciding where to add a Zettel in a slip box, the immediate predecessor provides the context (this would be a link in the FROM field in the digital ZK). The successor Zettels don't provide context (I gather). So for a digital ZK, the ID x of a new Zettel will be greater than any FROM link, but less than any TO link, which may be defined if there is a later Zettel inserted after the one with ID x. Perhaps that's a sanity check, but I'm no expert on sanity.

The other point was book keeping. There is supposed to be support for the four characteristics of Luhmann's ZK that Schmidt mentions. And a discipline to go with it. It would be easy to forget to decide "where" to add a new note if there is no space for predecessor IDs in the format. (Now I have to revisit my entire moribund ZK and add them.)

So that accounts for the Zettel format, but barely justifies the "bonus set algebra" (now that's funny--where I come from, wit transcends the ego of the target--the appropriate target in my case).

I could take the attitude of @Sascha and say that digital Zettelkasten don't have to model physical Zettelkasten, they are different instruments, much like MIDI keyboard synthesizers and pianos are different instruments. OK, I want to make the positive statement that the digital ZK has the characteristics Schmidt identified...at least enough of them to get my ostrich of a digital ZK to take off...

ctietze

Gooootcha, I read x to be a subset because of the use of operators; inferred that in From < x < To all must be of the same type, e.g. a set, but actually I don't know if that's the case. Programmer brain took over I guess

Still not sure what you'd actually be stating with that: The names From and To are misleading me a bit. My intuitive understanding is that these names are for incoming and outgoing links. But you don't actually say, so that's probably wrong. Furthermore, they cannot mean that: because if From < x < To is true, that means for any ID y , where x < y (e.g. y is created later in time in a date based ID approach like you mention), y cannot be part of From. (e.g. you cannot create a note today and link to a note from last week, because since From < x, and x < y, y cannot be part of From at the same time. One of these 3 statements would have to go.) That effect doesn't sound sensible, so I assume you have something rather different in mind, like, well, just stating that there's some kind of sequential order in the IDs, so (x, Before_x, After_x)? But wouldn't that be trivial when you say that "for the set of IDs there a strict increasing order"? (Or is this how you want to state that there's a strict order, and I'm wrongly looking for practical implications?)

ZettelDistraction

The body can have links to earlier Zettels. I need to state the update rule that from and to only get updated when a new note is created. There is no constraint on links in the body.. From and To record the choices that were made when a new zettel was created. They are a book keeping device and don't exhaust all the links out of a note. When x is added, the From of x gets the ID (or a set of them} that the user decides is the context for the note with ID x. Then for each y in From, the TO field of y is updated with x. The TO field of x is empty at the moment.

Only later, when some other note z is created, when we decide x is the context, is the TO field of x updated, in this case with z.

I'm doing this to record "where" the note x fit in at the time it was created, and where it led to when some other note was created that relies on x. Strictly speaking, none of this is needed, since the body can have any links at all, including these. So that's a trivial proof that digital Zettelkasten can emulate that aspect of Luhmann's system, in case one needed to convince a skeptic. My purpose was to help ensure that I really add these links.

If it is desirable or possible that a later note could provide context for an earlier one, then drop From < x and x < To and allow a the earlier note to record that it now also relies on the new note. This is a kind of reincarnation. The earlier note becomes reinterpreted as a new note, as a comment on what came after. It's FROM field gets the new x, and the TO field of the new x is now updated with the id of the earlier note (now that we are enlightened, we have learned enough to "truly" situate our note).

Fine, I will allow "reincarnation" in which an existing note is treated a new note that comments on something that was added later. Apply the two step procedure for introducing a new note into the ZK to an existing note, adding From elements from later notes.

I need to add illustrations.

Again, none of this extra structure is necessary, but it's a way to keep the user "honest" and to track the evolution of the ZK. The physical limitations of the slip box may have helped Luhmann...

The operator < is overloaded to allow comparisons between sets and elements.

ZettelDistraction

From and To could be misleading, since they indicate subsets of the links of notes that supply context for x, and for which x supplies some of the context.

So rename: FROM becomes CONTEXT and TO becomes PROTEXT. (which does have*a sense that I intend).

There are rules for addition, modification and reincarnation of a Zettel. These control when CONTEXT and PROTEXT can have links added.

In addition (or creation) of x, we decide what notes form the context and add their ids to the context of x. Then for each of the context id z, we update the PROTEXT of z with x.

For reincarnation of x, we do the same thing, only for an existing x. Essentially we choose some additional CONTEXT note IDs, some of which could be in "the future", and add them to x. Then for each of those new CONTEXT ID Z, we update the PROTEXT of Z with x.

For modification, we change the body or keywords as needed, but don't change the CONTEXT or PROTEXT portions. We can add links in the body if needed.

That's it.

Now the three-part structure header, body, footer by itself will do this anyway if this is your process. I wanted to write down something that would yield the properties that Schmidt documents. Now I have to show that the rules are adequate.

Completeness would be neat if it could be done also..

I will jettison the triples, since they don't adequately represent the dynamics. [The dynamics will have sequences of triples, where the ID is fixed, but the sets will form a non-decreasibg sequence ordered by subset inclusion over time, coordinate-wise. I will postpone this because I dont need it for the checklist.]

ZettelDistraction

Reincarnation is a silly and misleading name. (I'm almost experiencing auto-fremdscham thinking about it.) Let's call the operation reinterpret, which can happen if we need to supply more context for a note than it has. The operations are add, reinterpret, and modify. For bookkeeping, CON and PRO are long enough. CON is followed by a list of IDs that provide (some) context for a note, and PRO lists the IDs of notes for which the given note supplies some of the context.. These two lists do not necessarily exhaust the set of links of a note--the body can have its own links. The lists are there so that we follow Luhmann's practice of deciding where a note should go in the ZK. This is a local problem for which Luhmann provided a local solution, according Schmidt.

However, in this system, we can add context later if we want.

I'm going to need to quote Schmidt and rewrite the Zettel structure again, with very minor changes but with rules included, and with examples....

ZettelDistraction

Back to Folgezettel: that the list PRO(Z) (for the protext of Zettel Z) is ordered could be used to indicate distance of related notes to Z. The first ID of PRO(Z) is assumed to continue the line of thought of Z. If ID(Z) = 2/1, then the first ID of PRO(Z) corresponds to 2/2. The second ID of PRO(Z) corresponds to an alternative line of thought, say 2/1a.

Folgezettel orderings should be defined

Niklaus Luhmann might have formally defined his Folgezettel numberings as a language over an alphabet, together with a partial ordering called the Folgezettel order. But he left that for his successors and later scholarship--he had more important work to attend to. We, however, need to do this to prove that digital ZK can reproduce the dynamics afforded by Luhmann's Folgezettel ordering, and to obviate some unnecessary controversies and misleading narratives. (Life is too short for narratives.) As John von Neumann said to George Bernard Dantzig, "get to the point!"

Halfheartedly, the symbols of this language include the natural numbers N with their strict ordering <, along with a set X of doubly indexed symbols x^i_j, where the superscript i ranges over natural numbers N = {0,1,2,3...}, the subscript j ranges over the integers Z = {0, +1, -1, +2,-2,...}. The set X is partially ordered such that x^i_j < x^m_n if and only if i= m and j< n; otherwise the two are incomparable. Finally, natural numbers and doubly-indexed symbols are incomparable: for any n, for any i, j, n || x^i_j. We also include a punctuation symbol separator '/' to disambiguate certain words: e.g., the symbol 111 as a word of length 1 is distinct from the words 1/11 and 11/1 of length 2.

Define the Folgezettel partial order on the monoid generated by this Scheiß--I mean generated by N \cup X. This is the induced partial lexicographic order on words, where two words are incomparable if they differ at incomparable symbols... check this. I'm tired... let a_j := x^0_j, b_j := x^1_j
for j in Z, then we can illustrate insertions and new parallel branching threads

1/2 < 1/2a_1 and 1/2 < 1/3

1/2 < 1/2a_{-1} < 1/2a_1 and 1/2 < 1/3

1/2 < 1/2b_0 || 1/2a_j for any integer j...

Now we have linearizable partially ordered sequences of words with open-ended branching "over" common initial substrings. That these sequences are linearized in Luhmann's slip box only reflects the mathematical property that such Folgezettel orderings can be linearized (to assign each note a fixed position in the slip box). It has no semantic significance that the end of one or more branches off of 2/1 might be followed by 2/2 in the slip box.

Luhmann intended those branches to lead further away from the original note. We would not want to mistake a linearization of a Folgezettel order for the Folgezettel order.

The rest uses notation from an earlier draft...to be revisited.

Ok, so what? We want a Zettelkasten, here just a collection of notes with timestamp IDs, that can grow by inserting notes into an extensible array modeling our physical slip box at certain stages, with entries numbered 0, 1, 2,.... The length of the array is the stage of the evolution of the ZK. Sometimes we will insert notes in between others. We want to track this using words over the Folgezettel symbols, meaning words of the Folgezettel language.

The point is to show that at each stage, there is a mapping, called a Folgezettel numbering, from the array to words of our Folgezettel language such that each map extends the previous map in an appropriate sense, and respects the order of insertion of the notes. This will simulate the assignment of Folgezettel numbers to an evolving Zettelkasten.

If we can do this, which we can, our system will have the Folgezettel property...

The insertion business is handled by the simplicial monoid Delta of MacLane. Well now we have a one line proof

Our sequence of Folgezettel numberings
f_n: n \rightarrow \L
need to satisfy f_n = f_{n+1} \delta^n_i
where i is the index where we have inserted
a new note, and where
\delta^n_i: n\rightarrow n+1
is the i-th insertion map.

I should really keep my unclear, half baked thoughts to myself...

ZettelDistraction

Aha, the way Luhmann used natural numbers in his Folgezettel numbering should be encoded like this: a Folgezettel ID (a word) always begins with a doubly-indexed symbol x^n_0, which Luhmann's represented as the natural number n--with syntactic sugar. The remaining symbols don't abuse notation like this and represent themselves.

To avoid abusing notation, we could define the Folgezttel partial order on words so that leading numerals are either equal or incomparable, but otherwise the partial order is the induced partial lexicographic order as above. This is so that 1/1 < 1/2 < 1/2a but 11/1 is really the start of a new sequence. That's another source of potential confusion. No wonder it's hard to catch a break in this business. For proofs it it ok to work only with doubly-indexed symbols.

ZettelDistraction

20210926210413 A definition of Folgezettel numbers

A Folgezettel number is an expression of the form v/n, where v is a decimal outline number (e.g., 1, 1.1, 11.1.2, 4.3.5, 1.2.1.1,...) and where n is a positive integer. The Folgezettel order on Folgezettel numbers u/m, v/n is the partial order given by u/m << v/n if and only if u = v and m < n. If u ≠ v, then u/m and v/n are incomparable, denoted u/m || v/n. Given the Folgezettel number v/n, the decimal outline number v is called the branch number of v/n. There is a map from Folgezettel numbers to branch numbers, which are lexicographically ordered by a relation denoted by <<<. The preimage of v under this map is the branch represented by v. Given branch numbers v, v.n where n is a positive integer, we say that v.n is a branch of v. The lexicographic ordering <<< of the branches extends to a linear order of the Folgezettel numbers, called the linearization of <<. The linearization is defined by u/m <<< v/n if and only if either (u <<< v) or (u = v and u/m << v/n).

Remark. It is helpful to distinguish all three orders <, <<, and <<< when working with a physical Zettelkasten.

Examples
2/1< 2/2 < 2/3
2.1/1 < 2.1/2 < 2.1/3
2.2/1 < 2.2/2 < .... < 2.2/100
.... 2.2.1/1< 2.2.1/2...
2.50/1

A Folgezettel numbering differs from a decimal outline numbering by introducing forests of parallel branches at every level. This feature enabled Luhmann to continue one note with another note having the next Folgezettel number, or to start any number of independent parallel branches off of a given note. On a given branch, a note could be continued by a note with the next number on that branch, or any number of related but independent parallel branches could be added, and so on.

The intention was to either continue a thought, or else as pursue as many related but not necessarily hierarchically related thoughts as needed, with a notational system designed to indicate the relationship of a new note to existing notes at the time the new note was added.

Will

I find this vastly interesting. Not sure why? Exposing your thinking to the screen/note/page fascinates me like watching a kite get airborne then slowly stabilizing.

I'm intrigued by your equating a Folgezettel ID with a word. That may be mathematical jargon, but it struck me not as a simile but as literal. I can see in the notecard world using numbers to signify ideas would save writing space. In the modern era, though, computers erase this limiting factor, and words do a superior job of hinting at the idea contained on the note. Numbers are limiting and cryptic. No wonder Luhmann required such an extensive keyword list to equate words with numbers.

I never heard of #syntactic_sugar, but I love the metaphor.

ZettelDistraction

@Will said:
I find this vastly interesting. Not sure why? Exposing your thinking to the screen/note/page fascinates me like watching a kite get airborne then slowly stabilizing.

That is a great image.

The sight of those kites was missing from my association with professional mathematicians as a graduate student. You could speak with the faculty in the dining commons, but you wouldn't find out what they had been working on until they had a more or less finished product, a paper in hand. I was interested in the process of creation, in seeing the work take shape. At the time I found myself caught off guard and feeling somehow misled about the promise of collaborative intellectual development in the university. I experienced not seeing this process unfold—not even hearing about it—as a betrayal, although I was owed nothing. If my expectations about the intellectual atmosphere of the university had "needed to be managed," I would have taken that as a personal affront. No doubt the difference in ability and experience were insurmountable. That and untreated severe sleep apnea. Perhaps for those reasons I was unsuited to the competitive pressures that lead mathematicians and other researchers to guard their work until it is sufficiently developed. I was supposed to have learned that lesson, in retrospect. Here I am exposing my flaws.

As for the Folgezettel order, I haven't seen anyone write down a definition, either in the literature or in online forums. I have to be careful about the linearization of the order. What I wrote is a linearization, but not the "correct" one, which would be the linear order of the notes that (likely) would have ended up in Niklas Luhmann's Zettelkasten. [It occurs to me that "right" linearization arises by rewriting the slash '/' in the Folgezettel number v/n as a decimal point '.' to obtain the decimal outline number v.n, and using the lexicographic order.] In any case, Luhmann was keeping another order in mind--the Folgezettel order-- to facilitate his bottom up approach to writing, while he was "linearizing" the notes in the slip box. The linearization is different from the Folgezettel order. I think it's useful to see what those orders are, and how digital Zettelkasten can replicate them, improve on them, diverge from them, or obviate the need for them altogether. But you need, or at least I need, a definition to work with. I find the papers of Schmidt indispensable in all this.

ZettelDistraction

An extension. Given a Folgezettel number, u/n, add the pairs (u/n, u.n/1) to << and take the transitive closure. Now we get the full Folgezettel order...

So u/m << v/n if and only if (u = v and m < n)
or v = u.m. Then u/m << u.m/n for any m,n.
In particular, u/m << u.m/1 << u.m/2 << ...

Calculation on the screen less efficient than on paper...

ctietze

I agree, it's fun to follow along your project progress!

For readability of longer paragraphs, it'd be helpful if you would add some markup to help scanning what's what. E.g. backticks for code.

If you enjoy more TeX-like rendering, MathJax works too, if you want, so subscripts and superscripts render correctly; the syntax is $\(x = y^i_j\)$ for $(x = y^i_j)$ (note the slash before the parens)

Also, double dollar sign for single line:

$$\(u/m << v/n\)$$

produces

$$(u/m << v/n)$$

Doesn't work with the preview for some reason, sorry

ZettelDistraction

@ctietze said:
...
If you enjoy more TeX-like rendering, MathJax works too

Great! I'll use MathJax in the next post, when I wake up. Thanks for splitting this anti-antinet thread from the antinet thread!

ZettelDistraction

20210926113320 Folgezettel numbers defined

F Lengyel

Abstract

A formal definition of Niklas Luhmann's Folgezettel numbering has not yet appeared in the literature on Zettelkasten or in online forums. We give a definition of the set $(\mathcal{F})$ of Folgezettel numbers, together with a partial order $(\preceq)$ on $(\mathcal{F})$. The structure $((\mathcal{F}, \preceq))$ describes the open-ended branching of Luhmann's numbering of Zettels. The dynamics of Luhmann's Zettelkasten can be described by filtrations
$$( (\mathcal{L}_1, \preceq_1)\subset\dots\subset(\mathcal{L}_n, \preceq_n) )$$ of substructures of $((\mathcal{F}, \preceq))$, where the set $(\mathcal{L}_k\subset\mathcal{F})$ has cardinality $(k\in\mathbb{Z}^+)$ and where $(\preceq_k)$ is the restriction $(\left.\preceq\right|_{\mathcal{L}_k})$. Within a physical Zettelkasten, the partial order is linearized. The linearization we give of the partial order $(\preceq)$ models Luhmann's procedure for ordering notes in his Zettelkasten.

Definitions

Let $(\Sigma = \mathbb{Z}^+\cup\lbrace`.\text{'},`/ \text{'}\rbrace)$ be the set of symbols consisting of the positive integers, together with the symbols $(`.\text{'})$ and $(`/\text{'})$. The Kleene closure $(\Sigma^*)$ of $(\Sigma)$ is the set of words (finite sequences of symbols, including the empty word, denoted by $(\varepsilon)$) over $(\Sigma)$. Let $(\mathcal{D}\subset\Sigma^*)$ be the intersection of all subsets $(S)$ of $(\Sigma^*)$ such that $(\mathbb{Z}^+\subseteq S)$ and such that whenever $(v\in S)$ and $(n\in\mathbb{Z}^+)$, the word $(v.n\in S)$. A decimal outline number (a decimal for short) is an element of the set $(\mathcal{D})$.
Examples of decimals include $$(1, 1.1, 11.1.2, 4.3.5, 1.2.1.1,\ldots\,.)$$

A Folgezettel number is a word of $(\Sigma^*)$ of the form $(v/n)$, where $(v\in\mathcal{D})$ and $(n\in\mathbb{Z}^+)$. The set of Folgezettel numbers is denoted by $(\mathcal{F})$. Observe that for $(u/m, v/n\in\mathcal{F})$,
$$(u/m = v/n \Leftrightarrow u=v \land m=n\text{.})$$

The Folgezettel numbers have the structure of a partially ordered set $((\mathcal{F}, \preceq))$, where the partial order $(\preceq)$ on $(\mathcal{F})$ is the reflexive closure of the strict transitive, irreflexive relation $(\prec)$ on $(\mathcal{F})$, given by
$$(u/m \prec v/n \Leftrightarrow \left(u = v \land m < n\right)\lor (v=u.m))$$ for $(u/m, v/n\in\mathcal{F})$.

More terminology

Given the Folgezettel number $(v/n)$, the decimal number $(v)$ is called the branch number of $(v/n)$, and the positive integer $(n)$ is called the index of $(v/n)$. There is a map $(\mathcal{F}\rightarrow\mathcal{D})$ from Folgezettel numbers to decimals given by $(v/n\mapsto v)$. The preimage of $(v)$ under this map is the branch of $(v)$, denoted by $(v^{-1})$. For any $(v\in\mathcal{F},n\in\mathbb{Z}^+)$, the Folgezettel number $(v/n)$ satisfies $(v/n\in v^{-1})$ by definition; we say that $(v/n)$ is on the branch $(v^{-1})$, or simply that $(v/n)$ is on the $(v)$-branch. Likewise, given decimals $(v, v.n)$ (note the period!) where $(n\in\mathbb{Z}^+)$, we say that $((v.n)^{-1})$ is the $(n)$-th branch of $(v)$ and since for any $(k\in\mathbb{Z}^+)$, the Folgezettel number $(v.n/k)$ satisfies $(v.n/k\in(v.n)^{-1})$, we say that $(v.n/k)$ is on the $(n)$-th branch of $(v)$ (namely, $((v.n)^{-1})$).

Insert picture here

Linearization

The set $(\mathcal{D})$ of decimals is lexicographically ordered. The order $(\lll)$ is defined for $(x,y\in\mathcal{D})$ by
$$(x \lll y \Leftrightarrow
\begin{cases}
\exists a,b,c \in\mathcal{D}\cup\lbrace\varepsilon\rbrace,m,n\in\mathbb{Z}^+, x=a.m.b \land y=a.n.c \land m\lt n; \\
\exists a,b\in\mathcal{D}, x = a \land y = a.b.
\end{cases})$$ where $(\varepsilon\in\Sigma^*)$ is the empty word.

For example,
$$(3 \lll 3.1.4 \lll 3.1.4.1.5.9.2.6.5.3.5.8.9.7.9.3.2.3846.2643383279 \lll 3.2)$$

The set of decimals with this order is denoted $(\left(\mathcal{D}, \lll\right))$.

Proposition. There is an order-preserving bijection, called the linearization
$$(L:(\mathcal{F},\prec) \rightarrow \left(\mathcal{D},\lll\right))$$ given by $(L(v/n) = v.n)$. This map is not an order isomorphism.

Example

Note that $(2/1\prec 2.1/1)$ (this is a move to an adjacent branch) and $(2/1\prec 2/2)$ and $(2.1 \lll 2.1.1\lll 2.2)$. However—and this is the point— it is false that $(2.1/1 \prec 2/2)$. The Folgezettel numbers $(2.1/1)$ and $(2/2)$ are incomparable.

In this example, we imagine that Luhmann would have continued the note labeled $(2/1)$ with a note labeled $(2/2)$. He also pursued another thought related to $(2/1)$ on $(2.1/1)$. In the slip box, the order of the notes would be the lexicographic order $(2.1 \lll 2.1.1\lll 2.2)$. However, the assignment of Folgezettel numbers implies that the note labeled $(2.1/1)$ would not have been continued on $(2/2)$. A continuation of $(2.1/1)$ would be labeled $(2.1/2)$ (for definiteness), and although again $(2.1 \lll 2.1.1\lll 2.1.2 \lll 2.2)$ in the slip box, $(2.1/2)$ is not continued on $(2/2)$. We do have $(2/1\prec 2.1/1)$, which indicates the move from the 2-branch to the related $(2.1)$-branch. A change of adjacent branches indicates an alternative line of thought. A new topic would be started on $(3/1)$, for example.

It is plausible that when working with the slip box, Niklas Luhmann was mindful of two different orderings of notes like those described here: the Folgezettel partial order $(\preceq)$, to facilitate his bottom-up, local approach to writing; and the total linear lexicographic order $(\lll)$, while he was "linearizing" notes in the slip box.

A question: does Luhmann scholarship support this?
Add references to Schmidt, etc. Acknowledge your generous hosts, the ZK community, the non-existent funding agency...

To be continued.

ctietze

This was so unexpectedly over the top that I laughed more when reading this than during any movie I watched in 2021

Even my basic math is a bit rusty, so I stumbled a bit here:

@ZettelDistraction said:

Linearization

The set $(\mathcal{D})$ of decimals is lexicographically ordered. The order $(\lll)$ is defined for $(x,y\in\mathcal{D})$ by
$$(x \lll y \Leftrightarrow
\begin{cases}
\exists a,b,c \in\mathcal{D}\cup\lbrace\varepsilon\rbrace,m,n\in\mathbb{Z}^+, x=a.m.b \land y=a.n.c \land m\lt n; \\
\exists a,b\in\mathcal{D}, x = a \land y = a.b.
\end{cases})$$ where $(\varepsilon\in\Sigma^*)$ is the empty word.

For example,
$$(3 \lll 3.1.4 \lll 3.1.4.1.5.9.2.6.5.3.5.8.9.7.9.3.2.3846.2643383279 \lll 3.2)$$

In the 1st of the two cases listed, you introduce a,b,c and m,n but only end up making a comparison of $(x=a.m.b \land y=a.n.c \land m\lt n; )$ so, if I get this right, b and c are irrelevant placeholders. Which makes sense: 3.2.2 < 3.3.1 -- the 3rd and last digit is not important for the ordering. And in you definition, you effectively only require the middle digit to exist; x = a.m.b for example can resolve to 𝜀.m.𝜀, where 𝜀 is the empty word as you said.

Now couldn't that produce .m., e.g. .4.? Inclusion of the dots in the result doesn't seem to be conditional here.

Looking at the initial definitions of the decimals, i.e. words in $(\mathcal{D})$, I'm not sure if I get the effect right. The initial $(\Sigma)$ would allow .\..\. (just dots and slashes) to be a valid member of the Kleene closure. But you also say that $(\mathbb{Z}^+\subseteq S)$, which makes sure that at least all positive integers are covered and not omitted from $(S)$.

Ah nevermind!

[...] whenever $(v\in S)$ and $(n\in\mathbb{Z}^+)$, the word $(v.n\in S)$. [...]

That makes sure that in case we have a word with a literal dot ., that the part after the dot must be a number and cannot be the empty word.

But doesn't that still leave the part before the dot to be open to emptiness? Which condition forbids e.g. $(v = .5)$ ? I can continue speculating aloud, but maybe you can clarify this for a layman without me confusing everyone reading?

@ZettelDistraction said:
A Folgezettel number is a word of $(\Sigma^*)$ of the form $(v/n)$, where $(v\in\mathcal{D})$ and $(n\in\mathbb{Z}^+)$. The set of Folgezettel numbers is denoted by $(\mathcal{F})$. Observe that for $(u/m, v/n\in\mathcal{F})$,
$$(u/m = v/n \Leftrightarrow u=v \land m=n\text{.})$$

The Folgezettel numbers have the structure of a partially ordered set $((\mathcal{F}, \preceq))$, where the partial order $(\preceq)$ on $(\mathcal{F})$ is the reflexive closure of the strict transitive, irreflexive relation $(\prec)$ on $(\mathcal{F})$, given by
$$(u/m \prec v/n \Leftrightarrow \left(u = v \land m < n\right)\lor (v=u.m))$$ for $(u/m, v/n\in\mathcal{F})$.

In the last statement, it surprised me that you wrote $(u/m \prec v/n \Leftrightarrow \left(u = v \land m < n\right)\lor (v=u.m))$

To follow along: the right side of the $(\Leftrightarrow)$ is for all intents and purposes a definition of what the comparision of $(u/m \prec v/n)$ boils down to, i.e. "how do we compare these?". Now it's either 1️⃣ if the branch numbers are equal (u = v) and the indexes are in ascending order (and m < n), which is apparent in $(3/4 \prec 3/7)$. Or 2️⃣ $((v=u.m))$ is satisfied.

The latter doesn't yet make sense to me; the replacement of v with u.m, so that we would get e.g. $(3/4 \prec 3.4/9)$ (the 9 is n and doesn't actually appear to be constrained, i.e. it's irrelevant for this comparison). Another example that satisfies this is $(3/1 \prec 3.1/9)$ (again, 9 is a placeholder because it's irrelevant). So the literal 1 of the left hand side appears on the right hand side, but instead of a branch index, it becomes part of the branch decimal; 3/1 becomes 3.1. Why exactly 3.1 and not 3.x for any x? The specificity of this puzzles me.

ZettelDistraction

@ctietze said:
This was so unexpectedly over the top that I laughed more when reading this than during any movie I watched in 2021

Thank you! I'm trying to be my usual charming, affable self. This is a kind of comedy, especially with MathJax syntax.

Even my basic math is a bit rusty, so I stumbled a bit here:

...

(1) [...] whenever $(v\in S)$ and $(n\in\mathbb{Z}^+)$, the word $(v.n\in S)$. [...]

That makes sure that in case we have a word with a literal dot ., that the part after the dot must be a number and cannot be the empty word.

But doesn't that still leave the part before the dot to be open to emptiness? Which condition forbids e.g. $(v = .5)$ ? I can continue speculating aloud, but maybe you can clarify this for a layman without me confusing everyone reading?

I might have given a constructive definition, which is used to prove that $(\mathcal{D})$ is the smallest set closed under the rule (1) above. (I added the (1).) Consider this subset of $(\Sigma^*)$:
$$(S_0 = \left\lbrace n_1 \mathbf{.} \cdots \mathbf{.} n_k : k\in \mathbb{Z}^+\right\rbrace)$$. This set does not contain $(\varepsilon)$. It also contains $(\mathbb{Z}^+)$ (take $(k=1)$) and it is closed under (1), so $(\mathcal{D}\subseteq S_0)$. On the other hand, $(S_0\subseteq\mathcal{D})$ since every set of the intersection $(\mathcal{D})$ contains $(S_0)$, so the intersection $(\mathcal{D})$ does also. Therefore $(\mathcal{D}=S_0)$.

$(\mathcal{D})$ is the "smallest" such set.

@ZettelDistraction said:
A Folgezettel number is a word of $(\Sigma^*)$ of the form $(v/n)$, where $(v\in\mathcal{D})$ and $(n\in\mathbb{Z}^+)$. The set of Folgezettel numbers is denoted by $(\mathcal{F})$. Observe that for $(u/m, v/n\in\mathcal{F})$,
$$(u/m = v/n \Leftrightarrow u=v \land m=n\text{.})$$

The Folgezettel numbers have the structure of a partially ordered set $((\mathcal{F}, \preceq))$, where the partial order $(\preceq)$ on $(\mathcal{F})$ is the reflexive closure of the strict transitive, irreflexive relation $(\prec)$ on $(\mathcal{F})$, given by
$$(u/m \prec v/n \Leftrightarrow \left(u = v \land m < n\right)\lor (v=u\mathbf{.}m))$$ for $(u/m, v/n\in\mathcal{F})$.

In the last statement, it surprised me that you wrote $(u/m \prec v/n \Leftrightarrow \left(u = v \land m < n\right)\lor (v=u.m))$

To follow along: the right side of the $(\Leftrightarrow)$ is for all intents and purposes a definition of what the comparision of $(u/m \prec v/n)$ boils down to, i.e. "how do we compare these?". Now it's either 1️⃣ if the branch numbers are equal (u = v) and the indexes are in ascending order (and m < n), which is apparent in $(3/4 \prec 3/7)$. Or 2️⃣ $((v=u.m))$ is satisfied.

The latter doesn't yet make sense to me; the replacement of v with u.m, so that we would get e.g. $(3/4 \prec 3.4/9)$ (the 9 is n and doesn't actually appear to be constrained, i.e. it's irrelevant for this comparison). Another example that satisfies this is $(3/1 \prec 3.1/9)$ (again, 9 is a placeholder because it's irrelevant). So the literal 1 of the left hand side appears on the right hand side, but instead of a branch index, it becomes part of the branch decimal; 3/1 becomes 3.1. Why exactly 3.1 and not 3.x for any x? The specificity of this puzzles me.

Diagrams would be useful. Think of

3/1 $(\preceq)$ 3/2 $(\ldots)$
$(\downarrow)$
$(3\mathbf{.}1/1)$ $(\preceq)$ $(3\mathbf{.}1/2)$ $(\ldots)$

[I'm using a downarrow for a rotated $(\preceq)$. Maybe arrows would have been a better choice.] The idea is to move down to the branch underneath 3/1 (to pursue alternate related thoughts). Likewise we might have (I forced myself to use MathJaxological array environments...)

$$(\begin{array}{}
3/1 & \preceq &3/2 & \preceq &\ldots \\
\downarrow & & \downarrow \\
\vdots & & 3\mathbf.2/1 & \preceq &3\mathbf.2/2 & \preceq & \ldots \\
\downarrow \\
3\mathbf.1/1 & \preceq &3\mathbf.1/2 & \preceq & \ldots
\end{array})$$ I'm attempting to illustrate both branches coexisting. Anyway the idea is to jump to another branch. That alternative "$(v=u\mathbf{.}m)$" is included in the definition, because I wanted a connected tree, and I didn't think disconnected parallel branches expressed what Luhman was doing. Also in practice the trees are finite (that was the point of the "filtration"), you always know who your predecessor is, and you can always linearly order the tree using a combinatorial trick.

The trick of replacing a slash with a decimal point was my only independent thought.
(I feel as though I should do something mathematical on this forum, with so many active researchers here, even if it just barely escapes triviality.)

It occurs to me that this is a special case of the "super-Folgezettel numbers" where we could create outlines at any level, then branch from a specific note of an outline to a parallel branch that is itself an outline. Now that we have this simple case, we can do this by writing x/y, where both x and y are decimals. The case v/n with n a positive integer is a special case. Maybe the "super-Folgezettel numbers should be the Folgezettel numbers.

Thank you for taking the time to read through this. I think it does what it is supposed to do, so far. I'm sure it could be improved. I'm very grateful for the work that you and @Sascha are doing, and for the zettelkasten.de community. Not to mention the opportunity to work (ok not "work work") on this here.

ctietze

Hmm your elaborations sound like this is intentional: After 3/1 follows 3/2 follows 3/3 etc; but when you branch-off, you go from 3/1 to 3.1/1.

When you branch off from 3/27 you go to 3.27/1?

My confusion stems from how this looked similar to Luhmann's IDs, but in this case isn't; there's no hard and fast "slash rule". Example: https://niklas-luhmann-archiv.de/bestand/zettelkasten/zettel/ZK_2_NB_9-8a_V

One sequence is 9/8a, 9/8b, 9/8c etc, and branching-off you get 9/8a, 9/8a1, 9/8a2.

So that's totally on me and I mixed your design with the observations from Luhmann's IDs.

(Why doesn't the same visualization apply to 9/8b, 9/8b1, etc? even though there is a 9/8c? Why isn't there a branch but the 9/8b* stuff "inlined" into the sequence? I don't know.)

ZettelDistraction

@ctietze said:
Hmm your elaborations sound like this is intentional: After 3/1 follows 3/2 follows 3/3 etc; but when you branch-off, you go from 3/1 to 3.1/1.

When you branch off from 3/27 you go to 3.27/1?

Exactly.

Incidentally, I think our comments crossed.

My confusion stems from how this looked similar to Luhmann's IDs, but in this case isn't; there's no hard and fast "slash rule". Example: https://niklas-luhmann-archiv.de/bestand/zettelkasten/zettel/ZK_2_NB_9-8a_V

One sequence is 9/8a, 9/8b, 9/8c etc, and branching-off you get 9/8a, 9/8a1, 9/8a2.

So that's totally on me and I mixed your design with the observations from Luhmann's IDs.

It's on me because of my style of writing. I should have pointed out that the intention is to write down something calculational that does what Luhmann was doing, or at least close enough.

(Why doesn't the same visualization apply to 9/8b, 9/8b1, etc? even though there is a 9/8c? Why isn't there a branch but the 9/8b* stuff "inlined" into the sequence? I don't know.)

Well, I wanted something algorithmic, so that maintaining an expanding tree in a linear array with some maneuvering room would work without requiring thought (hardly any, but just at the limit of what I could do). It looks plausible to me now that what I called the super-Folgezettel numbers (numbers of the form u/v with u and v "decimals") are really closer to Luhmann's intentions. Either that or they generalize Luhmann's system. I thought of doing this originally, but decided to start with something simpler. I would have had to explain what outlines with branching parallel outlines at any point of any level, and so on would have meant...

What this doesn't capture is the semantics that Luhmann had in mind. It only hints at it.

ZettelDistraction

Several things occur to me. This will be brief since I have some work to do before I can elaborate.

1. This definition of Folgezettel numbers is programmable. In The Archive. I think I'm going to resuscitate my old iMac.
2. we could get a dense ordering from the "super-Folgezettel numbers" $(x/y,\,x,y\in\mathcal{D})$ by adding zero $(0)$ to $(\Sigma)$, provided decimals are identified under rightmost cancellation of zeros and assuming that $(x, y\ne0)$. Rightmost cancellation of zeros means $$(v = v\mathbf{.}0 = v\mathbf{.}0\mathbf{.}0 = v\mathbf{.}0\mathbf{.}0\mathbf{.}\cdots \mathbf{.}0)$$ Since I haven't thought this through, we'd want leftmost cancellation of zeros also.
3. Another representation is possible: decimals can be represented by the eventually zero functions. $$(f:\mathbb{N}\rightarrow\mathbb{N}.)$$ This representation spares us from cancelling zeros on the right. [not on the left, though--we'd have to shift maps eventually zero maps $(f)$ so that the first non-zero coordinate is $(f(0))$. That's too much work, just require $(f(0)\ne 0)$. Unless normalizing functions costs us something...] The eventually zero functions are closed under an ordinal addition $(f\oplus g.)$ There is a distinguished zero function $(\mathbf{0}:\mathbb{N}\rightarrow\mathbb{N}.)$ A dense Folgezettel number is a formal fraction $(f/g)$ with $(g\ne\mathbf{0})$. The linearization map is then $(L(f/g) = f\oplus g)$ from formal fractions to decimals.
4. The dense ordering of the dense Folgezettel numbers allows us to add notes in between others, if we wanted to.
5. Logicians and recursion theorists talk about reals, which are functions $(f:\mathbb{N}\rightarrow\mathbb{N}.)$
6. It's likely that this is well-known in other contexts...maybe not with regard to Niklas Luhmann. That's the trouble with the Internet: it short-circuits intermediaries. I could be adding to the grey goo Bill Joy warned about...

I haven't checked this. I should go back to the Zettel structure now...

ZettelDistraction

20210929122619 Folgezettel versus links: what information do Folgezettel IDs contain?

If you wish to educate a partner in communication, it will be good to provide him with independence from the beginning. A slip box, which has been made according to the suggestions just given can exhibit great independence. There may be equally apt ways to reach this goal. The described reduction to a fixed, but merely formal order of placement and the resulting combination of order and disorder is, however, one of these ways.
—Niklas Luhmann. Communicating with Slip Boxes: An Empirical Account

The links of a digital ZK can emulate any of the Folgezettel orderings. This is evident—the question then is "how to provide [the digital Zettelkasten] with independence from the beginning." Luhmann achieved a middle ground between order and chaos by assigning each note added to the slip box a Folgezettel ID depending on its contextual relationship to nearby notes. The assignment, following the conventions defined above for Folgezettel IDs of the form $(v/n)$ followed a rule of threes:

1. New. A note starting new topic receives a new branch number and an index; e.g., $(4/1)$.
2. Continuation. A continuation of an existing note receives the next Folgezettel ID; e.g., $(4/2)$.
3. Branching. A note related to an existing note but not a continuation (such as an extended footnote or an endnote) receives a Folgezettel ID indicating a new branch; eg., $(4.2/1)$

Folgezettel IDs contain information that timestamp IDs do not possess, in general.^† This information is the decision made about the local relationship of a note to nearby notes when the note was introduced into the slip box.^‡

If this information is worth preserving in a digital ZK, if its addition leads to a productive middle ground between predictable order and chaos, notes within a digital ZK could be designed to explicitly preserve it. Another possibility is to survey the literature on graphs with appropriate growth properties, and emulate such graphs in an evolving digital Zettelkasten.

Remark

The definition of Folgezettel IDs in 20210926113320 provides for a single alternative branch in the move from $(v/n)$ to $(v.n/1)$. This limitation (if it is one) could be addressed in several ways, for example, in the same system by numbering branches as follows: from $(v/n)$ to $(v.n,1/1)$ for the first note of the first branch of $(v/n)$; from$(v/n)$ to $(v.n.2/1)$ for the first note of the second branch of $(v/n)$, and so on. The partial order $(\preceq)$ would be modified accordingly. Another possibility is to allow decimals in the "denominator" with appropriate addition rules and numbering. No such limitation exists in a digital ZK, though the question remains of preserving in a new note those of its relationships to existing notes that will maintain the balance between predictable order and disorder within a dynamic ZK.

Keywords #zettelkasten #Luhmann #Folgezettel #link #digitalZK

Footnotes

†. There is a degenerate case: when all notes follow each other as continuations.
‡. The note 20210924172458 proposed a note format to record branch links and continuation links when a note is added to a digital ZK.

ZettelDistraction

20210929212721 Generalized Folgezettel IDs

F Lengyel

A generalized Folgezettel ID is the coordinate of node of an outline-like structure that may branch into any number of other such structures at any node. A definition of the generalized Folgezettel IDs generated by a set of decimals is given. A partial order on the generalized Folgezettel IDs is defined, with respect to which intervals within branches of Folgezettel IDs generated by dense sets of decimals are themselves dense. An order-preserving bijection defines a homomorphic linearization of the partial order into a lexicographically ordered set of decimals. The linearization is $(O(n))$.

Definitions

Symbols

Let $(\Sigma = \mathbb{N}\cup\left\lbrace`\mathbf{.}\text{'},`|_{n} \text{'}\right\rbrace_{n\in\mathbb{Z}^+})$ be the set of symbols consisting of the natural numbers, together with the set of constant symbols $(\left\lbrace`\mathbf{.}\text{'},`|_{n} \text{'}\right\rbrace_{n\in\mathbb{Z}^+})$. The Kleene closure (aka the free monoid) $(\Sigma^*)$ over $(\Sigma)$ is the set of words (finite sequences of symbols, including the empty word, denoted by $(\varepsilon)$) over $(\Sigma)$. If $(w\in\Sigma^*)$, $(|w|\in\mathbb{N})$ denotes the length of the word $(w)$. By definition, $(|\varepsilon|=0)$.

Decimals

Let $(\mathcal{D}_0\subset\Sigma^*)$ be the intersection of all subsets $(S)$ of $(\Sigma^*)$ such that $(\mathbb{N}\subseteq S)$ and such that whenever $(v\in S)$ and $(n\in\mathbb{N})$, the word $(v\mathbf{.}n\in S)$. A decimal is an element of the set $(\mathcal{D}_0)$. A decimal is nonzero if at least one of its integer symbols is nonzero; the set of nonzero decimals is denoted $(\mathcal{D}_{\ne0})$. A decimal is normalized if its first and last integer symbols are nonzero; the set of normalized decimals is denoted $(\mathcal{D})$. A decimal is positive if all of its integer symbols are nonzero; the set of positive decimals is denoted $(\mathcal{D}^+)$.

Edited by @ctietze: changed $(\mathbb{N}\cup\left\lbrace`\mathbf{.}\text{'}\right\rbrace\subseteq S)$ to $(\mathbb{N}\subseteq S)$ according to the errata

Examples

$$(
\begin{array}{}
0,0\mathbf{.}0, 0\mathbf{.}1\mathbf{.}0\mathbf{.}1, 4\mathbf{.}0\mathbf{.}0, 1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}_0 \\
1,400,25\mathbf{.}0\mathbf{.}0\mathbf{.}0.1,1\mathbf{.}0\mathbf{.}0\mathbf{.}1,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}\\
1\mathbf{.}1\mathbf{.}1\mathbf{.}1, 29\mathbf{.}396\mathbf{.}4\mathbf{.}8,1\mathbf{.}2\mathbf{.}1\mathbf{.}1\mathbf{.}4\in\mathcal{D}^+
\end{array}
)$$

Lexicographic order on decimals

An irreflexive transitive order $(\prec)$ on $(\mathcal{D}_0)$ is given by
$$(u \prec v \Leftrightarrow
\begin{cases} \left(\exists x,y,z \in\mathcal{D}_0\cup\left\lbrace\varepsilon\right\rbrace,
m,n\in\mathbb{N}\right)\\
\quad\left( \left(u=x\mathbf{.}m\mathbf{.}y\right) \land
\left(v=x\mathbf{.}n\mathbf{.}z\right) \land \left(m\lt n\right)\right); \\
\left(\exists x\in\mathcal{D}_0\right)v = u\mathbf{.}x.
\end{cases})$$ where $(u,v\in\mathcal{D}_0)$. Note that in the second alternative above, $(x\ne\varepsilon.)$ Define $(u \preceq v \Leftrightarrow u\prec v \lor (u = v))$ for $(u,v\in \mathcal{D}_0)$. Then
$$(\left(\mathcal{D}_0,\preceq\right),
\left(\mathcal{D},\preceq\left.\right|_{\mathcal{D}}\right),
\left(\mathcal{D}_{\ne0},\preceq\left.\right|_{\mathcal{D}_{\ne0}}\right),
\left(\mathcal{D}^+,\preceq\left.\right|_{\mathcal{D}^+}\right))$$ are lexicographically ordered by sub-orderings of $(\preceq)$ restricted to the respective domains. We denote each of the partial orders by $(\preceq)$.

Examples

$$(0\prec 0\mathbf{.}0\prec 0\mathbf{.}0\mathbf{.}0\mathbf{.}0\mathbf{.}0 \prec 0\mathbf{.}1\prec 0\mathbf{.} 1\mathbf{.}0\mathbf{.}0\mathbf{.}1\prec 1\mathbf{.}0\mathbf{.}0\mathbf{.}1 \prec 10\mathbf{.}1\prec10\mathbf{.}1\mathbf{.}1)$$ The nonzero condition eliminates "infinitesimals," which are decimals where every digit is zero. Nonzero decimals exhibit behavior such as $$(0\mathbf{.} 1\mathbf{.}0\mathbf{.}0\mathbf{.}1\prec 0\mathbf{.} 1\mathbf{.}0\mathbf{.}0\mathbf{.}1\mathbf{.}0.)$$ Nevertheless, infinitessimals and nonzero decimals are among the coordinates of the tree structures considered in the sequel.

Generalized Folgezettel IDs

A generalized (positive) Folgezettel ID or (or just a (positive) Folgezettel ID) is a word of $(\Sigma^*)$ of the form
$$(v_1 \left.\right|_{i_1} v_2 \left.\right|_{i_2} \cdots \left.\right|_{i_k} v_k)$$ where $(k\in\mathbb{Z}^+,i_1,\ldots, i_k\in\mathbb{Z}^+)$ and where the decimals $(v_1,\ldots, v_k\in\mathcal{D}\, (\mathcal{D}^+))$ are normalized (positive).

The set of (positive) Folgezettel IDs is denoted by $(\mathcal{F})$ (respectively $(\mathcal{F}^+)$).

It will be convenient to define classes of Folgezettel IDs inductively. The set $(\mathcal{F}(\mathcal{E}))$ of generalized Folgezettel IDs generated by a set $(\mathcal{E}\subseteq\mathcal{D}_0)$ of decimals is given as follows. Define the sets $(F_0,F_1,\ldots,F_n,\ldots )$ by
$$(
\begin{array}{}
F_0 = &\mathcal{E}\\
F_{n+1} =&\left\lbrace w\in \Sigma^{*} : \exists x\in F_{n}, k\in\mathbb{Z}^{+}, d\in F_0,
w= x \left.\right|_{k} d \right\rbrace
\end{array})$$ for $(n\in\mathbb{N}.)$ Then $$(\mathcal{F}(\mathcal{E}) = \bigcup_{n=0}^\infty F_n.)$$

The classes $(\mathcal{F}^+,\mathcal{F}_{\ne0},\mathcal{F}_0)$ of positive (nonzero, unrestricted) generalized Folgezettel IDs can be defined by setting $(\mathcal{E})$ in $(\mathcal{F}(\mathcal{E}))$ equal to $(\mathcal{D}^+,\mathcal{D}_{\ne0},\mathcal{D}_0)$, respectively.

For definiteness we work with the set $(\mathcal{F}(\mathcal{D}))$ of Folgezettel IDs (generated by the normalized decimals $(\mathcal{D})$), and we'll refer to them as Folgezettel IDs. However, the results hold for other classes of generalized Folgezettel IDs unless otherwise indicated.

The Folgezettel IDs have the structure of a partially ordered set $((\mathcal{F}, \preceq))$, where the partial order $(\preceq)$ on $(\mathcal{F})$ extends the relation $(\prec)$ on $(\mathcal{D})$, as follows.

$$( v \prec w \Leftrightarrow
\begin{cases}
\exists x\in\mathcal{F}, n\in\mathbb{Z}^{+}, c,d\in\mathcal{D},
v = x |_n c \land w = x |_n d \land c \prec d; \\
\exists n\in\mathbb{Z}^{+}, c,d\in\mathcal{D}, w = v |_n d
\end{cases})$$ for $(v,w\in\mathcal{F})$.

The two cases correspond to comparisons of Folgezettel IDs on the same branch, and to comparisons of IDs on a branch and a descendant branch. Note that $(x |_m c , x |_n d)$ are incomparable, where $(m,n\in\mathbb{Z}^+)$, $(m\ne n, x\in\mathcal{F})$, $(c,d\in\mathcal{D})$.

Luhmann IDs

A Folgezettel ID $(w)$ is a Luhmann ID if $(w=d |_{1} n)$, where $(d\in\mathcal{D}^+, n\in\mathbb{Z}^+)$. The Luhmann IDs allow a single descendent branch from a given ID. This is a special case of the unbounded parallel branching possible with the generalized Folgezettel IDs.

$$(\begin{array}{}
3|_1 1 & \preceq &3|_1 2 & \preceq &\ldots \\
\downarrow & & \downarrow \\
\vdots & & 3\mathbf.2|_1 1 & \preceq &3\mathbf.2|_1 2 & \preceq & \ldots \\
\downarrow \\
3\mathbf.1|_1 1 & \preceq &3\mathbf.1|_1 2 & \preceq & \ldots
\end{array})$$

[I need to produce better diagrams - FL.]

Linearization

Proposition. There is a bijective, order preserving map $(L)$ from the partially ordered set of Folgezettel IDs to the lexicographically ordered set of normalized decimals. $$(L: \left(\mathcal{F}(\mathcal{D}), \preceq\right)\rightarrow \left(\mathcal{D}, \preceq\right))$$The map $(L)$ is not an order isomorphism.

Proof. The map $(L)$ is the monoid homomorphism defined inductively by

$$(L(w) =
\begin{cases}
w,& w\in\mathcal{D};\\
L(x)\mathbf{.}n\mathbf{.} d, &\exists x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D},
w= x\left.\right|_n d.
\end{cases} )$$ for $(w\in \mathcal{F})$.

Edited by @ctietze: added missing $(n)$ in $(L(x).n.d)$ in the 2nd case according to the errata

The map $(L)$ immediately generalizes to IDs generated by any set $(\mathcal{E}\subseteq\mathcal{D}_0)$ of decimals closed under concatenation of decimals; i.e., if $(c,d\in\mathcal{E})$, then $(c\mathbf{.}d\in\mathcal{E})$. Then the following map is an order preserving bijection.
$$(L: \left(\mathcal{F}(\mathcal{E}), \preceq\right)\rightarrow \left(\mathcal{E}, \preceq\right).)$$ The map $(L)$ is not an order isomorphism.

Density of $((\mathcal{F}(\mathcal{D}_{\ne0}), \preceq))$

Since the nonzero decimals are dense, branches are dense. Taking $(\mathcal{F}=\mathcal{F}(\mathcal{E}), \mathcal{E}=\mathcal{D}_{\ne0})$,
$$(\forall x\in\mathcal{F},n\in\mathbb{Z}^+,c,d\in\mathcal{E}\left(c\lt d\right)\Rightarrow \exists e\in\mathcal{E}, x\left.\right|_n c
\prec x\left.\right|_n e \prec x\left.\right|_n d)$$

Sascha

I am both intrigued by the beauty and ignorant of the math.

Two questions:

1. I think I missed the beginning, so: What are you aiming at?
2. Can I assist you with non-math stuff?

ZettelDistraction

@Sascha said:
I am both intrigued by the beauty and ignorant of the math.

Two questions:

1. I think I missed the beginning, so: What are you aiming at?
2. Can I assist you with non-math stuff?

@Sascha, thank you. I will attempt to answer later—I'm in New York, in another time zone, well past my bedtime. The short answer is that I wanted to write down mathematically what Niklas Luhmann was doing, or at least capture the essence, even if he deviated from it on occasion. With a definition in hand, perhaps one could characterize the middle ground between predictable order and utter chaos that Luhmann was aiming for (and achieved) with his "communication partner." Writing these definitions seemed like low-hanging fruit to me. I haven't seen them elsewhere, and thought that they could obviate certain recurrent arguments, or possibly advance the discussion.There is more to this...

ZettelDistraction

Misery! In the definition of decimals, $(\mathbb{N}\cup\lbrace{`}\mathbf{.}\text{'}\rbrace)$ should be replaced with $(\mathbb{N})$. Too late to correct.

Sascha

@ZettelDistraction said:
Misery! In the definition of decimals, $(\mathbb{N}\cup\lbrace{`}\mathbf{.}\text{'}\rbrace)$ should be replaced with $(\mathbb{N})$. Too late to correct.

It's only done when its done. Nobody can loose if nobody gives up.

ZettelDistraction

Today is busy. But I do see the site is licensed under CC-BY-SA, which is good. That applies to whatever I have posted too. In the unlikely event that anyone would want to make use of the formulas, CC-BY-SA applies. Fine!

ZettelDistraction

20211001181142 The way of errata is endless

There are at least two self-inflicted misprints^† in 20210929212721 Generalized Folgezettel IDs. The first is an obvious error in the definition of decimals: $(\mathbb{N}\cup\lbrace{`}\mathbf{.}\text{'}\rbrace)$ should be replaced with $(\mathbb{N})$.

The second is an error a glaring, unforgivable omission in the definition of the map $(L)$. The definition omits what I had when writing, and still have rattling in my skull, which was $(L(x)\mathbf{.}n\mathbf{.} d)$, though I wrote the incorrect $(L(x)\mathbf{.}d.)$ The corrected definition of $(L)$ is
$$(L(w) =
\begin{cases}
w,& w\in\mathcal{D};\\
L(x)\mathbf{.}n\mathbf{.} d, &\exists x\in\mathcal{F}, n\in\mathbb{Z}^+,d\in\mathcal{D},
w= x\left.\right|_n d.
\end{cases} )$$ for $(w\in \mathcal{F})$.

#errata #recursion #folgezettel

^†The word 'misprint' is a defensive, self-protective, self-interested term, arising through what psychologists call "story editing," which is the creation of a coherent retrospective narrative (necessarily false—all narrative is false, especially personal narrative) in which the author is also a character, and which unfolds from the perspective of the author, who maintains the fiction of agency.