Mathematical definition of Folgezettel

scottscheper · February 2022

@ZettelDistraction said:

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.

This is really profound and interesting @ZettelDistraction.

Thank you for putting all of this together.

Do you plan to put this into a paper at some point? I think it would be a great addition to Zettelkästen literature.

ZettelDistraction · February 2022

@scottscheper said:

This is really profound and interesting @ZettelDistraction.

Thank you for putting all of this together.

Do you plan to put this into a paper at some point? I think it would be a great addition to Zettelkästen literature.

The last version is https://forum.zettelkasten.de/discussion/1982/mathematical-definition-of-folgezettel/p3

I'm going to be haunted by the prior revisions!

I don't plan to do anything with it. It's an exercise.

Luhmann's numbering has the limitation that the total degree of any note in the directed graph is at most 3, which means that you have at most 2 branches off a node. This doesn't include internal links. I suggested a more general numbering to get around this limitation. Others suggest identifying sequences of notes by listing their IDs in order in auxiliary notes called structure notes. So the Folgezttel numbering gives you a skeleton--a spanning tree, indicating the order in which notes were originally placed into the Zettelkasten.

As far as the directed graphs go, linking with a combination of Folgezttel and internal links or using structure notes is equivalent. What I mean by "equivalent" is that if you consider a diagram with dots for notes, and arrows between them for either Folgezttel or links, you won't be able to tell whether the graphs used structure notes and timestamps, or whether they used Folgezttel and internal links, or some combination. This abstraction erases the content of the notes and leaves only dots for notes and directed arrows (unidirectional links) visible.

I think it's important to be clear about what "equivalent" means in each case, since there are several possible sensible notions of equivalence. Here's a different notion of equivalence that a combinatorialist might use.

If you did use Folgezttel, you could take the same graph, but color the edges red and blue, red for a link defined by Folgezttel, and blue for internal links. Then right away you would notice that the red links form a spanning tree touching every note in the directed graph. That would show you where the note sequences were, or at least the sequences defined by Folgezttel. Also the note identifiers will tell you a path along red colored edges back to the root. (Pictures would probably help, but I'm pretty sure you're imagining what this looks like.)

Now it looks like a matter of personal preference what you choose. There may be some psychological advantages using Folgezttel, because over time the labeling will have some meaning and might facilitate visualization. The IDs might serve the function of locations or loci in a mnemonic system. The labeling might become familiar enough to carry around in one's head. This kind of shorthand encoded context is local to the note: it doesn't depend on seeing or listing any other adjacent notes in the graph. I find that it helps--I don't tend to memorize 14 digit timestamps. But perhaps there are people with eidetic memories who do, though they might not need a Zettelkasten.

I've seen it asserted (on medium.com I think) that it's always better to call out notes sequences in a separate structure note. If only there were studies to show this! In this subject it's hard to make universal declarations that hold for everyone. Maybe it's true, maybe not.

Finally I figured out how to accommodate both systems (Folgezettel only, timestamps only, a combination Folgezettel_timestamp ) in a backward compatible way in Zettlr. Other proposed systems for this didn't do what I wanted:

The new IDs should be recognized as IDs in software;
I wanted to be able to switch between ID types;
the IDs should be suitable as filenames (so {1, 4, d, 3, f} appended to a title would not work--I have seen this in the forums);
It should be straightforward to rename files to ID.md if desired.

At least in Zettlr it's possible to experiment--I don't have enough experience with other systems to say.

Sascha · February 2022

@ZettelDistraction said:
I've seen it asserted (on medium.com I think) that it's always better to call out notes sequences in a separate structure note. If only there were studies to show this! In this subject it's hard to make universal declarations that hold for everyone. Maybe it's true, maybe not.

I don't think that this is true. Do you remember what the justification for that claim is?

scottscheper · February 2022

@ZettelDistraction said:

@scottscheper said:

This is really profound and interesting @ZettelDistraction.

Thank you for putting all of this together.

Do you plan to put this into a paper at some point? I think it would be a great addition to Zettelkästen literature.

The last version is https://forum.zettelkasten.de/discussion/1982/mathematical-definition-of-folgezettel/p3

I'm going to be haunted by the prior revisions!

I don't plan to do anything with it. It's an exercise.

Luhmann's numbering has the limitation that the total degree of any note in the directed graph is at most 3, which means that you have at most 2 branches off a node. This doesn't include internal links. I suggested a more general numbering to get around this limitation. Others suggest identifying sequences of notes by listing their IDs in order in auxiliary notes called structure notes. So the Folgezttel numbering gives you a skeleton--a spanning tree, indicating the order in which notes were originally placed into the Zettelkasten.

Do you think you could create a diagram to better illustrate what you mean by this? I think I’m following you here but want to confirm. By two branches off a node, do you mean like: 12 > 12/1 and 12/1A

ZettelDistraction · February 2022

@ZettelDistraction said:
I've seen it asserted (on medium.com I think) that it's always better to call out note sequences in a separate structure note. If only there were studies to show this! In this subject it's hard to make universal declarations that hold for everyone. Maybe it's true, maybe not.

@Sascha said:
I don't think that this is true. Do you remember what the justification for that claim is?

This is from The Folgezettel Conundrum by Eva Thomas, who says that Folgezettel identifiers commit the user to a specific choice of predecessor ID (not location, but a name). Her recommendation is to call out note sequences with a separate "sequence note," in her terms:

Let’s take a look at the 3b1 note one more time. As we said, it is related to notes 3b, 4c11, 4d, and 7a10. We want to keep track of all these sequences, so we will use separate notes instead of ID nesting method. [sic] At the top of each sequence note, we add a short description of what topic or question the sequence is tackling. We add a list of notes that belong to the sequence, with their corresponding IDs and titles.

The Luhmann IDs aren't as flexible as the more general IDs I wrote about (the notation isn't optimal for defining filenames), so it's not that an ID encoding scheme cannot be made to work.

I'm swamped with work--I might not be able to respond so promptly.

ZettelDistraction · February 2022

@scottscheper said:

Do you think you could create a diagram to better illustrate what you mean by this? I think I’m following you here but want to confirm. By two branches off a node, do you mean like: 12 > 12/1 and 12/1A

I'm swamped at the moment. I might have more time over the weekend.

Sascha · February 2022

@ZettelDistraction said:

@ZettelDistraction said:
I've seen it asserted (on medium.com I think) that it's always better to call out note sequences in a separate structure note. If only there were studies to show this! In this subject it's hard to make universal declarations that hold for everyone. Maybe it's true, maybe not.

@Sascha said:
I don't think that this is true. Do you remember what the justification for that claim is?

This is from The Folgezettel Conundrum by Eva Thomas, who says that Folgezettel identifiers commit the user to a specific choice of predecessor ID (not location, but a name). Her recommendation is to call out note sequences with a separate "sequence note," in her terms:

Let’s take a look at the 3b1 note one more time. As we said, it is related to notes 3b, 4c11, 4d, and 7a10. We want to keep track of all these sequences, so we will use separate notes instead of ID nesting method. [sic] At the top of each sequence note, we add a short description of what topic or question the sequence is tackling. We add a list of notes that belong to the sequence, with their corresponding IDs and titles.

Aha! I think Eva is argueing on Folgezettel vs Structure Notes and just presents her approach without any attempt of generalisation. It might be just her wording that could give this impression (I don't have it).

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Mathematical definition of Folgezettel

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20210926113320 Folgezettel numbers defined

Abstract

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More terminology

Linearization

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